[ [ { "type": "image", "content": { "image_source": { "path": "images/ced86103fb59e0b03674d3fbef57fa32abd2c451b2a16f5c0f33f48c8f0d7d70.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 0, 0, 1000, 502 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "数学分析笔记" } ], "level": 2 }, "bbox": [ 33, 615, 289, 650 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "作者：Reichtum" } ] }, "bbox": [ 50, 701, 174, 718 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "组织：ZheJiang University" } ] }, "bbox": [ 50, 722, 243, 740 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "时间：September 22, 2022" } ] }, "bbox": [ 50, 744, 243, 762 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "版本：0.1" } ] }, "bbox": [ 52, 766, 129, 783 ] }, { "type": "image", "content": { "image_source": { "path": "images/f8e34492d364de6248e0c95112dac34ceb37621665dd12e77bb7f21e0c6e21aa.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 705, 733, 902, 869 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "There is nothing in the understanding which has not come from the senses, except the understanding itself, or the one who understands ——Gottfried Wilhelm Leibniz" } ] }, "bbox": [ 159, 878, 838, 913 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "目录" } ], "level": 2 }, "bbox": [ 468, 87, 531, 112 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "第1章 数学基础 1\n1.1 三角函数 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\n1.1.1 和差公式、万能公式、倍角公式、半角公式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\n1.1.2 和差化积与积化和差. . . .\n1.1.3 交替三角函数. . .\n1.1.4 重要的三角恒等式. . .\n1.1.5 反三角函数. . .\n1.1.6 三角不等式. . .\n1.2 恒等式. . .\n1.2.1 立方和、四次方和. . .\n1.2.2 最大值、最小值. . .\n1.2.3 次方和. . .\n1.3 不等式. . .\n1.3.1 基本不等式. . .\n1.3.2 Jensen 不等式. . .\n1.3.3 调和级数不等式. . .\n1.4 几何学基础. . .\n1.4.1 常见立体几何体积. . .\n第2章 数列极限 6\n2.1 数列极限的概念与性质. . .\n2.1.1 数列极限的概念. . .\n2.1.2 收敛数列的性质. . .\n2.2 一些重要极限结论. . .\n2.2.1 几个基本极限. . .\n2.2.2 Stirling 公式与 Wallis 公式. . .\n2.2.3 Euler 常数. . .\n2.3 夹逼准则. . .\n2.4 平均值定理. . .\n2.4.1 有限项幂次根号平均. . .\n2.4.2 平均值定理：无穷项算术平均与几何平均. \n2.4.3 技巧：一招解决 n 次根号与除以 n 10\n2.5 使用定积分定义. 10\n2.5.1 使用定积分计算极限. 10\n2.5.2 利用定积分定义构造等价无穷小 12\n2.6 单调有界定理 12\n2.6.1 单调有界定理及其推广 12\n2.6.2 相减法 13\n2.6.3 相除法 13\n2.6.4 不等式放缩 14\n2.7 递推问题 14" } ] }, "bbox": [ 92, 149, 910, 903 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2.7.1 极限压缩定理 14 \n2.7.2 压缩映射定理 15 \n2.8 Stolz 定理 15 \n2.9 积分形式极限：分段法 16 \n2.9.1 定积分形极限 16 \n2.9.2 反常积分形极限 16 \n2.9.3 周期函数积分极限 17 \n\n第3章 函数极限与连续性 18 \n3.1 函数的极限 18 \n3.1.1 函数极限定义 18 \n3.1.2 函数极限的性质 18 \n3.1.3 无穷大与无穷小 19 \n3.1.4 使用定义直接计算 19 \n3.2 函数极限的计算 19 \n3.2.1 等价无穷小公式 19 \n3.2.2 Taylor 展开公式 20 \n3.3 函数的连续性 20 \n3.3.1 函数连续的定义 20 \n3.3.2 间断点的概念 21 \n3.3.3 介值性定理 22 \n3.3.4 介值性定理的无连续推广 23 \n3.4 连续性的应用 23 \n3.5 一致连续 24 \n3.5.1 一致连续的概念 24 \n3.5.2 Cantor 定理及其推广 25 \n3.5.3 Lip 连续与一致连续 26 \n3.5.4 区间合并问题 26 \n3.5.5 四则运算、复合的一致连续性 27 \n3.6 一致连续的经典问题 27 \n\n第4章 一元微分学 28 \n4.1 导数与微分 28 \n4.1.1 导数概念与可导性 28 \n4.1.2 微分的概念 29 \n4.1.3 常用的导数公式 29 \n4.1.4 隐函数、反函数求导 30 \n4.1.5 高阶导数 31 \n4.1.6 使用导数证明恒等式 31 \n4.2 微分中值定理 32 \n4.2.1 三大中值定理 32 \n4.2.2 中值问题：构造辅助函数 33 \n4.2.3 带积分的微分中值问题 35 \n4.2.4 多个中值点的中值问题 35 \n4.2.5 中值等式问题 36" } ] }, "bbox": [ 102, 82, 910, 902 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "目录" } ] }, "bbox": [ 870, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "ii" } ] }, "bbox": [ 492, 954, 505, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "4.2.6 中值不等式 36 \n4.2.7 导数极限定理与导数介值定理 37 \n4.3 洛必达法则与Taylor公式 38 \n4.3.1 洛必达法则 38 \n4.3.2 Taylor公式 38 \n4.4 凸函数 39 \n4.4.1 函数的凹凸性 39 \n4.4.2 凸函数的基本性质 40 \n4.4.3 利用凹凸性证明不等式 41 \n4.5 一元极值与导数零点问题 41 \n4.5.1 一元极值的概念与判定 41 \n4.5.2 导数零点问题 41 \n\n第5章 不定积分 43 \n5.1 常用不定积分公式 43 \n5.2 不定积分基本方法 43 \n5.3 不定积分表选例 45 \n5.3.1 三角函数：重点 45 \n5.3.2 对数与指数 48 \n5.3.3 分式问题 48 \n5.3.4 根式问题 49 \n5.3.5 配对积分法 51 \n5.3.6 递推问题 51 \n\n第6章 定积分 52 \n6.1 定积分理论 52 \n6.1.1 可积性 52 \n6.1.2 Newton-Leibniz公式 53 \n6.1.3 四则运算与复合 54 \n6.1.4 定积分的保号性 55 \n6.1.5 积分中值定理 56 \n6.2 三角函数定积分计算 57 \n6.2.1 " }, { "type": "equation_inline", "content": "[0,\\frac{\\pi}{2}]" }, { "type": "text", "content": " 上三角函数积分公式 57 \n6.2.2 " }, { "type": "equation_inline", "content": "[0,\\pi],[0,2\\pi]" }, { "type": "text", "content": " 上三角函数积分 58 \n6.2.3 区间再现公式 59 \n6.2.4 Euler积分 60 \n6.2.5 Dirichlet积分 60 \n6.2.6 其他几个常见三角积分 61 \n6.3 积分(不)等式 61 \n6.3.1 整体放缩 61 \n6.3.2 同区域单调性放缩 61 \n6.3.3 利用Newton-Leibniz公式 62 \n6.3.4 利用Cauchy-Schwarz不等式 63 \n6.3.5 利用微分中值定理 63" } ] }, "bbox": [ 102, 82, 910, 884 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "第7章 反常积分" } ] }, "bbox": [ 90, 895, 226, 910 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "目录" } ] }, "bbox": [ 870, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "65" } ] }, "bbox": [ 884, 897, 905, 909 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "iii" } ] }, "bbox": [ 490, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "7.1 反常积分理论 65\n7.1.1 无穷积分与瑕积分概念 65\n7.1.2 反常积分比较判别法 66\n7.1.3 Dirichlet与Abel判别法 67\n7.1.4 函数在无穷远处的性质 69\n7.2 反常积分的计算 70\n7.2.1 两个常用的基础反常积分 70\n7.2.2 富如兰尼积分 70\n7.2.3 Dirichlet积分 71\n第8章 数项级数 73\n8.1 数项级数基本理论 73\n8.2 正项级数 74\n8.2.1 比较原则 74\n8.2.2 比式判别法与根式判别法 75\n8.2.3 积分判别法和拉贝判别法 77\n8.2.4 放缩证明敛散性 77\n8.3 任意项级数 78\n8.3.1 交错项级数 78\n8.3.2 Dirichlet与Abel判别法 79\n8.4 数项级数理论 79\n8.4.1 级数的加括号 79\n8.4.2 级数的重排 80\n8.4.3 正部与负部 81\n8.4.4 级数的乘积 82\n8.4.5 级数的相互控制 82\n第9章 函数列与函数项级数 84\n9.1 函数列的一致收敛 84\n9.1.1 一致收敛的概念 84\n9.1.2 一致收敛的基本判别法 84\n9.1.3 一致收敛判别练习 85\n9.1.4 四则运算、复合极限的一致收敛性 86\n9.2 函数列极限的性质 87\n9.2.1 极限函数的连续性 87\n9.2.2 极限函数的可积性 88\n9.2.3 极限函数的可微性 89\n9.2.4 连续+单调：推出一致收敛 90\n9.3 函数项级数的一致收敛性 90\n9.3.1 一致收敛的基本性质与判别 90\n9.3.2 Dirichlet与Abel判别法 91\n9.4 函数项级数极限函数的性质 92\n第10章 幂级数 93\n10.1 幂级数概念及其收敛区间 93\n10.1.1 收敛区间与收敛半径 93" } ] }, "bbox": [ 97, 82, 910, 915 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "目录" } ] }, "bbox": [ 870, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "iv" } ] }, "bbox": [ 490, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "10.1.2 幂级数的一致收敛性 ..... 94\n10.1.3 缺项幂级数的收敛区域 ..... 95\n10.2 幂级数展开式求和 ..... 95\n10.2.1 等比级数 ..... 96\n10.2.2 arctan x 级数 ..... 96\n10.2.3 " }, { "type": "equation_inline", "content": "e^x" }, { "type": "text", "content": " 级数 ..... 97\n10.3 幂级数求和式展开 ..... 97\n10.3.1 基本理论与直接展开 ..... 97\n10.3.2 求导与求积 ..... 98\n\n第 11 章 傅里叶级数 ..... 99\n11.1 以 " }, { "type": "equation_inline", "content": "2\\pi" }, { "type": "text", "content": " 为周期的傅里叶级数 ..... 99\n11.2 以 " }, { "type": "equation_inline", "content": "2l" }, { "type": "text", "content": " 为周期的傅里叶级数 ..... 99\n11.3 Parseval 等式 ..... 100\n\n第 12 章 多元极限与连续 ..... 101\n12.1 累次极限与重极限 ..... 101\n12.1.1 基本概念 ..... 101\n12.1.2 重极限的计算 ..... 102\n12.2 多元函数连续性 ..... 103\n12.2.1 多元函数连续性 ..... 103\n12.2.2 多元函数的一致连续性 ..... 103\n\n第 13 章 多元微分学 ..... 105\n13.1 多变量导数与微分 ..... 105\n13.1.1 全微分与偏导数 ..... 105\n13.1.2 可微性的证明 ..... 105\n13.1.3 梯度与方向导数 ..... 106\n13.1.4 可微、偏导、方向导数、连续关系总结 ..... 107\n13.1.5 复合求导：链式法则 ..... 108\n13.1.6 混合偏导 ..... 108\n13.2 多元中值定理与泰勒展开 ..... 109\n13.2.1 多元中值定理 ..... 109\n13.2.2 多元 Taylor 展开 ..... 110\n13.3 隐函数定理 ..... 110\n13.3.1 隐函数存在定理与可微定理 ..... 110\n13.3.2 隐函数组存在定理与微分定理 ..... 111\n13.4 多元无条件极值问题 ..... 112\n13.4.1 无条件极值 ..... 113\n13.4.2 最值问题：极值点 + 边界 ..... 114\n13.4.3 多元极值理论证明 ..... 114\n13.5 条件极值 ..... 115\n13.5.1 Lagrange 乘数法 ..... 115\n\n第 14 章 含参积分 ..... 117\n14.1 含参正常积分 ..... 117" } ] }, "bbox": [ 95, 82, 910, 908 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "目录" } ] }, "bbox": [ 870, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "V" } ] }, "bbox": [ 492, 954, 504, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "14.1.1 含参正常积分基本概念与定理 ..... 117\n14.1.2 常用含参正常积分的计算 ..... 118\n14.2 含参反常积分与一致收敛 ..... 119\n14.2.1 基本概念 ..... 119\n14.2.2 Dirichlet 与 Abel 判别法 ..... 120\n14.3 含参反常积分的性质 ..... 121\n14.3.1 含参反常积分的极限与连续性 ..... 121\n14.3.2 含参反常积分的可微性与可积性 ..... 121\n14.3.3 含参反常积分与函数项级数的关系 ..... 122\n\n第 15 章 多重积分 ..... 124\n15.1 二重积分 ..... 124\n15.1.1 二重积分的概念与计算 ..... 124\n15.1.2 二重积分可积性 ..... 125\n15.1.3 二重积分换元法 ..... 125\n15.1.4 极坐标换元 ..... 126\n15.2 三重积分 ..... 127\n15.2.1 投影法 ..... 127\n15.2.2 截面法 ..... 128\n15.2.3 三重积分换元法 ..... 128\n15.2.4 柱面坐标变换与球坐标变换 ..... 128\n\n第 16 章 曲线积分 ..... 130\n16.1 曲线的参数化 ..... 130\n16.2 第一型曲线积分 ..... 130\n16.2.1 第一型曲线积分的基本概念 ..... 130\n16.2.2 使用对称性 ..... 131\n16.2.3 曲线积分中值问题 ..... 132\n16.3 第二型曲线积分 ..... 132\n16.3.1 基本概念与参数计算 ..... 132\n16.3.2 使用对称性 ..... 133\n16.3.3 两类曲线积分的关系 ..... 135\n16.4 Green 公式 ..... 135\n16.4.1 Green 公式及其基本应用 ..... 135\n16.4.2 内部有瑕点 ..... 136\n16.4.3 曲线积分与路径无关性 ..... 137\n\n第 17 章 曲面积分 ..... 138\n17.1 曲面的面积 ..... 138\n17.2 第一型曲面积分 ..... 138\n17.2.1 基本概念与计算公式 ..... 138\n17.2.2 使用对称性 ..... 139\n17.3 第二型曲面积分 ..... 140\n17.3.1 基本概念、投影法、参数方程法 ..... 140\n17.3.2 利用对称性 ..... 141\n17.3.3 两类曲面积分的联系 ..... 141" } ] }, "bbox": [ 90, 83, 910, 915 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "目录" } ] }, "bbox": [ 870, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "vi" } ] }, "bbox": [ 490, 954, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "17.4 Gauss 公式 ..... 142\n17.4.1 Gauss 公式与基本使用 ..... 142\n17.4.2 内部有瑕点 ..... 143\n17.5 Stokes 公式 ..... 143\n17.5.1 Stokes 公式与基本使用 ..... 143\n17.5.2 空间第二型曲线积分与路径无关性 ..... 145\n18 章 实数完备性定理 ..... 146" } ] }, "bbox": [ 114, 84, 907, 223 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "目录" } ] }, "bbox": [ 870, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "vii" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第1章 数学基础" } ], "level": 2 }, "bbox": [ 394, 87, 620, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.1 三角函数" } ], "level": 2 }, "bbox": [ 95, 145, 243, 165 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1.1.1 和差公式、万能公式、倍角公式、半角公式" } ] }, "bbox": [ 92, 184, 536, 202 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "一、 " }, { "type": "equation_inline", "content": "\\sin x, \\cos x" }, { "type": "text", "content": " 和差公式" } ] }, "bbox": [ 89, 212, 278, 228 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin (\\alpha + \\beta) = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta", "math_type": "latex", "image_source": { "path": "images/9bf3eacd2bd6ef02558d16110ae33fcb227e0d92f980f1199b9accb57aa6f4b7.jpg" } }, "bbox": [ 361, 254, 635, 271 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin (\\alpha - \\beta) = \\sin \\alpha \\cos \\beta - \\cos \\alpha \\sin \\beta", "math_type": "latex", "image_source": { "path": "images/929ebf66f0f587a97c1576c4e7de46bf1efdeb8c89d80db894ed2ea01e36e978.jpg" } }, "bbox": [ 361, 275, 635, 293 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos (\\alpha + \\beta) = \\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta", "math_type": "latex", "image_source": { "path": "images/3a40942c88c8a2dd32fb019baddd19d6d33820bf1ddfb031f9d1d70d019c5bf8.jpg" } }, "bbox": [ 361, 298, 636, 315 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos (\\alpha - \\beta) = \\cos \\alpha \\cos \\beta + \\sin \\alpha \\sin \\beta", "math_type": "latex", "image_source": { "path": "images/e60ba28ca8d69993c0b2126d5e689980924558816d2201027bb27bb5c337222e.jpg" } }, "bbox": [ 361, 319, 636, 337 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "二、 " }, { "type": "equation_inline", "content": "\\tan x, \\cot x" }, { "type": "text", "content": " 和差公式" } ] }, "bbox": [ 90, 365, 280, 381 ] }, { "type": "equation_interline", "content": { "math_content": "\\tan (\\alpha + \\beta) = \\frac {\\tan \\alpha + \\tan \\beta}{1 - \\tan \\alpha \\tan \\beta}", "math_type": "latex", "image_source": { "path": "images/fc738b40c533683898ad3cc49d1e19bf2c475cd6e75f643720b0ba4f6b7e37fa.jpg" } }, "bbox": [ 388, 400, 606, 430 ] }, { "type": "equation_interline", "content": { "math_content": "\\tan (\\alpha - \\beta) = \\frac {\\tan \\alpha - \\tan \\beta}{1 + \\tan \\alpha \\tan \\beta}", "math_type": "latex", "image_source": { "path": "images/0fe4d381ffebdba14c9b86e0d96ebd534094cd8313f11c71ee297b1586cfb240.jpg" } }, "bbox": [ 389, 432, 606, 461 ] }, { "type": "equation_interline", "content": { "math_content": "\\cot (\\alpha + \\beta) = \\frac {\\cot \\alpha \\cot \\beta - 1}{\\cot \\beta + \\cot \\alpha}", "math_type": "latex", "image_source": { "path": "images/c9a5d648c6f56993d6eeba638839aa3aa1940e822475d95db2269aa6fa1c5aa9.jpg" } }, "bbox": [ 391, 463, 603, 492 ] }, { "type": "equation_interline", "content": { "math_content": "\\cot (\\alpha - \\beta) = \\frac {\\cot \\alpha \\cot \\beta + 1}{\\cot \\beta - \\cot \\alpha}", "math_type": "latex", "image_source": { "path": "images/7d09a2b9bdfda55728f480b346bbcaaea0b670bca96617aaa433bf391e208945.jpg" } }, "bbox": [ 391, 495, 603, 524 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "三、万能公式" } ] }, "bbox": [ 87, 526, 200, 542 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin \\alpha = \\frac {2 \\tan {\\frac {\\alpha}{2}}}{1 + \\tan^ {2} {\\frac {\\alpha}{2}}}", "math_type": "latex", "image_source": { "path": "images/924cc87867a88e899bb5c1cb7320f770a712970b24487640ea5b334fc146fa89.jpg" } }, "bbox": [ 425, 561, 569, 593 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos \\alpha = \\frac {1 - \\tan^ {2} {\\frac {\\alpha}{2}}}{1 + \\tan^ {2} {\\frac {\\alpha}{2}}}", "math_type": "latex", "image_source": { "path": "images/c5b4180a76284f74544eb4bc0e1df332ee0aa6df98b9c239c7825ed69f0e9742.jpg" } }, "bbox": [ 426, 595, 569, 629 ] }, { "type": "equation_interline", "content": { "math_content": "\\tan \\alpha = \\frac {2 \\tan {\\frac {\\alpha}{2}}}{1 - \\tan^ {2} {\\frac {\\alpha}{2}}}", "math_type": "latex", "image_source": { "path": "images/f61dc6d0a2f54f24008ef605e1992a90e0723bfd7b449c65ac3b024937973fdb.jpg" } }, "bbox": [ 426, 630, 571, 664 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "四、倍角公式" } ] }, "bbox": [ 90, 665, 200, 681 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sin 2\\alpha = 2 \\sin \\alpha \\cos \\alpha" } ] }, "bbox": [ 420, 709, 576, 722 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos 2 \\alpha = \\cos^ {2} \\alpha - \\sin^ {2} \\alpha = 2 \\cos^ {2} \\alpha - 1 = 1 - 2 \\sin^ {2} \\alpha", "math_type": "latex", "image_source": { "path": "images/e048737a2289970636851a86190fa948826323846c303a053c5b1f18d47dd641.jpg" } }, "bbox": [ 302, 728, 694, 745 ] }, { "type": "equation_interline", "content": { "math_content": "\\tan 2 \\alpha = \\frac {2 \\tan \\alpha}{1 - \\tan^ {2} \\alpha}", "math_type": "latex", "image_source": { "path": "images/fe6c45413082fcd4cf2a2f2956ee425d70de1aa1432e387d1de4d03f3e4ceefd.jpg" } }, "bbox": [ 420, 749, 574, 778 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "五、半角公式" } ] }, "bbox": [ 92, 780, 200, 795 ] } ], [ { "type": "equation_interline", "content": { "math_content": "\\sin {\\frac {\\alpha}{2}} = \\pm \\sqrt {\\frac {1 - \\cos \\alpha}{2}}", "math_type": "latex", "image_source": { "path": "images/c93b27026549f56f62e2e03c7fb60941321ff3796add92e30ca0645110989f41.jpg" } }, "bbox": [ 324, 102, 492, 133 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos {\\frac {\\alpha}{2}} = \\pm \\sqrt {\\frac {1 + \\cos \\alpha}{2}}", "math_type": "latex", "image_source": { "path": "images/d6ea403b6678d685c0a90286fdb1fd0641a1730eb363e90f7df5f8be25eea467.jpg" } }, "bbox": [ 324, 136, 492, 167 ] }, { "type": "equation_interline", "content": { "math_content": "\\tan {\\frac {\\alpha}{2}} = \\frac {\\sin \\alpha}{1 + \\cos \\alpha} = \\frac {1 - \\cos \\alpha}{\\sin \\alpha} = \\pm \\sqrt {\\frac {1 - \\cos \\alpha}{1 + \\cos \\alpha}}", "math_type": "latex", "image_source": { "path": "images/013c5aa387e90092a18e6b244861c0f00b90b71e0452a9977ae6f5637b7ce750.jpg" } }, "bbox": [ 322, 168, 673, 200 ] }, { "type": "equation_interline", "content": { "math_content": "\\cot {\\frac {\\alpha}{2}} = \\frac {1 + \\cos \\alpha}{\\sin \\alpha} = \\frac {\\sin \\alpha}{1 - \\cos \\alpha} = \\pm \\sqrt {\\frac {1 + \\cos \\alpha}{1 - \\cos \\alpha}}", "math_type": "latex", "image_source": { "path": "images/e7f37c4653e4213b3a907edbcc5dc6f107ad1063e95279b5dfe8a248e4d5d1c7.jpg" } }, "bbox": [ 324, 203, 673, 234 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.1.2 和差化积与积化和差" } ], "level": 2 }, "bbox": [ 94, 252, 332, 268 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "一、和差化积" } ], "level": 2 }, "bbox": [ 90, 280, 200, 294 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos \\alpha + \\cos \\beta = 2 \\cos {\\frac {\\alpha + \\beta}{2}} \\cos {\\frac {\\alpha - \\beta}{2}}", "math_type": "latex", "image_source": { "path": "images/553f15a99a3e43aa8ee327c2bdfd43d4ef6aee4fd5ce072d484bc017e39627e0.jpg" } }, "bbox": [ 356, 313, 633, 343 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos \\alpha - \\cos \\beta = - 2 \\sin {\\frac {\\alpha + \\beta}{2}} \\sin {\\frac {\\alpha - \\beta}{2}}", "math_type": "latex", "image_source": { "path": "images/42eb4dc176322e3946aed68547173e4c0157394480e39293d4c983dcefe9bfcc.jpg" } }, "bbox": [ 357, 344, 642, 372 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin \\alpha + \\sin \\beta = 2 \\sin {\\frac {\\alpha + \\beta}{2}} \\cos {\\frac {\\alpha - \\beta}{2}}", "math_type": "latex", "image_source": { "path": "images/d68fca6afc883974e39108369914f648be2aac81f2bb749b5a0813ab3268cde2.jpg" } }, "bbox": [ 356, 373, 628, 401 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin \\alpha - \\sin \\beta = 2 \\cos {\\frac {\\alpha + \\beta}{2}} \\sin {\\frac {\\alpha - \\beta}{2}}", "math_type": "latex", "image_source": { "path": "images/159de4047719425eab6fc4ff6dd77a90d6c275e3c34e2d8b9db654cd92a14489.jpg" } }, "bbox": [ 357, 401, 626, 430 ] }, { "type": "equation_interline", "content": { "math_content": "\\tan \\alpha + \\tan \\beta = \\frac {\\sin (\\alpha + \\beta)}{\\cos \\alpha \\cos \\beta}", "math_type": "latex", "image_source": { "path": "images/4190105fdaa1ce6131ee6f30ff09bac754e920feec87db684a4777c2905d6be1.jpg" } }, "bbox": [ 357, 432, 559, 462 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "二、积化和差：异 " }, { "type": "equation_inline", "content": "\\sin" }, { "type": "text", "content": " 同cos" } ], "level": 2 }, "bbox": [ 90, 464, 304, 480 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos \\alpha \\cos \\beta = \\frac {1}{2} [ \\cos (\\alpha + \\beta) + \\cos (\\alpha - \\beta) ]", "math_type": "latex", "image_source": { "path": "images/da76859f68fe5941a9b1b8ec05fec300ab2c2f97c165f37eed999e70915e845f.jpg" } }, "bbox": [ 344, 500, 645, 527 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin \\alpha \\sin \\beta = - \\frac {1}{2} [ \\cos (\\alpha + \\beta) - \\cos (\\alpha - \\beta) ]", "math_type": "latex", "image_source": { "path": "images/f4806057f23df2fed39f73be820f134950b825a98fc9d1948d9edd79f11d78fa.jpg" } }, "bbox": [ 344, 530, 653, 556 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin \\alpha \\cos \\beta = \\frac {1}{2} [ \\sin (\\alpha + \\beta) + \\sin (\\alpha - \\beta) ]", "math_type": "latex", "image_source": { "path": "images/f61579ad785d6a2345363d1854efbabe06555f76b6b76f6cadfb800e7e8c2466.jpg" } }, "bbox": [ 344, 558, 638, 585 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos \\alpha \\sin \\beta = \\frac {1}{2} [ \\sin (\\alpha + \\beta) - \\sin (\\alpha - \\beta) ]", "math_type": "latex", "image_source": { "path": "images/db32b1760bcaf9af1fab9809260be95b89304e541e68e1015d3b06502198273e.jpg" } }, "bbox": [ 344, 587, 638, 613 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.1.3 交替三角函数" } ], "level": 2 }, "bbox": [ 95, 632, 273, 648 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "(-1)^{n}\\sin nx=\\sin(x+\\pi)n" } ] }, "bbox": [ 110, 659, 334, 675 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. " }, { "type": "equation_inline", "content": "(-1)^{n}\\cos nx = \\cos (x + \\pi)n" } ] }, "bbox": [ 110, 678, 337, 694 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.1.4 重要的三角恒等式" } ], "level": 2 }, "bbox": [ 95, 715, 312, 731 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos \\frac {x}{2 ^ {n}} \\cdot \\cos \\frac {x}{2 ^ {n - 1}} \\dots \\cos \\frac {x}{2} = \\frac {\\sin x}{2 ^ {n} \\sin \\frac {x}{2 ^ {n}}}", "math_type": "latex", "image_source": { "path": "images/01f5e4345f5bb1d02b9987e94bfdbe3343adbf352d31c86e2b90656dbeb625e1.jpg" } }, "bbox": [ 110, 741, 413, 772 ] }, { "type": "equation_interline", "content": { "math_content": "2. \\sin x + \\sin 2 x + \\dots + \\sin n x = \\frac {\\cos \\frac {x}{2} - \\cos (n + \\frac {1}{2}) x}{2 \\sin \\frac {x}{2}}", "math_type": "latex", "image_source": { "path": "images/eee15ac4cc5f1e065f4333f15e5f9b67489f81236707c9562b681fe2386ac71b.jpg" } }, "bbox": [ 110, 772, 505, 803 ] }, { "type": "equation_interline", "content": { "math_content": "3. \\cos x + \\cos 2 x + \\dots + \\cos n x = \\frac {\\sin (n + \\frac {1}{2}) x - \\sin \\frac {x}{2}}{2 \\sin \\frac {x}{2}} = \\frac {\\sin (n + \\frac {1}{2}) x}{2 \\sin \\frac {x}{2}} - \\frac {1}{2}", "math_type": "latex", "image_source": { "path": "images/0aa8a77f1a55fee50c6cb02c1ea4a6a7abdabfe67e53a9afe4311699a9a7e881.jpg" } }, "bbox": [ 110, 802, 650, 837 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "4. " }, { "type": "equation_inline", "content": "\\arctan \\frac{1}{2k^2} = \\arctan \\frac{k}{k + 1} - \\arctan \\frac{k - 1}{k}" } ] }, "bbox": [ 110, 835, 442, 863 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 将右侧分母乘到左侧去，用二倍角即可。" } ] }, "bbox": [ 90, 863, 455, 877 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "1.1 三角函数" } ] }, "bbox": [ 806, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "2" } ] }, "bbox": [ 492, 953, 505, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 左边乘 " }, { "type": "equation_inline", "content": "\\sin \\frac{x}{2}" }, { "type": "text", "content": ", 得到" } ] }, "bbox": [ 122, 80, 295, 107 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin {\\frac {x}{2}} I = \\sum_ {k = 1} ^ {n} \\sin k x \\sin {\\frac {x}{2}} = - \\frac {1}{2} \\sum_ {k = 1} ^ {n} [ \\cos (k + \\frac {1}{2}) x - \\cos (k - \\frac {1}{2}) x ] = \\frac {1}{2} \\left[ \\cos {\\frac {x}{2}} - \\cos (n + \\frac {1}{2}) x \\right]", "math_type": "latex", "image_source": { "path": "images/85977bbdb9798121c45162ab259b97bb78518cc55707ef58770903d4f839cbae.jpg" } }, "bbox": [ 164, 110, 828, 146 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 和 (2) 类似" } ] }, "bbox": [ 124, 149, 231, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.1.5 反三角函数" } ], "level": 2 }, "bbox": [ 95, 187, 252, 204 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "定理 1.1 (反三角函数倒数关系)" } ] }, "bbox": [ 124, 219, 352, 235 ] }, { "type": "equation_interline", "content": { "math_content": "\\arctan x + \\arctan {\\frac {1}{x}} = \\frac {\\pi}{2}", "math_type": "latex", "image_source": { "path": "images/39e8d5908c057620570c301b27fb4bf222432fc7a28f7d23b9337613eb234cac.jpg" } }, "bbox": [ 406, 247, 589, 275 ] }, { "type": "equation_interline", "content": { "math_content": "\\operatorname{arccot} x + \\operatorname{arccot} \\frac {1}{x} = \\frac {\\pi}{2}", "math_type": "latex", "image_source": { "path": "images/2d73e6e101399f8a64bc1f16f811f74fb864bacdf0c4c8dfc3ee18ae9eaff294.jpg" } }, "bbox": [ 413, 275, 581, 304 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 画一个直角三角形，角 " }, { "type": "equation_inline", "content": "\\theta_{1}" }, { "type": "text", "content": " 对着边长 " }, { "type": "equation_inline", "content": "1" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\theta_{2} = \\frac{\\pi}{2} -\\theta_{1}" }, { "type": "text", "content": " 对着边长 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\tan \\theta_1 = \\frac{1}{x},\\tan \\theta_2 = x" }, { "type": "text", "content": " ，而两角互补。" } ] }, "bbox": [ 90, 316, 882, 344 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.1.6 三角不等式" } ], "level": 2 }, "bbox": [ 95, 362, 253, 379 ] }, { "type": "equation_interline", "content": { "math_content": "\\text { 定理 } 1. 2 (\\sin x \\text { 不等式 })", "math_type": "latex", "image_source": { "path": "images/aaa8edbfa35b3a1de74e8bde02e4d754139163b93265564a53f4a6b4e5535991.jpg" } }, "bbox": [ 122, 395, 290, 409 ] }, { "type": "equation_interline", "content": { "math_content": "x \\in (0, {\\frac {\\pi}{2}}) \\text { 时 }, {\\frac {2 x}{\\pi}} < \\sin x < x < \\tan x", "math_type": "latex", "image_source": { "path": "images/50f4d7511937e50a0cace29140271c1e2e96dd53c1861e7a784dac431e40e353.jpg" } }, "bbox": [ 114, 414, 400, 442 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 1.3 (反三角不等式)" } ], "level": 2 }, "bbox": [ 122, 466, 302, 481 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\arcsin x > x > \\arctan x" }, { "type": "text", "content": " ，当 x 为小正数时。" } ] }, "bbox": [ 114, 487, 431, 500 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.2 恒等式" } ], "level": 2 }, "bbox": [ 95, 544, 216, 563 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.2.1 立方和、四次方和" } ], "level": 2 }, "bbox": [ 95, 583, 312, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "定理 1.4 (立方和公式)" } ] }, "bbox": [ 122, 615, 285, 630 ] }, { "type": "equation_interline", "content": { "math_content": "a ^ {3} + b ^ {3} = (a + b) \\left(a ^ {2} - a b + b ^ {2}\\right), a ^ {3} - b ^ {3} = (a - b) \\left(a ^ {2} + a b + b ^ {2}\\right)", "math_type": "latex", "image_source": { "path": "images/ab3c613414d9cfe6ebc326bc84d0432f94532ceb0b83d0513f287eff8bbd17dc.jpg" } }, "bbox": [ 114, 634, 576, 653 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 1.5 (四次方和差公式)" } ], "level": 2 }, "bbox": [ 122, 677, 319, 692 ] }, { "type": "equation_interline", "content": { "math_content": "a ^ {4} + b ^ {4} = (a ^ {2} + b ^ {2}) ^ {2} - 2 a ^ {2} b ^ {2} = (a ^ {2} + b ^ {2} - \\sqrt {2} a b) (a ^ {2} + b ^ {2} - \\sqrt {2} a b)", "math_type": "latex", "image_source": { "path": "images/834e70dd61c62b07d79558688770b78c6af80df534d975366d47b5c4c5f89b6a.jpg" } }, "bbox": [ 114, 697, 594, 714 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.2.2 最大值、最小值" } ], "level": 2 }, "bbox": [ 95, 753, 292, 769 ] }, { "type": "equation_interline", "content": { "math_content": "\\min \\{a, b \\} = \\frac {1}{2} (a + b - | a - b |), \\max \\{a, b \\} = \\frac {1}{2} (a + b + | a - b |)", "math_type": "latex", "image_source": { "path": "images/841f42d6e27b1b284087830150fcba8b2bc92ab80267ab79438ea032ca2665db.jpg" } }, "bbox": [ 270, 793, 724, 821 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "1.2 恒等式" } ] }, "bbox": [ 823, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "3" } ] }, "bbox": [ 492, 954, 504, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.2.3 次方和" } ], "level": 2 }, "bbox": [ 95, 85, 213, 102 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} 1 + 2 + \\dots + n = \\frac {n (n - 1)}{2} \\\\ 1 ^ {2} + 2 ^ {2} + \\dots + n ^ {2} = \\frac {n (n + 1) (2 n + 1)}{6} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/c90a919cf511a3c48a04aedad16d915694ccf893f8e962f2665c4b4ede51b0d8.jpg" } }, "bbox": [ 339, 124, 638, 180 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.3 不等式" } ], "level": 2 }, "bbox": [ 95, 200, 218, 219 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.3.1 基本不等式" } ], "level": 2 }, "bbox": [ 94, 239, 253, 256 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "一、均值不等式：" } ], "level": 2 }, "bbox": [ 90, 267, 220, 282 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {n}{\\frac {1}{x _ {1}} + \\cdots + \\frac {1}{x _ {n}}} \\leq \\sqrt [ n ]{x _ {1} x _ {2} \\cdots x _ {n}} \\leq \\frac {x _ {1} + \\cdots + x _ {n}}{n} \\leq \\sqrt {\\frac {x _ {1} ^ {2} + \\cdots + x _ {n} ^ {2}}{n}}", "math_type": "latex", "image_source": { "path": "images/234dccefe2ca5f0ef7ccd9ac5287b906aef50d133ac416f83aa6ae05371cb780.jpg" } }, "bbox": [ 255, 298, 741, 335 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "二、对数不等式" } ], "level": 2 }, "bbox": [ 90, 341, 215, 357 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {x}{1 + x} \\leq \\ln (1 + x) \\leq x", "math_type": "latex", "image_source": { "path": "images/10945e2ec75df493fb91e09723f3a3047ab8d987faf827270a77af9ab6edd30a.jpg" } }, "bbox": [ 416, 375, 584, 401 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.3.2 Jensen 不等式" } ], "level": 2 }, "bbox": [ 95, 419, 277, 437 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 1.6 (Jensen 不等式)" } ], "level": 2 }, "bbox": [ 121, 451, 305, 467 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n} t_i = 1" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是 (下) 凸函数, 则" } ] }, "bbox": [ 115, 474, 378, 510 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {i = 1} ^ {n} t _ {i} f (x _ {i}) \\geq f (\\sum_ {i = 1} ^ {n} t _ {i} x _ {i})", "math_type": "latex", "image_source": { "path": "images/56e091a98d189643aaa5fee4b05c554e97b256787f232d9327675754a655beba.jpg" } }, "bbox": [ 405, 508, 589, 546 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 1.1Jensen 不等式的应用 (1)ZJU2021: 若 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n} p_{i} = 1, \\quad p_{i} \\in (0,1), \\quad x_{1}, \\cdots, x_{n} \\in R" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n} p_{i}(x_{i} - \\ln p_{i}) \\leq \\ln(\\sum_{i=1}^{n} e^{x_{i}})" } ] }, "bbox": [ 58, 558, 904, 630 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据 " }, { "type": "equation_inline", "content": "x_{i} = \\ln (e^{x_{i}})" }, { "type": "text", "content": "，因此即证明 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n} p_{i} (\\ln \\frac{e^{x_{i}}}{p_{i}}) \\leq \\ln (\\sum_{i=1}^{n} e^{x_{i}})" }, { "type": "text", "content": "，设 " }, { "type": "equation_inline", "content": "t_{i} = \\frac{e^{x_{i}}}{p_{i}}" }, { "type": "text", "content": "，不等式变为 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n} p_{i} \\ln (t_{i}) \\leq \\ln (\\sum_{i=1}^{n} t_{i} p_{i})" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "\\ln x" }, { "type": "text", "content": " 是凹函数，因此得证。" } ] }, "bbox": [ 87, 629, 905, 703 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.3.3 调和级数不等式" } ], "level": 2 }, "bbox": [ 95, 722, 294, 739 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 1.7 (调和级数不等式)" } ], "level": 2 }, "bbox": [ 122, 755, 321, 770 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\ln(n+1)<1+\\frac{1}{2}+\\cdots+\\frac{1}{n}<1+\\ln n," }, { "type": "text", "content": " 进一步地， " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{1+\\frac{1}{2}+\\cdots+\\frac{1}{n}}{\\ln n}=1" }, { "type": "text", "content": " (即 Euler 常数)" } ] }, "bbox": [ 110, 772, 794, 808 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "\\frac{x}{1 + x} \\leq \\ln (1 + x) \\leq x" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\frac{\\frac{1}{n}}{1 + \\frac{1}{n}} < \\ln (1 + \\frac{1}{n}) < \\frac{1}{n}" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "\\frac{1}{n + 1} < \\ln \\frac{n + 1}{n} < \\frac{1}{n}" }, { "type": "text", "content": "，求和得到：" } ] }, "bbox": [ 89, 816, 855, 852 ] }, { "type": "equation_interline", "content": { "math_content": "1 + \\frac {1}{2} + \\dots + \\frac {1}{n} < 1 + \\ln \\frac {2}{1} + \\dots + \\ln \\frac {n}{n - 1} = 1 + \\ln n", "math_type": "latex", "image_source": { "path": "images/4327be9307731cdf07679450dad2792a99b69daa6a3f1669d3803593d9161b4e.jpg" } }, "bbox": [ 295, 854, 700, 884 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "另一侧同理" } ] }, "bbox": [ 90, 885, 179, 900 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "1.3 不等式" } ] }, "bbox": [ 823, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "4" } ] }, "bbox": [ 492, 954, 505, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.4 几何学基础" } ], "level": 2 }, "bbox": [ 95, 87, 265, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1.4.1 常见立体几何体积" } ], "level": 2 }, "bbox": [ 95, 127, 312, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 1.8 (椭球面积)" } ], "level": 2 }, "bbox": [ 122, 160, 270, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "ax^{2} + by^{2} + cz^{2}\\leq r^{2}" }, { "type": "text", "content": " 的体积为 " }, { "type": "equation_inline", "content": "\\frac{4\\pi r^3}{3\\sqrt{abc}}" } ] }, "bbox": [ 114, 179, 396, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 做仿射变换 " }, { "type": "equation_inline", "content": "x' = \\sqrt{ax}, y' = \\sqrt{by}, z' = \\sqrt{cz}" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "S' = \\frac{4}{3}\\pi r^3" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 92, 225, 605, 250 ] }, { "type": "equation_interline", "content": { "math_content": "S = \\frac {S ^ {\\prime}}{\\sqrt {a b c}} = \\frac {4 \\pi r ^ {3}}{3 \\sqrt {a b c}}", "math_type": "latex", "image_source": { "path": "images/d61d365aa1812c91236192a2f33e85dda289b543af265e646a750f1c3955cd28.jpg" } }, "bbox": [ 421, 255, 573, 287 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "1.4 几何学基础" } ] }, "bbox": [ 789, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "5" } ] }, "bbox": [ 494, 954, 504, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 2 章 数列极限" } ], "level": 1 }, "bbox": [ 396, 87, 618, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.1 数列极限的概念与性质" } ], "level": 2 }, "bbox": [ 94, 145, 384, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.1.1 数列极限的概念" } ], "level": 2 }, "bbox": [ 94, 184, 292, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 2.1 (数列极限)" } ], "level": 2 }, "bbox": [ 122, 217, 270, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 是实数列， " }, { "type": "equation_inline", "content": "A\\in \\mathbb{R}" }, { "type": "text", "content": " ，若 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists N,\\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|a_n - A| < \\epsilon" }, { "type": "text", "content": " ，则称 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛到 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " ，记作 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = A" } ] }, "bbox": [ 112, 235, 825, 259 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.1 (极限的四则运算)" } ], "level": 2 }, "bbox": [ 122, 282, 321, 297 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n,\\lim_{n\\to \\infty}b_n" }, { "type": "text", "content": " 都存在(不存在时不能拆！)，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}(a_n\\odot b_n) = \\lim_{n\\to \\infty}a_n\\odot \\lim_{n\\to \\infty}b_n" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "\\odot" }, { "type": "text", "content": " 指任意运算(除法额外要求 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}b_n\\neq 0)" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 114, 302, 882, 337 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.2 (cauchy 收敛准则)" } ], "level": 2 }, "bbox": [ 122, 367, 322, 382 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数列 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛当且仅当 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall m, n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|a_{n} - a_{m}| < \\epsilon" } ] }, "bbox": [ 114, 387, 542, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.1 几个经典极限 (1) 证明 " }, { "type": "equation_inline", "content": "\\sin n" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 60, 423, 400, 437 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 证明 " }, { "type": "equation_inline", "content": "a_{n} = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n}" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 122, 438, 386, 462 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 显然区间 " }, { "type": "equation_inline", "content": "\\left[2k\\pi+\\frac{\\pi}{4},2k\\pi+\\frac{3\\pi}{4}\\right]" }, { "type": "text", "content": " 长度大于 1，一定包含一个整数，可以取 " }, { "type": "equation_inline", "content": "n\\in\\left[2k\\pi+\\frac{\\pi}{4},2k\\pi+\\frac{3\\pi}{4}\\right]" }, { "type": "text", "content": " ，取 " }, { "type": "equation_inline", "content": "m\\in\\left[2k\\pi+\\pi,2k\\pi+2\\pi\\right]" }, { "type": "text", "content": " ，则 n,m 可取无穷大， " }, { "type": "equation_inline", "content": "\\left|\\sin n-\\sin m\\right|\\geq\\frac{\\sqrt{2}}{2}" }, { "type": "text", "content": " ，因此根据 Cauchy 收敛准则知道发散。\n(2) " }, { "type": "equation_inline", "content": "\\forall N" }, { "type": "text", "content": " ，取n=N,m=2N，则 " }, { "type": "equation_inline", "content": "\\left|a_{n}-a_{m}\\right|=\\frac{1}{N+1}+\\cdots+\\frac{1}{2N}\\geq\\frac{N}{2N}=\\frac{1}{2}" }, { "type": "text", "content": " ，因此根据Cauchy收敛准则逆否命题可知发散" } ] }, "bbox": [ 89, 461, 905, 556 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.1.2 收敛数列的性质" } ], "level": 2 }, "bbox": [ 94, 577, 292, 594 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.3 (收敛数列的性质)" } ], "level": 2 }, "bbox": [ 124, 609, 321, 624 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = A" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 115, 630, 265, 646 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 有界性： " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 有界" } ] }, "bbox": [ 139, 648, 282, 662 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 保号性：若 " }, { "type": "equation_inline", "content": "A > a" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "n \\to \\infty" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "a_{n} > a" } ] }, "bbox": [ 139, 667, 458, 681 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 收敛当且仅当 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 所有子列都收敛到 " }, { "type": "equation_inline", "content": "A" } ] }, "bbox": [ 139, 684, 442, 699 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.4 (奇偶子列收敛)" } ], "level": 2 }, "bbox": [ 124, 724, 302, 738 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛到 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 当且仅当 " }, { "type": "equation_inline", "content": "a_{2k}, a_{2k + 1}" }, { "type": "text", "content": " 收敛到 " }, { "type": "equation_inline", "content": "A" } ] }, "bbox": [ 114, 744, 428, 758 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 只右推左：根据偶数列可知 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists K, \\forall 2k > K" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|a_{2k} - A| < \\epsilon" }, { "type": "text", "content": "，而对上面的 " }, { "type": "equation_inline", "content": "\\epsilon, \\exists K', \\forall 2k + 1 > k'" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|a_{2k+1} - A| < \\epsilon" }, { "type": "text", "content": "，因此取 " }, { "type": "equation_inline", "content": "N = \\max\\{K, K'\\}" }, { "type": "text", "content": " 即可。" } ] }, "bbox": [ 90, 778, 905, 814 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.2 一些重要极限结论" } ], "level": 2 }, "bbox": [ 94, 87, 337, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.2.1 几个基本极限" } ], "level": 2 }, "bbox": [ 94, 126, 273, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.5 (常用基本极限)" } ], "level": 2 }, "bbox": [ 122, 160, 302, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "a > 0, b > 1" }, { "type": "text", "content": "，则基本极限的关系为： " }, { "type": "equation_inline", "content": "\\log n < n^a < b^n < n! < n^n" }, { "type": "text", "content": "，更具体地" } ] }, "bbox": [ 115, 179, 685, 193 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 根式：(1) " }, { "type": "equation_inline", "content": "a > 0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{a} = 1(2)\\lim_{n\\to \\infty}\\sqrt[n]{n} = 1(3)\\lim_{n\\to \\infty}\\sqrt[n]{n!} = +\\infty" } ] }, "bbox": [ 139, 197, 628, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 比式：(1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a^n}{n!} = 0(2)a > 1,\\lim_{n\\to \\infty}\\frac{n^k}{a^n} = 0" } ] }, "bbox": [ 139, 219, 480, 247 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 对数：(1) " }, { "type": "equation_inline", "content": "\\forall k, \\lim_{x \\to \\infty} \\frac{\\ln x}{x^k} = 0" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\forall k, \\lim_{x \\to 0^+} x^k \\ln x = 0" } ] }, "bbox": [ 139, 248, 521, 273 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 指数： " }, { "type": "equation_inline", "content": "\\forall k, \\lim_{x\\to +\\infty}\\frac{x^k}{e^x} = 0" } ] }, "bbox": [ 139, 274, 341, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根式第一个设 " }, { "type": "equation_inline", "content": "a_{n} = \\sqrt[n]{a} = 1 + h_{n}" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "a = (1 + h_{n})^{n} \\geq 1 + nh_{n}" }, { "type": "text", "content": "，故 " }, { "type": "equation_inline", "content": "h_{n} \\leq \\frac{a - 1}{n} \\to 0" }, { "type": "text", "content": "。第二个设 " }, { "type": "equation_inline", "content": "a_{n} = \\sqrt[n]{n} = 1 + h_{n}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "n = (1 + h_{n})^{n} \\geq \\frac{n(n - 1)}{2} h_{n}^{2}" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "0 < h_{n} < \\frac{2}{n - 1}" }, { "type": "text", "content": "，夹逼准则即可。第三个用Stirling公式，等价于 " }, { "type": "equation_inline", "content": "\\frac{n}{e}" } ] }, "bbox": [ 90, 319, 878, 391 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.6 (重要极限)" } ], "level": 2 }, "bbox": [ 124, 403, 270, 417 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} (1 + \\frac {1}{n}) ^ {n} = e, \\lim _ {n \\to \\infty} (1 - \\frac {1}{n}) ^ {n} = \\frac {1}{e} 。 \\lim _ {n \\to \\infty} (1 + \\frac {1}{n}) ^ {n} = e - \\frac {e}{2 n} + o (\\frac {1}{n})", "math_type": "latex", "image_source": { "path": "images/1a3befb2d008b2e5e59dbb86f22b5f82c13da154ef5a8e9355e1d9520b0d255d.jpg" } }, "bbox": [ 115, 422, 648, 451 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\left(1 + \\frac{1}{n}\\right)^n = \\lim_{x\\to 0}(1 + x)^{\\frac{1}{x}} = \\lim_{x\\to 0}e^{\\frac{\\ln(1 + x)}{x}} = e^1 = e" } ] }, "bbox": [ 90, 462, 539, 492 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.2相关练习(1)计算 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\left(1 + \\frac{1}{n} +\\frac{1}{n^2}\\right)^n" } ] }, "bbox": [ 60, 498, 450, 529 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)等价于 " }, { "type": "equation_inline", "content": "e^{\\frac{\\ln(1 + x + x^2)}{x}} = e^{\\frac{2x + 1}{1 + x + x^2}} = e" } ] }, "bbox": [ 89, 529, 379, 550 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.2.2 Stirling 公式与 Wallis 公式" } ], "level": 2 }, "bbox": [ 94, 571, 386, 589 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.7 (Stirling 公式)" } ], "level": 2 }, "bbox": [ 122, 604, 295, 619 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "n\\to \\infty" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "n!\\approx \\sqrt{2\\pi n} (\\frac{n}{e})^n" }, { "type": "text", "content": " ，进一步地 " }, { "type": "equation_inline", "content": "\\sqrt[n]{n!}\\sim \\frac{n}{e}" } ] }, "bbox": [ 115, 623, 502, 651 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.8 (Wallis 公式)" } ], "level": 2 }, "bbox": [ 122, 674, 285, 688 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "n\\to \\infty" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\frac{(2n)!!}{(2n - 1)!!}\\sim \\sqrt{n\\pi}" }, { "type": "text", "content": " ，更具体地， " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\left[\\frac{(2n)!!}{(2n - 1)!!}\\right]\\frac{1}{2n + 1} = \\frac{\\pi}{2}" } ] }, "bbox": [ 115, 693, 682, 725 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 " }, { "type": "equation_inline", "content": "\\sin^{2n+1}x < \\sin^{2n}x < \\sin^{2n-1}x" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\frac{\\pi}{2}}\\sin^{2n+1}xdx < \\int_{0}^{\\frac{\\pi}{2}}\\sin^{2n}xdx < \\int_{0}^{\\frac{\\pi}{2}}\\sin^{2n-1}xdx" }, { "type": "text", "content": " ，根据 Wallis 积分公式下面不等式，并构造 " }, { "type": "equation_inline", "content": "A_{n}, B_{n}" } ] }, "bbox": [ 89, 738, 905, 784 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {(2 n) ! !}{(2 n + 1) ! !} < \\frac {(2 n - 1) ! !}{(2 n) ! !} \\frac {\\pi}{2} < \\frac {(2 n - 2) ! !}{(2 n - 1) ! !} \\Rightarrow A _ {n} := \\left[ \\frac {(2 n) ! !}{(2 n - 1) ! !} \\right] ^ {2} \\frac {1}{2 n + 1} < \\frac {\\pi}{2} < \\left[ \\frac {(2 n) ! !}{(2 n - 1) ! !} \\right] ^ {2} \\frac {1}{2 n} := B _ {n}", "math_type": "latex", "image_source": { "path": "images/b14c5d8c54d7dd959cce7811372df0707bb3386a1c220d9406c2234dc27286b2.jpg" } }, "bbox": [ 127, 787, 868, 821 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "此时 " }, { "type": "equation_inline", "content": "B_{n} - A_{n} = \\left[\\frac{(2n)!!}{(2n - 1)!!}\\right]^{2}\\frac{1}{2n(2n - 1)} = A_{n}\\cdot \\frac{1}{2n} < \\frac{\\pi}{2}\\frac{1}{2n}\\rightarrow 0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}A_n = \\lim_{n\\to \\infty}B_n = \\frac{\\pi}{2}" } ] }, "bbox": [ 90, 825, 810, 857 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.2 一些重要极限结论" } ] }, "bbox": [ 737, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "7" } ] }, "bbox": [ 492, 953, 504, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.2.3 Euler 常数" } ], "level": 2 }, "bbox": [ 94, 86, 245, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.9 (Euler 常数)" } ], "level": 2 }, "bbox": [ 122, 118, 280, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "极限 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\left[\\left(\\sum_{k = 1}^{n}\\frac{1}{k}\\right) - \\ln n\\right]" }, { "type": "text", "content": " 存在且为常数 " }, { "type": "equation_inline", "content": "\\gamma" }, { "type": "text", "content": " ，记作Euler常数。" } ] }, "bbox": [ 114, 137, 584, 177 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 2.1 (Euler 常数的应用)" } ], "level": 2 }, "bbox": [ 122, 202, 332, 217 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "n\\to \\infty" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\sum_{k = 1}^{n}\\frac{1}{k} = \\ln n + \\gamma = \\ln n + o(n)" } ] }, "bbox": [ 115, 223, 452, 261 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.3 夹逼准则" } ], "level": 2 }, "bbox": [ 94, 304, 240, 323 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.10 (夹逼准则)" } ], "level": 2 }, "bbox": [ 122, 344, 278, 360 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{n}, b_{n}, c_{n}" }, { "type": "text", "content": " 为三个序列，若 " }, { "type": "equation_inline", "content": "\\exists N" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "n \\geq N" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "a_{n} \\leq b_{n} \\leq c_{n}" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_{n} = \\lim_{n \\to \\infty} c_{n} = A" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} b_{n} = A" }, { "type": "text", "content": "。" } ] }, "bbox": [ 110, 362, 900, 393 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.3两道经典夹逼准则求极限(1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{1}{2}\\cdot \\frac{3}{4}\\dots \\frac{2n - 1}{2n}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sum_{k = n^2}^{(n + 1)^2}\\frac{1}{\\sqrt{k}}" }, { "type": "text", "content": " 解(1)可用Wallis公式，或者夹逼准则。记 " }, { "type": "equation_inline", "content": "a_{n} = \\frac{1}{2}\\cdot \\frac{3}{4}\\dots \\frac{2n - 1}{2n},b_{n} = \\frac{2}{3}\\cdot \\frac{4}{5}\\dots \\frac{2n}{2n + 1}" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "a_{n} < b_{n},a_{n}b_{n} = \\frac{1}{2n + 1}" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "a_n^2 < a_nb_n = \\frac{1}{2n + 1}\\rightarrow 0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = 0" } ] }, "bbox": [ 58, 420, 905, 513 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 用 " }, { "type": "equation_inline", "content": "\\frac{1}{n + 1} \\leq \\frac{1}{\\sqrt{k}} \\leq \\frac{1}{n}" }, { "type": "text", "content": "，从而求和后得到 " }, { "type": "equation_inline", "content": "\\frac{2n + 2}{n + 1} \\leq \\sum_{k=n^2}^{(n+1)^2} \\frac{1}{\\sqrt{k}} \\leq \\frac{2n + 2}{n}" }, { "type": "text", "content": "，取极限得到 2。" } ] }, "bbox": [ 122, 513, 781, 551 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.4整体放缩or局部放缩(1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\left(\\frac{1}{\\sqrt{n^2 + 1}} +\\frac{1}{\\sqrt{n^2 + 2}} +\\dots +\\frac{1}{\\sqrt{n^2 + n}}\\right)" } ] }, "bbox": [ 58, 554, 680, 585 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)如果是 " }, { "type": "equation_inline", "content": "\\frac{1}{\\sqrt{n^2 + i^2}}" }, { "type": "text", "content": " 形式则是用定积分，这里看上去就是 " }, { "type": "equation_inline", "content": "\\sum_{k=1}^{n} \\frac{1}{n} = 1" }, { "type": "text", "content": "，因此放缩为 " }, { "type": "equation_inline", "content": "\\frac{n}{\\sqrt{n^2 + n}} \\leq I \\leq \\frac{n}{\\sqrt{n^2}}" }, { "type": "text", "content": " 即可知结果为 1。" } ] }, "bbox": [ 87, 586, 904, 634 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.4 平均值定理" } ], "level": 2 }, "bbox": [ 94, 662, 265, 680 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.4.1 有限项幂次根号平均" } ], "level": 2 }, "bbox": [ 94, 701, 332, 717 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.11 (根号平均)" } ], "level": 2 }, "bbox": [ 122, 733, 278, 747 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{1},a_{2},\\cdots,a_{m}" }, { "type": "text", "content": " 是 m 个正数，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{a_{1}^{n}+\\cdots+a_{m}^{n}}=\\max\\{a_{1},\\cdots,a_{m}\\}" }, { "type": "text", "content": " (注意这里不是 " }, { "type": "equation_inline", "content": "\\sqrt[n]{a_{1}^{n}+\\cdots+a_{n}^{n}}" }, { "type": "text", "content": "，与平均值定理不同)" } ] }, "bbox": [ 112, 752, 900, 794 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "A = \\max\\{a_{1}, a_{2}, \\cdots, a_{m}\\}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\sqrt[n]{A^{n}} \\leq \\sqrt[n]{a_{1}^{n} + a_{2}^{n} + \\cdots + a_{m}^{n}} \\leq \\sqrt[n]{mA^{n}}" }, { "type": "text", "content": " ，两侧都趋于 A，因此极限为 A。" } ] }, "bbox": [ 87, 807, 902, 841 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.5 根号平均的两个经典推广 (1) 正数列 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛到 a > 0，证明 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{a_{n}} = 1" } ] }, "bbox": [ 58, 863, 700, 879 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 非负且有界 (可能是可数个)，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{a_{1}^{n}+a_{2}^{n}+\\cdots+a_{n}^{n}}=\\sup_{n\\geq1}a_{n}" }, { "type": "text", "content": "证明 (1) 根据保号性， " }, { "type": "equation_inline", "content": "\\exists N, n > N" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\frac{1}{2} a \\leq a_n \\leq \\frac{3}{2} a" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "n > N" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\sqrt[n]{\\frac{1}{2} a} \\leq \\sqrt[n]{a_n} \\leq \\sqrt[n]{\\frac{3}{2} a}" }, { "type": "text", "content": "，由夹逼准则得到结果为 1。" } ] }, "bbox": [ 122, 883, 668, 908 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.3夹逼准则" } ] }, "bbox": [ 805, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "8" } ] }, "bbox": [ 492, 954, 504, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 89, 82, 905, 126 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 记 " }, { "type": "equation_inline", "content": "a = \\sup_{n \\geq 1} a_n" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "a_n \\leq a" }, { "type": "text", "content": " 且 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists k, a_k > a - \\epsilon" }, { "type": "text", "content": " 。当 " }, { "type": "equation_inline", "content": "n \\geq k" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "a_k^n \\leq a_1^n + a_2^n + \\cdots + a_n^n \\leq na^n" }, { "type": "text", "content": " ，两侧开 n 次根号得到 " }, { "type": "equation_inline", "content": "a - \\epsilon < \\sqrt[n]{a_1^n + a_2^n + \\cdots + a_n^n} \\leq a\\sqrt[n]{n}" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a\\sqrt[n]{n} = a" }, { "type": "text", "content": " ，对上述 " }, { "type": "equation_inline", "content": "\\epsilon, \\exists N, \\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "a\\sqrt[n]{n} < a + \\epsilon" }, { "type": "text", "content": " ，根据夹逼准则可证。" } ] }, "bbox": [ 90, 131, 907, 189 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.4.2 平均值定理：无穷项算术平均与几何平均" } ], "level": 2 }, "bbox": [ 92, 211, 515, 228 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.12 (平均值定理)" } ], "level": 2 }, "bbox": [ 121, 244, 295, 258 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "已知 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = a" }, { "type": "text", "content": " （有限数或正负无穷），则" } ] }, "bbox": [ 115, 263, 425, 284 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. 算术平均： " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{a_{1}+a_{2}+\\cdots+a_{n}}{n}=a" } ] }, "bbox": [ 134, 282, 453, 305 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. 几何平均：若 " }, { "type": "equation_inline", "content": "a_{n} > 0" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\sqrt[n]{a_{1}a_{2}\\cdots a_{n}} = a" } ] }, "bbox": [ 134, 304, 515, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 只证 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " 为有限数的情况。 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, n > N" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|a_{n} - a| < \\epsilon" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 90, 347, 611, 363 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} \\left| \\frac {a _ {1} + a _ {2} + \\cdots + a _ {n}}{n} - a \\right| = & \\left| \\frac {(a _ {1} - a) + \\cdots + (a _ {N} - a) + (a _ {N + 1} - a) + \\cdots + (a _ {n} - a)}{n} \\right| \\\\ & \\leq \\frac {| a _ {1} - a | + \\cdots + | a _ {N} - a |}{n} + \\frac {| a _ {N + 1} - a | + \\cdots + | a _ {n} - a |}{n} \\leq 2 \\epsilon \\end{array}", "math_type": "latex", "image_source": { "path": "images/2a859ade91b8b60080c585935697c89ba60816655f9842f5438a8c79e4b44ec6.jpg" } }, "bbox": [ 179, 365, 818, 429 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 取对数则转换为 (1)" } ] }, "bbox": [ 122, 430, 295, 444 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.6 奇偶子列推广 已知 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}a_{2n+1}=a,\\lim_{n\\to\\infty}a_{2n}=b" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{a_{1}+a_{2}+\\cdots+a_{n}}{n}=\\frac{a+b}{2}" }, { "type": "text", "content": " \n证明 分为奇偶数项进行讨论：" } ] }, "bbox": [ 58, 461, 803, 499 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\lim _ {n \\to \\infty} \\frac {a _ {1} + a _ {2} + \\cdots + a _ {2 n}}{2 n} \\\\ = \\frac {1}{2} \\lim _ {n \\to \\infty} \\frac {a _ {1} + a _ {3} + \\cdots + a _ {2 n - 1}}{n} + \\frac {1}{2} \\lim _ {n \\to \\infty} \\frac {a _ {2} + a _ {4} + \\cdots + a _ {2 n}}{n} = \\frac {a + b}{n} \\end{array}", "math_type": "latex", "image_source": { "path": "images/1f30e3e411d1ed16dd71bea5ff27059ef61ee7e4ed5235e809f16561cc1dfaba.jpg" } }, "bbox": [ 238, 502, 759, 560 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "另一方面" } ] }, "bbox": [ 90, 560, 163, 574 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\lim _ {n \\to \\infty} \\frac {a _ {1} + a _ {2} + \\cdots + a _ {2 n + 1}}{2 n + 1} \\\\ = \\lim _ {n \\to \\infty} \\frac {a _ {1} + a _ {2} + \\cdots + a _ {2 n + 1}}{n} \\cdot \\frac {n}{2 n - 1} + \\lim _ {n \\to \\infty} \\frac {a _ {2} + a _ {4} + \\cdots + a _ {2 n - 2}}{n - 1} \\cdot \\frac {n - 1}{2 n - 1} = \\frac {a + b}{2} \\end{array}", "math_type": "latex", "image_source": { "path": "images/ea1c3bd0d5e95d4aa5409bc955162ab1a36582e513d7722ec65f37ca4089f1cf.jpg" } }, "bbox": [ 183, 577, 815, 636 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此得到结论成立。" } ] }, "bbox": [ 90, 638, 238, 652 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.7倒乘推广 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = a,\\lim_{n\\to \\infty}b_n = b" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_1b_n + a_2b_{n - 1} + \\cdots + a_nb_1}{n} = ab" }, { "type": "text", "content": " 证明由于 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = a" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 有界，即 " }, { "type": "equation_inline", "content": "\\exists M" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|a_n| < M" }, { "type": "text", "content": " ，故：" } ] }, "bbox": [ 58, 668, 757, 711 ] }, { "type": "equation_interline", "content": { "math_content": "\\vert a _ {k} b _ {n - k + 1} - a b \\vert < \\vert a _ {k} \\vert \\cdot \\vert b _ {n - k + 1} - b \\vert + \\vert b \\vert \\cdot \\vert a _ {k} - a \\vert \\leq M \\vert b _ {n - k + 1} - b \\vert + \\vert b \\vert \\cdot \\vert a _ {k} - a \\vert", "math_type": "latex", "image_source": { "path": "images/0286a1b0d0faf757802434ca47bd57c78553d7456dc2f90b1051ae4b3626425a.jpg" } }, "bbox": [ 194, 715, 801, 733 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此得到：" } ] }, "bbox": [ 90, 739, 169, 753 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\left| \\frac {a _ {1} b _ {n} + \\cdots + a _ {n} b _ {1}}{n} - a b \\right| \\\\ \\leq \\frac {| a _ {1} b _ {n} - a b | + \\cdots + | a _ {n} b _ {1} - a b |}{n} \\\\ \\leq M \\cdot \\frac {| b _ {n} - b | + \\cdot + | b _ {1} - b |}{n} + | b | \\cdot \\frac {| a _ {1} - a | + \\cdots + | a _ {n} - a |}{n} \\end{array}", "math_type": "latex", "image_source": { "path": "images/fe6edaf77fdf939f40e0c11f0577b993f0ccd06823cf40aca1462e3954e98c2b.jpg" } }, "bbox": [ 280, 753, 715, 845 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "有平均值定理可知右侧极限为0，因此 " }, { "type": "equation_inline", "content": "\\left|\\frac{a_{1}b_{n}+\\cdots+a_{n}b_{1}}{n}-ab\\right|\\leq0" }, { "type": "text", "content": " ，得到目标极限为0。" } ] }, "bbox": [ 90, 848, 742, 879 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.4 平均值定理" } ] }, "bbox": [ 788, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "9" } ] }, "bbox": [ 492, 953, 504, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.4.3 技巧：一招解决 n 次根号与除以 n" } ], "level": 2 }, "bbox": [ 92, 85, 452, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.13 (n 次根号与除以 n)" } ], "level": 2 }, "bbox": [ 122, 118, 339, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 为数列，则" } ] }, "bbox": [ 114, 137, 226, 153 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}(a_n - a_{n - 1}) = a" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{n} = a" } ] }, "bbox": [ 139, 153, 465, 178 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{a_{n - 1}} = a" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{a_n} = a" } ] }, "bbox": [ 139, 177, 433, 203 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 由于 " }, { "type": "equation_inline", "content": "a_{n}=a_{1}+(a_{2}-a_{1})+\\cdots+(a_{n}-a_{n-1})" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{a_{n}}{n}=\\lim_{n\\to\\infty}\\frac{a_{1}+(a_{2}-a_{1})+\\cdots+(a_{n}-a_{n-1})}{n}" }, { "type": "text", "content": " ，根据平均值定理即 a" } ] }, "bbox": [ 90, 216, 902, 255 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据 " }, { "type": "equation_inline", "content": "a_{n} = a_{1} \\cdot \\frac{a_{2}}{a_{1}} \\cdots \\frac{a_{n}}{a_{n-1}}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\sqrt[n]{a_n} = \\lim_{n \\to \\infty} \\sqrt[n]{a_1 \\cdot \\frac{a_2}{a_1} \\cdots \\frac{a_n}{a_{n-1}}}" }, { "type": "text", "content": ", 根据平均值定理即 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": "." } ] }, "bbox": [ 122, 253, 791, 284 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.8n 次根号 计算极限 (1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{a}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{n}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{n!}" } ] }, "bbox": [ 58, 294, 561, 318 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\frac{a_n}{a_{n-1}} = 1" }, { "type": "text", "content": "，因此极限为 " }, { "type": "equation_inline", "content": "1" }, { "type": "text", "content": "。" } ] }, "bbox": [ 90, 315, 327, 341 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\frac{a_{n}}{a_{n-1}}=\\frac{n}{n-1}\\rightarrow1" }, { "type": "text", "content": " ，极限为1" } ] }, "bbox": [ 122, 338, 366, 368 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\frac{a_n}{a_{n-1}} = n \\to \\infty" }, { "type": "text", "content": "，因此不收敛" } ] }, "bbox": [ 122, 368, 364, 393 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.9 除以 n 重点：计算极限 (1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{\\sqrt[n]{n!}}{n}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{\\sqrt[n]{n(n+1)\\cdots(2n-1)}}{n}" } ] }, "bbox": [ 58, 398, 658, 430 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 " }, { "type": "equation_inline", "content": "(1)a_{n} = \\frac{n!}{n^{n}}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{a_{n - 1}} = \\frac{\\frac{n!}{n^n}}{\\frac{(n - 1)!}{(n - 1)^{n - 1}}} = \\frac{1}{(1 - \\frac{1}{n})^{n - 1}} = \\frac{1}{\\left[(1 - \\frac{1}{n})^n\\right]^{\\frac{n - 1}{n}}} = \\frac{1}{e}" }, { "type": "text", "content": " 因此极限为 " }, { "type": "equation_inline", "content": "\\frac{1}{e}" } ] }, "bbox": [ 87, 428, 771, 466 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "a_{n}=\\frac{n(n+1)\\cdots(2n-1)}{n^{n}}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 122, 464, 391, 494 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} \\frac {a _ {n}}{a _ {n - 1}} = \\lim _ {n \\to \\infty} \\frac {2 (2 n - 1) (n - 1) ^ {n - 1}}{n ^ {n}} = (4 - \\frac {2}{n}) \\left((1 - \\frac {1}{n}) ^ {n}\\right) ^ {\\frac {n - 1}{n}} = \\frac {4}{e}", "math_type": "latex", "image_source": { "path": "images/b9121190cbf8fce36061ede7f56de07cdc39003c5c4ae1ba9a4a6b5f5f2f29d5.jpg" } }, "bbox": [ 233, 498, 761, 532 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此极限为 " }, { "type": "equation_inline", "content": "\\frac{4}{e}" } ] }, "bbox": [ 92, 533, 196, 563 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.5 使用定积分定义" } ], "level": 2 }, "bbox": [ 92, 583, 312, 602 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.5.1 使用定积分计算极限" } ], "level": 2 }, "bbox": [ 92, 621, 332, 639 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.14 (用定积分定义计算极限)" } ], "level": 2 }, "bbox": [ 121, 653, 379, 670 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续，则" } ] }, "bbox": [ 115, 675, 302, 690 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\rightarrow \\infty} \\frac {b - a}{n} \\left[ f \\left(a + \\frac {b - a}{n}\\right) + f \\left(a + \\frac {2 (b - a)}{n}\\right) + \\dots + f \\left(a + \\frac {n (b - a)}{n}\\right)\\right] = \\int_ {a} ^ {b} f (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/f17c2d6b2231f30a6ecd93641b8e09766319de2c3ba9574dba83b1c9ca44d386.jpg" } }, "bbox": [ 171, 693, 825, 728 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 如果是 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{1}{n + a}\\sum_{k = 1}^{n}f\\left(\\frac{k}{n}\\right)" }, { "type": "text", "content": " ，则可以加一项得到 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{n}{n + a}\\cdot \\frac{1}{n}\\sum_{k = 1}^{n}f\\left(\\frac{k}{n}\\right)" }, { "type": "text", "content": " 再转换为定积分。" } ] }, "bbox": [ 58, 739, 821, 776 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.10 直接判断积分 (1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty} n\\left(\\frac{1}{n^{2}+1}+\\frac{1}{n^{2}+2}+\\cdots+\\frac{1}{n^{2}+n^{2}}\\right)" } ] }, "bbox": [ 60, 778, 610, 808 ] }, { "type": "equation_interline", "content": { "math_content": "(2) \\lim _ {n \\to \\infty} n \\left[ \\frac {1}{(n + 1) ^ {2}} + \\frac {1}{(n + 2) ^ {2}} + \\dots + \\frac {1}{(n + n) ^ {2}} \\right]", "math_type": "latex", "image_source": { "path": "images/862db8fc611d14fa37f6aebda15740ddfb1cba2cd0491bc4153197e6996335b5.jpg" } }, "bbox": [ 126, 809, 487, 840 ] }, { "type": "equation_interline", "content": { "math_content": "(3) \\lim _ {n \\rightarrow \\infty} \\left(\\frac {1}{\\sqrt {n ^ {2} + 1 ^ {2}}} + \\frac {1}{\\sqrt {n ^ {2} + 2 ^ {2}}} + \\dots + \\frac {1}{\\sqrt {n ^ {2} + n ^ {2}}}\\right)", "math_type": "latex", "image_source": { "path": "images/18f61ff6c198768fc1b1b7ef4a8d1e23d8a021d28d32fa35452e629f22abf756.jpg" } }, "bbox": [ 122, 839, 507, 869 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sum_{k = 1}^{n}\\frac{1}{\\sqrt{nk}}" } ] }, "bbox": [ 124, 871, 252, 904 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.5 使用定积分定义" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "10" } ] }, "bbox": [ 490, 953, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)等价于 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{1}{1 + x^2}\\mathrm{d}x = \\frac{\\pi}{4}" }, { "type": "text", "content": " (2)等价于 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{1}{(1 + x)^2}\\mathrm{d}x = \\frac{1}{2}" }, { "type": "text", "content": " (3)等价于 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{1}{\\sqrt{1 + x^2}}\\mathrm{d}x = \\ln (1 + \\sqrt{2})" }, { "type": "text", "content": " (4)转换为 " }, { "type": "equation_inline", "content": "\\frac{1}{n}\\sum_{k = 1}^{n}\\frac{1}{\\sqrt{\\frac{k}{n}}}" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{1}{\\sqrt{x}}\\mathrm{d}x" }, { "type": "text", "content": " ，极限为2" } ] }, "bbox": [ 87, 82, 480, 215 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.11对数 " }, { "type": "equation_inline", "content": "+" }, { "type": "text", "content": " 积分重点：取对数 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{\\sqrt[n]{n(n + 1)\\cdots(2n - 1)}}{n}" }, { "type": "text", "content": " 解等价于 " }, { "type": "equation_inline", "content": "\\exp \\left[\\frac{1}{n}\\ln \\left(\\frac{n(n + 1)\\cdots(2n - 1)}{n^n}\\right)\\right]" }, { "type": "text", "content": "，指数积分即 " }, { "type": "equation_inline", "content": "\\int_0^1\\ln (1 + x)\\mathrm{d}x = [(1 + x)\\ln (1 + x) - x]\\bigg|_0^1 = \\ln 4 - 1," }, { "type": "text", "content": " 因此极限为 " }, { "type": "equation_inline", "content": "\\frac{4}{e}" }, { "type": "text", "content": " 或者本题也在 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 次根号部分出现了(更推荐 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 次根号的做法)" } ] }, "bbox": [ 58, 219, 907, 319 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.12改变有限项(1)重点： " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\left(\\frac{1}{n} +\\frac{1}{n + 1} +\\dots +\\frac{1}{n + (n - 1)}\\right)" }, { "type": "text", "content": " 解(1)可改变有限项不影响极限，变为 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{1}{n + 1} +\\dots +\\frac{1}{n + n}" }, { "type": "text", "content": " ，因此极限为 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{1}{1 + x}\\mathrm{d}x = \\ln 2" } ] }, "bbox": [ 58, 330, 806, 394 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.13 经典夹逼 + 积分 (1) 重点: " }, { "type": "equation_inline", "content": "I = \\lim_{n \\to \\infty} \\left( \\frac{\\sin \\frac{\\pi}{n}}{n + \\frac{1}{n}} + \\frac{\\sin \\frac{2}{n}\\pi}{n + \\frac{2}{n}} + \\cdots + \\frac{\\sin \\pi}{n + 1} \\right)" } ] }, "bbox": [ 58, 400, 680, 432 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点: " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\left( \\frac{1}{n^2 + n + 1} + \\frac{2}{n^2 + n + 2} + \\cdots + \\frac{n}{n^2 + n + n} \\right)" } ] }, "bbox": [ 122, 432, 591, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点: " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\left( \\frac{1}{n^2 + n + 1^2} + \\frac{2}{n^2 + n + 2^2} + \\cdots + \\frac{n}{n^2 + n + n^2} \\right)" } ] }, "bbox": [ 122, 464, 611, 492 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\rightarrow \\infty} \\left(\\frac {\\sin \\frac {x}{n}}{n + 1} + \\frac {2 \\sin \\frac {2}{n} x}{2 n + 1} + \\dots + \\frac {n \\sin \\frac {n}{n} x}{n ^ {2} + 1}\\right), x \\in (0, \\pi)", "math_type": "latex", "image_source": { "path": "images/b26e76518bf1c47581fc70ca08a65d1f2673c4f6b7302e66f443000469c0a58c.jpg" } }, "bbox": [ 144, 493, 537, 523 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "I = \\lim_{n\\to \\infty}\\frac{1}{n}\\left(\\frac{\\sin\\frac{\\pi}{n}}{1 + \\frac{1}{n^2}} +\\frac{\\sin\\frac{2}{n}\\pi}{1 + \\frac{2}{n^2}} +\\dots +\\frac{\\sin\\pi}{1 + \\frac{n}{n^2}}\\right)" }, { "type": "text", "content": " ，进行 " }, { "type": "equation_inline", "content": "\\frac{\\sin\\frac{i}{n}\\pi}{n + 1}\\leq \\frac{\\sin\\frac{i}{n}\\pi}{n + \\frac{i}{n}}\\leq \\frac{\\sin\\frac{i}{n}\\pi}{n}" }, { "type": "text", "content": " 放缩： \n " }, { "type": "equation_inline", "content": "\\left\\{ \\begin{array}{l}\\lim_{n\\to \\infty}\\frac{1}{n}\\sum_{i = 1}^{n}\\sin \\frac{i}{n}\\pi = \\int_0^1\\sin \\pi x\\mathrm{d}x = \\frac{2}{\\pi}\\\\ \\lim_{n\\to \\infty}\\frac{1}{n + 1}\\sum_{n = 1}^{\\infty}\\sin \\frac{i}{n}\\pi = \\lim_{n\\to \\infty}\\frac{n}{n + 1}\\cdot \\frac{1}{n}\\sum_{i = 1}^{n}\\sin \\frac{i}{n}\\pi = \\frac{2}{\\pi} \\end{array} \\right." } ] }, "bbox": [ 89, 523, 776, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此根据夹逼可知极限为 " }, { "type": "equation_inline", "content": "\\frac{2}{\\pi}" } ] }, "bbox": [ 90, 636, 297, 658 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 进行放缩: " }, { "type": "equation_inline", "content": "\\frac{1 + 2 + \\cdots + n}{n^2 + n + n} \\leq I \\leq \\frac{1 + 2 + \\cdots + n}{n^2 + n}" }, { "type": "text", "content": " (或者 " }, { "type": "equation_inline", "content": "\\frac{i}{n^2 + n + n} \\leq \\frac{i}{n^2 + n + i} \\leq \\frac{i}{n^2}" }, { "type": "text", "content": " 也行)。左右侧分别写为\n " }, { "type": "equation_inline", "content": "\\left\\{\\begin{aligned}&\\sum_{k=1}^{n}\\frac{k}{n^2+2n}=\\frac{1}{n+2}\\sum_{k=1}^{n}\\frac{k}{n}=\\frac{n}{n+2}\\cdot\\frac{1}{n}\\sum_{k=1}^{n}\\frac{k}{n}\\\\&\\sum_{k=1}^{n}\\frac{k}{n^2+n}=\\frac{n}{n+1}\\cdot\\frac{1}{n}\\sum_{k=1}^{n}\\frac{k}{n}\\end{aligned}\\right." } ] }, "bbox": [ 90, 656, 905, 774 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此两侧均等于积分 " }, { "type": "equation_inline", "content": "\\int_0^1 x\\mathrm{d}x = \\frac{1}{2}" }, { "type": "text", "content": " (3)一方面 " }, { "type": "equation_inline", "content": "\\frac{i}{n^2 + n + i^2}\\leq \\frac{i}{n^2 + i^2} = \\frac{1}{n}\\cdot \\frac{\\frac{i}{n}}{1 + (\\frac{i}{n})^2}" }, { "type": "text", "content": " ，另一方面 " }, { "type": "equation_inline", "content": "\\frac{i}{n^2 + n + i}\\geq \\frac{i}{(n + 1)^2 + i^2} = \\frac{1}{n + 1}" }, { "type": "equation_inline", "content": "\\frac{\\frac{i}{n + 1}}{1 + \\left(\\frac{i}{n + 1}\\right)^2}" }, { "type": "text", "content": " ，结果等价于 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{x}{1 + x^2}\\mathrm{d}x = \\frac{1}{2}\\ln 2" } ] }, "bbox": [ 90, 778, 907, 889 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.5 使用定积分定义" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "11" } ] }, "bbox": [ 490, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 看上去等价于 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sum_{k = 1}^{n}\\frac{k\\sin\\frac{x}{n}}{kn} = \\int_0^1\\sin tx\\mathrm{d}t = \\frac{1 - \\cos x}{x}" }, { "type": "text", "content": "，放缩为 " }, { "type": "equation_inline", "content": "\\frac{k\\sin\\frac{k}{n}x}{kn + k}\\leq \\frac{k\\sin\\frac{k}{n}x}{kn + 1}\\leq \\frac{k\\sin\\frac{k}{n}x}{kn}" }, { "type": "text", "content": " 再求和即可。 \n笔记一般不能直接写成积分形式的都需要进行放缩。" } ] }, "bbox": [ 87, 82, 904, 152 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.5.2 利用定积分定义构造等价无穷小" } ], "level": 2 }, "bbox": [ 92, 173, 433, 190 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.15 (利用积分构造等价无穷小)" } ], "level": 2 }, "bbox": [ 121, 205, 396, 221 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于正整数 k， " }, { "type": "equation_inline", "content": "1^{k} + 2^{k} + \\cdots + n^{k} \\sim \\frac{1}{k+1} n^{k+1}" } ] }, "bbox": [ 114, 227, 465, 254 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 考虑 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{1}{n^{k+1}}\\left(1^{k}+\\cdots+n^{k}\\right)=\\int_{0}^{1}x^{k}\\mathrm{d}x=\\frac{1}{k+1}" }, { "type": "text", "content": " 即可。" } ] }, "bbox": [ 90, 267, 561, 298 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.14 利用积分构造等价无穷小 计算 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{\\ln n}{\\ln(1^{2022}+2^{2022}+\\cdots+n^{2022})}" } ] }, "bbox": [ 60, 307, 653, 335 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{1}{n^{2023}}\\left(1^{2022}+2^{2022}+\\cdots+n^{2022}\\right)=\\int_{0}^{1}x^{2022}\\mathrm{d}x=\\frac{1}{2023}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 87, 335, 620, 366 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\rightarrow \\infty} \\frac {\\ln n}{\\ln \\left(1 ^ {2 0 2 2} + 2 ^ {2 0 2 2} + \\cdots + n ^ {2 0 2 2}\\right)} = \\lim _ {n \\rightarrow \\infty} \\frac {\\ln n}{2 0 2 3 \\ln n + \\ln \\frac {1}{n ^ {2 0 2 3}} \\left(1 ^ {2 0 2 2} + 2 ^ {2 0 2 2} + \\dots + n ^ {2 0 2 2}\\right)} = \\frac {1}{2 0 2 3}", "math_type": "latex", "image_source": { "path": "images/aea854978192d8a4f393f7aeb8413952416abf069b3340df999ec0b62b63902b.jpg" } }, "bbox": [ 132, 369, 862, 400 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.16 (已知极限计算求和极限)" } ], "level": 2 }, "bbox": [ 122, 422, 379, 437 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = A" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sum_{k = 1}^{n}\\frac{a_{n + k}}{n + k} = A\\ln 2" } ] }, "bbox": [ 114, 442, 443, 479 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 首先显然 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sum_{k = 1}^{n}\\frac{A}{n + k} = A\\ln 2" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = A" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists N,\\forall n > N,|a_n - A| < \\epsilon" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "n > N" }, { "type": "text", "content": " 时：" } ] }, "bbox": [ 89, 493, 878, 530 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\sum_ {k = 1} ^ {n} \\frac {a _ {n + k}}{n + k} - \\sum_ {k = 1} ^ {n} \\frac {A}{n + k} \\right| \\leq \\sum_ {k = 1} ^ {n} \\frac {| a _ {n + k} - A |}{n + k} < \\sum_ {k = 1} ^ {n} \\frac {\\epsilon}{n} = \\epsilon", "math_type": "latex", "image_source": { "path": "images/484e0c96db60f3b0e008d01717554180772b017ad01d7e79f80452634b704803.jpg" } }, "bbox": [ 297, 533, 700, 573 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.6 单调有界定理" } ], "level": 2 }, "bbox": [ 94, 599, 289, 618 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.6.1 单调有界定理及其推广" } ], "level": 2 }, "bbox": [ 94, 637, 352, 653 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.17 (单调有界定理)" } ], "level": 2 }, "bbox": [ 122, 670, 312, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若数列 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单调且有界，则其必收敛。" } ] }, "bbox": [ 115, 690, 388, 706 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.15 单调有界定理理论推广 (1) " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单增， " }, { "type": "equation_inline", "content": "b_{n}" }, { "type": "text", "content": " 单减， " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}(b_{n}-a_{n})=0" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}a_{n},\\lim_{n\\to\\infty}b_{n}" }, { "type": "text", "content": " 存在且相等\n\n(2) " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 为有界数列，记 " }, { "type": "equation_inline", "content": "\\overline{a}=\\sup\\{a_{n},a_{n+1},\\cdots\\},\\underline{a}_{n}=\\inf\\{a_{n},a_{n+1},\\cdots\\}" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\overline{a}_{n},\\lim_{n\\to\\infty}\\underline{a}_{n}" }, { "type": "text", "content": " 存在，且 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}a_{n}" }, { "type": "text", "content": " 存在的充要条件为 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\overline{a}_{n}=\\lim_{n\\to\\infty}\\underline{a}_{n}" }, { "type": "text", "content": "\n\n证明 (1) 由于 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}(b_{n}-a_{n})=0,\\exists N,\\forall n>N" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|b_{n}-a_{n}|<1" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "a_{n} N" }, { "type": "text", "content": " 有" } ] }, "bbox": [ 90, 837, 905, 873 ] }, { "type": "equation_interline", "content": { "math_content": "a - \\epsilon < \\underline {{a}} _ {n} \\leq \\overline {{a}} _ {n} < a + \\epsilon", "math_type": "latex", "image_source": { "path": "images/b3ebdd9230d1279ede52de1f052c6702375007e0a2b73055526b7bc867e2b559.jpg" } }, "bbox": [ 406, 881, 589, 894 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "取极限可知极限均为 a." } ] }, "bbox": [ 90, 901, 263, 915 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.6 单调有界定理" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "12" } ] }, "bbox": [ 490, 954, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.16两道经典循环根式(1)重点： " }, { "type": "equation_inline", "content": "a_1 = \\sqrt{c}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "a_{n + 1} = \\sqrt{a_n + c}" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n" } ] }, "bbox": [ 58, 102, 670, 123 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点: " }, { "type": "equation_inline", "content": "a_{n} = \\sqrt{1 + \\sqrt{2 + \\cdots + \\sqrt{n}}}" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 122, 123, 510, 153 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)由于 " }, { "type": "equation_inline", "content": "a_{n}=\\sqrt{c+\\sqrt{c+\\cdots+\\sqrt{c}}}" }, { "type": "text", "content": " ，根据该通项的形式，每次 " }, { "type": "equation_inline", "content": "a_{n+1}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 基础上往最里层加入一个 " }, { "type": "equation_inline", "content": "\\sqrt{c}" }, { "type": "text", "content": " ，因此整体单增。若极限存在，则显然极限为 " }, { "type": "equation_inline", "content": "a=\\frac{1+\\sqrt{1+4c}}{2}" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "a=\\sqrt{a+c}" }, { "type": "text", "content": " ，因此若 " }, { "type": "equation_inline", "content": "a_{n}\\ln(n+1)-\\ln(n)>0" }, { "type": "text", "content": " (调和级数不等式)，且根据 " }, { "type": "equation_inline", "content": "\\ln(1+x)\\geq\\frac{x}{1+x}" }, { "type": "text", "content": " 得到\n\n " }, { "type": "equation_inline", "content": "\\frac{1}{n+1}<\\ln(1+\\frac{1}{n})" } ] }, "bbox": [ 58, 378, 882, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "于是 " }, { "type": "equation_inline", "content": "a_{n + 1} - a_n = \\frac{1}{n + 1} -\\ln (1 + \\frac{1}{n}) < 0" }, { "type": "text", "content": " ，这说明单减有下界，故收敛。" } ] }, "bbox": [ 92, 464, 618, 495 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.18 交替级数 " }, { "type": "equation_inline", "content": "S_{n}=1-\\frac{1}{2}+\\frac{1}{3}-\\cdots+\\frac{(-1)^{n+1}}{n}" }, { "type": "text", "content": " ，计算 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}S_{n}" }, { "type": "text", "content": " 解由 Leibniz 定理可知收敛。只需要考虑偶数项" } ] }, "bbox": [ 58, 500, 594, 541 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "S_{2n} = (1 + \\frac{1}{2} +\\dots +\\frac{1}{2n}) - 2(\\frac{1}{2} +\\frac{1}{4} +\\dots +\\frac{1}{2n}) = (1 + \\frac{1}{2} +\\dots +\\frac{1}{2n}) - (1 + \\frac{1}{2} +\\dots +\\frac{1}{n}) = (\\frac{1}{n + 1} +\\dots +\\frac{1}{2n})\\to \\ln 2" }, { "type": "text", "content": " 由于收敛数列的奇偶子列收敛到同一极限，因此极限为 " }, { "type": "equation_inline", "content": "\\ln 2" } ] }, "bbox": [ 87, 542, 905, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.19经典相减法(1)重点： " }, { "type": "equation_inline", "content": "A > 0, a_{1} > 0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "a_{n + 1} = \\frac{1}{2} (a_n + \\frac{A}{a_n})" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛，并求极限解(1)首先 " }, { "type": "equation_inline", "content": "a_{n + 1} = \\frac{1}{2} (a_n + \\frac{A}{a_n}) > \\sqrt{A}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "n \\geq 2" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "a_{n} > \\sqrt{A}" }, { "type": "text", "content": " ，而前后项相减得到" } ] }, "bbox": [ 58, 604, 783, 662 ] }, { "type": "equation_interline", "content": { "math_content": "a _ {n + 1} - a _ {n} = \\frac {1}{2} (\\frac {A}{a _ {n}} - a _ {n}) = \\frac {1}{2} \\cdot \\frac {A - a _ {n} ^ {2}}{a _ {n}}", "math_type": "latex", "image_source": { "path": "images/a3dda4746906400321e7ef52946acf7bf370a532eebcc4b79a7e7235d3e04ebf.jpg" } }, "bbox": [ 352, 664, 642, 695 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "结合 " }, { "type": "equation_inline", "content": "n \\geq 2" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "a_{n} > \\sqrt{A}" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "a_{n + 1} - a_{n} < 0" }, { "type": "text", "content": " ，因此单减，显然 " }, { "type": "equation_inline", "content": "a_{n} > 0" }, { "type": "text", "content": " ，因此收敛。两侧取极限得到 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = \\sqrt{A}" } ] }, "bbox": [ 87, 697, 905, 725 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.6.3 相除法" } ], "level": 2 }, "bbox": [ 92, 740, 213, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.20两道经典相除法(1)重点： " }, { "type": "equation_inline", "content": "a_1 = 1, a_{n + 1} = \\sqrt{2a_n}" }, { "type": "text", "content": "，求 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n" } ] }, "bbox": [ 58, 768, 611, 787 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点: " }, { "type": "equation_inline", "content": "c > 0, 0 < x_{1} < \\frac{1}{c}, x_{n+1} = x_{n}(2 - cx_{n})" }, { "type": "text", "content": ", 证明: " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 收敛并求其极限" } ] }, "bbox": [ 119, 785, 700, 814 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\frac{a_{n+1}}{a_n} = \\sqrt{\\frac{2}{a_n}}" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "a_n < 2" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "a_{n+1} > a_n" }, { "type": "text", "content": "。根据 " }, { "type": "equation_inline", "content": "a_{n+1} = \\sqrt{2a_n}" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "a_n < 2" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "a_{n+1} < 2" }, { "type": "text", "content": "，单调有界故收敛，两侧同取极限得到极限为 " }, { "type": "equation_inline", "content": "A = 2" }, { "type": "text", "content": "。" } ] }, "bbox": [ 89, 816, 905, 854 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\frac{x_{n + 1}}{x_n} = 2 - cx_n" }, { "type": "text", "content": " ，当 " }, { "type": "equation_inline", "content": "x_{n} < \\frac{1}{c}" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "x_{n + 1} > x_n" }, { "type": "text", "content": " 。设 " }, { "type": "equation_inline", "content": "f(x) = 2x - cx^2, f'(x) = 2 - 2cx" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = \\frac{1}{c}" }, { "type": "text", "content": " 取最大，因此 " }, { "type": "equation_inline", "content": "x_{n} < \\frac{1}{c}" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "x_{n + 1} < \\frac{1}{c}" }, { "type": "text", "content": " 有界，根据单调有界收敛，两侧取极限得到极限为 " }, { "type": "equation_inline", "content": "\\frac{1}{c}" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 92, 856, 905, 910 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.6 单调有界定理" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "13" } ] }, "bbox": [ 490, 954, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.6.4 不等式放缩" } ], "level": 2 }, "bbox": [ 92, 85, 253, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.21 两道经典双数列 + 不等式放缩 (1) 重点: " }, { "type": "equation_inline", "content": "a_1 > b_1 > 0" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "a_{n+1} = \\frac{a_n + b_n}{2}, b_{n+1} = \\sqrt{a_nb_n}" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_n, \\lim_{n \\to \\infty} b_n" }, { "type": "text", "content": " 存在且相等。" } ] }, "bbox": [ 60, 115, 905, 156 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点: " }, { "type": "equation_inline", "content": "a_1 > b_1 > 0" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "a_{n+1} = \\frac{a_n + b_n}{2}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "b_{n+1} = \\frac{2a_nb_n}{a_n + b_n}" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_n, \\lim_{n \\to \\infty} b_n" }, { "type": "text", "content": " 存在, 并求其值" } ] }, "bbox": [ 104, 156, 836, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)根据均值不等式 " }, { "type": "equation_inline", "content": "a_{n + 1}\\geq b_{n + 1}" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "a_1 > b_1" }, { "type": "text", "content": " 可知 " }, { "type": "equation_inline", "content": "\\forall n,a_n > b_n" }, { "type": "text", "content": " ，从而" } ] }, "bbox": [ 89, 185, 636, 200 ] }, { "type": "equation_interline", "content": { "math_content": "a _ {n + 1} = \\frac {a _ {n} + b _ {n}}{2} \\leq \\frac {a _ {n} + a _ {n}}{2} = a _ {n}, b _ {n + 1} = \\sqrt {a _ {n} b _ {n}} \\geq \\sqrt {b _ {n} b _ {n}} = b _ {n}", "math_type": "latex", "image_source": { "path": "images/660482a4d34015fb3df00d3f0c479629a4ec241661ab569947b92fa60a922ba1.jpg" } }, "bbox": [ 260, 204, 732, 229 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单减， " }, { "type": "equation_inline", "content": "b_{n}" }, { "type": "text", "content": " 单增，即 " }, { "type": "equation_inline", "content": "a_{1} \\geq a_{2} \\geq \\cdots \\geq a_{n} \\geq b_{n} \\geq b_{n-1} \\cdots \\geq b_{1}" }, { "type": "text", "content": " ，这说明 " }, { "type": "equation_inline", "content": "a_{n}, b_{n}" }, { "type": "text", "content": " 均单调有界，故均收敛。 " }, { "type": "equation_inline", "content": "a_{n+1} = \\frac{a_{n} + b_{n}}{2}" }, { "type": "text", "content": " 两侧取极限可知极限相等。" } ] }, "bbox": [ 89, 231, 907, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据均值不等式可知 " }, { "type": "equation_inline", "content": "a_{n} \\geq b_{n}" }, { "type": "text", "content": ", 由于 " }, { "type": "equation_inline", "content": "a_{1} > b_{1}" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "\\forall n, a_{n} \\geq b_{n}" }, { "type": "text", "content": ", 因此" } ] }, "bbox": [ 121, 271, 662, 286 ] }, { "type": "equation_interline", "content": { "math_content": "a _ {n + 1} = \\frac {a _ {n} + b _ {n}}{2} \\leq \\frac {a _ {n} + a _ {n}}{2} = a _ {n}, b _ {n + 1} = \\frac {2}{\\frac {1}{a _ {n}} + \\frac {1}{b _ {n}}} \\geq \\frac {2}{\\frac {1}{b _ {n}} + \\frac {1}{b _ {n}}} = b _ {n}", "math_type": "latex", "image_source": { "path": "images/a8d9586d639601f72dafe1375c65380e04a391e6f318562c7784accfa1618d97.jpg" } }, "bbox": [ 250, 288, 742, 323 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单减， " }, { "type": "equation_inline", "content": "b_{n}" }, { "type": "text", "content": " 单增，且 " }, { "type": "equation_inline", "content": "a_{1} \\geq a_{2} \\geq \\cdots \\geq a_{n} \\geq b_{n} \\geq \\cdots \\geq b_{1}" }, { "type": "text", "content": " ，单调有界故均收敛。 " }, { "type": "equation_inline", "content": "a_{n+1} = \\frac{a_{n} + b_{n}}{2}" }, { "type": "text", "content": " 两侧取极限可知极限满足 " }, { "type": "equation_inline", "content": "a = \\frac{a + b}{2}" }, { "type": "text", "content": " ，故相等。而" } ] }, "bbox": [ 89, 326, 905, 376 ] }, { "type": "equation_interline", "content": { "math_content": "a _ {n + 1} b _ {n + 1} = \\frac {a _ {n} + b _ {n}}{2} \\cdot \\frac {2 a _ {n} b _ {n}}{a _ {n} + b _ {n}} = a _ {n} b _ {n}", "math_type": "latex", "image_source": { "path": "images/9671e2473679e833359bc0e1fc6b2a13e34eccc40e3fc822b95b4c7e6db1c0ab.jpg" } }, "bbox": [ 354, 380, 640, 410 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "ab = a_{1}b_{1}" }, { "type": "text", "content": " ，即极限为 " }, { "type": "equation_inline", "content": "a = b = \\sqrt{a_1b_1}" } ] }, "bbox": [ 90, 414, 400, 430 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.22 前后项不等式放缩 (1) 重点: " }, { "type": "equation_inline", "content": "0 < a_{n} < 2" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "(2 - a_{n})a_{n+1} \\geq 1" }, { "type": "text", "content": " , 证明 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛, 并求其极限" } ] }, "bbox": [ 58, 451, 813, 467 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点: " }, { "type": "equation_inline", "content": "x_{n} > 0" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "x_{n+1} + \\frac{4}{x_{n}} < 4" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 收敛并求 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} x_{n}" } ] }, "bbox": [ 122, 467, 584, 493 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)显然 " }, { "type": "equation_inline", "content": "a_{n}(2 - a_{n})\\leq \\left[\\frac{a_{n} + (2 - a_{n})}{2}\\right]^{2} = 1\\leq a_{n + 1}(2 - a_n)" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "a_{n}\\leq a_{n + 1}" }, { "type": "text", "content": " ，根据单调有界定理可知收敛，两侧取极限得到极限为1" } ] }, "bbox": [ 89, 493, 905, 537 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 显然 " }, { "type": "equation_inline", "content": "x_{n} + \\frac{4}{x_{n}} \\geq 2\\sqrt{x_{n} \\cdot \\frac{4}{x_{n}}} = 4 > x_{n+1} + \\frac{4}{x_{n}}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "x_{n} > x_{n+1}" }, { "type": "text", "content": "，单减。另外有 " }, { "type": "equation_inline", "content": "0 < x_{n+1} < x_{n+1} + \\frac{4}{x_{n}} < 4" }, { "type": "text", "content": " 可知 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 有界，故收敛，两侧取极限得到极限为 2。" } ] }, "bbox": [ 90, 538, 905, 583 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 前后项不等式先用同项进行放缩，可得出单调性结论" } ] }, "bbox": [ 87, 583, 519, 599 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.7 递推问题" } ], "level": 2 }, "bbox": [ 92, 627, 240, 645 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.7.1 极限压缩定理" } ], "level": 2 }, "bbox": [ 94, 665, 273, 683 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.18 (极限压缩定理)" } ], "level": 2 }, "bbox": [ 121, 697, 312, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "A \\in \\mathbb{R}" }, { "type": "text", "content": ", 若 " }, { "type": "equation_inline", "content": "\\exists r \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|a_n - A| < r |a_{n-1} - A|" }, { "type": "text", "content": ", 则数列 " }, { "type": "equation_inline", "content": "a_n" }, { "type": "text", "content": " 收敛到 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": "。" } ] }, "bbox": [ 114, 718, 655, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.23经典极限压缩定理题(1)设 " }, { "type": "equation_inline", "content": "x_{n + 1} = \\cos x_n, x_1 \\in [0, \\frac{\\pi}{3}]" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} x_n" }, { "type": "text", "content": " 存在且极限为 " }, { "type": "equation_inline", "content": "\\cos x - x = 0" }, { "type": "text", "content": " 的根解(1)考虑 " }, { "type": "equation_inline", "content": "f(x) = \\cos x - x" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f'(x) = \\sin x - 1 \\leq 0" }, { "type": "text", "content": " 单减，因此有唯一解，设 " }, { "type": "equation_inline", "content": "\\cos a = a" }, { "type": "text", "content": "。由于" } ] }, "bbox": [ 58, 771, 902, 806 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| x _ {n + 1} - a \\right| = \\left| \\cos x _ {n} - a \\right| = \\left| \\cos x _ {n} - \\cos a \\right| = \\left| \\sin \\xi \\right| \\cdot \\left| x _ {n} - a \\right|", "math_type": "latex", "image_source": { "path": "images/ecdb4efd9718aa1b4f31d22fe703db6ffc6f93aaecc612909587b4892a444556.jpg" } }, "bbox": [ 270, 814, 726, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而由于 " }, { "type": "equation_inline", "content": "\\xi \\in [0, \\frac{\\pi}{3}]" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|\\sin \\xi| < \\frac{\\sqrt{3}}{2}" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "|x_{n} - a| \\to 0" } ] }, "bbox": [ 90, 835, 475, 863 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.7 递推问题" } ] }, "bbox": [ 803, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "14" } ] }, "bbox": [ 490, 954, 509, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.7.2 压缩映射定理" } ], "level": 2 }, "bbox": [ 94, 85, 273, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.19 (压缩映射定理)" } ], "level": 2 }, "bbox": [ 122, 118, 312, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若存在 " }, { "type": "equation_inline", "content": "r \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|a_{n} - a_{n-1}| \\leq r |a_{n} - a_{n-1}|" }, { "type": "text", "content": "，则数列 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛。" } ] }, "bbox": [ 115, 137, 613, 155 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 2.2 (导数判断压缩映射)" } ], "level": 2 }, "bbox": [ 122, 179, 337, 193 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "a_{n + 1} = f(a_n)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 可导，且 " }, { "type": "equation_inline", "content": "\\exists r\\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f'(x)|\\leq r < 1" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛。" } ] }, "bbox": [ 115, 198, 668, 215 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.24 两个经典循环式 证明收敛并求极限:" } ] }, "bbox": [ 58, 253, 431, 267 ] }, { "type": "equation_interline", "content": { "math_content": "(1) x _ {1} = 2, x _ {2} = 2 + \\frac {1}{2}, \\dots , x _ {n + 1} = 2 + \\frac {1}{a _ {n}}", "math_type": "latex", "image_source": { "path": "images/4d9c5c867fb8d964f93accba3bde77d59e62900a0d7e4330897d355e9311e923.jpg" } }, "bbox": [ 122, 268, 431, 293 ] }, { "type": "equation_interline", "content": { "math_content": "(2) x _ {1} = 1, x _ {2} = \\sqrt {2}, \\dots , x _ {n + 1} = \\sqrt {2 \\sqrt {x _ {n}}}", "math_type": "latex", "image_source": { "path": "images/bd2110f1e37b1f31de10f8ccac8a03388295ea995304bab098025dec14ac4c91.jpg" } }, "bbox": [ 122, 296, 423, 313 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "|x_{n+1}-x_{n}|=\\left|\\frac{1}{x_{n}}-\\frac{1}{x_{n-1}}\\right|=\\frac{|x_{n-1}-x_{n}|}{|x_{n}x_{n-1}|}\\leq\\frac{1}{4}|x_{n}-x_{n-1}|" }, { "type": "text", "content": " ，因此收敛，取极限得到极限为 " }, { "type": "equation_inline", "content": "1+\\sqrt{2}" } ] }, "bbox": [ 87, 315, 855, 346 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 设 " }, { "type": "equation_inline", "content": "f(x) = \\sqrt{2x}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|f'(x)| = \\frac{\\sqrt{2}}{2\\sqrt{x}} \\leq \\frac{\\sqrt{2}}{2} < 1" }, { "type": "text", "content": "。" } ] }, "bbox": [ 122, 347, 470, 376 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.8 Stolz 定理" } ], "level": 2 }, "bbox": [ 94, 400, 248, 418 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.20 (Stolz 定理)" } ], "level": 2 }, "bbox": [ 122, 439, 284, 454 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Stolz 定理分为以下两种，下面的 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 可以为无穷。" } ] }, "bbox": [ 114, 460, 472, 473 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "\\frac{\\infty}{\\infty}" }, { "type": "text", "content": " 型： " }, { "type": "equation_inline", "content": "b_{n}" }, { "type": "text", "content": " 严格单增趋于 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n - a_{n - 1}}{b_n - b_{n - 1}} = A" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{b_n} = A" } ] }, "bbox": [ 139, 474, 682, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "\\frac{0}{0}" }, { "type": "text", "content": " 型： " }, { "type": "equation_inline", "content": "b_{n}" }, { "type": "text", "content": " 严格单减趋于0， " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 也收敛到0，且 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n - a_{n - 1}}{b_n - b_{n - 1}} = A" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{b_n} = A" } ] }, "bbox": [ 139, 502, 786, 530 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.25 直接使用 用 Stolz 定理证明平均值定理" } ] }, "bbox": [ 58, 568, 452, 582 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "a_{n}=x_{1}+x_{2}+\\cdots+x_{n}, b_{n}=n" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{a_{n}-a_{n-1}}{b_{n}-b_{n-1}}=x_{n}\\to a" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\frac{a_{n}}{b_{n}}=\\frac{x_{1}+x_{2}+\\cdots+x_{n}}{n}\\to a" } ] }, "bbox": [ 62, 581, 873, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 由于 Stolz 定理可以证出平均值定理，在使用平均值定理时就不需要使用放缩法了。" } ] }, "bbox": [ 55, 612, 736, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.26 几道经典题目 (1) ZJU2022.2: " }, { "type": "equation_inline", "content": "x_{0} > 0, x_{n} = \\arctan x_{n-1}" }, { "type": "text", "content": " ，证明：" } ] }, "bbox": [ 58, 633, 625, 648 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(a) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}x_n = 0" } ] }, "bbox": [ 124, 652, 236, 668 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(b) " }, { "type": "equation_inline", "content": "\\sqrt[n]{nx_{n}}" }, { "type": "text", "content": " 收敛，并求极限" } ] }, "bbox": [ 122, 669, 312, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)(a)由于 " }, { "type": "equation_inline", "content": "x_{n} = \\arctan x_{n - 1} < x_{n - 1}" }, { "type": "text", "content": " 因此单减，根据单调有界定理则极限存在，且两侧取极限得到零" } ] }, "bbox": [ 87, 688, 853, 703 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(b) 考虑 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}nx_n^2" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "nx_{n}^{2} = \\frac{n}{\\frac{1}{x_{n}^{2}}}" }, { "type": "text", "content": " ，根据Stolz定理即" } ] }, "bbox": [ 122, 703, 529, 734 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} \\frac {1}{\\frac {1}{x _ {n} ^ {2}} - \\frac {1}{x _ {n - 1} ^ {2}}} = \\lim _ {n \\to \\infty} \\frac {x _ {n - 1} ^ {2} x _ {n} ^ {2}}{x _ {n - 1} ^ {2} - x _ {n} ^ {2}} = \\lim _ {n \\to \\infty} \\frac {x _ {n - 1} ^ {2} \\arctan^ {2} x _ {n - 1}}{x _ {n - 1} ^ {2} - \\arctan^ {2} x _ {n - 1}} = \\frac {x _ {n - 1} ^ {2} (x _ {n - 1} ^ {2} + \\frac {2}{3} x _ {n - 1} ^ {4} + \\frac {1}{9} x _ {n - 1} ^ {6})}{- \\frac {2}{3} x _ {n - 1} ^ {4} - \\frac {1}{9} x _ {n - 1} ^ {6}} \\frac {3}{2}", "math_type": "latex", "image_source": { "path": "images/73628130ae6aba4dda3cde20659f502fae45a19419e4e378f3ec696f65f494ee.jpg" } }, "bbox": [ 131, 736, 863, 772 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt{n} x_n = \\frac{\\sqrt{6}}{2}" } ] }, "bbox": [ 92, 780, 263, 809 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.8 Stolz 定理" } ] }, "bbox": [ 801, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "15" } ] }, "bbox": [ 490, 954, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.9 积分形式极限：分段法" } ], "level": 2 }, "bbox": [ 92, 87, 386, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.9.1 定积分形极限" } ], "level": 2 }, "bbox": [ 92, 126, 273, 143 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.27 基础拆分法 求下列极限:" } ] }, "bbox": [ 58, 155, 349, 168 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\int_0^{\\frac{\\pi}{2}}\\sin^n x\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\int_0^{\\frac{\\pi}{2}}(1 - \\sin x)^n\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\int_0^1\\frac{x^n}{1 + \\sqrt{x}}\\mathrm{d}x" }, { "type": "text", "content": " (4) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\int_0^1 e^{x^n}\\mathrm{d}x" } ] }, "bbox": [ 147, 170, 848, 206 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 显然 " }, { "type": "equation_inline", "content": "x \\in [0, \\frac{\\pi}{2})" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\sin^n x = 0" }, { "type": "text", "content": "，但 " }, { "type": "equation_inline", "content": "x = \\frac{\\pi}{2}" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\sin x = 1" }, { "type": "text", "content": "，因此做以下拆分：" } ] }, "bbox": [ 87, 208, 726, 233 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {\\frac {\\pi}{2}} \\sin^ {n} x \\mathrm{d} x = \\int_ {0} ^ {\\frac {\\pi}{2} - \\delta} \\sin^ {n} x \\mathrm{d} x + \\int_ {\\frac {\\pi}{2} - \\delta} ^ {\\frac {\\pi}{2}} \\sin^ {n} x \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/a6e6446fb4c381f8f4a126bc47dec931d71519f928de8840762f8cd7e916b237.jpg" } }, "bbox": [ 319, 234, 675, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "前者 " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2} -\\delta}\\sin^n x\\mathrm{d}x\\leq \\frac{\\pi}{2}\\sin^n (\\frac{\\pi}{2} -\\delta)\\to 0" }, { "type": "text", "content": " ，后者 " }, { "type": "equation_inline", "content": "\\int_{\\frac{\\pi}{2} -\\delta}^{\\frac{\\pi}{2}}\\sin^n x\\mathrm{d}x\\leq \\int_{\\frac{\\pi}{2} -\\delta}^{\\frac{\\pi}{2}}1\\mathrm{d}x = \\delta" }, { "type": "text", "content": " ，因此积分极限等于0。" } ] }, "bbox": [ 89, 273, 850, 309 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)(3) 同理都是 0" } ] }, "bbox": [ 122, 307, 253, 321 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 可以考虑 " }, { "type": "equation_inline", "content": "\\int_{0}^{1} e^{x^{n}} - 1 dx" }, { "type": "text", "content": " ，结果为 1." } ] }, "bbox": [ 122, 321, 408, 353 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.28用分段法证明命题(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 连续，证明 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\int_0^1 f(\\sqrt[n]{x})\\mathrm{d}x = f(1)" } ] }, "bbox": [ 58, 357, 697, 391 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.9.2 反常积分形极限" } ], "level": 2 }, "bbox": [ 92, 409, 294, 426 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 2.21 (函数形式平均值定理)" } ], "level": 2 }, "bbox": [ 121, 441, 364, 457 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 有定义且在任意有限区间可积， " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = A" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 112, 461, 623, 481 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\to + \\infty} \\frac {1}{x} \\int_ {0} ^ {x} f (t) \\mathrm{d} t = A", "math_type": "latex", "image_source": { "path": "images/ec566204afc566df0b2095cd52aeed37133f02b3b0617e64b7c8fc4918cfbc52.jpg" } }, "bbox": [ 408, 487, 588, 518 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 先斩后奏，已知 " }, { "type": "equation_inline", "content": "A = \\lim_{x\\to +\\infty}\\frac{1}{x}\\int_{a}^{x}Adt" }, { "type": "text", "content": " ，因此只需证" } ] }, "bbox": [ 89, 529, 517, 560 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\rightarrow + \\infty} \\left(\\frac {1}{x} \\int_ {a} ^ {x} f (t) \\mathrm{d} t - \\frac {1}{x} \\int_ {a} ^ {x} A \\mathrm{d} t\\right) = \\lim _ {x \\rightarrow + \\infty} \\frac {1}{x} \\int_ {a} ^ {x} (f (t) - A) \\mathrm{d} t = 0", "math_type": "latex", "image_source": { "path": "images/5cd291916856f2c0b09a6b544699729928bc32e8e15ec1582a1c6a73f4d51488.jpg" } }, "bbox": [ 247, 562, 749, 595 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists M, x \\geq M" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x) - A| < \\epsilon" }, { "type": "text", "content": "，因此可有 " }, { "type": "equation_inline", "content": "N, x > N" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\frac{1}{x} \\int_{a}^{M} |f(t) - A| \\mathrm{d}t < \\epsilon" }, { "type": "text", "content": "，得到" } ] }, "bbox": [ 87, 598, 732, 630 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\frac {1}{x} \\int_ {a} ^ {x} (f (t) - A) \\mathrm{d} t \\right| \\leq \\frac {1}{x} \\int_ {a} ^ {M} | f (t) - A | \\mathrm{d} t + \\frac {1}{x} \\int_ {M} ^ {x} | f (t) - A | \\mathrm{d} t < \\epsilon + \\frac {x - M}{x} \\cdot \\epsilon < 2 \\epsilon", "math_type": "latex", "image_source": { "path": "images/8611ca1e8e82d402634873dfde78e2a1c119710c7442c11d4e0b1314e1d49bac.jpg" } }, "bbox": [ 186, 633, 808, 667 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此结论成立" } ] }, "bbox": [ 90, 669, 196, 683 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此结论成立笔记如果有说明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 连续，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{x}f(t)\\mathrm{d}t" }, { "type": "text", "content": " 可导，可用洛必达法则证明。" } ] }, "bbox": [ 55, 675, 613, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.29相关练习已知 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 非负，对 " }, { "type": "equation_inline", "content": "\\forall A > 0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "xf(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, A]" }, { "type": "text", "content": " 可积，且 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 可积，证明： " }, { "type": "equation_inline", "content": "\\lim_{A \\to +\\infty} \\frac{1}{A} \\int_0^A xf(x) \\mathrm{d}x = 0" } ] }, "bbox": [ 58, 715, 907, 776 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛，因此 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists M,\\forall A^{\\prime \\prime}\\geq A^{\\prime}\\geq M" }, { "type": "text", "content": " 都有 " }, { "type": "equation_inline", "content": "0\\leq \\int_{A^{\\prime}}^{A^{\\prime \\prime}}f(x)\\mathrm{d}x < \\epsilon" }, { "type": "text", "content": " ，固定上面的 " }, { "type": "equation_inline", "content": "M" }, { "type": "equation_inline", "content": "\\exists N,\\forall A > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "0 < \\frac{1}{A}\\int_0^M xf(x)\\mathrm{d}x < \\epsilon" }, { "type": "text", "content": " ，进而 " }, { "type": "equation_inline", "content": "A > N" }, { "type": "text", "content": " 时" } ] }, "bbox": [ 87, 775, 907, 840 ] }, { "type": "equation_interline", "content": { "math_content": "0 < \\frac {1}{A} \\int_ {0} ^ {A} x f (x) \\mathrm{d} x = \\frac {1}{A} \\int_ {0} ^ {M} x f (x) \\mathrm{d} x + \\frac {1}{A} \\int_ {M} ^ {A} x f (x) \\mathrm{d} x < \\epsilon + \\frac {1}{A} \\int_ {M} ^ {A} A f (x) \\mathrm{d} x < 2 \\epsilon", "math_type": "latex", "image_source": { "path": "images/d86dd887c30712f85a709abb7f1eb2c07d3a65542c12f0ff4261b13120a67e06.jpg" } }, "bbox": [ 186, 843, 808, 878 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.9 积分形式极限：分段法" } ] }, "bbox": [ 704, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "16" } ] }, "bbox": [ 490, 954, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2.9.3 周期函数积分极限" } ], "level": 2 }, "bbox": [ 92, 85, 315, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.30 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上周期为 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": " 的可积函数，证明： " }, { "type": "equation_inline", "content": "\\forall a\\in \\mathbb{R},\\int_0^T f(x)\\mathrm{d}x = \\int_a^{a + T}f(x)\\mathrm{d}x" } ] }, "bbox": [ 58, 112, 742, 145 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 令 " }, { "type": "equation_inline", "content": "x = T + u" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\int_{T}^{a + T} f(x) \\mathrm{d}x = \\int_{0}^{a} f(T + u) \\mathrm{d}(T + u) = \\int_{0}^{a} f(x) \\mathrm{d}x" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 87, 143, 702, 177 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {a + T} f (x) \\mathrm{d} x = \\int_ {a} ^ {0} f (x) \\mathrm{d} x + \\int_ {0} ^ {T} f (x) \\mathrm{d} x + \\int_ {T} ^ {T + a} f (x) \\mathrm{d} x = \\int_ {0} ^ {T} f (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/1115c1ab4d274c6c2d49adbb3b8cae718921dd9cdce608513d84ab7a87ba321b.jpg" } }, "bbox": [ 231, 179, 764, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习2.31 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是周期为 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": " 的连续函数，证明： " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}\\frac{1}{x}\\int_0^x f(t)\\mathrm{d}t = \\frac{1}{T}\\int_0^T f(t)\\mathrm{d}t" }, { "type": "text", "content": " 证明对 " }, { "type": "equation_inline", "content": "\\forall x > 0" }, { "type": "text", "content": " ，都 " }, { "type": "equation_inline", "content": "\\exists n_x\\in \\mathbb{N}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "x - n_{x}T\\in [0,T)" }, { "type": "text", "content": " ，此时 " }, { "type": "equation_inline", "content": "\\lim_{x\\to \\infty}\\frac{n_x}{x} = \\frac{1}{T}" }, { "type": "text", "content": " 。根据周期性可知" } ] }, "bbox": [ 58, 223, 754, 280 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\rightarrow + \\infty} \\frac {1}{x} \\int_ {0} ^ {n _ {x} T} f (t) \\mathrm{d} t = \\frac {n _ {x}}{x} \\int_ {0} ^ {T} f (t) \\mathrm{d} t = \\frac {1}{T} \\int_ {0} ^ {T} f (t) \\mathrm{d} t", "math_type": "latex", "image_source": { "path": "images/d6aed4093a6ac4f4909502c5f86b6c619d67b18e077ef242ead099dca02c7ce8.jpg" } }, "bbox": [ 297, 282, 697, 316 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 " }, { "type": "equation_inline", "content": "||f(x)| - |f(y)|| \\leq |f(x) - f(y)|" }, { "type": "text", "content": "，从振幅角度来看，若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 可积，则 " }, { "type": "equation_inline", "content": "|f(x)|" }, { "type": "text", "content": " 也可积，且显然 " }, { "type": "equation_inline", "content": "|f(x)|" }, { "type": "text", "content": " 也是周期函数。因此 " }, { "type": "equation_inline", "content": "\\int_0^T |f(x)|\\mathrm{d}x" }, { "type": "text", "content": " 有界，此时：" } ] }, "bbox": [ 89, 318, 905, 362 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {n _ {x} T} ^ {x} f (t) \\mathrm{d} t \\leq \\int_ {n _ {x} T} ^ {(n _ {x} + 1) T} f (x) \\mathrm{d} x \\leq \\int_ {0} ^ {T} | f (x) | \\mathrm{d} x \\leq M", "math_type": "latex", "image_source": { "path": "images/c959bbaa102c34d85034e9936dd8bcaf3ae70a5dc3a038a677d9a6d91a2cf267.jpg" } }, "bbox": [ 300, 367, 694, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\lim_{x\\to \\infty}\\frac{1}{x}\\int_{n_xT}^x f(x)\\mathrm{d}x = 0" } ] }, "bbox": [ 90, 404, 315, 436 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 2.32 进阶难度 " }, { "type": "equation_inline", "content": "f(x) \\geq 0" }, { "type": "text", "content": " ，周期为 T，且连续，证明：" } ] }, "bbox": [ 58, 449, 512, 463 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明" } ] }, "bbox": [ 90, 502, 131, 517 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\rightarrow \\infty} n \\int_ {n} ^ {+ \\infty} \\frac {f (x)}{x ^ {2}} \\mathrm{d} x = \\frac {1}{T} \\int_ {0} ^ {T} f (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/cf6af53a2adfb54fb6615108d17c4a3bb9741244300ed71cb7404b631ef66746.jpg" } }, "bbox": [ 357, 467, 640, 500 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "2.9 积分形式极限：分段法" } ] }, "bbox": [ 704, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "17" } ] }, "bbox": [ 490, 953, 507, 964 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/8eb8fb666bbaa9a816146c4a4a4c8ebfda7fa464a434b62b0448bbe8a2aa3b16.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 464, 78, 483 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 3 章 函数极限与连续性" } ], "level": 1 }, "bbox": [ 337, 87, 675, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.1 函数的极限" } ], "level": 2 }, "bbox": [ 92, 145, 265, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.1.1 函数极限定义" } ], "level": 2 }, "bbox": [ 92, 184, 273, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.1 (函数极限)" } ], "level": 2 }, "bbox": [ 119, 217, 270, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "x_0 \\in \\mathbb{R}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 邻域有定义 ( " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处可以没有), " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta" }, { "type": "text", "content": ", s.t. " }, { "type": "equation_inline", "content": "\\forall x \\in [x_0 - \\delta, x_0 + \\delta] - \\{x_0\\}" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x) - A| < \\epsilon" }, { "type": "text", "content": ", 则称 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处极限。" } ] }, "bbox": [ 114, 235, 897, 277 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.1Riemann函数Riemann函数定义如下，证明对 " }, { "type": "equation_inline", "content": "\\forall x_0\\in [0,1]" }, { "type": "text", "content": " ，有 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}R(x) = 0" } ] }, "bbox": [ 58, 309, 724, 329 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "R(x) = \\left\\{ \\begin{array}{ll}\\frac{1}{q}, & x = \\frac{p}{q} (\\frac{p}{q}\\text{是既约真分数})\\\\ 0, & x\\text{是}(0,1)\\text{中的无理数或者}0,1\\text{本身} \\end{array} \\right." } ] }, "bbox": [ 315, 335, 677, 386 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "\\forall\\epsilon" }, { "type": "text", "content": " ，满足 " }, { "type": "equation_inline", "content": "\\frac{1}{q}\\geq\\epsilon" }, { "type": "text", "content": " 的正整数 q 只有有限个，从而 " }, { "type": "equation_inline", "content": "\\frac{p}{q}" }, { "type": "text", "content": " 有限，令这些数为 " }, { "type": "equation_inline", "content": "x_{1},\\cdots,x_{k}" }, { "type": "text", "content": " ，取" } ] }, "bbox": [ 87, 388, 737, 420 ] }, { "type": "equation_interline", "content": { "math_content": "\\delta = \\min \\left\\{\\left| x _ {i} - x _ {0} \\right|: i = 1, 2, \\dots , k \\right\\}", "math_type": "latex", "image_source": { "path": "images/7e602bd17a137f5ca438f277564044cfc16a764f6e579b79d57eb2375ad59ed3.jpg" } }, "bbox": [ 366, 420, 626, 437 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 属于 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 的 " }, { "type": "equation_inline", "content": "\\delta" }, { "type": "text", "content": " 邻域时，显然 " }, { "type": "equation_inline", "content": "R(x) < \\epsilon" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}R(x) = 0" } ] }, "bbox": [ 90, 444, 552, 462 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 上述练习说明 Riemann 函数在每个点极限为 0，在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 的无理点和 0,1 连续，在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 有理点间断。这说明 Riemann 函数是几乎处处连续的，根据实变可知其 Riemann 可积。" } ] }, "bbox": [ 87, 469, 905, 504 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.1 (极限的唯一性)" } ], "level": 2 }, "bbox": [ 121, 513, 305, 527 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若函数的极限存在，则它是唯一的。" } ] }, "bbox": [ 115, 533, 379, 548 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 反设有两个极限，取 " }, { "type": "equation_inline", "content": "\\epsilon" }, { "type": "text", "content": " 为差距的一半，根据定义可证。" } ] }, "bbox": [ 90, 567, 529, 583 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.1.2 函数极限的性质" } ], "level": 2 }, "bbox": [ 92, 605, 292, 621 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.2 (函数极限四则运算)" } ], "level": 2 }, "bbox": [ 122, 637, 337, 652 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处极限存在， " }, { "type": "equation_inline", "content": "f + g, f - g, f \\times g, \\frac{f}{g}" }, { "type": "text", "content": " 极限存在且极限为对应极限的运算 (除法要求分母极限非 0)。" } ] }, "bbox": [ 112, 657, 884, 700 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.2函数绝对值的极限设 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}f(x) = A" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}|f(x)| = |A|" }, { "type": "text", "content": " ，反之不一定成立。" } ] }, "bbox": [ 58, 719, 729, 739 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}f(x) = A" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists \\delta ,\\forall x\\in U(x_0,\\delta)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x) - A| < \\epsilon" }, { "type": "text", "content": " ，而 " }, { "type": "equation_inline", "content": "||f(x)| - |A||\\leq |f(x) - A|" }, { "type": "text", "content": " ，因此极限存在。反之有反例 " }, { "type": "equation_inline", "content": "f(x) = \\mathrm{sgn}(x)" } ] }, "bbox": [ 75, 736, 905, 777 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.3 (归结原则)" } ], "level": 2 }, "bbox": [ 122, 787, 270, 801 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\lim_{x\\to x_{0}}f(x)=A" }, { "type": "text", "content": " 当且仅当任意收敛到 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 的数列 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " ，有 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}f(x_{n})=A" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 112, 807, 638, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 左推右: " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, |x - x_0| < \\delta" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x) - A| < \\epsilon" }, { "type": "text", "content": ", 对于 " }, { "type": "equation_inline", "content": "x_n" }, { "type": "text", "content": ", 由于 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} x_n = x_0" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "\\exists N, \\forall n > N, |x_n - x_0| < \\delta" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "|f(x_n) - A| < \\epsilon" } ] }, "bbox": [ 90, 848, 905, 884 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 右推左：反设 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_{0}}f(x)\\neq A" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\exists\\epsilon_{0},\\forall\\delta,\\exists x\\in U(x_{0},\\delta)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|f(x)-A|>\\epsilon_{0}" }, { "type": "text", "content": " ，每次取 " }, { "type": "equation_inline", "content": "\\delta'=\\delta,\\frac{\\delta}{2},\\cdots,\\frac{\\delta}{n}" }, { "type": "text", "content": " ，每个 " }, { "type": "equation_inline", "content": "\\delta'" }, { "type": "text", "content": " 邻域对应一个 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " ，但是总有 " }, { "type": "equation_inline", "content": "|f(x_{n})-A|>\\epsilon_{0}" }, { "type": "text", "content": " ，这与 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}f(x_{n})=A" }, { "type": "text", "content": " 矛盾。" } ] }, "bbox": [ 90, 82, 905, 129 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.3用归结原则证明不收敛 证明： " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}\\sin \\frac{1}{x}" }, { "type": "text", "content": " 不存在。" } ] }, "bbox": [ 58, 140, 515, 165 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 令 " }, { "type": "equation_inline", "content": "x_{n} = \\frac{1}{(2n + \\frac{1}{2})\\pi}, x_{n}^{\\prime} = \\frac{1}{2n\\pi}" }, { "type": "text", "content": "，两者均收敛至0，但是 " }, { "type": "equation_inline", "content": "f(x_{n}) \\to 1" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f(x_{n}^{\\prime}) \\to 0" }, { "type": "text", "content": "，从而无极限。" } ] }, "bbox": [ 90, 166, 818, 197 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.1.3 无穷大与无穷小" } ], "level": 2 }, "bbox": [ 92, 216, 292, 233 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.2 (O 与 o)" } ], "level": 2 }, "bbox": [ 122, 249, 250, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f, g" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 附近有定义， " }, { "type": "equation_inline", "content": "g(x) \\neq 0" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\lim_{x \\to x_0} \\frac{|f(x)|}{|g(x)|} \\leq M" }, { "type": "text", "content": "，则记 " }, { "type": "equation_inline", "content": "f(x) = O(g(x))" }, { "type": "text", "content": "；若 " }, { "type": "equation_inline", "content": "\\lim_{x \\to x_0} \\frac{|f(x)|}{|g(x)|} = 0" }, { "type": "text", "content": "，则记 " }, { "type": "equation_inline", "content": "f(x) = o(g(x))" } ] }, "bbox": [ 112, 267, 884, 313 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.3 (Ω)" } ], "level": 2 }, "bbox": [ 122, 338, 216, 351 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f, g" }, { "type": "text", "content": " 是两个算法，若 " }, { "type": "equation_inline", "content": "\\exists C \\in \\mathbb{R}^{+}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "n > n_{0}" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "0 \\leq Cg(n) \\leq f(n)" }, { "type": "text", "content": "，则记 " }, { "type": "equation_inline", "content": "f(n) = \\Omega(g(n))" } ] }, "bbox": [ 112, 357, 752, 374 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.4 ( " }, { "type": "equation_inline", "content": "\\Theta" }, { "type": "text", "content": " )" } ], "level": 2 }, "bbox": [ 122, 398, 216, 413 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(n)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\Omega (g(n))" }, { "type": "text", "content": " 且 " }, { "type": "equation_inline", "content": "O(g(n))" }, { "type": "text", "content": " 的，则记 " }, { "type": "equation_inline", "content": "f(n) = \\Theta (g(n))" } ] }, "bbox": [ 114, 418, 527, 434 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 记忆方法：全以多项式为例，O 表示阶数 (次数) 大于等于，o 表示阶数大于， " }, { "type": "equation_inline", "content": "\\Omega" }, { "type": "text", "content": " 表示阶数小于等于， " }, { "type": "equation_inline", "content": "\\Theta" }, { "type": "text", "content": " 表示阶数等于。" } ] }, "bbox": [ 89, 453, 905, 486 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.1.4 使用定义直接计算" } ], "level": 2 }, "bbox": [ 92, 508, 314, 525 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.4 几个经典函数极限 (1) 计算 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sin^{2}(\\pi\\sqrt{n^{2}+n})" } ] }, "bbox": [ 58, 538, 515, 556 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}\\frac{[xf(x)]}{x}" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = a" } ] }, "bbox": [ 122, 557, 418, 587 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\sin^2 (\\pi \\sqrt{n^2 + n}) = \\sin^2 (\\pi \\sqrt{n^2 + n} -\\pi n) = \\sin^2 (\\frac{n\\pi}{\\sqrt{n^2 + n} + n})" }, { "type": "text", "content": " ，极限为 " }, { "type": "equation_inline", "content": "\\sin^2 (\\frac{\\pi}{2}) = 1" } ] }, "bbox": [ 89, 585, 749, 611 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于 " }, { "type": "equation_inline", "content": "xf(x) - 1 < [xf(x)] \\leq xf(x)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "f(x) - \\frac{1}{x} < \\frac{[xf(x)]}{x} \\leq f(x)" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} \\left( f(x) - \\frac{1}{x} \\right) = \\lim_{x \\to +\\infty} f(x) = a" }, { "type": "text", "content": "，因此由夹逼准则，极限为 " }, { "type": "equation_inline", "content": "a" } ] }, "bbox": [ 90, 611, 905, 661 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.5几个经典递推问题 " }, { "type": "equation_inline", "content": "(1)f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, +\\infty)" }, { "type": "text", "content": " 单调增， " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}\\frac{f(2x)}{f(x)} = 1" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "\\forall a > 0" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}\\frac{f(ax)}{f(x)} = 1" }, { "type": "text", "content": " 证明" } ] }, "bbox": [ 58, 669, 882, 714 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.2 函数极限的计算" } ], "level": 2 }, "bbox": [ 92, 741, 314, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3.2.1 等价无穷小公式" } ] }, "bbox": [ 92, 780, 294, 797 ] }, { "type": "equation_interline", "content": { "math_content": "1 - \\cos x \\sim \\frac {1}{2} x ^ {2} \\tan x \\sim x", "math_type": "latex", "image_source": { "path": "images/c3c326aca837bb33bc617b19791bfd4b2116433800ef37293dbf0cc155b9fa5f.jpg" } }, "bbox": [ 509, 819, 717, 846 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "反三角： " }, { "type": "equation_inline", "content": "\\arcsin x\\sim x" } ] }, "bbox": [ 243, 847, 455, 862 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\arctan x \\sim x" } ] }, "bbox": [ 509, 850, 606, 860 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "指数、对数： " }, { "type": "equation_inline", "content": "\\ln (1 + x)\\sim x" } ] }, "bbox": [ 243, 865, 465, 879 ] }, { "type": "equation_interline", "content": { "math_content": "e ^ {x} - 1 \\sim x", "math_type": "latex", "image_source": { "path": "images/fcb1b03481c70305328f15aaa67fb838a8bfbbe16519e485342c4a0c76ace1bd.jpg" } }, "bbox": [ 509, 866, 593, 879 ] }, { "type": "equation_interline", "content": { "math_content": "a ^ {x} - 1 \\sim x \\ln a", "math_type": "latex", "image_source": { "path": "images/a663cbcaa24f030fe7ae10832e502c6f06615268a729b2be25d44a6d9d66416b.jpg" } }, "bbox": [ 640, 866, 754, 879 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "其他： " }, { "type": "equation_inline", "content": "(1 + x)^{\\alpha} - 1\\sim \\alpha x" } ] }, "bbox": [ 243, 884, 500, 898 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "19" } ] }, "bbox": [ 490, 954, 507, 964 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.2 函数极限的计算" } ] }, "bbox": [ 754, 55, 905, 68 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.2.2 Taylor 展开公式" } ], "level": 2 }, "bbox": [ 92, 86, 295, 103 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "指数、对数:" } ] }, "bbox": [ 97, 137, 194, 155 ] }, { "type": "equation_interline", "content": { "math_content": "e ^ {x} = \\sum_ {n = 0} ^ {\\infty} \\frac {x ^ {n}}{n !}", "math_type": "latex", "image_source": { "path": "images/b39eefef52be5321914c6efd0e9350b1245e78628e1304df892de7db2f3cc744.jpg" } }, "bbox": [ 215, 130, 309, 167 ] }, { "type": "equation_interline", "content": { "math_content": "\\ln (1 + x) = \\sum_ {n = 1} ^ {\\infty} (- 1) ^ {n - 1} \\frac {x ^ {n}}{n}", "math_type": "latex", "image_source": { "path": "images/8e9126f6ef1edf5dae3ab4a5e48c4eafc4e017b4dca3908e9a0e0add614c3cc8.jpg" } }, "bbox": [ 618, 130, 821, 165 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "三角：" } ] }, "bbox": [ 97, 174, 144, 189 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin x = \\sum_ {n = 0} ^ {\\infty} (- 1) ^ {n} \\frac {x ^ {2 n + 1}}{(2 n + 1) !}", "math_type": "latex", "image_source": { "path": "images/98528848081b9f420868f2f0912fb0f8f833c8dab6549086769ee1908d543211.jpg" } }, "bbox": [ 213, 165, 413, 200 ] }, { "type": "equation_interline", "content": { "math_content": "\\cos x = \\sum_ {n = 0} ^ {\\infty} (- 1) ^ {n} \\frac {x ^ {2 n}}{(2 n) !}", "math_type": "latex", "image_source": { "path": "images/ee28a7fe160e0839e39781d0e51e59ada127a404ef694ccd500d524358d767ec.jpg" } }, "bbox": [ 621, 166, 793, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "反三角：" } ] }, "bbox": [ 99, 208, 161, 224 ] }, { "type": "equation_interline", "content": { "math_content": "\\arcsin x = \\sum_ {n = 0} ^ {\\infty} \\left(\\frac {(2 n) !}{2 ^ {2 n} (n !) ^ {2}}\\right) \\frac {x ^ {2 n + 1}}{2 n + 1} = x + \\frac {1}{6} x ^ {3} + o \\left(x ^ {3}\\right)", "math_type": "latex", "image_source": { "path": "images/2933f6ea19874fa9c1a71fc265865fd118c7af05cffe28585b7f1f4785bc95ea.jpg" } }, "bbox": [ 213, 199, 608, 235 ] }, { "type": "equation_interline", "content": { "math_content": "\\arctan x = \\sum_ {n = 0} ^ {\\infty} \\frac {(- 1) ^ {n} x ^ {2 n + 1}}{2 n + 1} = x - \\frac {1}{3} x ^ {3} + o (x ^ {3})", "math_type": "latex", "image_source": { "path": "images/286f7018abbf9c8e1cfe92bd731fc570ea9b97f282adc7e211c47f2ad8cac71b.jpg" } }, "bbox": [ 621, 199, 966, 235 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "分式:" } ] }, "bbox": [ 97, 242, 144, 259 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {1}{1 + x}} = \\sum_ {n = 0} ^ {\\infty} (- 1) ^ {n} x ^ {n}", "math_type": "latex", "image_source": { "path": "images/5dfb2cf5dd9f97c5b16e0a7584aed159d0f217df0a293f840bee82a74297c439.jpg" } }, "bbox": [ 215, 236, 374, 269 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {1}{1 - x}} = \\sum_ {n = 0} ^ {\\infty} x ^ {n}", "math_type": "latex", "image_source": { "path": "images/06c8d945b07e973730ea89029e01aeb04146cb8466f4bbb865293721690aa5b1.jpg" } }, "bbox": [ 621, 236, 739, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "其他：" } ] }, "bbox": [ 97, 277, 144, 293 ] }, { "type": "equation_interline", "content": { "math_content": "(1 + x) ^ {\\alpha} = 1 + \\alpha x + \\frac {\\alpha (\\alpha - 1)}{2} x ^ {2} + o (x ^ {2})", "math_type": "latex", "image_source": { "path": "images/77748c82f14a3c5c180a91805f3b6502c887ecafb59c6cbbcf7b7edf4974043b.jpg" } }, "bbox": [ 215, 269, 517, 300 ] }, { "type": "equation_interline", "content": { "math_content": "(1 + x) ^ {\\alpha} = \\sum_ {n = 0} ^ {\\infty} {\\binom {\\alpha} {n}} x ^ {n}", "math_type": "latex", "image_source": { "path": "images/3b29fff19662c1d9f03f2b55b5fdf23583aa688e0436f4630ea4d7853c4cb9cf.jpg" } }, "bbox": [ 621, 269, 791, 306 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.3 函数的连续性" } ], "level": 2 }, "bbox": [ 92, 326, 289, 347 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.3.1 函数连续的定义" } ], "level": 2 }, "bbox": [ 92, 366, 292, 384 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.5 (单点连续)" } ], "level": 2 }, "bbox": [ 121, 398, 270, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f:[a,b]\\to \\mathbb{R},x_0\\in [a,b]" }, { "type": "text", "content": " 若 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}f(x) = f(x_0)" }, { "type": "text", "content": " ，则称 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处连续。" } ] }, "bbox": [ 112, 418, 653, 439 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 有时也可以展开用 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\forall x \\in (x_0 - \\delta, x_0 + \\delta)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|f(x) - f(x_0)| < \\epsilon" }, { "type": "text", "content": " 证明。例如证明 Riemann 函数在无理点连续。" } ] }, "bbox": [ 87, 460, 905, 494 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.6 (单点左右连续)" } ], "level": 2 }, "bbox": [ 121, 502, 304, 519 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x_0^+) = f(x_0)" }, { "type": "text", "content": " 则称 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 右连续。左连续同理。" } ] }, "bbox": [ 114, 524, 485, 541 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 注意右连续是指右极限等于当前点的函数值，而非左侧极限靠过来等于当前点的函数值！" } ] }, "bbox": [ 87, 558, 778, 575 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.6Dirichlet 函数的连续性 判断下面 Dirichlet 函数的连续性、单调性、周期性" } ] }, "bbox": [ 58, 596, 700, 612 ] }, { "type": "equation_interline", "content": { "math_content": "D (x) = {\\left\\{ \\begin{array}{l l} {1,} & {x \\text {为有理数}} \\\\ {0} & {x \\text {为无理数}} \\end{array} \\right.}", "math_type": "latex", "image_source": { "path": "images/e97277a79e50104ae0e521bdef614f24c1cb5d982d6ce317559b77f8e064be9d.jpg" } }, "bbox": [ 401, 615, 591, 659 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)周期: 任意有理数(显然)" } ] }, "bbox": [ 87, 664, 324, 678 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 连续性：处处不连续，因为根据有理数和无理数的稠密性，总能在小邻域中找到相差 1 的点，因此不连续。" } ] }, "bbox": [ 89, 683, 905, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.7Riemann 函数的连续性 证明 Riemann 函数在无理点连续，在有理点间断。" } ] }, "bbox": [ 58, 738, 692, 753 ] }, { "type": "equation_interline", "content": { "math_content": "R (x) = {\\left\\{ \\begin{array}{l l} { {\\frac {1}{q}},} & {x = {\\frac {p}{q}} \\text {为既约分数}} \\\\ {0,} & {x \\text {为无理数}} \\end{array} \\right.}", "math_type": "latex", "image_source": { "path": "images/b39f58e7c265f812f1523302ed86805fec9ba6a56449da4d7df91c4f7ae2ff9f.jpg" } }, "bbox": [ 374, 756, 618, 806 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 有理点: 设 " }, { "type": "equation_inline", "content": "x_0 = \\frac{p}{q}" }, { "type": "text", "content": " 为有理点, " }, { "type": "equation_inline", "content": "R(x_0) = \\frac{1}{q}" }, { "type": "text", "content": ", 无理点列趋于 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 时, 极限为 0 , 与函数值不同。" } ] }, "bbox": [ 87, 810, 831, 839 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 无理点： " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 是无理点， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\frac{1}{q} > \\epsilon" }, { "type": "text", "content": " 的个数有限，设这些点为 " }, { "type": "equation_inline", "content": "x_{1}, \\cdots, x_{n}" }, { "type": "text", "content": "，取 " }, { "type": "equation_inline", "content": "\\delta < \\min\\{|x_{0} - x_{i}| : i = 1 \\sim n\\}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x \\in (x_{0} - \\delta, x_{0} + \\delta) - \\{x_{0}\\}" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x_{0}) - f(x)| < \\epsilon" } ] }, "bbox": [ 87, 841, 905, 881 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.8 初等函数的连续性 f, g, h 连续，证明： " }, { "type": "equation_inline", "content": "\\max\\{f(x), g(x)\\}, \\min\\{f(x), g(x)\\}, |f(x)|" }, { "type": "text", "content": " 连续" } ] }, "bbox": [ 58, 901, 778, 917 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.3 函数的连续性" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "20" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 最大、最小：根据 " }, { "type": "equation_inline", "content": "\\max \\{f(x), g(x)\\} = \\frac{1}{2}[f(x) + g(x) + |f(x) - g(x)|]" } ] }, "bbox": [ 90, 82, 662, 109 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 绝对值: " }, { "type": "equation_inline", "content": "|f(x)| = \\sqrt{|f(x)|^2}" }, { "type": "text", "content": " 连续" } ] }, "bbox": [ 122, 109, 389, 124 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.3.2 间断点的概念" } ], "level": 2 }, "bbox": [ 92, 146, 273, 162 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.7 (第一类间断点)" } ], "level": 2 }, "bbox": [ 121, 179, 304, 193 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "跳跃点与可去间断点统称为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的第一类间断点" } ] }, "bbox": [ 112, 199, 480, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 跳跃点：若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处左右极限 " }, { "type": "equation_inline", "content": "f(x_0^-), f(x_0^+)" }, { "type": "text", "content": " 存在但不相等，" } ] }, "bbox": [ 139, 217, 623, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 可去间断点：若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处左右极限存在且相等，即 " }, { "type": "equation_inline", "content": "f(x_0^-) = f(x_0^+)" } ] }, "bbox": [ 139, 235, 678, 252 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.9第一类间断点的性质(1)若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 至多有第一类间断点，证明： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 有界证明(1)若没有间断点，则连续，显然有界。否则反设无界，则对 " }, { "type": "equation_inline", "content": "M = 1" }, { "type": "text", "content": " ，存在 " }, { "type": "equation_inline", "content": "x_{1} \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(x_1)| > 1" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "M = 2" }, { "type": "text", "content": " 存在 " }, { "type": "equation_inline", "content": "x_{2} \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(x_2)| > 2" }, { "type": "text", "content": " ，以此类推构造 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 。由于 " }, { "type": "equation_inline", "content": "x_{n} \\in [a,b]" }, { "type": "text", "content": " ，根据致密性定理，存在收敛子列 " }, { "type": "equation_inline", "content": "x_{n_k}" }, { "type": "text", "content": " ，而由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 只有第一类间断点，因此 " }, { "type": "equation_inline", "content": "\\lim_{k \\to +\\infty} f(x_{n_k})" }, { "type": "text", "content": " 存在，但其趋于无穷，矛盾。" } ] }, "bbox": [ 58, 290, 905, 365 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.8 (第二类间断点)" } ], "level": 2 }, "bbox": [ 121, 387, 304, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x_0^+)" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "f(x_0^-)" }, { "type": "text", "content": " 至少一个不存在或者不是有限的数，则称 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的第二类间断点。" } ] }, "bbox": [ 114, 407, 766, 424 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.10 判断间断点类型 (1) 已知 " }, { "type": "equation_inline", "content": "f(x) = \\text{sgn}(x)" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "g(x) = x - [x]" }, { "type": "text", "content": " ，讨论 " }, { "type": "equation_inline", "content": "f(g(x))" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "g(f(x))" }, { "type": "text", "content": " 的间断点类型\n\n解 (1) " }, { "type": "equation_inline", "content": "f(g(x))" }, { "type": "text", "content": " 在 x 为整数时等于 0，在 x 非整数时大于 0，因此在整数处取可去间断点。 " }, { "type": "equation_inline", "content": "g(f(x)) = 0" }, { "type": "text", "content": " ，因此连续" } ] }, "bbox": [ 58, 462, 904, 497 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.11 单调函数的间断点 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 邻域单增， " }, { "type": "equation_inline", "content": "f(x_{0}^{-}), f(x_{0}^{+})" }, { "type": "text", "content": " 均存在，证明如下性质，即单调函数的间断点都是跳跃间断点" } ] }, "bbox": [ 58, 517, 905, 550 ] }, { "type": "equation_interline", "content": { "math_content": "f (x _ {0} ^ {-}) = \\sup _ {x \\in U _ {-} ^ {o} (x _ {0})} f (x), f (x _ {0} ^ {+}) = \\inf _ {x \\in U _ {+} ^ {o} (x _ {0})} f (x)", "math_type": "latex", "image_source": { "path": "images/368bebeec756c93225128352a67d9ee9f95542a8437ab840d3926afe22513d5c.jpg" } }, "bbox": [ 329, 554, 665, 579 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 取 " }, { "type": "equation_inline", "content": "x_{1} \\in U_{+}^{o}(x_{0})" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\forall x \\in U_{-}^{o}(x_{0})" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f(x) \\leq f(x_{1})" }, { "type": "text", "content": "，从而根据确界原理有上确界 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": "，从而 " }, { "type": "equation_inline", "content": "f(x) \\leq A" }, { "type": "text", "content": " 且 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists x'" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(x') > A - \\epsilon" }, { "type": "text", "content": "，从而 " }, { "type": "equation_inline", "content": "\\forall x \\in (x_{0} - \\delta, x_{0})" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x) - A| < \\epsilon" }, { "type": "text", "content": " 笔记 单调函数左右极限均存在是一个会在考题中使用的结论。" } ] }, "bbox": [ 75, 589, 902, 642 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.12单调函数右连续 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上单调函数，定义 " }, { "type": "equation_inline", "content": "g(x) = f(x + 0)" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 每一点右连续证明只需要证明 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "|x - x_0| < \\delta" }, { "type": "text", "content": " 时，满足" } ] }, "bbox": [ 58, 663, 847, 697 ] }, { "type": "equation_interline", "content": { "math_content": "| g (x) - g (x _ {0}) | < \\epsilon", "math_type": "latex", "image_source": { "path": "images/5f9835047d108ad45e1b77dda477497fd1c66920a11ea08cc06ed46bd9f2844c.jpg" } }, "bbox": [ 431, 706, 566, 722 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 " }, { "type": "equation_inline", "content": "g(x_0) = \\lim_{y\\to x_0^+}f(y)" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "y\\in (x_0,x_0 + \\delta)" }, { "type": "text", "content": " 时有 " }, { "type": "equation_inline", "content": "|f(y) - g(x_0)| < \\epsilon" }, { "type": "text", "content": " ，令 " }, { "type": "equation_inline", "content": "y\\to x^{+}" }, { "type": "text", "content": " 即可得到" } ] }, "bbox": [ 89, 728, 757, 753 ] }, { "type": "equation_interline", "content": { "math_content": "| g (x) - g (x _ {0}) | < \\epsilon", "math_type": "latex", "image_source": { "path": "images/ff2901bd3fa04dcd8522ed90fde1449e33de3157cc3362d77e04a4b09f990998.jpg" } }, "bbox": [ 431, 757, 566, 772 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.13极限的连续性 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 只有可去间断点， " }, { "type": "equation_inline", "content": "g(x) = \\lim_{y\\to x}f(y)" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 是连续函数证明可去间断点说明极限存在， " }, { "type": "equation_inline", "content": "g(x_0) = \\lim_{y\\to x_0}f(y)" }, { "type": "text", "content": "，即对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, y \\in U^o(x_0, \\delta)" }, { "type": "text", "content": " 时：" } ] }, "bbox": [ 58, 797, 736, 840 ] }, { "type": "equation_interline", "content": { "math_content": "g (x _ {0}) - \\epsilon < f (y) < g (x _ {0}) + \\epsilon", "math_type": "latex", "image_source": { "path": "images/8382129c4df536f6a2f103e5faf388190c41541e1aa0e87dc5060c577c5c04f6.jpg" } }, "bbox": [ 389, 845, 606, 860 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 " }, { "type": "equation_inline", "content": "g(x) = \\lim_{y\\to x}f(y)" }, { "type": "text", "content": " 和保不等式性得到" } ] }, "bbox": [ 90, 866, 388, 888 ] }, { "type": "equation_interline", "content": { "math_content": "g (x _ {0}) - \\epsilon \\leq g (x) \\leq g (x _ {0}) + \\epsilon", "math_type": "latex", "image_source": { "path": "images/76a6da078d81ff424be986656dda9fbe121e8b4d575a33607556ded4551356de.jpg" } }, "bbox": [ 389, 892, 605, 907 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "21" } ] }, "bbox": [ 489, 954, 507, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.3 函数的连续性" } ] }, "bbox": [ 771, 55, 904, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 连续，由 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 任意性可知结论成立。" } ] }, "bbox": [ 90, 85, 462, 99 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.14 第一类间断点的经典题目 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 至多只有第一类间断点，且对 " }, { "type": "equation_inline", "content": "\\forall x,y\\in(a,b)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f\\left(\\frac{x+y}{2}\\right)\\leq\\frac{f(x)+f(y)}{2}" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 连续" } ] }, "bbox": [ 58, 116, 905, 166 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "\\forall x_{0}\\in(a,b),\\forall h" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "x_{0}\\pm h\\in(a,b)" }, { "type": "text", "content": " 时，根据条件可知：" } ] }, "bbox": [ 87, 167, 539, 184 ] }, { "type": "equation_interline", "content": { "math_content": "f (x _ {0}) \\leq \\frac {f (x _ {0} + h) + f (x _ {0} - h)}{2} \\quad f (x _ {0} + \\frac {h}{2}) \\leq \\frac {f (x _ {0}) + f (x _ {0} + h)}{2}", "math_type": "latex", "image_source": { "path": "images/10b12ed658bf1def5f4fa75a19cf3f35d0aa848900d16324f218d0c790fa355a.jpg" } }, "bbox": [ 255, 187, 737, 216 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "前者取 " }, { "type": "equation_inline", "content": "h \\rightarrow 0^{+}" }, { "type": "text", "content": " ，后者分别取 " }, { "type": "equation_inline", "content": "h \\rightarrow 0^{+}, h \\rightarrow 0^{-}" }, { "type": "text", "content": " 则" } ] }, "bbox": [ 90, 218, 450, 231 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} f (x _ {0}) \\leq \\frac {f (x _ {0} + 0) + f (x _ {0} - 0)}{2} \\\\ f (x _ {0} + 0) \\leq \\frac {f (x _ {0}) + f (x _ {0} + 0)}{2} \\\\ f (x _ {0} - 0) \\leq \\frac {f (x _ {0}) + f (x _ {0} - 0)}{2} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/181f6351788ab33c4708d434dca6bbafad8c450356671a98fd4f11c86fdddd3f.jpg" } }, "bbox": [ 368, 235, 611, 313 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "综合得到 " }, { "type": "equation_inline", "content": "f(x_0 - 0) = f(x_0 + 0) = f(x_0)" }, { "type": "text", "content": "，因此在任意点连续。" } ] }, "bbox": [ 90, 316, 549, 332 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.3.3 介值性定理" } ], "level": 2 }, "bbox": [ 94, 353, 253, 370 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.4 (零点存在定理)" } ], "level": 2 }, "bbox": [ 121, 385, 304, 400 ] }, { "type": "equation_interline", "content": { "math_content": "f (a) f (b) < 0", "math_type": "latex", "image_source": { "path": "images/9000471368bf58498dd6c06f38f1b1d3c5b335c7235bf85b96cbf03dfe30c682.jpg" } }, "bbox": [ 312, 407, 408, 420 ] }, { "type": "equation_interline", "content": { "math_content": "\\exists \\xi \\in (a, b)", "math_type": "latex", "image_source": { "path": "images/b3cb484c797a019f4bb6c8235dc4d7b4a30d41e467266142a5fd840b2fa97d67.jpg" } }, "bbox": [ 442, 407, 517, 420 ] }, { "type": "equation_interline", "content": { "math_content": "f (\\xi) = 0", "math_type": "latex", "image_source": { "path": "images/87ca9fa4902982172c25500b72fcd5df1cc3822320c1e1845162ecbca687978c.jpg" } }, "bbox": [ 559, 407, 623, 420 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 使用闭区间套定理进行证明：取 " }, { "type": "equation_inline", "content": "a_1 = a, b_1 = b, c_1 = \\frac{a_1 + b_1}{2}" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "f(c_1)" }, { "type": "text", "content": " 正负性选取 " }, { "type": "equation_inline", "content": "a_2, b_2" }, { "type": "text", "content": "，以此类推，由于 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}|a_n - b_n| = 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f(a_n)f(b_n) < 0" }, { "type": "text", "content": "，根据闭区间套 " }, { "type": "equation_inline", "content": "\\exists \\xi" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f^2 (\\xi)\\leq 0" }, { "type": "text", "content": "，因此得出 " }, { "type": "equation_inline", "content": "f(\\xi) = 0" } ] }, "bbox": [ 90, 438, 905, 476 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 使用有限覆盖定理证明：根据连续函数局部保号性， " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b]" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists \\delta_x" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x - \\delta_x, x + \\delta_x) \\cap [a, b]" }, { "type": "text", "content": " 中保号，取遍 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 显然是一族开覆盖，根据有限覆盖定理可找到有限覆盖，不妨设这有限覆盖相互有重叠，则全部等号，这与 " }, { "type": "equation_inline", "content": "f(a)f(b) < 0" }, { "type": "text", "content": " 矛盾。" } ] }, "bbox": [ 90, 479, 905, 530 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.5 (介值性定理)" } ], "level": 2 }, "bbox": [ 121, 542, 287, 557 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 中的连续函数， " }, { "type": "equation_inline", "content": "a\\leq x_1 < x_2\\leq b" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f(x_{1})\\neq f(x_{2})" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可取遍 " }, { "type": "equation_inline", "content": "f(x_{1}),f(x_{2})" }, { "type": "text", "content": " 间任意值。" } ] }, "bbox": [ 112, 561, 894, 596 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.15 构造辅助函数 (1) " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 是 [0,1] 上的连续函数，且在 [0,1] 有相同最大值，证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi) = g(\\xi)" } ] }, "bbox": [ 58, 632, 905, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[0, 2a]" }, { "type": "text", "content": " 连续函数, 且 " }, { "type": "equation_inline", "content": "f(0) = f(2a)" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in [0, a]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi) = f(\\xi + a)" } ] }, "bbox": [ 121, 669, 789, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 上连续函数， " }, { "type": "equation_inline", "content": "f(0) = f(1) = 0" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\forall n \\in \\mathbb{N}^{+}, \\exists \\xi \\in [0,1]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi + \\frac{1}{n}) = f(\\xi)" } ] }, "bbox": [ 121, 687, 870, 706 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 上非负连续函数, " }, { "type": "equation_inline", "content": "f(0) = f(1) = 0" }, { "type": "text", "content": ", 证明: 对 " }, { "type": "equation_inline", "content": "\\forall a \\in (0,1), \\exists \\xi \\in [0,1]" }, { "type": "text", "content": ", 使得 " }, { "type": "equation_inline", "content": "f(\\xi) = f(\\xi + a)" }, { "type": "text", "content": " 证明 (1) 问题在于最大值取值点不一定一样: 若最大值对应 " }, { "type": "equation_inline", "content": "f(x_1), g(x_2)" }, { "type": "text", "content": ", 若 " }, { "type": "equation_inline", "content": "x_1 = x_2" }, { "type": "text", "content": ", 则显然成立。若 " }, { "type": "equation_inline", "content": "x_1 \\neq x_2" }, { "type": "text", "content": ", 则取 " }, { "type": "equation_inline", "content": "F(x) = f(x) - g(x)" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "F(x_1) \\geq 0, F(x_2) \\leq 0" }, { "type": "text", "content": ", 从而由零点存在定理成立。" } ] }, "bbox": [ 87, 707, 904, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 取 " }, { "type": "equation_inline", "content": "F(x) = f(x) - f(x + a)" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "F(0) = f(0) - f(a)" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "F(a) = f(a) - f(2a) = f(a) - f(0)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "F(0) = -F(a)" }, { "type": "text", "content": "，显然成立。" } ] }, "bbox": [ 90, 763, 902, 795 ] }, { "type": "equation_interline", "content": { "math_content": "F (x) = f \\left(x + \\frac {1}{n}\\right) - f (x), F (0) = f \\left(\\frac {1}{n}\\right) - f (0), F \\left(\\frac {1}{n}\\right) = f \\left(\\frac {2}{n}\\right) - f \\left(\\frac {1}{n}\\right), \\dots , F \\left(\\frac {n - 1}{n}\\right) = f (1) - f \\left(\\frac {n - 1}{n}\\right)", "math_type": "latex", "image_source": { "path": "images/4cb20353f00f8cdaf4930d4d9da49e9bf010b4536891322a9271a85d5fda3684.jpg" } }, "bbox": [ 169, 793, 899, 821 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到 " }, { "type": "equation_inline", "content": "F(0)+F(\\frac{1}{n})+\\cdots+F(\\frac{n-1}{n})=f(1)-f(0)=0" }, { "type": "text", "content": " ，要么这些全部为零，要么有正有负，根据定理可知。" } ] }, "bbox": [ 90, 820, 890, 841 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 取 " }, { "type": "equation_inline", "content": "F(x) = f(x) - f(x + a)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "F(0) = f(0) - f(a) = -f(a)" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "F(1 - a) = f(1 - a) - f(1) = f(1 - a)" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 非负，因此 " }, { "type": "equation_inline", "content": "F(0)F(1 - a) \\leq 0" }, { "type": "text", "content": "。" } ] }, "bbox": [ 89, 843, 905, 878 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.3 函数的连续性" } ] }, "bbox": [ 771, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "22" } ] }, "bbox": [ 489, 953, 509, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.6 (连续函数平均值定理)" } ], "level": 2 }, "bbox": [ 121, 87, 354, 101 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "x_{1},\\dots ,x_{n}\\in [a,b]" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in [a,b]" }, { "type": "text", "content": " ，使得" } ] }, "bbox": [ 114, 105, 584, 123 ] }, { "type": "equation_interline", "content": { "math_content": "f (\\xi) = \\frac {1}{n} [ f (x _ {1}) + \\dots + f (x _ {n}) ]", "math_type": "latex", "image_source": { "path": "images/b7ebeed293c1c4db3b43fb63bd0b701781833569c052d1f3ef34af004de96868.jpg" } }, "bbox": [ 384, 126, 610, 153 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 不妨设 " }, { "type": "equation_inline", "content": "f(x_{1}),\\cdots,f(x_{n})" }, { "type": "text", "content": " 中最大值为 " }, { "type": "equation_inline", "content": "f(x_{n})" }, { "type": "text", "content": " ，最小值为 " }, { "type": "equation_inline", "content": "f(x_{1})" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 89, 171, 611, 187 ] }, { "type": "equation_interline", "content": { "math_content": "f (x _ {1}) \\leq \\frac {1}{n} [ f (x _ {1}) + \\dots + f (x _ {n}) ] \\leq f (x _ {n})", "math_type": "latex", "image_source": { "path": "images/5377da7f52a38c74aa06ee45f81dfe25505d519c237e75dba4a73aa2bdd4b9d5.jpg" } }, "bbox": [ 347, 190, 647, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x_{1}) = \\frac{1}{n}\\sum_{i = 1}^{n}f(x_{i})" }, { "type": "text", "content": " 或 " }, { "type": "equation_inline", "content": "f(x_{n}) = \\frac{1}{n}\\sum_{i = 1}^{n}f(x_{i})" }, { "type": "text", "content": "，则取 " }, { "type": "equation_inline", "content": "\\xi = x_{1},x_{n}" }, { "type": "text", "content": " 。否则由连续函数介值性定理可证。" } ] }, "bbox": [ 89, 219, 818, 258 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.3.4 介值性定理的无连续推广" } ], "level": 2 }, "bbox": [ 92, 275, 374, 293 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "有时没有连续的条件却能推出介值性定理的结论，常见的条件如单调。" } ] }, "bbox": [ 122, 305, 640, 319 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.16单调介值定理 核心： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在[0,1]上单增， " }, { "type": "equation_inline", "content": "f(0) > 0,f(1) < 1" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "k\\in \\mathbb{N}^{+}" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "\\exists x_0\\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(x_0) = x_0^k" }, { "type": "text", "content": " \n证明取 " }, { "type": "equation_inline", "content": "F(x) = f(x) - x^k" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "F(0) = f(0) > 0,F(1) = f(1) - 1 < 0" }, { "type": "text", "content": " ，不停地二分取闭区间套，有 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}|a_n-" }, { "type": "equation_inline", "content": "b_{n}| = 0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = \\lim_{n\\to \\infty}b_n = x_0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "F(a_{n}) < 0,F(b_{n}) > 0\\Rightarrow f(a_{n}) > a_{n}^{k},f(b_{n}) < b_{n}^{k}" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 单增， " }, { "type": "equation_inline", "content": "a_{n}^{k} < f(a_{n})\\leq f(x_{0})\\leq f(b_{n}) < b_{n}^{k}" }, { "type": "text", "content": " ，因此取极限可知 " }, { "type": "equation_inline", "content": "x_0^k\\leq f(x_0)\\leq x_0^k" }, { "type": "text", "content": " 得证。" } ] }, "bbox": [ 58, 341, 907, 437 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 单调介值性定理不能将极限取进函数，但是用了两侧夹逼的方式。" } ] }, "bbox": [ 55, 435, 610, 451 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.17 证明：闭区间连续函数一定有界" } ] }, "bbox": [ 58, 474, 403, 489 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据局部有界性以及有限开覆盖可证。" } ] }, "bbox": [ 89, 492, 413, 507 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.4 连续性的应用" } ], "level": 2 }, "bbox": [ 92, 535, 287, 554 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.18几道经典题目(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "f(2x) = f(x)" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是常值函数" } ] }, "bbox": [ 58, 569, 719, 585 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 x=0,1 连续， " }, { "type": "equation_inline", "content": "f(x^{2})=f(x)" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是常值函数" } ] }, "bbox": [ 121, 588, 589, 604 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在 R 上连续，且在每个有理点有 " }, { "type": "equation_inline", "content": "f(x) = g(x)" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 恒等于 " }, { "type": "equation_inline", "content": "g(x)" } ] }, "bbox": [ 122, 607, 731, 621 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 容易被吓到： " }, { "type": "equation_inline", "content": "f(x) \\in C[a, b]" }, { "type": "text", "content": " ，且对 " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b]" }, { "type": "text", "content": " 都有 " }, { "type": "equation_inline", "content": "y \\in [a, b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(y)| \\leq \\frac{1}{2} |f(x)|" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 存在零点\n\n证明 (1) " }, { "type": "equation_inline", "content": "f(x) = f\\left(\\frac{x}{2}\\right) = \\cdots = f\\left(\\frac{x}{2n}\\right) = f(0)" }, { "type": "text", "content": " ，因此为常值" } ] }, "bbox": [ 87, 624, 905, 681 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)x 为正直接 " }, { "type": "equation_inline", "content": "f(x)=f(x^{\\frac{1}{2}})=\\cdots=f(x^{\\frac{1}{2^{n}}})=f(1)" }, { "type": "text", "content": " ，x 为负平方一下转换为正，x=0 根据连续性可知等于附近，因此整体常数" } ] }, "bbox": [ 89, 684, 905, 717 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 根据有理点稠密性 + 连续性" } ] }, "bbox": [ 122, 722, 354, 736 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 假设无零点，根据闭区间连续函数有最小值，这与 " }, { "type": "equation_inline", "content": "|f(y)| \\leq \\frac{1}{2} |f(x)|" }, { "type": "text", "content": " 矛盾\n\n练习 3.19 几道经典最值问题 (1) 重点：设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 连续，且 " }, { "type": "equation_inline", "content": "\\lim_{x \\to a^{+}} f(x) = \\lim_{x \\to b^{-}} f(x) = A" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 可以有限，也可以为 " }, { "type": "equation_inline", "content": "\\pm \\infty" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 上有最大或最小值" } ] }, "bbox": [ 58, 739, 905, 794 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上连续， " }, { "type": "equation_inline", "content": "\\lim_{x\\to+\\infty}f(x)=\\lim_{x\\to-\\infty}f(x)=A" }, { "type": "text", "content": " ，这里 A 可以有限，也可以为 " }, { "type": "equation_inline", "content": "\\pm\\infty" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上有最大或最小值" } ] }, "bbox": [ 87, 797, 905, 832 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} f(x) = A" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 为有限数，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 存在最大值或最小值" } ] }, "bbox": [ 89, 835, 905, 872 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 对 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 进行分类讨论:" } ] }, "bbox": [ 89, 873, 309, 889 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.4 连续性的应用" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "23" } ] }, "bbox": [ 489, 953, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(a) 先考虑 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 为有限值。如果 " }, { "type": "equation_inline", "content": "f(x) \\equiv A" }, { "type": "text", "content": "，则显然。若 " }, { "type": "equation_inline", "content": "\\exists x_0, f(x_0) > A" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\exists \\delta, \\forall x \\in (a, a + \\delta) \\cup (b - \\delta, b)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(x) < f(x_0)" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a + \\delta, b - \\delta]" }, { "type": "text", "content": " 上连续，因此 " }, { "type": "equation_inline", "content": "\\exists \\xi" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a + \\delta, b - \\delta]" }, { "type": "text", "content": " 最大，而 " }, { "type": "equation_inline", "content": "\\forall x \\in (a, a + \\delta] \\cup [b - \\delta, b)" }, { "type": "text", "content": "，自然也有 " }, { "type": "equation_inline", "content": "f(\\xi) \\geq f(x_0) \\geq f(x)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "f(\\xi)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 最大。若 " }, { "type": "equation_inline", "content": "f(x_0) < A" }, { "type": "text", "content": " 也同理有最小值。" } ] }, "bbox": [ 90, 84, 905, 137 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(b) 若 " }, { "type": "equation_inline", "content": "A = +\\infty" }, { "type": "text", "content": "，则对 " }, { "type": "equation_inline", "content": "\\forall x_0 \\in (a, b)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f(x_0)" }, { "type": "text", "content": " 为有限数， " }, { "type": "equation_inline", "content": "\\exists \\delta, \\forall x \\in (a, a + \\delta) \\cup (b - \\delta, b)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f(x) > f(x_0)" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a + \\delta, b - \\delta]" }, { "type": "text", "content": " 连续，因此有最小值，且最小值比 " }, { "type": "equation_inline", "content": "f(x_0)" }, { "type": "text", "content": " 还小，因此在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 上取最小值。" } ] }, "bbox": [ 89, 140, 905, 174 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 和 (1) 同理，只是把 " }, { "type": "equation_inline", "content": "\\exists \\delta" }, { "type": "text", "content": " 换成 " }, { "type": "equation_inline", "content": "\\exists N" } ] }, "bbox": [ 122, 177, 388, 191 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 若 " }, { "type": "equation_inline", "content": "f(x) \\equiv A" }, { "type": "text", "content": "，则显然。若 " }, { "type": "equation_inline", "content": "\\exists x_0, f(x_0) > A" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\exists M, \\forall x > M" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f(x_0) \\geq f(x)" }, { "type": "text", "content": "，由于在 " }, { "type": "equation_inline", "content": "[a, M]" }, { "type": "text", "content": " 连续，有最大值 " }, { "type": "equation_inline", "content": "f(\\xi)" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\forall x > M, f(\\xi) > f(x_0) > f(x)" }, { "type": "text", "content": "，因此有最大值。最小值同理" } ] }, "bbox": [ 89, 195, 905, 230 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.5 一致连续" } ], "level": 2 }, "bbox": [ 92, 256, 240, 274 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.5.1 一致连续的概念" } ], "level": 2 }, "bbox": [ 92, 294, 292, 312 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 3.9 (一致连续)" } ], "level": 2 }, "bbox": [ 122, 326, 270, 342 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 是任意区间 (任意开闭，有界无界)， " }, { "type": "equation_inline", "content": "f: I \\to \\mathbb{R}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta" }, { "type": "text", "content": " 对 " }, { "type": "equation_inline", "content": "\\forall x_1, x_2 \\in I" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|x_1 - x_2| < \\delta" }, { "type": "text", "content": " 时有 " }, { "type": "equation_inline", "content": "|f(x_1) - f(x_2)| < \\epsilon" }, { "type": "text", "content": "，则称 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 一致连续" } ] }, "bbox": [ 112, 344, 897, 386 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.7 (常用充要条件)" } ], "level": 2 }, "bbox": [ 122, 406, 304, 420 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上有定义，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 一致连续当且仅当任意数列 " }, { "type": "equation_inline", "content": "x_{n}^{\\prime}, x_{n}^{\\prime \\prime}" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}x_n' - x_n'' = 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}[f(x_n') - f(x_n')] = 0" } ] }, "bbox": [ 112, 424, 885, 468 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 左推右: 显然" } ] }, "bbox": [ 90, 485, 263, 500 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 右推左：反设不一致连续， " }, { "type": "equation_inline", "content": "\\exists \\epsilon_0, \\forall \\delta, \\exists x', x''" }, { "type": "text", "content": "，虽然 " }, { "type": "equation_inline", "content": "|x' - x''| < \\delta" }, { "type": "text", "content": "，但是 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| \\geq \\epsilon_0" }, { "type": "text", "content": "。取 " }, { "type": "equation_inline", "content": "\\delta = 1, \\frac{1}{2}, \\cdots, \\frac{1}{n}, \\cdots, \\exists x_n', x_n''" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|x_n' - x_n''| < \\frac{1}{n}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|f(x_n') - f(x_n'')| \\geq \\epsilon_0" }, { "type": "text", "content": "，这与条件矛盾。" } ] }, "bbox": [ 89, 502, 905, 545 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 证明不一致连续绝大多数情况都用上述定理。" } ] }, "bbox": [ 87, 548, 460, 562 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.20连续但不一致连续的例子： " }, { "type": "equation_inline", "content": "(1)x^{2},(0, + \\infty)(2)\\sin x^{2},[0, + \\infty)(3)\\frac{x + 2}{x + 1}\\sin \\frac{1}{x},(0,1)(4)\\frac{1}{x},x\\in (0,1)" }, { "type": "text", "content": " 证明 " }, { "type": "equation_inline", "content": "(1)x_{n} = \\sqrt{n},y_{n} = \\sqrt{n + 1},\\lim_{n\\to \\infty}(x_n - y_n) = 0,\\lim_{n\\to \\infty}[f(x_n) - f(y_n)] = -1" } ] }, "bbox": [ 58, 579, 860, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 取 " }, { "type": "equation_inline", "content": "x_{n} = \\sqrt{2n\\pi}, y_{n} = \\sqrt{2n\\pi + \\frac{\\pi}{2}}" } ] }, "bbox": [ 122, 627, 384, 656 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 取 " }, { "type": "equation_inline", "content": "x_{n} = \\frac{1}{2n\\pi}, y_{n} = \\frac{1}{2n\\pi + \\frac{\\pi}{2}}" } ] }, "bbox": [ 122, 657, 359, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 取 " }, { "type": "equation_inline", "content": "x_{n} = \\frac{1}{n}, y_{n} = \\frac{1}{n + 1}" } ] }, "bbox": [ 122, 684, 319, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.21一致连续的证明 (1) 证明 " }, { "type": "equation_inline", "content": "f(x) = \\cos \\sqrt{x}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 一致连续" } ] }, "bbox": [ 58, 727, 611, 741 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 证明 " }, { "type": "equation_inline", "content": "f(x) = \\sqrt{x}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 一致连续" } ] }, "bbox": [ 122, 745, 423, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 证明 " }, { "type": "equation_inline", "content": "f(x) = \\sin x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 一致连续" } ] }, "bbox": [ 122, 764, 431, 778 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 方法 1: 不妨设 " }, { "type": "equation_inline", "content": "x_{1} > x_{2}" }, { "type": "text", "content": ", 则" } ] }, "bbox": [ 89, 783, 369, 797 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} | f (x _ {1}) - f (x _ {2}) | = | \\cos \\sqrt {x _ {1}} - \\cos \\sqrt {x _ {2}} | = 2 \\left| \\sin \\frac {\\sqrt {x _ {1}} + \\sqrt {x _ {2}}}{2} \\sin \\frac {\\sqrt {x _ {1}} - \\sqrt {x _ {2}}}{2} \\right| \\\\ \\leq 2 \\left| \\sin \\frac {\\sqrt {x _ {1}} - \\sqrt {x _ {2}}}{2} \\right| \\leq | \\sqrt {x _ {1}} - \\sqrt {x _ {2}} | = \\frac {| x _ {1} - x _ {2} |}{\\sqrt {x _ {1}} + \\sqrt {x _ {2}}} \\\\ = \\frac {x _ {1} - x _ {2}}{\\sqrt {(x _ {1} - x _ {2}) + x _ {2}} + \\sqrt {x _ {2}}} \\leq \\frac {x _ {1} - x _ {2}}{\\sqrt {x _ {1} - x _ {2}}} = \\sqrt {x _ {1} - x _ {2}} \\end{array}", "math_type": "latex", "image_source": { "path": "images/85e740a4abc4400265582368a97c5459b3d191c843216a62f467a88dcf5bf74e.jpg" } }, "bbox": [ 221, 800, 771, 897 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "\\delta = \\epsilon^2" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "|x_1 - x_2| < \\delta" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x_1) - f(x_2)| < \\epsilon" }, { "type": "text", "content": " ，因此一致连续方法 2: " }, { "type": "equation_inline", "content": "(\\cos \\sqrt{x})' = \\sin \\sqrt{x} \\frac{1}{2 \\sqrt{x}}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[1, +\\infty)" }, { "type": "text", "content": " 有界，而 " }, { "type": "equation_inline", "content": "[0, 1]" }, { "type": "text", "content": " 用 Cantor 定理，因此 Lip 连续推出一致连续 (2) 分段 +Lip 连续" } ] }, "bbox": [ 90, 901, 576, 917 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.5 一致连续" } ] }, "bbox": [ 806, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "24" } ] }, "bbox": [ 489, 954, 509, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 122, 83, 878, 126 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.5.2 Cantor 定理及其推广" } ], "level": 2 }, "bbox": [ 92, 147, 339, 164 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.8 (Cantor 定理)" } ], "level": 2 }, "bbox": [ 122, 180, 292, 193 ] }, { "type": "table", "content": { "image_source": { "path": "images/63cc111394a953545ebb78d304cc469e5eb6e754232aa5c90b7056bf49befcdc.jpg" }, "table_caption": [], "table_footnote": [], "html": "
定理 3.8 (Cantor 足理)
若 f(x) 在闭区间 [a,b] 连续,则其在 [a,b] 上一致连续。
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f(x) 在 (a,b) 连续,则 f(x) 在 (a,b) 一致连续的充要条件是 $\\lim_{x\\to a^{+}}f(x),\\lim_{x\\to b^{-}}f(x)$ 存在(有限)
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 107, 451, 902, 492 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 左推右：根据 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 一致连续， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, \\forall x', x'', |x' - x''| < \\delta" }, { "type": "text", "content": "，有 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| < \\epsilon" }, { "type": "text", "content": "。取 " }, { "type": "equation_inline", "content": "x', x'' \\in (a, a + \\delta)" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| < \\epsilon" }, { "type": "text", "content": "，即右极限的 Cauchy 准则，另一侧同理。" } ] }, "bbox": [ 90, 502, 905, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 右推左：补充定义两个端点，根据 Cantor 定理可知。" } ] }, "bbox": [ 122, 539, 532, 554 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.22 开区间一致连续应用 (1)(东北师大) 设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 上一致连续，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 上有界\n\n证明 (1) 由于一致连续，因此 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 a,b 的分别存在右、左极限，补充定义为 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上连续有界，且 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 相等，因此 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 有界。" } ] }, "bbox": [ 58, 576, 907, 630 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.10 (无穷区间一致连续)" } ], "level": 2 }, "bbox": [ 122, 655, 346, 670 ] }, { "type": "table", "content": { "image_source": { "path": "images/d6cbdc1fb91fffbb43593ae91955c2e9169fa002b44b1cbd630e88ba5887ba25.jpg" }, "table_caption": [], "table_footnote": [], "html": "
定理3.10(元为区间一致连续)
f(x)在[a,+∞)连续,limx→+∞f(x)=A(有限),则f(x)在[a,+∞)一致连续
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 105, 664, 902, 707 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = A" }, { "type": "text", "content": " ，根据Cauchy收敛准则， " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists M,\\forall x^{\\prime},x^{\\prime \\prime} > M" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x^{\\prime}) - f(x^{\\prime \\prime})| < \\epsilon" }, { "type": "text", "content": " 。根据Cantor定理， " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,M + 1]" }, { "type": "text", "content": " 一致连续，对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists \\delta \\in (0,1)" }, { "type": "text", "content": " ，只要 " }, { "type": "equation_inline", "content": "|x^{\\prime} - x^{\\prime \\prime}| < \\delta" }, { "type": "text", "content": " 时，两点要么同时落在 " }, { "type": "equation_inline", "content": "[a,M + 1]" }, { "type": "text", "content": " ，要么同时落在 " }, { "type": "equation_inline", "content": "[M, + \\infty)" }, { "type": "text", "content": " ，根据上面的命题可知一致连续。" } ] }, "bbox": [ 89, 715, 905, 771 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 一致连续并不代表无穷处 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 极限一定存在，只是一个充分条件，例如 " }, { "type": "equation_inline", "content": "f(x) = x" }, { "type": "text", "content": " 极限不存在，但一致连续。" } ] }, "bbox": [ 87, 774, 905, 807 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.23 开区间一致连续 证明：(1) " }, { "type": "equation_inline", "content": "x \\sin \\frac{1}{x}" }, { "type": "text", "content": " 在 (0,1) 一致连续 (2) " }, { "type": "equation_inline", "content": "\\sin \\frac{1}{x}" }, { "type": "text", "content": " 在 (0,1) 不一致连续证明 (1) 因为 " }, { "type": "equation_inline", "content": "\\lim_{x \\to 0} x \\sin \\frac{1}{x} = 1" }, { "type": "text", "content": "，因此一致连续" } ] }, "bbox": [ 58, 822, 754, 876 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 因为 " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}\\sin \\frac{1}{x}" }, { "type": "text", "content": " 不存在，故不一致连续。" } ] }, "bbox": [ 122, 877, 438, 902 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "25" } ] }, "bbox": [ 489, 954, 507, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.5 一致连续" } ] }, "bbox": [ 805, 55, 905, 68 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.5.3 Lip 连续与一致连续" } ], "level": 2 }, "bbox": [ 92, 85, 329, 103 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.11 (Lip 连续与一致连续)" } ], "level": 2 }, "bbox": [ 121, 118, 359, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上满足Lip条件，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致连续" } ] }, "bbox": [ 114, 137, 532, 154 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 Lip 条件即 " }, { "type": "equation_inline", "content": "\\forall x', x''" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| \\leq L|x' - x''|" }, { "type": "text", "content": " ，取 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\delta = \\frac{\\epsilon}{L}" }, { "type": "text", "content": " 即可。" } ] }, "bbox": [ 89, 170, 650, 195 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "推论 3.1 (导数有界一定一致连续)" } ] }, "bbox": [ 119, 205, 371, 221 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f'(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上有界，则根据Lagrange中值定理可知Lip连续，从而一致连续" } ] }, "bbox": [ 114, 225, 689, 241 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 3.2 (导数在无穷处趋于无穷则不一致连续)" } ], "level": 2 }, "bbox": [ 122, 266, 472, 281 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f'(x) = \\infty" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, + \\infty)" }, { "type": "text", "content": " 不一致连续" } ] }, "bbox": [ 115, 286, 505, 307 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}|f'(x)| = \\infty" }, { "type": "text", "content": " 可知 " }, { "type": "equation_inline", "content": "|x' - x''|" }, { "type": "text", "content": " 控制不住导数的增长。" } ] }, "bbox": [ 89, 326, 576, 348 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 注意上述两个推论都是充分条件。前者例如 " }, { "type": "equation_inline", "content": "\\sqrt{x}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 一致连续，其导数 " }, { "type": "equation_inline", "content": "\\frac{1}{2\\sqrt{x}}" }, { "type": "text", "content": " 却不有界。后者例如 " }, { "type": "equation_inline", "content": "x^2" }, { "type": "text", "content": " 在无穷处导数趋于无穷，则一定不一致连续。" } ] }, "bbox": [ 87, 350, 905, 391 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.24 基础练习 判断一致连续性：(1) " }, { "type": "equation_inline", "content": "\\sin x, x \\in R" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "x^{k}, x \\in (1, +\\infty)" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\cos \\sqrt{x}, x \\in (1, +\\infty)" }, { "type": "text", "content": " (4) " }, { "type": "equation_inline", "content": "x \\ln x, x \\in (0, +\\infty)" } ] }, "bbox": [ 58, 412, 905, 445 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)一致连续，因为导数有界" } ] }, "bbox": [ 89, 450, 324, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 若 " }, { "type": "equation_inline", "content": "k \\leq 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{x \\to \\infty} f(x)" }, { "type": "text", "content": " 存在，补充定义 " }, { "type": "equation_inline", "content": "x = 1" }, { "type": "text", "content": "，根据 Cantor 定理的无穷推广可知一致连续。若 " }, { "type": "equation_inline", "content": "0 < k \\leq 1" }, { "type": "text", "content": "，则导数有界，因此一致连续。若 " }, { "type": "equation_inline", "content": "k > 1" }, { "type": "text", "content": "，因为导数在无穷处趋于无穷，则不一致连续" } ] }, "bbox": [ 89, 468, 904, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 导数有界，一致连续" } ] }, "bbox": [ 122, 506, 305, 520 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 导数在无穷处趋于无穷，则不一致连续。" } ] }, "bbox": [ 122, 524, 445, 539 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.5.4 区间合并问题" } ], "level": 2 }, "bbox": [ 94, 561, 273, 577 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.12 (一致连续区间合并定理)" } ], "level": 2 }, "bbox": [ 121, 593, 379, 608 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设区间 " }, { "type": "equation_inline", "content": "I_{1}" }, { "type": "text", "content": " 右端点为 " }, { "type": "equation_inline", "content": "c \\in I_{1}(\\text{闭})" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "I_{2}" }, { "type": "text", "content": " 左端点为 " }, { "type": "equation_inline", "content": "c \\in I_{2}(\\text{闭})" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I_{1}, I_{2}" }, { "type": "text", "content": " 均一致连续，则其在 " }, { "type": "equation_inline", "content": "I_{1} \\cup I_{2}" }, { "type": "text", "content": " 也一致连续" } ] }, "bbox": [ 114, 613, 895, 649 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I_1, I_2" }, { "type": "text", "content": "，对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon" }, { "type": "text", "content": "，存在公共 " }, { "type": "equation_inline", "content": "\\delta_1" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "x', x''" }, { "type": "text", "content": " 同时落于 " }, { "type": "equation_inline", "content": "I_1" }, { "type": "text", "content": " 或 " }, { "type": "equation_inline", "content": "I_2" }, { "type": "text", "content": " 时，若 " }, { "type": "equation_inline", "content": "|x' - x''| < \\delta_1" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| < \\epsilon" }, { "type": "text", "content": "。若 " }, { "type": "equation_inline", "content": "x', x''" }, { "type": "text", "content": " 分别落于 " }, { "type": "equation_inline", "content": "I_1, I_2" }, { "type": "text", "content": " 中，则 " }, { "type": "equation_inline", "content": "\\exists \\delta_2, |x' - c| < \\delta_2" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "|f(x') - f(c)| < \\frac{\\epsilon}{2}" }, { "type": "text", "content": " 且 " }, { "type": "equation_inline", "content": "|x'' - c| < \\delta_2" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x'') - f(c)| < \\frac{\\epsilon}{2}" }, { "type": "text", "content": "，取 " }, { "type": "equation_inline", "content": "\\delta = \\min\\{\\delta_1, \\delta_2\\}" }, { "type": "text", "content": "，得出 " }, { "type": "equation_inline", "content": "\\forall x', x'' \\in I_1 \\cup I_2" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "|x' - x''| < \\delta" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| < \\epsilon" } ] }, "bbox": [ 90, 663, 907, 721 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 如果边界点为开区间点，则可以补上定义。" } ] }, "bbox": [ 87, 721, 443, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.25区间合并练习(1)考虑 " }, { "type": "equation_inline", "content": "f(x) = \\frac{|\\sin x|}{x}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(-1,0)\\cup (0,1)" }, { "type": "text", "content": " 上是否一致连续证明(1)由于 " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0^{+}}f(x) = 1,\\lim_{x\\to 0^{-}}f(x) = -1,\\lim_{x\\to 1^{-}}f(x),\\lim_{x\\to -1^{+}}f(x)" }, { "type": "text", "content": " 显然存在，因此 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(-1,0),(0,1)" }, { "type": "text", "content": " 显然一致连续。但是由于0处左右极限不相同，取 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 单减趋于 " }, { "type": "equation_inline", "content": "0,y_{n}" }, { "type": "text", "content": " 单增趋于0，则 " }, { "type": "equation_inline", "content": "|x_{n} - y_{n}|\\rightarrow 0" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "|f(x_n) - f(y_n)|\\to 2" }, { "type": "text", "content": "，因此不一致连续。" } ] }, "bbox": [ 58, 752, 905, 832 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.5 一致连续" } ] }, "bbox": [ 805, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "26" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.5.5 四则运算、复合的一致连续性" } ], "level": 2 }, "bbox": [ 92, 85, 413, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.13 (四则运算的一致连续性)" } ], "level": 2 }, "bbox": [ 122, 118, 379, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致连续，则" } ] }, "bbox": [ 115, 137, 401, 153 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 数乘、加减： " }, { "type": "equation_inline", "content": "\\forall k \\in \\mathbb{R}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "kf(x), f(x) \\pm g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 一致连续" } ] }, "bbox": [ 139, 156, 569, 171 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 乘法： " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 为有限区间时， " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 一致连续。 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 为无穷区间时， " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 不一定一致连续" } ] }, "bbox": [ 139, 175, 875, 190 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 除法： " }, { "type": "equation_inline", "content": "f(x) \\neq 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\frac{1}{f(x)}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\frac{g(x)}{f(x)}" }, { "type": "text", "content": " 不一定一致连续" } ] }, "bbox": [ 139, 191, 495, 219 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "证明 (1) 直接三角不等式放缩" } ], "level": 2 }, "bbox": [ 90, 240, 312, 255 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "|f(x')g(x')-f(x'')g(x'')|\\leq|f(x')g'(x)-f(x')g(x'')|+|f(x')g(x'')-f(x'')g(x'')|\\leq|f(x')||g(x')-g(x'')|+|g(x'')||f(x')-f(x'')|<2M\\epsilon" }, { "type": "text", "content": " ，因此若M有限显然成立，M不有限时不一定。" } ] }, "bbox": [ 89, 258, 904, 293 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "例题3.1四则运算反例(1)乘法： " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(1, + \\infty)" }, { "type": "text", "content": " 一致连续，但是 " }, { "type": "equation_inline", "content": "x^{2}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(1, + \\infty)" }, { "type": "text", "content": " 不一致连续(2)倒数： " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 一致连续，但是 " }, { "type": "equation_inline", "content": "\\frac{1}{x}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 不一致连续" } ] }, "bbox": [ 89, 313, 905, 356 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.26四则运算练习 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 一致连续， " }, { "type": "equation_inline", "content": "\\lim_{x \\to \\infty} [f(x) - g(x)] = 0" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 上一致连续" } ] }, "bbox": [ 60, 370, 905, 404 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "f(x) - g(x)" }, { "type": "text", "content": " 在无穷远处极限存在，其一致连续。 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 一致连续，根据四则运算可知 " }, { "type": "equation_inline", "content": "f(x) = [f(x) - g(x)] + g(x)" }, { "type": "text", "content": " 一致连续。" } ] }, "bbox": [ 87, 407, 905, 443 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 上述练习本身也是一个重要的结论。" } ] }, "bbox": [ 57, 443, 391, 458 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 3.14 (复合的一致连续性)" } ], "level": 2 }, "bbox": [ 121, 486, 344, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I_{1}" }, { "type": "text", "content": " 一致连续， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I_{2}" }, { "type": "text", "content": " 一致连续， " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 值域在 " }, { "type": "equation_inline", "content": "I_{2}" }, { "type": "text", "content": " 中，则 " }, { "type": "equation_inline", "content": "g(f(x))" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I_{1}" }, { "type": "text", "content": " 一致连续" } ] }, "bbox": [ 114, 506, 769, 521 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据定义套即可。外层取 " }, { "type": "equation_inline", "content": "\\epsilon, \\delta" }, { "type": "text", "content": " ，内层取 " }, { "type": "equation_inline", "content": "\\delta, \\delta'" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 89, 542, 452, 557 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 3.27 复合一致连续性 设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 满足 Lip 条件， " }, { "type": "equation_inline", "content": "\\alpha \\in (0, 1)" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "f(x^{\\alpha})" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 一致连续证明 " }, { "type": "equation_inline", "content": "\\alpha \\in (0, 1)" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "x^{\\alpha}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 一致连续，而 f 满足 Lip 条件也一致连续，根据复合一致连续即可。" } ] }, "bbox": [ 58, 579, 877, 613 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.6 一致连续的经典问题" } ], "level": 2 }, "bbox": [ 92, 640, 361, 658 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习3.28设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上一致连续，证明 " }, { "type": "equation_inline", "content": "\\exists a, b > 0" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(x)| \\leq a|x| + b" } ] }, "bbox": [ 58, 674, 626, 689 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 一致连续，对 " }, { "type": "equation_inline", "content": "\\epsilon = 1, \\exists \\delta" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "|x' - x''| < \\delta" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| \\leq 1" }, { "type": "text", "content": "，不妨设" } ] }, "bbox": [ 87, 692, 741, 708 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x) | \\leq M, x \\in [ - \\delta , \\delta ]", "math_type": "latex", "image_source": { "path": "images/ffebb01164c652e93f8da6bcd5dba4af07a7dc22e252e37d30d5b45cc2035ce3.jpg" } }, "bbox": [ 413, 715, 581, 732 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对 " }, { "type": "equation_inline", "content": "\\forall x > \\delta, \\exists n_x" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "x - n_x\\delta = x_0 \\in [0, \\delta)" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "n_x = \\frac{x - x_0}{\\delta} \\leq \\frac{|x|}{\\delta}" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 90, 736, 636, 762 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x) | \\leq \\sum_ {k = 0} ^ {n - 1} | f (x - k \\delta) - f (x - (k + 1) \\delta) | + | f (x - n _ {x} \\delta) | \\leq n _ {x} + M \\leq \\frac {| x |}{\\delta} + M", "math_type": "latex", "image_source": { "path": "images/b7c64ff0b2f9a03f8828d9a6281fa2f36e70330aef908cf530787fc6aeb11d19.jpg" } }, "bbox": [ 205, 766, 789, 803 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理 " }, { "type": "equation_inline", "content": "\\forall x < -\\delta" }, { "type": "text", "content": " 也有 " }, { "type": "equation_inline", "content": "|f(x)| < \\frac{|x|}{\\delta} + M" }, { "type": "text", "content": "，综上，取 " }, { "type": "equation_inline", "content": "a = \\frac{1}{\\delta}, b = M" }, { "type": "text", "content": " 即可。" } ] }, "bbox": [ 90, 807, 596, 837 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "3.6 一致连续的经典问题" } ] }, "bbox": [ 721, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "27" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/c122ebe756b85cc934a9cc595d279b73472d6fd550c0d6f9bccc6b54fa5a2a43.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 57, 285, 80, 304 ] }, { "type": "image", "content": { "image_source": { "path": "images/17fd49c281bf8c586d120b9c43414c3c152dbfa67f7288b38082014cd6080b4a.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 780, 80, 800 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 4 章 一元微分学" } ], "level": 2 }, "bbox": [ 379, 87, 633, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.1 导数与微分" } ], "level": 2 }, "bbox": [ 92, 145, 265, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.1.1 导数概念与可导性" } ], "level": 2 }, "bbox": [ 92, 184, 314, 202 ] }, { "type": "table", "content": { "image_source": { "path": "images/cb0cb82adcfedbbd3225fe35deca2cf88f479879c66b07bb44ae7f3078c7942d.jpg" }, "table_caption": [], "table_footnote": [], "html": "
定义4.1(导数、左右导数)
导数定义为 $\\lim_{x\\to x_0}\\frac{f(x)-f(x_0)}{x-x_0}$ ,左右导数定义为 $\\lim_{x\\to x_0^-}\\frac{f(x)-f(x_0)}{x-x_0},\\lim_{x\\to x_0^+}\\frac{f(x)-f(x_0)}{x-x_0}$
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 100, 218, 905, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处可导当且仅当其在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处左右导数存在且相等。" } ] }, "bbox": [ 55, 288, 566, 306 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.1 用定义求导函数 设 " }, { "type": "equation_inline", "content": "g(0)=g'(0)=0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 定义如下，求 " }, { "type": "equation_inline", "content": "f'(0)" } ] }, "bbox": [ 58, 326, 618, 343 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = \\left\\{ \\begin{array}{l l} g (x) \\sin {\\frac {1}{x}}, & x \\neq 0 \\\\ 0, & x = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/e2e21f527baaf21ed00bb2b3838e16e68bfb2167161fef417fb01c37ebaa81ed.jpg" } }, "bbox": [ 391, 347, 605, 392 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 根据定义可知 " }, { "type": "equation_inline", "content": "f'(0) = \\lim_{\\Delta x \\to 0} \\frac{f(\\Delta x) - f(0)}{\\Delta x} = \\lim_{\\Delta x \\to 0} \\frac{g(\\Delta x) \\sin \\frac{1}{\\Delta x}}{\\Delta x} = \\lim_{\\Delta x \\to 0} \\frac{g(\\Delta x) - g(0)}{\\Delta x - 0} \\sin \\frac{1}{\\Delta x} = 0" } ] }, "bbox": [ 87, 397, 835, 430 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.2可导概念推广(1)重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}\\frac{f(2x) - f(x)}{x} = A" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f^{\\prime}(0)" }, { "type": "text", "content": " 存在且 " }, { "type": "equation_inline", "content": "f^{\\prime}(0) = A" } ] }, "bbox": [ 58, 436, 878, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点：若 " }, { "type": "equation_inline", "content": "\\lim_{\\Delta x \\to 0} \\frac{f(x_0 + \\Delta x) - f(x_0 - \\Delta x)}{\\Delta x}" }, { "type": "text", "content": "，并不能推出 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处导数存在，例如 " }, { "type": "equation_inline", "content": "|x|" }, { "type": "text", "content": "。但反命题成立" } ] }, "bbox": [ 121, 462, 860, 492 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据极限存在可知 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, |x| < \\delta" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "A - \\frac{\\epsilon}{2} < \\frac{f(2x) - f(x)}{x} < A + \\frac{\\epsilon}{2}" }, { "type": "text", "content": "，同理可以得到：" } ] }, "bbox": [ 87, 489, 781, 518 ] }, { "type": "equation_interline", "content": { "math_content": "A - \\frac {\\epsilon}{2} < \\frac {f (\\frac {x}{2 ^ {k - 1}}) - f (\\frac {x}{2 ^ {k}})}{x / 2 ^ {k}} < A + \\frac {\\epsilon}{2} \\Rightarrow \\frac {1}{2 ^ {k}} (A - \\frac {\\epsilon}{2}) < \\frac {f (\\frac {x}{2 ^ {k - 1}}) - f (\\frac {x}{2 ^ {k}})}{x} < \\frac {1}{2 ^ {k}} (A + \\frac {\\epsilon}{2})", "math_type": "latex", "image_source": { "path": "images/f28a4877645abe0e19d3bc17dea9db79b05afec4b7d0a4e9e8e3d794a0cb4377.jpg" } }, "bbox": [ 191, 520, 806, 551 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "取 " }, { "type": "equation_inline", "content": "x_{k}=\\frac{x}{2^{k}}" }, { "type": "text", "content": " ，将 " }, { "type": "equation_inline", "content": "k=1,2,\\cdots,n" }, { "type": "text", "content": " 全部相加得到 " }, { "type": "equation_inline", "content": "(1-\\frac{1}{2^{n}})(A-\\frac{\\epsilon}{2})<\\frac{f(x)-f(x_{n})}{x}<(1-\\frac{1}{2^{n}})(A+\\frac{\\epsilon}{2})" }, { "type": "text", "content": " ，令 " }, { "type": "equation_inline", "content": "n\\to\\infty" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "A-\\frac{\\epsilon}{2}<\\frac{f(x)-f(0)}{x} 0" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\frac{af(b) - bf(a)}{a - b} = f(\\xi) - \\xi f'(\\xi)" } ] }, "bbox": [ 60, 398, 904, 438 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 实例： " }, { "type": "equation_inline", "content": "0 < a < b < \\infty" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "\\exists\\theta \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "ae^{b} - be^{a} = (1 - \\theta)e^{\\theta}(a - b)" } ] }, "bbox": [ 122, 441, 687, 456 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 由于左侧有交叉项，考虑分子分母同除以 " }, { "type": "equation_inline", "content": "ab" }, { "type": "text", "content": "，凑 Cauchy 中值定理形式即 " }, { "type": "equation_inline", "content": "\\frac{\\left(\\frac{f(x)}{x}\\right)'}{\\left(\\frac{1}{x}\\right)'}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": " 处取值，因此即证明：" } ] }, "bbox": [ 87, 457, 905, 502 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {\\frac {f (b)}{b} - \\frac {f (a)}{a}}{\\frac {1}{b} - \\frac {1}{a}} = \\frac {\\left(\\frac {f (x)}{x}\\right) ^ {\\prime}}{\\left(\\frac {1}{x}\\right) ^ {\\prime}}", "math_type": "latex", "image_source": { "path": "images/9101256b3d2ecd6619dc346a3a2a309bf5657223a4cae5065ce77cadb28f6d5b.jpg" } }, "bbox": [ 418, 500, 578, 538 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 Cauchy 中值定理即可证明。" } ] }, "bbox": [ 90, 545, 329, 558 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 等价于 " }, { "type": "equation_inline", "content": "\\frac{\\frac{e^{b}}{b}-\\frac{e^{a}}{a}}{\\frac{1}{b}-\\frac{1}{a}}=e^{\\theta}-\\theta e^{\\theta}" } ] }, "bbox": [ 122, 560, 349, 596 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.2.2 中值问题：构造辅助函数" } ], "level": 2 }, "bbox": [ 94, 614, 373, 631 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.9 (直接积分法)" } ], "level": 2 }, "bbox": [ 122, 646, 287, 662 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "要构造 " }, { "type": "equation_inline", "content": "f^{\\prime}(\\xi) = g(\\xi)" }, { "type": "text", "content": " ，则构造 " }, { "type": "equation_inline", "content": "F(x) = f(x) - \\int_0^x g(t)\\mathrm{d}t" }, { "type": "text", "content": " 即可" } ] }, "bbox": [ 114, 665, 554, 697 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.10 (常用中值定理辅助函数)" } ], "level": 2 }, "bbox": [ 122, 722, 379, 737 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "看到 " }, { "type": "equation_inline", "content": "f'(\\xi) + f(\\xi)g(\\xi)" }, { "type": "text", "content": " ，需要构造 " }, { "type": "equation_inline", "content": "f(x)e^{\\int g(x)dx}" }, { "type": "text", "content": " ，因为 " }, { "type": "equation_inline", "content": "[f(x)e^{\\int g(x)dx}]' = [f'(x) + f(x)g(x)]e^{\\int g(x)dx}" }, { "type": "text", "content": " ，具体有下面这些技巧：" } ] }, "bbox": [ 112, 741, 880, 777 ] }, { "type": "equation_interline", "content": { "math_content": "m f (\\xi) + n f ^ {\\prime} (\\xi): f (x) e ^ {\\frac {m}{n} x}", "math_type": "latex", "image_source": { "path": "images/8377a7f6388c35cc3da61804458121ec3711b005ad17c16d27cc6da760181b9c.jpg" } }, "bbox": [ 139, 781, 352, 797 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "mf(\\xi) + n\\xi f'(\\xi)" }, { "type": "text", "content": ": " }, { "type": "equation_inline", "content": "x^{m}f^{n}(x)" }, { "type": "text", "content": " (注意不是 " }, { "type": "equation_inline", "content": "x^{\\frac{m}{n}}f(x)!" }, { "type": "text", "content": ")" } ] }, "bbox": [ 139, 799, 521, 815 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "mf(\\xi) - n\\xi f'(\\xi)" }, { "type": "text", "content": ": " }, { "type": "equation_inline", "content": "\\frac{f^n(x)}{x^m}" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "\\frac{x^m}{f^n(x)}" }, { "type": "text", "content": "，但此时一般要配上 " }, { "type": "equation_inline", "content": "g(x) = \\frac{1}{x^m}" }, { "type": "text", "content": "，用 Cauchy 中值定理消去分母" } ] }, "bbox": [ 139, 814, 880, 844 ] }, { "type": "equation_interline", "content": { "math_content": "\\left[ \\frac {f ^ {n} (x)}{x ^ {m}} \\right] = \\frac {f ^ {n - 1} (x)}{x ^ {m + 1}} [ n x f ^ {\\prime} (x) - m f (x) ], \\quad \\left[ \\frac {x ^ {m}}{f ^ {n} (x)} \\right] ^ {\\prime} = \\frac {x ^ {m - 1}}{f ^ {n + 1} (x)} [ m f (x) - n x f ^ {\\prime} (x) ]", "math_type": "latex", "image_source": { "path": "images/88e1f550e945e2ed1d3f413b67648b4981393a379a487583c897d6a4bc8c7970.jpg" } }, "bbox": [ 220, 847, 818, 881 ] }, { "type": "equation_interline", "content": { "math_content": "n f ^ {\\prime} (\\xi) f (1 - \\xi) - m f (\\xi) f ^ {\\prime} (1 - \\xi): f ^ {n} (x) f ^ {m} (1 - x)", "math_type": "latex", "image_source": { "path": "images/4d7ef137e53168b54e358bdf752a888f4956df1680f4901f1f3f4cb81b7f802b.jpg" } }, "bbox": [ 141, 883, 532, 898 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.2 微分中值定理" } ] }, "bbox": [ 769, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "33" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\bullet m f ^ {\\prime} (x) g (x) + n f (x) g ^ {\\prime} (x): f ^ {m} (x) g ^ {n} (x) \\\\ \\bullet m f ^ {\\prime} (x) g (x) - n f (x) g ^ {\\prime} (x): \\frac {f ^ {m} (x)}{g ^ {n} (x)} \\\\ \\left[ \\frac {f ^ {m} (x)}{g ^ {n} (x)} \\right] ^ {\\prime} = \\frac {f ^ {m - 1} (x)}{g ^ {n + 1} (x)} [ m f ^ {\\prime} (x) g (x) - n f (x) g ^ {\\prime} (x) ] \\\\ \\bullet f (\\xi) g ^ {\\prime \\prime} (\\xi) - g (\\xi) f ^ {\\prime \\prime} (\\xi): f ^ {\\prime} (x) g (x) - f (x) g ^ {\\prime} (x) \\end{array}", "math_type": "latex", "image_source": { "path": "images/441aa211edb5c0f879c01bf77248923cbded589338cef0ce7c0362bf13f85afc.jpg" } }, "bbox": [ 137, 93, 700, 197 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 主要看高阶导数项的系数是啥，是常数的话肯定是 " }, { "type": "equation_inline", "content": "e^{ax}f(x)" }, { "type": "text", "content": " ，是 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": " 的话是 " }, { "type": "equation_inline", "content": "x^{a}f(x)" }, { "type": "text", "content": " ，是 " }, { "type": "equation_inline", "content": "-\\xi" }, { "type": "text", "content": " 的话是 " }, { "type": "equation_inline", "content": "\\frac{f(x)}{x^{a}}" }, { "type": "text", "content": " ，是 " }, { "type": "equation_inline", "content": "(1-\\xi)" }, { "type": "text", "content": " 的话是 " }, { "type": "equation_inline", "content": "(1-x)^{a}f(x)" } ] }, "bbox": [ 55, 210, 902, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.18行列式求导 " }, { "type": "equation_inline", "content": "f(x), g(x), h(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 可导，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 下面等式成立。若 " }, { "type": "equation_inline", "content": "h(x) = 1" }, { "type": "text", "content": " 则为Cauchy中值定理，若 " }, { "type": "equation_inline", "content": "h(x) = 1, g(x) = x" }, { "type": "text", "content": " 则为Lagrange中值定理。" } ] }, "bbox": [ 58, 269, 904, 304 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\begin{array}{c c c} f (a) & g (a) & h (a) \\\\ f (b) & g (b) & h (b) \\\\ f ^ {\\prime} (\\xi) & g ^ {\\prime} (\\xi) & h ^ {\\prime} (\\xi) \\end{array} \\right| = 0", "math_type": "latex", "image_source": { "path": "images/bbacc322ac3c6c4c1f4c0ab90f51d47be7e64e6c1fcd5da5ed53e313599474ad.jpg" } }, "bbox": [ 400, 309, 601, 365 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 构造下面的 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": "，注意到 " }, { "type": "equation_inline", "content": "F(a) = F(b) = 0" }, { "type": "text", "content": " 以及行列式求导即对一行求导即可。" } ] }, "bbox": [ 87, 369, 717, 385 ] }, { "type": "equation_interline", "content": { "math_content": "F (x) = \\left| \\begin{array}{c c c} f (a) & g (a) & h (a) \\\\ f (b) & g (b) & h (b) \\\\ f (x) & g (x) & h (x) \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/19dce8935400da82ac9bd8c283ed5fd966fcc8edff2f7d9032c0a3309a8ae113.jpg" } }, "bbox": [ 386, 390, 601, 447 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.19直接积分法(1)重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在[1,2]连续， " }, { "type": "equation_inline", "content": "(1,2)" }, { "type": "text", "content": " 可微，证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (1,2)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(2) - f(1) = \\frac{1}{2}\\xi^2 f'\\left(\\xi\\right)" }, { "type": "text", "content": " (2)重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a,b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "2\\xi [f(b) - f(a)] = (b^{2} - a^{2})f^{\\prime}(\\xi)" } ] }, "bbox": [ 58, 456, 892, 494 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "2\\xi[f(b) - f(a)] = (b^2 - a^2)f'(\\xi)" }, { "type": "text", "content": "\n\n证明 (1) 由于只有单项，只需要直接用积分法：所求等价于 " }, { "type": "equation_inline", "content": "f'(\\xi) = \\frac{2}{\\xi^2}[f(2) - f(1)]" }, { "type": "text", "content": "，构造 " }, { "type": "equation_inline", "content": "F(x) = f(x) + \\frac{2}{x}[f(2) - f(1)]" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "F(1) = F(2) = 2f(2) - f(1)" }, { "type": "text", "content": "，因此根据 Rolle 中值定理可知 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (1, 2), F'(\\xi) = 0" } ] }, "bbox": [ 84, 476, 894, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 直接用积分法： " }, { "type": "equation_inline", "content": "F(x) = (b^{2} - a^{2})f(x) - x^{2}[f(b) - f(a)]" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "F(b) = F(a)" }, { "type": "text", "content": " 以及 Rolle 中值定理可知。" } ] }, "bbox": [ 121, 537, 890, 554 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.20 简单的辅助函数法 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "f(a) < 0, f(b) < 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists c \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(c) > 0" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\xi \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi) + f'(\\xi) = 0" } ] }, "bbox": [ 58, 574, 905, 608 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 阶连续可微，在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 上存在 " }, { "type": "equation_inline", "content": "n + 1" }, { "type": "text", "content": " 阶导数， " }, { "type": "equation_inline", "content": "f^{(k)}(a) = f^{(k)}(b), k = 0, 1, \\cdots, n" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi) = f^{(n+1)}(\\xi)" }, { "type": "text", "content": "\n\n(3) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, 1]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "(0, 1)" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "f(0) = f(1) = 0, f(\\frac{1}{2}) = 1" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\forall \\lambda > 0, \\exists \\xi \\in (0, 1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f'(\\xi) - \\lambda[f(\\xi) - \\xi] = 1" } ] }, "bbox": [ 90, 609, 907, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 可微, " }, { "type": "equation_inline", "content": "f(0) = 0" }, { "type": "text", "content": ", 对 " }, { "type": "equation_inline", "content": "\\forall x \\in (0,1), f(x) \\neq 0" }, { "type": "text", "content": ", 证明: " }, { "type": "equation_inline", "content": "\\forall n, \\exists \\xi \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "n\\frac{f'(\\xi)}{f(\\xi)} = \\frac{f'(1 - \\xi)}{f(1 - \\xi)}" }, { "type": "text", "content": " 证明 (1) 构造 " }, { "type": "equation_inline", "content": "F(x) = e^x f(x)" }, { "type": "text", "content": ", 则 " }, { "type": "equation_inline", "content": "F(a) < 0, F(b) < 0, F(c) > 0" }, { "type": "text", "content": ", 根据连续函数零点存在定理可知有两个零点, 再根据 Rolle 中值定理可知。" } ] }, "bbox": [ 87, 684, 905, 744 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 构造 " }, { "type": "equation_inline", "content": "F(x) = [f(x) + f'(x) + \\cdots + f^{(n)}(x)]e^{-x}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "F(a) = F(b) = 0" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "F'(\\xi) = 0" }, { "type": "text", "content": " 可以得到结论" } ] }, "bbox": [ 122, 747, 855, 763 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 考虑配为 " }, { "type": "equation_inline", "content": "[f'(\\xi) - 1] - \\lambda[f(\\xi) - \\xi] = 0" }, { "type": "text", "content": "，故构造 " }, { "type": "equation_inline", "content": "F(x) = [f(x) - x]e^{-\\lambda x}" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "F(0) = 0, F(\\frac{1}{2}) > 0, F(1) < 0" }, { "type": "text", "content": "，在 " }, { "type": "equation_inline", "content": "[\\frac{1}{2}, 1]" }, { "type": "text", "content": " 中 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 有一个零点，根据 Rolle 可知。" } ] }, "bbox": [ 89, 766, 905, 812 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 即 " }, { "type": "equation_inline", "content": "nf'(\\xi)f(1-\\xi)-f(\\xi)f'(1-\\xi)=0" }, { "type": "text", "content": "，因此构造 " }, { "type": "equation_inline", "content": "F(x)=f^n(x)f(1-x)" } ] }, "bbox": [ 119, 810, 675, 827 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.21 进阶难度的辅助函数法 (1)(上交 2021) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, c]" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "f_{-}^{\\prime}(c) = 0" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, c)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f^{\\prime}(\\xi) = 2(f(\\xi) - f(a))" } ] }, "bbox": [ 58, 847, 905, 883 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 二阶可导， " }, { "type": "equation_inline", "content": "f(0)=f(1)" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\exists\\xi\\in(0,1)" }, { "type": "text", "content": " ，使得 " }, { "type": "equation_inline", "content": "f''(\\xi)=\\frac{3f'(\\xi)}{1-\\xi}" }, { "type": "text", "content": "证明 (1) 首先显然最终等式可写为 " }, { "type": "equation_inline", "content": "[f(x) - f(a)]' - 2(f(x) - f(a)) = 0" }, { "type": "text", "content": "，因此构造 " }, { "type": "equation_inline", "content": "F(x) = (f(x) - f(a))e^{-2x}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "F'(x) = [f'(x) - 2(f(x) - f(a))]e^{-2x}" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "F(a) = 0" }, { "type": "text", "content": "，但另一侧信息不足。根据条件可知 " }, { "type": "equation_inline", "content": "F_{-}'(c) = -2[f(c) - f(a)]e^{-2c}" }, { "type": "text", "content": "，下面对 " }, { "type": "equation_inline", "content": "F(c)" }, { "type": "text", "content": " 情况进行讨论。(a) 若 " }, { "type": "equation_inline", "content": "F(c) = 0" }, { "type": "text", "content": " 则显然成立 (b) 若 " }, { "type": "equation_inline", "content": "F(c) \\neq 0" }, { "type": "text", "content": "，不妨设 " }, { "type": "equation_inline", "content": "F(c) > 0" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "F(c) = -2F_{-}'(c)" }, { "type": "text", "content": "（根据表达式）可看出 " }, { "type": "equation_inline", "content": "F_{-}'(c) < 0" }, { "type": "text", "content": "，根据 Lagrange 中值定理：" } ] }, "bbox": [ 121, 881, 719, 910 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.2 微分中值定理" } ] }, "bbox": [ 769, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "34" } ] }, "bbox": [ 489, 953, 509, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 89, 84, 905, 156 ] }, { "type": "equation_interline", "content": { "math_content": "0 < F (c) = F (c) - F (a) = F ^ {\\prime} (\\eta) (c - a) \\Rightarrow F ^ {\\prime} (\\eta) > 0", "math_type": "latex", "image_source": { "path": "images/46de13498bf7440b03d8e8c3e9ba3e188426d64ba85695c5b0d5abc5a3b4fced.jpg" } }, "bbox": [ 300, 162, 694, 179 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据导数介值性定理可知结论成立。" } ] }, "bbox": [ 90, 186, 356, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 等价于 " }, { "type": "equation_inline", "content": "(1 - \\xi)f''(\\xi) - 3f'(\\xi) = 0" }, { "type": "text", "content": "，因此构造 " }, { "type": "equation_inline", "content": "F(x) = (1 - x)^3 f'(x)" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "f(0) = f(1) = 0" }, { "type": "text", "content": "，根据 Rolle 可知 " }, { "type": "equation_inline", "content": "\\exists \\eta" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f'(\\eta) = 0" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "F(\\eta) = 0" }, { "type": "text", "content": "，再根据 " }, { "type": "equation_inline", "content": "F(1) = 0" }, { "type": "text", "content": " 可知 " }, { "type": "equation_inline", "content": "\\exists \\xi" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "F'(\\xi) = 0" } ] }, "bbox": [ 89, 204, 905, 240 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.2.3 带积分的微分中值问题" } ], "level": 2 }, "bbox": [ 92, 259, 354, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于变限积分 " }, { "type": "equation_inline", "content": "\\int_{a}^{x}f(t)\\mathrm{d}t" }, { "type": "text", "content": " 的导数为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " ，该性质常常会应用于微分中值定理。" } ] }, "bbox": [ 122, 285, 690, 318 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.22基础问题 (1)重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在[0,1]连续，(0,1)可导， " }, { "type": "equation_inline", "content": "\\int_0^1 xf(x)\\mathrm{d}x = f(1)" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f^{\\prime}(\\xi) = -\\frac{f(\\xi)}{\\xi}" } ] }, "bbox": [ 58, 318, 905, 374 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)ZJU2021: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "g(x) = f(x)\\int_0^x f(t)\\mathrm{d}t" }, { "type": "text", "content": " 单减，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 恒为0" } ] }, "bbox": [ 121, 373, 690, 405 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 上连续函数, 证明: " }, { "type": "equation_inline", "content": "\\exists c \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\int_{0}^{c} f(x) \\mathrm{d}x = (1 - c)f(c)" }, { "type": "text", "content": " 证明 (1) 所证形式即 " }, { "type": "equation_inline", "content": "\\xi f'(\\xi) + f(\\xi) = 0" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "F(x) = xf(x)" }, { "type": "text", "content": ", 根据积分第一中值定理可知" } ] }, "bbox": [ 87, 407, 749, 447 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {1} x f (x) d x = \\eta f (\\eta) \\cdot 1 = F (\\eta) = f (1)", "math_type": "latex", "image_source": { "path": "images/ff3f0e61e0da414336ccbdea2fcd26787a4b5eb6a03c6c10410f112af607ff22.jpg" } }, "bbox": [ 352, 451, 642, 483 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "F(1)=f(1)" }, { "type": "text", "content": " ，因此根据 Rolle 中值定理可知。" } ] }, "bbox": [ 90, 486, 440, 500 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "F(x)=\\frac{1}{2}(\\int_{0}^{x}f(t)\\mathrm{d}t)^{2}" }, { "type": "text", "content": " ，根据条件 " }, { "type": "equation_inline", "content": "F'(x)" }, { "type": "text", "content": " 单减， " }, { "type": "equation_inline", "content": "F'(0)=0" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "F(0)=0" }, { "type": "text", "content": " 得出 " }, { "type": "equation_inline", "content": "F(x)\\leq0" }, { "type": "text", "content": " ，但 " }, { "type": "equation_inline", "content": "F(x)\\geq0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 恒为0，推出 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 恒为0。" } ] }, "bbox": [ 90, 502, 904, 543 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 所求即 " }, { "type": "equation_inline", "content": "\\int_0^c f(x)\\mathrm{d}x - (1 - c)f(c) = 0" }, { "type": "text", "content": " ，构造 " }, { "type": "equation_inline", "content": "F(x) = (1 - x)\\int_0^x f(t)\\mathrm{d}t" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "F(0) = F(1) = 0" }, { "type": "text", "content": " ，根据 Rolle 可知。" } ] }, "bbox": [ 90, 544, 905, 585 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.23进阶问题 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在[0,2]可导，(0,2)上三阶可导， " }, { "type": "equation_inline", "content": "f(0) = f'(0) = 0, \\int_0^2 f(x) \\mathrm{d}x = 8 \\int_0^1 f(x) \\mathrm{d}x" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (0,2)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f'''(\\xi) = 0" }, { "type": "text", "content": "。证明构造 " }, { "type": "equation_inline", "content": "F(x) = \\int_0^{2x} f(t) dt - 8 \\int_0^x f(t) dt" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "F(0) = F(1) = 0" }, { "type": "text", "content": "，故 " }, { "type": "equation_inline", "content": "\\exists \\xi_1, F'(\\xi_1) = 0" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "F'(x) = 2f(2x) - 8f(x)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "F'(0) = 0" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "\\exists \\xi_2" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "F''(\\xi_2) = 0" }, { "type": "text", "content": "。再根据 " }, { "type": "equation_inline", "content": "F''(x) = 4f'(2x) - 8f'(x)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "F''(0) = 0" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "\\exists \\xi_3" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "F'''(\\xi_3) = 0" }, { "type": "text", "content": "。而 " }, { "type": "equation_inline", "content": "F'''(x) = 8f''(2x) - 8f''(x)" }, { "type": "text", "content": "，代入 " }, { "type": "equation_inline", "content": "\\xi_3" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "f''(2\\xi_3) = f''(\\xi_3)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\exists \\xi" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f'''(\\xi) = 0" } ] }, "bbox": [ 58, 600, 909, 711 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.2.4 多个中值点的中值问题" } ], "level": 2 }, "bbox": [ 92, 730, 354, 746 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.24经典两个中值点的问题设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在[0,1]连续，(0,1)可导， " }, { "type": "equation_inline", "content": "f(0) = 0,f(1) = 1" }, { "type": "text", "content": " ，证明：" } ] }, "bbox": [ 58, 758, 789, 772 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\exists\\xi\\in(0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f(\\xi)=1-\\xi" } ] }, "bbox": [ 122, 777, 357, 791 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 存在互异的 " }, { "type": "equation_inline", "content": "\\eta_1, \\eta_2 \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f'(\\eta_1)f'(\\eta_2) = 1" } ] }, "bbox": [ 122, 795, 497, 810 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 存在互异的 " }, { "type": "equation_inline", "content": "\\eta_1, \\eta_2 \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f'(\\eta_1)(f'(\\eta_2) + 1) = 2" } ] }, "bbox": [ 122, 813, 542, 828 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 构造 " }, { "type": "equation_inline", "content": "F(x) = f(x) - 1 + x" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "F(0) = -1, F(1) = 1" }, { "type": "text", "content": "，根据连续零点存在定理可得" } ] }, "bbox": [ 89, 832, 754, 847 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 对 (1) 中的 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": " ，在 " }, { "type": "equation_inline", "content": "[0, \\xi]" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "[\\xi, 1]" }, { "type": "text", "content": " 上用 Lagrange 中值定理，得到 " }, { "type": "equation_inline", "content": "\\eta_{1}, \\eta_{2}" }, { "type": "text", "content": " 满足" } ] }, "bbox": [ 121, 851, 663, 866 ] }, { "type": "equation_interline", "content": { "math_content": "f ^ {\\prime} (\\eta_ {1}) = \\frac {f (\\xi) - 0}{\\xi} = \\frac {1 - \\xi}{\\xi}, f ^ {\\prime} (\\eta_ {2}) = \\frac {f (1) - f (\\xi)}{1 - \\xi} = \\frac {\\xi}{1 - \\xi}", "math_type": "latex", "image_source": { "path": "images/94e12e025b6e05e988b080252fda4c319dc5ae93b4d3a8683a5e975eb7cc934d.jpg" } }, "bbox": [ 277, 868, 717, 900 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "f^{\\prime}(\\eta_{1})f^{\\prime}(\\eta_{2}) = 1" } ] }, "bbox": [ 90, 902, 252, 917 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "35" } ] }, "bbox": [ 489, 954, 507, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.2 微分中值定理" } ] }, "bbox": [ 771, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 提示：先找 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(\\xi) = 2(1 - \\xi)" }, { "type": "text", "content": "，再用两次 Lagrange 中值定理。" } ] }, "bbox": [ 121, 85, 633, 99 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.2.5 中值等式问题" } ], "level": 2 }, "bbox": [ 92, 122, 275, 137 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "中值等式问题本质很像多项式插值中 Cauchy 常数的构造方法，根据目标等式的形式，将某个常数换成 x，以此得出辅助函数在某些位置相等或等于 0。" } ] }, "bbox": [ 87, 149, 905, 184 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.25 利用等式构造辅助函数 (1) 重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 二阶可导, " }, { "type": "equation_inline", "content": "f(a) = f(b) = 0" }, { "type": "text", "content": ", 证明对 " }, { "type": "equation_inline", "content": "\\forall x \\in (a, b)" }, { "type": "text", "content": ", 都存在 " }, { "type": "equation_inline", "content": "\\xi \\in (a, b)" }, { "type": "text", "content": ", 使得 " }, { "type": "equation_inline", "content": "f(x) = \\frac{f''(\\xi)}{2}(x - a)(x - b)" } ] }, "bbox": [ 58, 205, 904, 243 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 变形： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 二阶可导，a < c < b，证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 121, 244, 618, 260 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {f (b) - f (a)}{b - a} - \\frac {f (c) - f (a)}{c - a} = \\frac {1}{2} f ^ {\\prime \\prime} (\\xi) (b - c)", "math_type": "latex", "image_source": { "path": "images/6f420a6cf69506c5f1d3c34154cb0f4d24da9c0261eeb5590da51b5829c8aed6.jpg" } }, "bbox": [ 337, 262, 660, 293 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 固定 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": "，记常数 " }, { "type": "equation_inline", "content": "k = \\frac{2f(x)}{(x - a)(x - b)}" }, { "type": "text", "content": "，构造 " }, { "type": "equation_inline", "content": "F(t) = f(t) - \\frac{k}{2}(t - a)(t - b)" }, { "type": "text", "content": "，满足 " }, { "type": "equation_inline", "content": "F(x) = F(a) = F(b) = 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists \\xi_1 \\in (a, x), \\xi_2 \\in (x, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "F'(\\xi_1) = F'(\\xi_2) = 0" }, { "type": "text", "content": "，根据 Rolle 定理可知 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (\\xi_1, \\xi_2)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "F''(\\xi) = 0" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "f''(\\xi) = k" }, { "type": "text", "content": "，结论成立。" } ] }, "bbox": [ 87, 297, 907, 357 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 构造常数 " }, { "type": "equation_inline", "content": "k = \\frac{(c - a)[f(b) - f(a)] - (b - a)[f(c) - f(a)]}{\\frac{1}{2}(b - c)(b - a)(c - a)}" }, { "type": "text", "content": "，辅助函数将 " }, { "type": "equation_inline", "content": "c" }, { "type": "text", "content": " 用 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 代替" } ] }, "bbox": [ 121, 356, 749, 388 ] }, { "type": "equation_interline", "content": { "math_content": "F (x) = (x - a) [ f (b) - f (a) ] - (b - a) [ f (x) - f (a) ] - \\frac {k}{2} (b - x) (b - a) (x - a)", "math_type": "latex", "image_source": { "path": "images/1dae8b0ec7f94effeee4212aa660c6ef06a1f9b34fcf1298076f8af0b2f4a776.jpg" } }, "bbox": [ 218, 392, 776, 419 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到 " }, { "type": "equation_inline", "content": "F(a)=F(b)=F(c)=0" }, { "type": "text", "content": " ，因此根据两次 Rolle 得到 " }, { "type": "equation_inline", "content": "F''(\\xi)=0" }, { "type": "text", "content": " ，从而得到结论。" } ] }, "bbox": [ 87, 420, 709, 437 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.26另外几道经典题(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 三阶可导，证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a,b)" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 58, 458, 642, 474 ] }, { "type": "equation_interline", "content": { "math_content": "f (b) = f (a) + \\frac {1}{2} (b - a) [ f ^ {\\prime} (a) + f ^ {\\prime} (b) ] - \\frac {1}{1 2} (b - a) ^ {3} f ^ {\\prime \\prime \\prime} (\\xi)", "math_type": "latex", "image_source": { "path": "images/d571acc1a681b6f8ba4ce86a866b52b2e6bb44358180ed11739dfb9914f9c7b4.jpg" } }, "bbox": [ 287, 476, 707, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 配合拉格朗日插值或拉格朗日余项的 Taylor 展开： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 二阶可导，证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 121, 506, 850, 521 ] }, { "type": "equation_interline", "content": { "math_content": "f (b) - 2 f (\\frac {a + b}{2}) + f (a) = \\frac {1}{4} (b - a) ^ {2} f ^ {\\prime \\prime} (\\xi)", "math_type": "latex", "image_source": { "path": "images/11defa7b1f5b4e60e49f68c3b0f5efab17dc48fdc2320f38b283ce89807d7051.jpg" } }, "bbox": [ 339, 525, 655, 552 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 需配合 Taylor 展开: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上二阶可导, 证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": ", 使得" } ] }, "bbox": [ 122, 554, 663, 570 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} f (x) \\mathrm{d} x = (b - a) f \\left(\\frac {a + b}{2}\\right) + \\frac {1}{2 4} (b - a) ^ {3} f ^ {\\prime \\prime} (\\xi)", "math_type": "latex", "image_source": { "path": "images/57048488c256800cbfb73652c7de76f0ba24935303d64c6ac4f8e13a26934d01.jpg" } }, "bbox": [ 310, 573, 685, 605 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (3) 设 " }, { "type": "equation_inline", "content": "k = \\frac{\\int_{a}^{b} f(x) \\, \\mathrm{d}x - (b - a)f\\left(\\frac{a + b}{2}\\right)}{\\frac{1}{24}(b - a)^{3}}" }, { "type": "text", "content": "，构造" } ] }, "bbox": [ 87, 609, 468, 645 ] }, { "type": "equation_interline", "content": { "math_content": "F (x) = \\int_ {a} ^ {x} f (t) \\mathrm{d} t - (x - a) f \\left(\\frac {x + a}{2}\\right) + \\frac {1}{2 4} (x - a) ^ {3} k", "math_type": "latex", "image_source": { "path": "images/9c159d25514e7f47ac9965259a82dc4a6afb17d1e41b8f719dccaa30a07c5ba6.jpg" } }, "bbox": [ 297, 648, 699, 678 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "F(a)=F(b)=0" }, { "type": "text", "content": " ，根据 Rolle 得到 " }, { "type": "equation_inline", "content": "\\exists\\eta\\in(a,b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "F'(\\eta)=0" }, { "type": "text", "content": " ，即" } ] }, "bbox": [ 90, 682, 571, 697 ] }, { "type": "equation_interline", "content": { "math_content": "f (\\eta) - f \\left(\\frac {a + \\eta}{2}\\right) - f ^ {\\prime} \\left(\\frac {a + \\eta}{2}\\right) \\frac {\\eta - a}{2} - \\frac {k}{8} (\\eta - a) ^ {2} = 0", "math_type": "latex", "image_source": { "path": "images/2dfb8dbe9de0e8065379d9fd417a3447c276fa4efda22f25016c5ec9fb69ac82.jpg" } }, "bbox": [ 292, 701, 705, 731 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而根据 " }, { "type": "equation_inline", "content": "\\eta" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\frac{a+\\eta}{2}" }, { "type": "text", "content": " 的 Taylor 展开， " }, { "type": "equation_inline", "content": "\\exists\\xi\\in\\left(\\frac{a+\\eta}{2},\\eta\\right)" } ] }, "bbox": [ 90, 736, 468, 765 ] }, { "type": "equation_interline", "content": { "math_content": "f (\\eta) = f \\left(\\frac {a + \\eta}{2}\\right) + f ^ {\\prime} \\left(\\frac {a + \\eta}{2}\\right) \\frac {\\eta - a}{2} + \\frac {f ^ {\\prime \\prime} (\\xi)}{2} \\left(\\frac {\\eta - a}{2}\\right) ^ {2}", "math_type": "latex", "image_source": { "path": "images/891fb5be9d414cc9f2265e17265023263fd029ef24d74bb6de312f442cc2fb27.jpg" } }, "bbox": [ 280, 770, 715, 802 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此对比得到 " }, { "type": "equation_inline", "content": "k = f''(\\xi)" } ] }, "bbox": [ 90, 806, 272, 821 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.2.6 中值不等式" } ], "level": 2 }, "bbox": [ 92, 843, 253, 859 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.27基于Taylor展开(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 二阶可导， " }, { "type": "equation_inline", "content": "|f(x)| \\leq A, |f''(x)| \\leq B" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "|f'(x)| \\leq \\frac{2A}{b - a} + \\frac{B}{2}(b - a)" } ] }, "bbox": [ 58, 869, 905, 898 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 特例 1: 区间 " }, { "type": "equation_inline", "content": "[0,2]" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|f(x)| \\leq 1" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|f''(x)| \\leq 2" }, { "type": "text", "content": " 证明 " }, { "type": "equation_inline", "content": "|f'(x)| \\leq 3" } ] }, "bbox": [ 121, 896, 586, 913 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "36" } ] }, "bbox": [ 489, 954, 509, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.2 微分中值定理" } ] }, "bbox": [ 769, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 特例 2：区间 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "|f(x)| \\leq A, |f''(x)| \\leq B" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "|f'(x)| \\leq 2A + \\frac{1}{2}B" } ] }, "bbox": [ 122, 83, 667, 108 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 推论：将 " }, { "type": "equation_inline", "content": "|f(x)| \\leq A" }, { "type": "text", "content": " 的条件改为 " }, { "type": "equation_inline", "content": "f(a) = f(b)" }, { "type": "text", "content": "，则得到 " }, { "type": "equation_inline", "content": "|f'(x)| \\leq \\frac{B}{2}(b - a)" } ] }, "bbox": [ 122, 110, 685, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(6) 推论特例 1: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 二阶可导, " }, { "type": "equation_inline", "content": "f(0) = f(1) = f(\\frac{1}{2}) = 0" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "|f''(x)| \\leq M" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "|f'(x)| \\leq \\frac{M}{2}" }, { "type": "text", "content": "\n\n证明 注意这里是已知点在 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 处展开, 并使用 Lagrange 余项! 不要写成 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "a, b" }, { "type": "text", "content": " 展开, 或用 Peano 余项。" } ] }, "bbox": [ 90, 133, 853, 173 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 根据 Taylor 展开得到 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, x), \\eta \\in (x, b)" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 122, 175, 499, 191 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} f (a) = f (x) + f ^ {\\prime} (x) (a - x) + \\frac {f ^ {\\prime \\prime} (\\xi)}{2} (a - x) ^ {2} \\\\ f (b) = f (x) + f ^ {\\prime} (x) (b - x) + \\frac {f ^ {\\prime \\prime} (\\eta)}{2} (b - x) ^ {2} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/1a29dd331f65fb2aae6979cd8a9a305c69c6e8fc665b7892393323a0cea96656.jpg" } }, "bbox": [ 317, 193, 660, 247 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "两式相减得到 " }, { "type": "equation_inline", "content": "f(b)-f(a)=f'(x)(b-a)+\\frac{f''(\\eta)}{2}(b-x)^{2}-\\frac{f''(\\xi)}{2}(a-x)^{2}" }, { "type": "text", "content": " ，结合 A, B 得到" } ] }, "bbox": [ 90, 250, 764, 279 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f ^ {\\prime} (x) \\right| \\leq \\frac {2 A}{b - a} + \\frac {B}{2 (b - a)} \\left[ (a - x) ^ {2} + (b - x) ^ {2} \\right]", "math_type": "latex", "image_source": { "path": "images/868114173da29192226033498c134a6ef5634eade7542a10b653e40201bd528b.jpg" } }, "bbox": [ 324, 281, 670, 310 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "(a - x)^2 +(b - x)^2 < (b - a)^2" }, { "type": "text", "content": " （用二次函数最值)，因此得到 " }, { "type": "equation_inline", "content": "|f'(x)| < \\frac{2A}{b - a} +\\frac{B}{2} (b - a)" } ] }, "bbox": [ 89, 315, 747, 341 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 由于 " }, { "type": "equation_inline", "content": "f(b) - f(a) = f'(x)(b - a) + \\frac{f''(\\eta)}{2} (b - x)^2 - \\frac{f''(\\xi)}{2} (a - x)^2 = 0" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 项被消除，后面同理。" } ] }, "bbox": [ 121, 340, 907, 370 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.28 几道经典配凑放缩 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "f(a) = f(b), |f'(x)| \\leq 1" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\forall x_1, x_2" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|f(x_1) - f(x_2)| \\leq \\frac{b - a}{2}" } ] }, "bbox": [ 60, 384, 905, 424 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 首先根据 Lagrange 中值定理得到 " }, { "type": "equation_inline", "content": "\\left|f(x_{1})-f(x_{2})\\right|\\leq x_{2}-x_{1}" }, { "type": "text", "content": " ，再根据" } ] }, "bbox": [ 87, 424, 660, 439 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f \\left(x _ {1}\\right) - f \\left(x _ {2}\\right) \\right| = \\left| f \\left(x _ {1}\\right) - f (a) + f (b) - f \\left(x _ {2}\\right) \\right| \\leq \\left(x _ {1} - a\\right) + \\left(b - x _ {2}\\right)", "math_type": "latex", "image_source": { "path": "images/4ad96d60db9299b0e27c7426a3d992b62653aaf7d5891a4b43c93a36d9f800f7.jpg" } }, "bbox": [ 240, 445, 754, 463 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此相加得到 " }, { "type": "equation_inline", "content": "\\left|f(x_{1})-f(x_{2})\\right|\\leq\\frac{b-a}{2}" } ] }, "bbox": [ 90, 466, 374, 493 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.2.7 导数极限定理与导数介值定理" } ], "level": 2 }, "bbox": [ 92, 511, 413, 527 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.11 (导数介值定理：达布定理)" } ], "level": 2 }, "bbox": [ 121, 543, 396, 558 ] }, { "type": "table", "content": { "image_source": { "path": "images/3783c0874b99c547e766ea447319b6ef265256fa2f854f4a029111f07cd666d1.jpg" }, "table_caption": [], "table_footnote": [], "html": "
定理4.11(等数升值定理:达布定理)
f(x)在[a,b]可导,f′+(a)≠f′-b),k介于f′+(a),f′-b之间,则∃ξ∈(a,b)使得f′(ξ)=k
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 105, 552, 902, 589 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 构造 " }, { "type": "equation_inline", "content": "F(x) = f(x) - kx" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "F'(x) = f'(x) - k" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "k" }, { "type": "text", "content": " 介于 " }, { "type": "equation_inline", "content": "f_{+}'(a), f_{-}'(b)" }, { "type": "text", "content": " 之间，因此 " }, { "type": "equation_inline", "content": "F_{+}'(a)F_{-}'(b) < 0" }, { "type": "text", "content": "，不妨设 " }, { "type": "equation_inline", "content": "F_{+}'(a) > 0, F_{-}'(b) < 0" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 89, 600, 905, 634 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\to a ^ {+}} \\frac {F (x) - F (a)}{x - a} > 0, \\lim _ {x \\to a ^ {-}} \\frac {F (x) - F (b)}{x - b} < 0", "math_type": "latex", "image_source": { "path": "images/b8c9f54c8dda2bef1ef3b59b4d57a33d8f58ee847ee8dfc07a69d7fccdd38697.jpg" } }, "bbox": [ 324, 638, 673, 668 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据极限保号性得到 " }, { "type": "equation_inline", "content": "\\exists x_{1},x_{2}\\in (a,b),x_{1} < x_{2}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\frac{F(x_1) - F(a)}{x_1 - a} >0,\\frac{F(x_2) - F(b)}{x_2 - b} < 0" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "F(x_{1}) > F(a),F(x_{2})>" }, { "type": "equation_inline", "content": "F(b)" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 连续，存在 " }, { "type": "equation_inline", "content": "\\xi \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "F(\\xi)" }, { "type": "text", "content": " 最大值，根据费马定理得到 " }, { "type": "equation_inline", "content": "F^{\\prime}(\\xi) = 0" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "f^{\\prime}(\\xi) = k" }, { "type": "text", "content": " 笔记导数介值定理在ZJU2020中出现过" } ] }, "bbox": [ 87, 671, 927, 732 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.3 洛必达法则与 Taylor 公式" } ] }, "bbox": [ 687, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "37" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.3 洛必达法则与 Taylor 公式" } ], "level": 2 }, "bbox": [ 92, 87, 416, 108 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.3.1 洛必达法则" } ], "level": 2 }, "bbox": [ 92, 126, 253, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.12 (洛必达法则)" } ], "level": 2 }, "bbox": [ 121, 159, 295, 175 ] }, { "type": "table", "content": { "image_source": { "path": "images/ed8db0b53f71e51c5cc9d644b137880b6d507194d33ff01dd70c2c327da6cd5f.jpg" }, "table_caption": [], "table_footnote": [], "html": "
f,g满足(1) $\\lim_{x\\to x_0}f(x)=\\lim_{x\\to x_0}g(x)=0$ 或者 $\\lim_{x\\to x_0}g(x)=\\infty(2)x_0$ 空心邻域中 $f'(x),g'(x)$ 有定义(非常重要的条件!)则 $\\lim_{x\\to x_0}\\frac{f(x)}{g(x)}=\\lim_{x\\to x_0}\\frac{f'(x)}{g'(x)}$
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 104, 170, 902, 252 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 空心邻域中 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 有定义且可导条件非常重要！" } ] }, "bbox": [ 87, 262, 529, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.29 空心邻域中没定义则只能用定义 已知 " }, { "type": "equation_inline", "content": "g(0)=g'(0)=0, g''(0)=3" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "f'(0)" } ] }, "bbox": [ 58, 299, 715, 315 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = \\left\\{ \\begin{array}{l l} \\frac {g (x)}{x}, & x \\neq 0 \\\\ 0, & x = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/226603d18c0c5fe1ab6d857ec40f49dd0e22428415f7dc4422cebe5eaf5d1427.jpg" } }, "bbox": [ 408, 319, 586, 365 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 可以直接用 Peano 余项的 Taylor 展开。但这里展现其证明过程，先代入并用一次洛必达，" } ] }, "bbox": [ 87, 370, 781, 386 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\rightarrow 0} \\frac {f (x) - f (0)}{x - 0} = \\lim _ {x \\rightarrow 0} \\frac {g (x)}{x ^ {2}} = \\lim _ {x \\rightarrow 0} \\frac {g ^ {\\prime} (x)}{2 x}", "math_type": "latex", "image_source": { "path": "images/73a6a55025bb04998faed36e3a5944a7744ca38a95e58f43f2ef554074e2bc0a.jpg" } }, "bbox": [ 344, 388, 650, 418 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "注意后面不能继续用洛必达了，原因在于 " }, { "type": "equation_inline", "content": "g'(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 空心邻域中不一定有定义，极限 " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}g''(x)" }, { "type": "text", "content": " 没有意义。因此需要用导数定义： " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}\\frac{g'(x) - g'(0)}{2x} = \\frac{1}{2}\\lim_{x\\to 0}g''(x) = \\frac{3}{2}" } ] }, "bbox": [ 87, 420, 905, 470 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 该证明过程是 Peano 余项的 Taylor 公式的证明思路，极度重要，保证最后一次不能用洛必达法则。" } ] }, "bbox": [ 87, 472, 843, 488 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.3.2 Taylor 公式" } ], "level": 2 }, "bbox": [ 94, 508, 253, 526 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.13 (Peano 余项 Taylor 公式)" } ], "level": 2 }, "bbox": [ 119, 541, 381, 556 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 存在直至 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 阶导数 (只需要单点导数，比Lagrange要求低很多)，则" } ] }, "bbox": [ 112, 562, 689, 577 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = f (x _ {0}) + f ^ {\\prime} (x _ {0}) (x - x _ {0}) + \\dots + \\frac {f ^ {(n)} (x _ {0})}{n !} (x - x _ {0}) ^ {n} + o ((x - x _ {0}) ^ {n})", "math_type": "latex", "image_source": { "path": "images/b54ab173c59c881be2966c528ad2cb3e93caaad107fa50b9c64ed76c8bd8c4a5.jpg" } }, "bbox": [ 231, 580, 764, 611 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "T_{n}(x)=f(x_{0})+f'(x_{0})(x-x_{0})+\\cdots+\\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "R_{n}(x)=f(x)-T_{n}(x)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "Q_{n}(x)=(x-x_{0})^{n}" }, { "type": "text", "content": " ，先做 n-1 次洛必达，再用一次导数定义可以得到：" } ] }, "bbox": [ 87, 623, 905, 663 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\to x _ {0}} \\frac {R _ {n} (x)}{Q _ {n} (x)} = \\lim _ {x \\to x _ {0}} \\frac {R _ {n} ^ {(n - 1)} (x)}{Q _ {n} ^ {(n - 1)} (x)} = \\frac {1}{n !} \\lim _ {x \\to x _ {0}} \\left[ \\frac {f ^ {(n - 1)} (x) - f ^ {(n - 1)} (x _ {0})}{x - x _ {0}} - f ^ {(n)} (x _ {0}) \\right] = 0", "math_type": "latex", "image_source": { "path": "images/e87d8abc449b067ff7f244ece13ebe9b4c790cda1770ba409e3688f532fed4a5.jpg" } }, "bbox": [ 191, 665, 805, 701 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.14 (Lagrange 余项 Taylor 公式)" } ], "level": 2 }, "bbox": [ 122, 718, 406, 733 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 存在直至 " }, { "type": "equation_inline", "content": "n + 1" }, { "type": "text", "content": " 阶导数，且在 " }, { "type": "equation_inline", "content": "(a,b)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 阶连续可导(条件比Peano强得多)，则" } ] }, "bbox": [ 114, 737, 806, 753 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = f \\left(x _ {0}\\right) + f ^ {\\prime} \\left(x _ {0}\\right) \\left(x - x _ {0}\\right) + \\dots + \\frac {f ^ {(n)} \\left(x _ {0}\\right)}{n !} \\left(x - x _ {0}\\right) ^ {n} + \\frac {f ^ {(n + 1)} (\\xi)}{(n + 1) !} \\left(x - x _ {0}\\right) ^ {n + 1}", "math_type": "latex", "image_source": { "path": "images/cb149e0069fff5784915c16e75a6a8e980904d84a4f7008919b1aa7335bacfef.jpg" } }, "bbox": [ 196, 756, 798, 788 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "T_{n}(x)=f(x_{0})+f'(x_{0})(x-x_{0})+\\cdots+\\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}" }, { "type": "text", "content": " ，构造 " }, { "type": "equation_inline", "content": "F(t)=f(t)-T_{n}(t), G(t)=(x-t)^{n+1}" }, { "type": "text", "content": " ，即证明 " }, { "type": "equation_inline", "content": "\\frac{F(x_{0})}{G(x_{0})}=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}" }, { "type": "text", "content": " ，因为 " }, { "type": "equation_inline", "content": "F(x)=G(x)=0, F'(t)=-\\frac{f^{(n+1)}(t)}{n!}(x-t)^{n}, G'(t)=-(n+1)(x-t)^{n}" }, { "type": "text", "content": " ，因此根据 Cauchy 中值定理可知" } ] }, "bbox": [ 87, 803, 905, 873 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {F (x _ {0})}{G (x _ {0})} = \\frac {F (x _ {0}) - F (x)}{G (x _ {0}) - G (x)} = \\frac {F ^ {\\prime} (\\xi)}{G ^ {\\prime} (\\xi)} = \\frac {f ^ {(n + 1)} (\\xi)}{(n + 1) !}", "math_type": "latex", "image_source": { "path": "images/c7f08d7bb3d2f8ff263bf7d664602a4a3e8d67c8beb8218ab85e881e8020b29c.jpg" } }, "bbox": [ 327, 877, 667, 910 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.3 洛必达法则与 Taylor 公式" } ] }, "bbox": [ 687, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "38" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.4 凸函数" } ], "level": 2 }, "bbox": [ 92, 87, 216, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.4.1 函数的凹凸性" } ], "level": 2 }, "bbox": [ 92, 126, 273, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 4.3 (凸函数)" } ], "level": 2 }, "bbox": [ 122, 159, 253, 174 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上的函数，若对 " }, { "type": "equation_inline", "content": "\\forall x_{1},x_{2}\\in I" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "\\forall \\lambda \\in (0,1)" }, { "type": "text", "content": " 满足以下条件，则称 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为凸函数" } ] }, "bbox": [ 112, 179, 766, 193 ] }, { "type": "equation_interline", "content": { "math_content": "f (\\lambda x _ {1} + (1 - \\lambda) x _ {2}) \\leq \\lambda f (x _ {1}) + (1 - \\lambda) f (x _ {2})", "math_type": "latex", "image_source": { "path": "images/095d3aa5edb630d9f867c42c4e2188de009ed05152e892b905471aafc512b68d.jpg" } }, "bbox": [ 332, 203, 663, 219 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "引理 4.1 (定比分点公式)" } ], "level": 2 }, "bbox": [ 124, 243, 304, 258 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于一点 " }, { "type": "equation_inline", "content": "x \\in [x_1, x_2]" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 可由其与 " }, { "type": "equation_inline", "content": "x_1, x_2" }, { "type": "text", "content": " 的距离唯一表出：" } ] }, "bbox": [ 114, 263, 566, 278 ] }, { "type": "equation_interline", "content": { "math_content": "x = \\frac {x _ {2} - x}{x _ {2} - x _ {1}} x _ {1} + \\frac {x - x _ {1}}{x _ {2} - x _ {1}} x _ {2}", "math_type": "latex", "image_source": { "path": "images/a72e224d2cd80ed329480ef83d5f0b930cf84e3a230be7410fcb1e55667b29e2.jpg" } }, "bbox": [ 391, 282, 603, 309 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "记忆方式：首先保证分子都是正的，而且 " }, { "type": "equation_inline", "content": "x_{1}" }, { "type": "text", "content": " 对应的权重是 x 到 " }, { "type": "equation_inline", "content": "x_{2}" }, { "type": "text", "content": " 的距离" } ] }, "bbox": [ 114, 312, 657, 326 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.15 (凸函数的等价条件)" } ], "level": 2 }, "bbox": [ 122, 351, 344, 366 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上的凸函数与下面条件等价：" } ] }, "bbox": [ 112, 370, 376, 385 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 一阶可导最常用： " }, { "type": "equation_inline", "content": "\\forall x_0 \\in I" }, { "type": "text", "content": "，有 " }, { "type": "equation_inline", "content": "f(x) \\geq f(x_0) + f'(x_0)(x - x_0)" }, { "type": "text", "content": "。" } ] }, "bbox": [ 137, 390, 616, 404 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若不可导，上述命题等价于： " }, { "type": "equation_inline", "content": "f(x) \\geq f(x_0) + k(x - x_0)" } ] }, "bbox": [ 139, 407, 566, 423 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 可导，则等价于 " }, { "type": "equation_inline", "content": "f'(x)" }, { "type": "text", "content": " 单增" } ] }, "bbox": [ 139, 426, 416, 441 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 二阶可导：若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 二阶可导，则等价于 " }, { "type": "equation_inline", "content": "f''(x) \\geq 0" } ] }, "bbox": [ 139, 445, 532, 460 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.30 不二阶可导凸性质的应用 (1) 重点：函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(0)=0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f'(x)" }, { "type": "text", "content": " 单增，证明： " }, { "type": "equation_inline", "content": "F(x)=\\frac{f(x)}{x}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0,+\\infty)" }, { "type": "text", "content": " 单增" } ] }, "bbox": [ 58, 494, 905, 533 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "f'(x)" }, { "type": "text", "content": " 单增等价于可导，而没说二阶可导，因此取 y > x > 0，只能用 " }, { "type": "equation_inline", "content": "f(y) \\geq f(x) + f'(x)(y - x)" }, { "type": "text", "content": " 。此时" } ] }, "bbox": [ 87, 535, 905, 551 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} F (y) - F (x) = \\frac {x f (y) - y f (x)}{x y} \\geq \\frac {x f (x) + x f ^ {\\prime} (x) (y - x) - (y - x) f (x) - x f (x)}{x y} \\\\ \\quad = \\frac {(y - x) (x f ^ {\\prime} (x) - f (x))}{x y} = \\frac {(y - x) (f ^ {\\prime} (x) - \\frac {f (x) - f (0)}{x - 0})}{y} \\\\ \\quad = \\frac {(y - x) (f ^ {\\prime} (x) - f ^ {\\prime} (\\xi))}{y} \\geq 0 \\end{array}", "math_type": "latex", "image_source": { "path": "images/147385ef370d2a557d8b5cc588b59f49214635b0b8f8302c32a7b0600e546f9f.jpg" } }, "bbox": [ 206, 554, 786, 652 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.31 凸函数等价条件应用 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 R 二阶连续可导，证明 " }, { "type": "equation_inline", "content": "f''(x) \\geq 0" }, { "type": "text", "content": " 当且仅当 " }, { "type": "equation_inline", "content": "\\forall x \\in \\mathbb{R}, \\forall h, f(x + h) + f(x - h) - 2f(x) \\geq 0" } ] }, "bbox": [ 58, 671, 905, 706 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 I 上凸函数当且仅当 " }, { "type": "equation_inline", "content": "\\forall x_{1}, x_{2} \\in I" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "\\varphi(\\lambda) = f(\\lambda x_{1} + (1 - \\lambda)x_{2})" }, { "type": "text", "content": " 为 [0,1] 上的凸函数" } ] }, "bbox": [ 119, 709, 821, 724 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "f''(x) \\geq 0" }, { "type": "text", "content": " 等价于凸函数，等价于 " }, { "type": "equation_inline", "content": "\\frac{f(x+h)-f(x)}{h} \\geq \\frac{f(x)-f(x-h)}{h}" }, { "type": "text", "content": "，即得到结论。" } ] }, "bbox": [ 87, 724, 774, 745 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2.1) 右推左：要证 " }, { "type": "equation_inline", "content": "f(\\lambda x_1 + (1 - \\lambda)x_2) \\leq \\lambda f(x_1) + (1 - \\lambda)f(x_2)" }, { "type": "text", "content": "，等价于 " }, { "type": "equation_inline", "content": "\\varphi(\\lambda) \\leq \\lambda \\varphi(1) + (1 - \\lambda)\\varphi(0)" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\varphi(\\lambda) = \\varphi(\\lambda \\cdot 1 + (1 - \\lambda) \\cdot 0)" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "\\varphi" }, { "type": "text", "content": " 的凸性可直接得出。" } ] }, "bbox": [ 89, 747, 902, 783 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2.2) 左推右：要证 " }, { "type": "equation_inline", "content": "\\varphi(\\alpha t_1 + (1 - \\alpha)t_2) \\leq \\alpha \\varphi(t_1) + (1 - \\alpha)\\varphi(t_2)" }, { "type": "text", "content": "，等价于" } ] }, "bbox": [ 121, 785, 653, 801 ] }, { "type": "equation_interline", "content": { "math_content": "f \\left(\\left(\\alpha t _ {1} + (1 - \\alpha) t _ {2}\\right) x _ {1} + \\left(1 - \\alpha t _ {1} - (1 - \\alpha) t _ {2}\\right) x _ {2}\\right) \\leq \\alpha f \\left(t _ {1} x _ {1} + \\left(1 - t _ {1}\\right) x _ {2}\\right) + (1 - \\alpha) f \\left(t _ {2} x _ {1} + \\left(1 - t _ {2}\\right) x _ {2}\\right)", "math_type": "latex", "image_source": { "path": "images/6600e121b7b56d7ce0f3099435cf3f8f2d71484dc8637365e2a367a73b9a2e63.jpg" } }, "bbox": [ 114, 809, 878, 824 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的凸性，右侧大于等于左侧 (可能有点麻烦)。" } ] }, "bbox": [ 90, 832, 495, 846 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.4 凸函数" } ] }, "bbox": [ 821, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "39" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.16 (凸性质的保持)" } ], "level": 2 }, "bbox": [ 122, 86, 310, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "考虑凸函数的四则运算、复合、最值：" } ] }, "bbox": [ 115, 105, 396, 120 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 凸函数的数乘和加法保持凸性质，差、积、商不一定" } ] }, "bbox": [ 139, 124, 547, 139 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 复合：若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为凸函数， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 为凸增函数，则 " }, { "type": "equation_inline", "content": "g(f(x))" }, { "type": "text", "content": " 为凸函数，去掉增性质则不一定" } ] }, "bbox": [ 139, 143, 803, 158 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 最值：若 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 均为 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上凸函数，则 " }, { "type": "equation_inline", "content": "\\max \\{f(x), g(x)\\}" }, { "type": "text", "content": " 为凸函数" } ] }, "bbox": [ 139, 162, 655, 177 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (3) 最值： " }, { "type": "equation_inline", "content": "F(\\lambda x_{1} + (1 - \\lambda)x_{2}) = \\max \\{f(\\lambda x_{1} + (1 - \\lambda)x_{2}), g(\\lambda x_{1} + (1 - \\lambda)x_{2})\\} \\leq \\max \\{\\lambda f(x_{1}) + (1 - \\lambda)f(x_{2}), \\lambda g(x_{1}) + (1 - \\lambda)g(x_{2})\\}" }, { "type": "text", "content": "，进行放缩 " }, { "type": "equation_inline", "content": "F \\geq f, g" }, { "type": "text", "content": " 得到" } ] }, "bbox": [ 89, 197, 904, 231 ] }, { "type": "equation_interline", "content": { "math_content": "F (\\lambda x _ {1} + (1 - \\lambda) x _ {2}) \\leq \\lambda F (x _ {1}) + (1 - \\lambda) F (x _ {2})", "math_type": "latex", "image_source": { "path": "images/928c7da45ff59e610f4ab658405bdb534164670c473a0172bead1033b81b312e.jpg" } }, "bbox": [ 327, 239, 668, 255 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 4.4 (拐点)" } ], "level": 2 }, "bbox": [ 124, 285, 236, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "y = f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 两侧分别为严格凸和严格凹的，则称 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的拐点。若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 二阶可导，则 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 是拐点的必要条件为 " }, { "type": "equation_inline", "content": "f''(x_0) = 0" } ] }, "bbox": [ 112, 305, 880, 338 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.4.2 凸函数的基本性质" } ], "level": 2 }, "bbox": [ 95, 378, 312, 395 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.17 (凸函数基本性质)" } ], "level": 2 }, "bbox": [ 122, 411, 329, 425 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上为凸函数，则" } ] }, "bbox": [ 115, 431, 363, 445 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 中任意一点有左右导函数" } ] }, "bbox": [ 139, 449, 416, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 区间内部连续，端点未知，若是开区间，则在整个区间连续" } ] }, "bbox": [ 139, 467, 668, 483 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 只证右导数: " }, { "type": "equation_inline", "content": "F(h) = \\frac{f(x_0 + h) - f(x_0)}{h}" }, { "type": "text", "content": " 为增函数, 根据有界的单调函数有左右极限可知左右导数存在。" } ] }, "bbox": [ 90, 497, 904, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据左右导数的存在性可知左右连续 (否则导数定义中分母趋于 0，分子不趋于 0，显然无极限)，因此在内部点连续，端点单边连续。" } ] }, "bbox": [ 90, 539, 904, 574 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.18 (凸函数的最值)" } ], "level": 2 }, "bbox": [ 122, 600, 312, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上的凸函数，则 " }, { "type": "equation_inline", "content": "f(x) \\leq \\max \\{f(a), f(b)\\}" }, { "type": "text", "content": "，即凸函数的最值只在端点取到。" } ] }, "bbox": [ 115, 619, 771, 636 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "\\forall x\\in [a,b]" }, { "type": "text", "content": " ，设 " }, { "type": "equation_inline", "content": "x = \\lambda a + (1 - \\lambda)b" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为凸函数可知：" } ] }, "bbox": [ 90, 656, 576, 671 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) \\leq \\lambda f (a) + (1 - \\lambda) f (b) \\leq \\max \\{f (a), f (b) \\}", "math_type": "latex", "image_source": { "path": "images/2dd7089239075bff5594863fa12051fa9521d972555bc298d42c1b1d713882c7.jpg" } }, "bbox": [ 324, 678, 668, 695 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.19 (Jensen 不等式)" } ], "level": 2 }, "bbox": [ 122, 725, 314, 741 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上的凸函数，则对 " }, { "type": "equation_inline", "content": "\\forall x_{i}\\in [a,b],\\lambda_{i} > 0,\\sum_{i = 1}^{n}\\lambda_{i} = 1" }, { "type": "text", "content": " ，有" } ] }, "bbox": [ 114, 747, 606, 782 ] }, { "type": "equation_interline", "content": { "math_content": "f \\left(\\sum_ {i = 1} ^ {n} \\lambda_ {i} x _ {i}\\right) \\leq \\sum_ {i = 1} ^ {n} \\lambda_ {i} f (x _ {i})", "math_type": "latex", "image_source": { "path": "images/c7b64ede0377d85c47c7c1d766dcdcf2a91d6d395a95d2521bd8a46b59d22f81.jpg" } }, "bbox": [ 394, 785, 601, 825 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.4 凸函数" } ] }, "bbox": [ 821, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "40" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.32Jensen 不等式经典题 正数 " }, { "type": "equation_inline", "content": "p_{1}, \\cdots, p_{n}" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n} p_{i} = 1" }, { "type": "text", "content": " ，对 " }, { "type": "equation_inline", "content": "\\forall x_{1}, \\cdots, x_{n} \\in R" }, { "type": "text", "content": " ，证明：" } ] }, "bbox": [ 58, 80, 761, 120 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {i = 1} ^ {n} p _ {i} \\left(x _ {i} - \\ln p _ {i}\\right) \\leq \\ln \\left(\\sum_ {i = 1} ^ {n} e ^ {x _ {i}}\\right)", "math_type": "latex", "image_source": { "path": "images/a91a37deb0121177c4677c092442953b3c91cd91a4f1ac29edad6c01f4063af5.jpg" } }, "bbox": [ 374, 123, 616, 162 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 即证明 " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{n}p_{i}\\ln\\frac{e^{x_{i}}}{p_{i}}\\leq\\ln\\left(\\sum_{i=1}^{n}e^{x_{i}}\\right)" }, { "type": "text", "content": " ，视 " }, { "type": "equation_inline", "content": "e^{x_{i}}=t_{i}" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "\\ln t" }, { "type": "text", "content": " 的凹性即可得到结论。" } ] }, "bbox": [ 87, 164, 726, 203 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.4.3 利用凹凸性证明不等式" } ], "level": 2 }, "bbox": [ 92, 222, 354, 239 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习4.33对中点位置的估计(1)经典：设 " }, { "type": "equation_inline", "content": "\\varphi (x)" }, { "type": "text", "content": " 是[0,1]上连续凸函数，证明 " }, { "type": "equation_inline", "content": "\\varphi \\left(\\frac{1}{2}\\right)\\leq \\int_0^1\\varphi (x)\\mathrm{d}x\\leq \\frac{\\varphi(0) + \\varphi(1)}{2}" } ] }, "bbox": [ 58, 250, 902, 281 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 太经典： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上连续凸函数，证明 " }, { "type": "equation_inline", "content": "f\\left(\\frac{a + b}{2}\\right) \\leq \\frac{1}{b - a} \\int_{a}^{b} f(x) \\, \\mathrm{d}x \\leq \\frac{f(a) + f(b)}{2}" }, { "type": "text", "content": " 证明 (1) 由于 " }, { "type": "equation_inline", "content": "x = x \\cdot 1 + (1 - x) \\cdot 0" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 87, 282, 800, 326 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {1} \\varphi (x) \\mathrm{d} x \\leq \\int_ {0} ^ {1} [ (1 - x) \\varphi (0) + x \\varphi (1) ] = \\frac {\\varphi (0) + \\varphi (1)}{2}", "math_type": "latex", "image_source": { "path": "images/a0d9a9adbe2d1cf20a55b618f505002aea6489c7f4da5339c66b490dc92c576c.jpg" } }, "bbox": [ 295, 330, 697, 363 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "另一方面做变量替换 " }, { "type": "equation_inline", "content": "x = 1 - t" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\int_0^1\\varphi (x)\\mathrm{d}x = \\int_0^1\\varphi (1 - x)\\mathrm{d}x" }, { "type": "text", "content": " （区间再现），得到" } ] }, "bbox": [ 90, 365, 695, 398 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {1} \\varphi (x) \\mathrm{d} x = \\int_ {0} ^ {1} \\frac {1}{2} \\varphi (x) + \\frac {1}{2} \\varphi (1 - x) \\mathrm{d} x \\geq \\int_ {0} ^ {1} \\varphi \\left(\\frac {x}{2} + \\frac {1 - x}{2}\\right) \\mathrm{d} x = \\varphi \\left(\\frac {1}{2}\\right)", "math_type": "latex", "image_source": { "path": "images/98fe7564cfe5773a27e85343d66df484f2344f462512168bd4cb8a947c9951e4.jpg" } }, "bbox": [ 220, 400, 773, 434 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 作替换 " }, { "type": "equation_inline", "content": "x = \\lambda a + (1 - \\lambda)b = b - (b - a)\\lambda" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\frac{1}{b - a}\\int_{a}^{b}f(x)\\mathrm{d}x = \\int_{0}^{1}\\varphi (\\lambda)\\mathrm{d}\\lambda" }, { "type": "text", "content": "，转换为 (1)" } ] }, "bbox": [ 121, 437, 800, 470 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.5 一元极值与导数零点问题" } ], "level": 2 }, "bbox": [ 92, 491, 410, 511 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.5.1 一元极值的概念与判定" } ], "level": 2 }, "bbox": [ 92, 530, 352, 546 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 4.20 (一元极值三大充分条件)" } ], "level": 2 }, "bbox": [ 122, 562, 379, 577 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 可导，且满足必要条件 " }, { "type": "equation_inline", "content": "f'(x_{0})=0" }, { "type": "text", "content": " ，则有三大充分条件：" } ] }, "bbox": [ 112, 583, 564, 598 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "U^o (x_0,\\delta)" }, { "type": "text", "content": " 可导，若左邻域 " }, { "type": "equation_inline", "content": "f^{\\prime}(x)\\leq 0" }, { "type": "text", "content": " ，右邻域 " }, { "type": "equation_inline", "content": "f^{\\prime}(x)\\geq 0" }, { "type": "text", "content": " ，则为极小值。反之为极大值" } ] }, "bbox": [ 134, 601, 880, 633 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. 若在 " }, { "type": "equation_inline", "content": "U(x_{0},\\delta)" }, { "type": "text", "content": " 二阶可导 " }, { "type": "equation_inline", "content": "f''(x_{0})\\neq0" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "f''(x_{0})<0" }, { "type": "text", "content": " 时取极大值， " }, { "type": "equation_inline", "content": "f''(x_{0})>0" }, { "type": "text", "content": " 时取极小值" } ] }, "bbox": [ 132, 638, 803, 653 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3. 若在 " }, { "type": "equation_inline", "content": "U(x_{0}, \\delta)" }, { "type": "text", "content": " 存在 n 阶导数，且 " }, { "type": "equation_inline", "content": "f'(x_{0}), \\cdots, f^{(n-1)}(x_{0}) = 0, f^{(n)}(x_{0}) \\neq 0" }, { "type": "text", "content": " 。则 n 为偶时， " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 为极值点， " }, { "type": "equation_inline", "content": "f^{(n)}(x_{0}) < 0" }, { "type": "text", "content": " 极大， " }, { "type": "equation_inline", "content": "f^{(n)}(x_{0}) > 0" }, { "type": "text", "content": " 时极小，n 为奇时不取极值。" } ] }, "bbox": [ 134, 655, 887, 692 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 只证明第三充分条件。用 Peano 余项的 Taylor 展开，根据前面导数为 0 得到 " }, { "type": "equation_inline", "content": "f(x)-f(x_{0})=\\left[\\frac{f^{(n)}(x_{0})}{n!}+o(1)\\right](x-x_{0})^{n}" }, { "type": "text", "content": " 。若 n 为偶且导数大于 0，则显然为极小值，极大值同理。若 n 为奇数，则 " }, { "type": "equation_inline", "content": "(x-x_{0})^{n}" }, { "type": "text", "content": " 符号不定，因此无法确定。" } ] }, "bbox": [ 87, 703, 936, 768 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4.5.2 导数零点问题" } ], "level": 2 }, "bbox": [ 92, 789, 275, 806 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "要善于取出区间中的极值点或最值点，利用 " }, { "type": "equation_inline", "content": "f'(x_0) = 0" }, { "type": "text", "content": " 的性质加上中值定理或 Taylor 展开研究更高阶的导数。" } ] }, "bbox": [ 90, 816, 905, 852 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 4.34 几道经典导数零点问题 (1) 经典: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续, 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 二阶可导, " }, { "type": "equation_inline", "content": "f(a) = f(b) = 0" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "\\exists c \\in (a, b)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(c) > 0" }, { "type": "text", "content": ", 证明 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f''(\\xi) < 0" } ] }, "bbox": [ 58, 873, 905, 908 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "41" } ] }, "bbox": [ 489, 954, 507, 964 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.5 一元极值与导数零点问题" } ] }, "bbox": [ 687, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "f(a) = f(b) = 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f(c) > 0" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 连续，总能找到最大值 " }, { "type": "equation_inline", "content": "f(x_0)" }, { "type": "text", "content": "，由于是最值点， " }, { "type": "equation_inline", "content": "f'(x_0) = 0" }, { "type": "text", "content": "，考虑 " }, { "type": "equation_inline", "content": "f(a)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处 Lagrange 余项的 Taylor 展开：" } ] }, "bbox": [ 87, 84, 902, 118 ] }, { "type": "equation_interline", "content": { "math_content": "f (a) = f (x _ {0}) + \\frac {1}{2} f ^ {\\prime \\prime} (\\xi) (a - x _ {0}) ^ {2}", "math_type": "latex", "image_source": { "path": "images/0f2834d16d1ecd8363a1bfe4e44ac41b388690b0d6f3574fad0e5d5f8d7fa7b5.jpg" } }, "bbox": [ 376, 120, 618, 149 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 " }, { "type": "equation_inline", "content": "f(a)=0" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "f''(\\xi)<0" } ] }, "bbox": [ 92, 151, 322, 166 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "4.5 一元极值与导数零点问题" } ] }, "bbox": [ 687, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "42" } ] }, "bbox": [ 489, 954, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第5章 不定积分" } ], "level": 2 }, "bbox": [ 396, 87, 618, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.1 常用不定积分公式" } ], "level": 2 }, "bbox": [ 92, 145, 337, 165 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "必背部分:" } ] }, "bbox": [ 122, 180, 203, 195 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "普通三角：" } ] }, "bbox": [ 119, 219, 200, 236 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\sin x \\mathrm{d} x = - \\cos x + c", "math_type": "latex", "image_source": { "path": "images/da59b38407278da4496f4d6c90e6d00d46ad03df19c147b7829634d45fd48eaf.jpg" } }, "bbox": [ 221, 214, 401, 243 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "sec、csc:" } ] }, "bbox": [ 119, 248, 193, 262 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\sec^ {2} x \\mathrm{d} x = \\tan x + c", "math_type": "latex", "image_source": { "path": "images/24e49b056be2c5c11a758f64c3c798c8f09ecf801191a532929e41e29afa9f6b.jpg" } }, "bbox": [ 231, 242, 400, 268 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\cos x \\mathrm{d} x = \\sin x + c", "math_type": "latex", "image_source": { "path": "images/0e24215ff59ecd8e05e6362094b1c5288f46af9ebffc5588c6211191b928f681.jpg" } }, "bbox": [ 524, 214, 690, 243 ] }, { "type": "equation_interline", "content": { "math_content": "\\csc^ {2} x \\mathrm{d} x = - \\cot x + c", "math_type": "latex", "image_source": { "path": "images/79495dd7cbf5bb06eeeb8a6a55965183dda6f087c3bb817be905f47e387ba8ad.jpg" } }, "bbox": [ 539, 243, 714, 263 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} \\sec x \\mathrm{d} x = \\ln | \\sec x + \\tan x |", "math_type": "latex", "image_source": { "path": "images/6642bdb1cc7c82561b9dc7b06a62108c6c06032ca3d8592b6162f81f2b99ed85.jpg" } }, "bbox": [ 225, 267, 442, 297 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\sec^ {3} x \\mathrm{d} x = \\frac {1}{2} \\tan x \\sec x + \\frac {1}{2} \\ln | \\tan x + \\sec x |", "math_type": "latex", "image_source": { "path": "images/a6b910e996ddc3b96d72a6082031ec1a57c2dc5fa6fbd9dabbb09f5a9e380d2c.jpg" } }, "bbox": [ 532, 267, 875, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "乘积三角：" } ] }, "bbox": [ 119, 299, 200, 315 ] }, { "type": "equation_interline", "content": { "math_content": "\\sec x \\tan x \\mathrm{d} x = \\sec x + c", "math_type": "latex", "image_source": { "path": "images/28a39e9e8c0070b1ba59a2b266f149d67125da098100d6ce473b15e8d14c2cc4.jpg" } }, "bbox": [ 236, 297, 425, 317 ] }, { "type": "equation_interline", "content": { "math_content": "\\csc x \\cot x \\mathrm{d} x = - \\csc x + c", "math_type": "latex", "image_source": { "path": "images/625f1806979ba009774409ef27de1e5f4dfa5d1d8600bd551a8dbf5d6b485c1d.jpg" } }, "bbox": [ 541, 298, 742, 317 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "分式:" } ] }, "bbox": [ 119, 325, 166, 341 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {1}{x ^ {2} + a ^ {2}}} \\mathrm{d} x = \\frac {1}{a} \\arctan {\\frac {x}{a}} + c", "math_type": "latex", "image_source": { "path": "images/4d45cde1f4fb5fbf688dcfd8e2aab070d9650ec8ce0b48b82b2b83bbb58c1e3c.jpg" } }, "bbox": [ 225, 319, 453, 348 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} \\frac {1}{x ^ {2} - a ^ {2}} \\mathrm{d} x = \\frac {1}{2 a} \\ln | \\frac {x - a}{x + a} | + c", "math_type": "latex", "image_source": { "path": "images/238ad55fcc19c9808b209a1f784c87ac086508f28e4527c9ddbf2c6c6f2dd91f.jpg" } }, "bbox": [ 534, 319, 773, 349 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {c} \\frac {1}{\\sqrt {x ^ {2} + a ^ {2}}} \\mathrm{d} x = \\ln | x + \\sqrt {x ^ {2} + a ^ {2}} | + c", "math_type": "latex", "image_source": { "path": "images/9cec29d2352516d536bfa91d61fee9b5accf77d7e6307de3b284d33a552d8fd9.jpg" } }, "bbox": [ 223, 346, 515, 376 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {1}{\\sqrt {x ^ {2} - a ^ {2}}}} \\mathrm{d} x = \\ln | x + \\sqrt {x ^ {2} - a ^ {2}} | + c", "math_type": "latex", "image_source": { "path": "images/a1f1471e26d1f762bb06852204e8b752aaf2104135c5d31dc0bc97c0c0c068e7.jpg" } }, "bbox": [ 526, 347, 820, 376 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} \\frac {1}{\\sqrt {a ^ {2} - x ^ {2}}} \\mathrm{d} x = \\arcsin {\\frac {x}{a}} + c", "math_type": "latex", "image_source": { "path": "images/bbbb1bf2d8b78bd950f77e5f537af22acca65b24e16c77d09b39d86884301f66.jpg" } }, "bbox": [ 223, 376, 448, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对数:" } ] }, "bbox": [ 119, 406, 166, 420 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\ln x \\mathrm{d} x = x \\ln x - x + c", "math_type": "latex", "image_source": { "path": "images/5863c12f13dd1fb1993eff5e0ed652984680c5a757dbd42a1bd307a6060a67d7.jpg" } }, "bbox": [ 228, 403, 413, 428 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\ln (1 + x) \\mathrm{d} x = (1 + x) \\ln (1 + x) - x + c", "math_type": "latex", "image_source": { "path": "images/b00d55047ad3a7b63faa3825cdc2ce1db0da4679a77d0b26c97447c9d68118f8.jpg" } }, "bbox": [ 524, 398, 836, 430 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "会算部分:" } ] }, "bbox": [ 122, 431, 203, 447 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "三角：" } ] }, "bbox": [ 122, 472, 159, 488 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\tan x \\mathrm{d} x = - \\ln | \\cos x | + c", "math_type": "latex", "image_source": { "path": "images/59d3cb1c17eba7d2d2d9d86dfc13ade8a7ac18e42bbbb2aafd636e8d310a22b2.jpg" } }, "bbox": [ 213, 466, 426, 494 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\cot x \\mathrm{d} x = \\ln | \\sin x | + c", "math_type": "latex", "image_source": { "path": "images/0a0dcee953f134622a08eddb43dc5ceccdf18c23a0065d2d7fb00bbc3e60f915.jpg" } }, "bbox": [ 668, 464, 860, 494 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "平方三角：" } ] }, "bbox": [ 112, 498, 191, 514 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} \\sin^ {2} x \\mathrm{d} x = \\frac {x}{2} - \\frac {\\sin 2 x}{4} + c", "math_type": "latex", "image_source": { "path": "images/538d939a2e39c2c0531d1bfd681d1a10589daba2647654a4556fb1d0dcf168e3.jpg" } }, "bbox": [ 213, 493, 428, 521 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\cos^ {2} x \\mathrm{d} x = \\frac {x}{2} + \\frac {\\sin 2 x}{4} + c", "math_type": "latex", "image_source": { "path": "images/67b2c80729ddaefb7cf69322414a6bc35c725df95cc59395b40ccea45a2c5beb.jpg" } }, "bbox": [ 668, 493, 884, 521 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\tan^ {2} x \\mathrm{d} x = \\tan x - x + c", "math_type": "latex", "image_source": { "path": "images/77b4ce3f15bb1824224b9e32a8b464c5a1f89aff5888f9f5ccc9666702185e07.jpg" } }, "bbox": [ 213, 520, 421, 545 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\cot^ {2} x \\mathrm{d} x = - \\cot x - x + c", "math_type": "latex", "image_source": { "path": "images/f91fdbdc8df3818e62aa49ca6f5a5340c6340466ce02a611c7f5686a2a5b87c9.jpg" } }, "bbox": [ 665, 520, 885, 548 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "复合三角：" } ] }, "bbox": [ 112, 551, 191, 567 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} e ^ {a x} \\cos b x \\mathrm{d} x = \\frac {e ^ {a x}}{a ^ {2} + b ^ {2}} (a \\cos b x + b \\sin b x) + c", "math_type": "latex", "image_source": { "path": "images/1f83e13e45d7951052d313e7427901c0add38a9aa545ff3a70fedb5a8483f0de.jpg" } }, "bbox": [ 213, 545, 571, 574 ] }, { "type": "equation_interline", "content": { "math_content": "\\int e ^ {a x} \\sin b x \\mathrm{d} x = \\frac {e ^ {a x}}{a ^ {2} + b ^ {2}} (a \\sin b x - b \\cos b x) + c", "math_type": "latex", "image_source": { "path": "images/7a5be8b054ef136acf40b19843954a525535dfc228471d89e06d57f1bf5b86f9.jpg" } }, "bbox": [ 215, 571, 569, 600 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} x \\cos n x \\mathrm{d} x = \\frac {1}{n ^ {2}} \\cos n x + \\frac {x}{n} \\sin n x + c", "math_type": "latex", "image_source": { "path": "images/c81f3a2e679b60cf6f1641c7a075c42c1d4ae5f3fcc526fb533b5615653a85a0.jpg" } }, "bbox": [ 213, 598, 517, 627 ] }, { "type": "equation_interline", "content": { "math_content": "\\int x \\sin n x \\mathrm{d} x = \\frac {1}{n ^ {2}} \\sin n x - \\frac {x}{n} \\cos n x + c", "math_type": "latex", "image_source": { "path": "images/f53db76fded4ef7689070a7e5ede4a416cda4d0ae3d789436c7669eb529f6580.jpg" } }, "bbox": [ 213, 627, 515, 652 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根式:" } ] }, "bbox": [ 112, 656, 159, 671 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {1} \\sqrt {x ^ {2} \\pm a ^ {2}} \\mathrm{d} x = \\frac {1}{2} \\left[ x \\sqrt {x ^ {2} \\pm a ^ {2}} \\pm a ^ {2} \\ln | x + \\sqrt {x ^ {2} \\pm a ^ {2}} | \\right] + c", "math_type": "latex", "image_source": { "path": "images/2c95d96a73a5602ea28d80581fb872e2dea3ef6ef2e1f11d5f234e8401db42ca.jpg" } }, "bbox": [ 213, 651, 658, 678 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\sqrt {a ^ {2} - x ^ {2}}} \\mathrm{d} x = \\frac {1}{2} \\left[ x \\sqrt {a ^ {2} - x ^ {2}} + a ^ {2} \\arcsin {\\frac {x}{a}} \\right] + c", "math_type": "latex", "image_source": { "path": "images/1124fc78be1e30760f2762dc4b64ac1b06253f8b5e30f3dea0cac26093218b52.jpg" } }, "bbox": [ 213, 677, 591, 706 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.2 不定积分基本方法" } ], "level": 2 }, "bbox": [ 94, 728, 337, 747 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.1 (分部积分公式)" } ], "level": 2 }, "bbox": [ 122, 768, 304, 784 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "函数 " }, { "type": "equation_inline", "content": "u, v" }, { "type": "text", "content": " 的积分满足：" } ] }, "bbox": [ 115, 789, 280, 803 ] }, { "type": "equation_interline", "content": { "math_content": "\\int u \\mathrm{d} v = u v - \\int v \\mathrm{d} u", "math_type": "latex", "image_source": { "path": "images/ac3926f287bc1ea4d9dacbee19ecf9166b01c9880722484ec9a540592c39355b.jpg" } }, "bbox": [ 416, 799, 579, 829 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.2 (表格法)" } ], "level": 2 }, "bbox": [ 122, 86, 253, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "函数 u, v 的积分 " }, { "type": "equation_inline", "content": "\\int uv^{(n+1)}\\mathrm{d}x = uv^{(n)} - u'v^{(n-1)} + u''v^{(n-2)} - \\cdots + (-1)^{n}u^{(n)}v + (-1)^{n+1} \\int u^{(n+1)}v\\mathrm{d}x" }, { "type": "text", "content": " ，具体操作时可列以下表格：" } ] }, "bbox": [ 115, 108, 882, 149 ] }, { "type": "table", "content": { "image_source": { "path": "images/98059397853c4ee6e58e23c248cf60ba2313e3f9bcc186a1a41d0682c63aa193.jpg" }, "table_caption": [], "table_footnote": [], "html": "
u的各阶导数u $u'$ $u''$ $u'''$ ... $u^{(n+1)}$
$v^{(n+1)}$ 的各阶原函数 $v^{(n+1)}$ $v^{(n)}$ $v^{(n-1)}$ $v^{(n-2)}$ ...v
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 231, 156, 764, 196 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算方法：以 " }, { "type": "equation_inline", "content": "u" }, { "type": "text", "content": " 为起点，左上右下相乘，符号先正后负交替，最后一项为最后一列两个的积分 " }, { "type": "equation_inline", "content": "(-1)^{n + 1}\\int u^{(n + 1)}v\\mathrm{d}x" } ] }, "bbox": [ 114, 204, 882, 246 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 " }, { "type": "equation_inline", "content": "\\int P_{n}(x)e^{kx}\\mathrm{d}x,\\int P_{n}(x)\\sin ax\\mathrm{d}x,\\int P_{n}(x)\\cos ax\\mathrm{d}x" }, { "type": "text", "content": " 都可以使用表格法快速算出" } ] }, "bbox": [ 55, 258, 709, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 " }, { "type": "equation_inline", "content": "5.1P_{n}(x)e^{kx}" }, { "type": "text", "content": " 型 计算 " }, { "type": "equation_inline", "content": "\\int(x^{3}+2x+6)e^{2x}\\mathrm{d}x" } ] }, "bbox": [ 58, 297, 443, 325 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 先列表格:" } ] }, "bbox": [ 90, 324, 191, 338 ] }, { "type": "table", "content": { "image_source": { "path": "images/24317722ca95f84eab9df580854fba2241192cc4a0c7ae0edeb80463a7a22e33.jpg" }, "table_caption": [], "table_footnote": [], "html": "
$x^{3} + 2x + 6$ $3x^{2} + 2$ $6x$ $6$ $0$
$e^{2x}$ $\\frac{1}{2}e^{2x}$ $\\frac{1}{4}e^{2x}$ $\\frac{1}{8}e^{2x}$ $\\frac{1}{16}e^{2x}$
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 317, 347, 678, 392 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到结果为 " }, { "type": "equation_inline", "content": "(x^{3} + 2x + 6)\\frac{1}{2} e^{2x} - (3x^{2} + 2)\\frac{1}{4} e^{2x} + 6x\\left(\\frac{1}{8} e^{2x}\\right) - 6\\left(\\frac{1}{16} e^{2x}\\right) + \\int 0\\cdot \\left(\\frac{1}{16} e^{2x}\\right)\\mathrm{d}x" } ] }, "bbox": [ 90, 399, 749, 429 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.3 (积分有理分解)" } ], "level": 2 }, "bbox": [ 124, 439, 302, 455 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若被积函数为 " }, { "type": "equation_inline", "content": "R(x)=\\frac{P(x)}{Q(x)}" }, { "type": "text", "content": " ，且 " }, { "type": "equation_inline", "content": "Q(x)" }, { "type": "text", "content": " 的次数大于等于 3，若 " }, { "type": "equation_inline", "content": "Q(x)=(x-a_{1})^{\\lambda_{1}}\\cdots(x-a_{k})^{\\lambda_{k}}(x^{2}+p_{1}x+q_{1})^{\\mu_{1}}\\cdots(x^{2}+p_{l}x+q_{l})^{\\mu_{l}}" }, { "type": "text", "content": " ，对于 " }, { "type": "equation_inline", "content": "(x-a_{1})^{\\lambda_{1}}" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "(x^{2}+p_{1}x+q_{1})^{\\mu_{1}}" }, { "type": "text", "content": " 项，对应展开式为：" } ] }, "bbox": [ 114, 461, 880, 506 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\frac {A _ {1}}{x - a _ {1}} + \\frac {A _ {2}}{(x - a _ {1}) ^ {2}} + \\dots + \\frac {A _ {\\lambda_ {1}}}{(x - a) ^ {\\lambda_ {1}}} \\\\ \\frac {B _ {1} x + C _ {1}}{x ^ {2} + p _ {1} x + q _ {1}} + \\dots + \\frac {B _ {\\mu_ {1}} x + C _ {\\mu_ {1}}}{(x ^ {2} + p _ {1} x + q _ {1}) ^ {\\mu_ {1}}} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/aad04b14b1cc2b4e07b3ff3642ab5b0550b6271397584f9eec3dffd99760e559.jpg" } }, "bbox": [ 332, 508, 643, 565 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "其中 " }, { "type": "equation_inline", "content": "\\frac{A_{\\lambda_l}}{(x - a)^{\\lambda_l}}" }, { "type": "text", "content": " 的积分好算，而 " }, { "type": "equation_inline", "content": "\\frac{B_{\\mu_l}x + C_{\\mu_l}}{(x^2 + p_1x + q_1)^{\\mu_l}}" }, { "type": "text", "content": " 可配方化为 " }, { "type": "equation_inline", "content": "\\int \\frac{Lu + N}{(u^2 + r^2)^j}\\mathrm{d}u" }, { "type": "text", "content": " (因为判别式一定小于0)，第一部分 " }, { "type": "equation_inline", "content": "L\\int \\frac{u}{(u^2 + r^2)^j}\\mathrm{d}u" }, { "type": "text", "content": " 好算，第二部分 " }, { "type": "equation_inline", "content": "N\\int \\frac{1}{(u^2 + r^2)^j}\\mathrm{d}u" }, { "type": "text", "content": " 用 " }, { "type": "equation_inline", "content": "u = r\\tan v" }, { "type": "text", "content": " 替换，变为形如 " }, { "type": "equation_inline", "content": "\\int \\cos^k vv\\mathrm{d}v" }, { "type": "text", "content": " 的不定积分，有递推公式辅助求解。" } ] }, "bbox": [ 114, 567, 882, 640 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.4 (三角有理式转化为有理式)" } ], "level": 2 }, "bbox": [ 122, 683, 386, 699 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "R(\\sin x, \\cos x)" }, { "type": "text", "content": " 为关于 " }, { "type": "equation_inline", "content": "\\sin x, \\cos x" }, { "type": "text", "content": " 的有理式，记 " }, { "type": "equation_inline", "content": "t = \\tan \\frac{x}{2}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "x = 2\\arctan t, \\mathrm{d}x = \\frac{2}{1 + t^2}\\mathrm{d}t" }, { "type": "text", "content": "，以及万能公式" } ] }, "bbox": [ 114, 703, 880, 741 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin x = \\frac {2 t}{1 + t ^ {2}}, \\cos x = \\frac {1 - t ^ {2}}{1 + t ^ {2}}, \\tan x = \\frac {2 t}{1 - t ^ {2}}", "math_type": "latex", "image_source": { "path": "images/be8f0e79a3d36eb771499ab9cc7dc73da9ae7023415dc28575b3f2377e8369bb.jpg" } }, "bbox": [ 329, 739, 662, 771 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此" } ] }, "bbox": [ 115, 776, 154, 790 ] }, { "type": "equation_interline", "content": { "math_content": "\\int R (\\sin x, \\cos x) \\mathrm{d} x = \\int R \\left(\\frac {2 t}{1 + t ^ {2}}, \\frac {1 - t ^ {2}}{1 + t ^ {2}}\\right) \\frac {2}{1 + t ^ {2}} \\mathrm{d} t", "math_type": "latex", "image_source": { "path": "images/59fa2bdd25f659ad8a520d9e65e82aa982c2375d4c8d3d399a258bc228ff67ce.jpg" } }, "bbox": [ 300, 787, 695, 819 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.2 不定积分基本方法" } ] }, "bbox": [ 737, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "44" } ] }, "bbox": [ 489, 954, 509, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.5 (两类无理根式的不定积分)" } ], "level": 2 }, "bbox": [ 122, 86, 388, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "\\int R\\left(x,\\sqrt[n]{\\frac{ax + b}{cx + d}}\\right)\\mathrm{d}x" }, { "type": "text", "content": " ，只需要令 " }, { "type": "equation_inline", "content": "t = \\sqrt[n]{\\frac{ax + b}{cx + d}}" }, { "type": "text", "content": " 即可化为有理函数。对于 " }, { "type": "equation_inline", "content": "\\int R(x,\\sqrt{ax^2 + bx + c})" }, { "type": "text", "content": " 要么进行配方，要么用Euler变换。" } ] }, "bbox": [ 112, 107, 885, 159 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ], "level": 2 }, "bbox": [ 92, 200, 312, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下面假设必背部分都会背，且不做计算证明" } ] }, "bbox": [ 124, 235, 448, 250 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.1 三角函数：重点" } ], "level": 2 }, "bbox": [ 92, 273, 292, 290 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "练习 5.2 核心会算" } ], "level": 2 }, "bbox": [ 58, 300, 226, 315 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} \\int \\tan x \\mathrm{d} x = - \\ln | \\cos x | + c & \\int \\cot x \\mathrm{d} x = \\ln | \\sin x | + c \\\\ \\int \\sec x \\mathrm{d} x = \\ln | \\sec x + \\tan x | + c & \\int \\csc x \\mathrm{d} x = - \\ln | \\csc x + \\cot x | + c \\end{array}", "math_type": "latex", "image_source": { "path": "images/2034ce7cd42cc884f511eed722d0a7a1db38568f2a9029cd8d74284d7ee299ef.jpg" } }, "bbox": [ 235, 318, 761, 372 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 " }, { "type": "equation_inline", "content": "\\int \\tan x\\mathrm{d}x = -\\int \\frac{\\mathrm{d}\\cos x}{\\cos x},\\int \\cot x\\mathrm{d}x" }, { "type": "text", "content": " 同理。另外两个要记住" } ] }, "bbox": [ 87, 376, 563, 406 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\sec x \\mathrm{d} x = \\int \\frac {\\sec x (\\sec x + \\tan x)}{\\sec x + \\tan x} \\mathrm{d} x = \\int \\frac {\\mathrm{d} (\\sec x + \\tan x)}{\\sec x + \\tan x} = \\ln | \\sec x + \\tan x | + c", "math_type": "latex", "image_source": { "path": "images/25e6b195b2929f6a8b0c67c376f6050740bb0379bb47ab83b2ecae210afa986e.jpg" } }, "bbox": [ 191, 407, 803, 439 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.3简单平方(1) " }, { "type": "equation_inline", "content": "\\int \\sin^2 x\\mathrm{d}x(2)" }, { "type": "text", "content": " 重点： " }, { "type": "equation_inline", "content": "\\int \\tan^2 x\\mathrm{d}x" }, { "type": "text", "content": " (3)超重点： " }, { "type": "equation_inline", "content": "\\int \\sec^3 x\\mathrm{d}x" }, { "type": "text", "content": " 解(1)直接用半角公式" } ] }, "bbox": [ 58, 449, 663, 492 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 这个要记住， " }, { "type": "equation_inline", "content": "\\int \\tan^2 x \\mathrm{d}x = \\int \\sec^2 x - 1 \\mathrm{d}x = \\tan x - x + c" } ] }, "bbox": [ 121, 489, 586, 519 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 这个经常会用到!" } ] }, "bbox": [ 122, 518, 280, 533 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{c} \\int \\sec^ {3} x \\mathrm{d} x = \\int \\sec x \\mathrm{d} \\tan x = \\sec x \\tan x - \\int \\tan^ {2} x \\sec x \\mathrm{d} x \\\\ = \\sec x \\tan x - \\int \\sec^ {3} x \\mathrm{d} x + \\int \\sec x \\mathrm{d} x \\end{array}", "math_type": "latex", "image_source": { "path": "images/9de5065a87efa57dbf01637397cb53f6653ceb7489f2d7b92ae260d919511cdd.jpg" } }, "bbox": [ 273, 533, 721, 595 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "I = \\frac{1}{2}\\sec x\\tan x + \\ln |\\sec x + \\tan x| + c" } ] }, "bbox": [ 90, 598, 423, 626 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 " }, { "type": "equation_inline", "content": "5.4\\sin x, \\cos x" }, { "type": "text", "content": " 加常数分式积分 (1) 重点: " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{1 \\pm \\cos x}" }, { "type": "text", "content": " (2) 重点: " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{1 \\pm \\sin x}" } ] }, "bbox": [ 58, 633, 678, 664 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 极度重点，含参积分大量用到： " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{1 + a\\cos x}, |a| \\leq 1" } ] }, "bbox": [ 122, 663, 544, 690 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)根据 " }, { "type": "equation_inline", "content": "1+\\cos x=2\\cos^{2}\\frac{x}{2},1-\\cos x=2\\sin^{2}\\frac{x}{2}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 89, 689, 522, 714 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\int \\frac {\\mathrm{d} x}{1 + \\cos x} = \\frac {1}{2} \\int \\sec^ {2} \\frac {x}{2} \\mathrm{d} x = \\tan \\frac {x}{2} + c \\\\ \\int \\frac {\\mathrm{d} x}{1 - \\cos x} = \\frac {1}{2} \\int \\csc^ {2} \\frac {x}{2} \\mathrm{d} x = - \\cot \\frac {x}{2} + c \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/52f405c59f12f030ea2b90df0841ebc549c43d579558a7ae75f442ae4fa193b3.jpg" } }, "bbox": [ 324, 715, 657, 772 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据诱导公式 " }, { "type": "equation_inline", "content": "\\sin x = \\cos \\left(\\frac{\\pi}{2} - x\\right)" } ] }, "bbox": [ 122, 776, 401, 799 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\int \\frac {\\mathrm{d} x}{1 + \\sin x} = \\int \\frac {\\mathrm{d} x}{1 + \\cos \\left(\\frac {\\pi}{2} - x\\right)} = - \\tan \\left(\\frac {\\pi}{4} - \\frac {x}{2}\\right) + c \\\\ \\int \\frac {\\mathrm{d} x}{1 - \\sin x} = \\int \\frac {\\mathrm{d} x}{1 - \\cos \\left(\\frac {\\pi}{2} - x\\right)} = \\cot \\left(\\frac {\\pi}{4} - \\frac {x}{2}\\right) + c \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/12c4eeacb05dc5bce4734b40c4f17cbfc77c5b587fed81fed505c3fc98916d87.jpg" } }, "bbox": [ 277, 802, 702, 863 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "45" } ] }, "bbox": [ 489, 954, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "a = \\pm 1" }, { "type": "text", "content": " 时即(1)。用万能公式，令 " }, { "type": "equation_inline", "content": "t = \\tan \\frac{x}{2}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\mathrm{dx} = \\frac{2}{1 + t^2}\\mathrm{dt}" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 122, 80, 663, 111 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int \\frac {1}{1 + a \\frac {1 - t ^ {2}}{1 + t ^ {2}}} \\cdot \\frac {2}{1 + t ^ {2}} \\mathrm{d} t = \\int \\frac {2}{(1 + t ^ {2}) + a (1 - t ^ {2})} \\mathrm{d} t = \\int \\frac {2}{(1 - a) t ^ {2} + (1 + a)} \\mathrm{d} t = \\frac {2}{1 - a} \\sqrt {\\frac {1 - a}{1 + a}} \\arctan \\left(\\sqrt {\\frac {1 - a}{1 + a}} t\\right) + c", "math_type": "latex", "image_source": { "path": "images/598c73f09de99910b8fc612dbbcf5a81087828ff55f3d900eda8ab87f0d4d156.jpg" } }, "bbox": [ 90, 114, 971, 149 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到结果为 " }, { "type": "equation_inline", "content": "I = \\frac{2}{\\sqrt{1 - a^2}}\\arctan \\left(\\sqrt{\\frac{1 - a}{1 + a}}\\tan \\frac{x}{2}\\right) + c" } ] }, "bbox": [ 90, 155, 490, 192 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.6 ( " }, { "type": "equation_inline", "content": "\\frac{1}{a\\sin^{2}x + b\\cos^{2}x}" }, { "type": "text", "content": " 积分)" } ], "level": 2 }, "bbox": [ 121, 225, 371, 255 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "形如 " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{a\\sin^2x + b\\cos^2x}" }, { "type": "text", "content": " 的积分都采用上下同除以 " }, { "type": "equation_inline", "content": "\\cos^2 x" }, { "type": "text", "content": " ，转换为" } ] }, "bbox": [ 114, 259, 608, 288 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {\\sec^ {2} x \\mathrm{d} x}{a \\tan^ {2} x + b}}", "math_type": "latex", "image_source": { "path": "images/da5b9988f52cc83a6909f498866dce54609cc3bdb6d3dd47ebb37ca0a2dbe88e.jpg" } }, "bbox": [ 440, 291, 554, 323 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "直接用 " }, { "type": "equation_inline", "content": "\\mathrm{d}(\\tan x) = \\sec^2 x\\mathrm{d}x" }, { "type": "text", "content": " 即可。" } ] }, "bbox": [ 117, 324, 361, 341 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理5.7 " }, { "type": "equation_inline", "content": "\\left(\\frac{1}{a\\sin^2x + b\\sin x\\cos x + c\\cos^2x + d}\\right." }, { "type": "text", "content": " 积分）" } ], "level": 2 }, "bbox": [ 121, 367, 500, 397 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "形如 " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{a\\sin^2x + b\\sin x\\cos x + c\\cos^2x + d}" }, { "type": "text", "content": " 的积分首先上下同除以 " }, { "type": "equation_inline", "content": "\\cos^2 x" }, { "type": "text", "content": " ，得到" } ] }, "bbox": [ 114, 400, 705, 430 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {\\sec^ {2} x \\mathrm{d} x}{a \\tan^ {2} x + b \\tan x + c + d \\sec^ {2} x}}", "math_type": "latex", "image_source": { "path": "images/728b9ff4a0fb1547036ff61e79c659a7acd871bab9989d5b2681e147545cb660.jpg" } }, "bbox": [ 369, 432, 623, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "再利用 " }, { "type": "equation_inline", "content": "\\sec^2 x = \\tan^2 x + 1" }, { "type": "text", "content": " ，全部转换为 " }, { "type": "equation_inline", "content": "\\tan x" }, { "type": "text", "content": " ，再分母配方即可" } ] }, "bbox": [ 114, 464, 586, 482 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 " }, { "type": "equation_inline", "content": "5.5\\sin x, \\cos x" }, { "type": "text", "content": " 无常数、无分子分式 (1) 重点: " }, { "type": "equation_inline", "content": "\\int \\frac{dx}{\\sin^{2}x + 2\\cos^{2}x}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int \\frac{1}{\\sin^{2}x\\cos x} dx" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int \\frac{dx}{\\sin x\\cos^{2}x}" }, { "type": "text", "content": " \n解(1)同除以 " }, { "type": "equation_inline", "content": "\\cos^{2}x" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "\\int\\frac{\\sec^{2}x\\mathrm{d}x}{2+\\tan^{2}x}" }, { "type": "text", "content": " ，易算" } ] }, "bbox": [ 58, 512, 873, 570 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 同乘 " }, { "type": "equation_inline", "content": "\\cos x" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "I = \\int \\frac{\\mathrm{d}(\\sin x)}{\\sin^2 x (1 - \\sin^2 x)}" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "\\int \\frac{1}{\\sin^2 x} \\mathrm{d}(\\sin x) + \\int \\frac{1}{1 - \\sin^2 x} \\mathrm{d}x" }, { "type": "text", "content": "，前者直接算，后者用 " }, { "type": "equation_inline", "content": "\\int \\frac{1}{x^2 - a} \\mathrm{d}x = \\frac{1}{2a} \\ln \\left|\\frac{x - a}{x + a}\\right|" }, { "type": "text", "content": "。\n(3) 与 (2) 同理" } ] }, "bbox": [ 87, 568, 905, 644 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 5.6sin x, cos x 的高次 (1) " }, { "type": "equation_inline", "content": "\\int(\\sin^{4}x + \\cos^{4}x)\\mathrm{d}x" }, { "type": "text", "content": " (2) 重点: " }, { "type": "equation_inline", "content": "\\int\\frac{\\sin x\\cos x}{\\sin^{4}x + \\cos^{4}x}\\mathrm{d}x" }, { "type": "text", "content": " (3) 重点: " }, { "type": "equation_inline", "content": "\\int\\frac{\\mathrm{d}x}{\\sin^{4}x + \\cos^{4}x}" }, { "type": "text", "content": " \n\n(4) " }, { "type": "equation_inline", "content": "\\int\\sin^{6}x + \\cos^{6}x\\mathrm{d}x" }, { "type": "text", "content": " (5) " }, { "type": "equation_inline", "content": "\\int\\frac{\\mathrm{d}x}{\\sin^{6}x + \\cos^{6}x}" } ] }, "bbox": [ 58, 657, 890, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)降到一次： " }, { "type": "equation_inline", "content": "\\sin^4 x + \\cos^4 x = (\\sin^2 x + \\cos^2 x)^2 - 2\\sin^2 x\\cos^2 x = 1 - \\frac{1}{2}\\sin^2 2x = 1 - \\frac{1}{4}(1 - \\cos 4x)" } ] }, "bbox": [ 89, 714, 853, 741 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 降到二次，并切换 " }, { "type": "equation_inline", "content": "\\sin x, \\cos x: \\int \\frac{\\frac{1}{2} \\sin 2x}{1 - \\frac{1}{2} \\sin^2 2x}" }, { "type": "text", "content": "，要善于切换正余弦：" } ] }, "bbox": [ 122, 739, 657, 772 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {\\sin 2 x}{2 - \\sin^ {2} 2 x}} \\mathrm{d} x = - \\frac {1}{2} \\int {\\frac {\\mathrm{d} (\\cos 2 x)}{1 + \\cos^ {2} 2 x}} = - \\frac {1}{2} \\arctan (\\cos 2 x) + c", "math_type": "latex", "image_source": { "path": "images/b0964b4540fe08a412d358bfcda1d94fed3813eb9d9e5704541acf603d07e228.jpg" } }, "bbox": [ 265, 777, 731, 807 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 直接降到二次，并转换为 " }, { "type": "equation_inline", "content": "\\tan x, \\sec x" }, { "type": "text", "content": "：使用 " }, { "type": "equation_inline", "content": "\\sec^2 x = 1 + \\tan^2 x" } ] }, "bbox": [ 122, 810, 613, 825 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int {\\frac {\\mathrm{d} x}{1 - \\frac {1}{2} \\sin^ {2} 2 x}} = \\int {\\frac {\\sec^ {2} 2 x}{\\sec^ {2} 2 x - \\frac {1}{2} \\tan^ {2} 2 x}} = \\frac {1}{2} \\int {\\frac {\\mathrm{d} (\\tan 2 x)}{1 + \\frac {1}{2} \\tan^ {2} 2 x}} = \\frac {1}{\\sqrt {2}} \\arctan {\\frac {\\tan 2 x}{\\sqrt {2}}} + c", "math_type": "latex", "image_source": { "path": "images/653b20f1c47de1b0b62c94e396f5732a6faac6543c782da39cdc7f0a96e75ba8.jpg" } }, "bbox": [ 163, 828, 833, 860 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "\\sin^6 x + \\cos^6 x = (\\sin^2 x + \\cos^2 x)(\\sin^4 - \\sin^2 \\cos^2 x + \\cos^4 x)" }, { "type": "text", "content": " 转化为四次，再配方：" } ] }, "bbox": [ 122, 863, 751, 881 ] }, { "type": "equation_interline", "content": { "math_content": "(\\sin^ {2} x + \\cos^ {2} x) ^ {2} - 3 \\sin^ {2} x \\cos^ {2} x = 1 - \\frac {3}{4} \\sin^ {2} 2 x = 1 - \\frac {3}{8} (1 - \\cos 4 x)", "math_type": "latex", "image_source": { "path": "images/f525b3686a4f572c3d048aee8bed59553082195bfdb28d1d788117839bc256da.jpg" } }, "bbox": [ 240, 883, 754, 912 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "46" } ] }, "bbox": [ 489, 954, 509, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "结果为 " }, { "type": "equation_inline", "content": "\\frac{5}{8} x + \\frac{3}{32}\\sin 4x + c" } ] }, "bbox": [ 90, 80, 289, 109 ] }, { "type": "equation_interline", "content": { "math_content": "(5) I = \\int {\\frac {1}{1 - \\frac {3}{4} \\sin^ {2} 2 x}} \\mathrm{d} x = \\int {\\frac {\\sec^ {2} x}{\\sec^ {2} 2 x - \\frac {3}{4} \\tan^ {2} 2 x}} \\mathrm{d} x = 2 \\int {\\frac {\\mathrm{d} (\\tan 2 x)}{\\tan^ {2} 2 x + 4}}", "math_type": "latex", "image_source": { "path": "images/4ba99007853ff0aaced16ad6ec2116a23ac00ed33f66946e49a7c2102d8f0d30.jpg" } }, "bbox": [ 122, 107, 653, 141 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 高次问题，如果分母为 1 则降到一次直接积分，分子为 1 则降到二次转换为 " }, { "type": "equation_inline", "content": "\\sec, \\tan" }, { "type": "text", "content": "，均不为 1 则降到齐次转化为有理式。降次要用立方和公式，要么用配方。" } ] }, "bbox": [ 87, 143, 905, 179 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.8 ( " }, { "type": "equation_inline", "content": "\\frac{a\\sin x + b\\cos x}{c\\sin x + d\\cos x}" }, { "type": "text", "content": " 型计算技巧)" } ], "level": 2 }, "bbox": [ 122, 205, 405, 234 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\frac{a\\sin x + b\\cos x}{c\\sin x + d\\cos x}" }, { "type": "text", "content": " 型的积分把分子写为 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 倍分母加上 " }, { "type": "equation_inline", "content": "B" }, { "type": "text", "content": " 倍分母导数的形式，用待定系数法进行计算" } ] }, "bbox": [ 115, 237, 833, 271 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 5.1 ( " }, { "type": "equation_inline", "content": "\\tan x, \\cot x" }, { "type": "text", "content": " 分式型)" } ], "level": 2 }, "bbox": [ 121, 290, 337, 306 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "类似于 " }, { "type": "equation_inline", "content": "\\int \\frac{a\\mathrm{d}x}{b + c\\tan x},\\int \\frac{a\\mathrm{d}x}{b + c\\cot x}" }, { "type": "text", "content": " 都可以转换回 " }, { "type": "equation_inline", "content": "\\frac{a\\sin x + b\\cos x}{c\\sin x + d\\cos x}" }, { "type": "text", "content": " 型积分进行计算" } ] }, "bbox": [ 115, 309, 721, 342 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 5.7 " }, { "type": "equation_inline", "content": "\\frac{a\\sin x+b\\cos x}{c\\sin x+d\\cos x}" }, { "type": "text", "content": " 型计算 (1) 重点: " }, { "type": "equation_inline", "content": "\\int\\frac{\\sin x}{\\sin x+\\cos x}\\mathrm{d}x" }, { "type": "text", "content": " (2) 重点: " }, { "type": "equation_inline", "content": "\\int\\frac{\\mathrm{d}x}{3+5\\tan x}" } ] }, "bbox": [ 58, 370, 712, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)设 " }, { "type": "equation_inline", "content": "\\sin x = A(\\sin x + \\cos x) + B(\\cos x - \\sin x)" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "A = \\frac{1}{2}, B = -\\frac{1}{2}" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 87, 400, 687, 425 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int {\\frac {{\\frac {1}{2}} (\\sin x + \\cos x) - {\\frac {1}{2}} \\mathrm{d} (\\sin x + \\cos x)}{\\sin x + \\cos x}} \\mathrm{d} x = {\\frac {1}{2}} x - {\\frac {1}{2}} \\ln | \\sin x + \\cos x | + c", "math_type": "latex", "image_source": { "path": "images/81fb8777fc4e06da81df25063d18b79fe1edc41e14a677f1938d7da9f0840624.jpg" } }, "bbox": [ 220, 428, 776, 460 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于 " }, { "type": "equation_inline", "content": "\\frac{1}{3 + 5\\tan x} = \\frac{\\cos x}{5\\sin x + 3\\cos x}" }, { "type": "text", "content": "，设 " }, { "type": "equation_inline", "content": "\\cos x = A(5\\sin x + 3\\cos x) + B(5\\cos x - 3\\sin x)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "5A = 3B, 3A + 5B = 1" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "A = \\frac{3}{34}, B = \\frac{5}{34}" }, { "type": "text", "content": "，得到" } ] }, "bbox": [ 87, 463, 905, 516 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {3}{3 4} x + \\frac {5}{3 4} \\ln | 5 \\sin x + 3 \\cos x | + c", "math_type": "latex", "image_source": { "path": "images/8ffeff5ed13030487f83a1bc162e7d1296d37f22e5fefb419aaa0563aa586a17.jpg" } }, "bbox": [ 359, 517, 638, 546 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 5.9 ( " }, { "type": "equation_inline", "content": "\\frac{1}{a\\sin x + b\\cos x + c}" }, { "type": "text", "content": " 积分)" } ], "level": 2 }, "bbox": [ 122, 571, 381, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "类似于 " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{a\\sin x + b\\cos x + c}" }, { "type": "text", "content": " 的积分采用二倍角公式转换为二次，得到" } ] }, "bbox": [ 114, 604, 642, 633 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {1}{2 a \\sin {\\frac {x}{2}} \\cos {\\frac {x}{2}} + b (2 \\cos^ {2} {\\frac {x}{2}} - 1) + c}} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/d202b259beb32491a4668f73fba491ece10713352c218b46dc8f691b8218e452.jpg" } }, "bbox": [ 349, 636, 647, 668 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "使用前面的积分方法，上下同时除以 " }, { "type": "equation_inline", "content": "\\cos^2\\frac{x}{2}" }, { "type": "text", "content": "，再根据 " }, { "type": "equation_inline", "content": "\\sec^2 x = \\tan^2 x + 1" }, { "type": "text", "content": " 进行配方即可。" } ] }, "bbox": [ 112, 670, 759, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 5.8 相关训练 (1) " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{\\sin x+2\\cos x+3}" } ] }, "bbox": [ 58, 725, 403, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\sin x + 2\\cos x + 3 = 1 + 2\\sin \\frac{x}{2}\\cos \\frac{x}{2} +4\\cos^2{\\frac{x}{2}}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 89, 750, 534, 778 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int {\\frac {\\sec^ {2} {\\frac {x}{2}}}{4 + 2 \\tan {\\frac {x}{2}} + \\sec^ {2} x}} = 2 \\int {\\frac {d (\\tan {\\frac {x}{2}} + 1)}{(\\tan {\\frac {x}{2}} + 1) ^ {2} + 4}} = \\arctan {\\frac {\\tan {\\frac {x}{2}} + 1}{2}} + c", "math_type": "latex", "image_source": { "path": "images/8643637a6f2c64949d1b580989a8a7618b99ae40bc9c1cb5d13befbd364f5ac0.jpg" } }, "bbox": [ 225, 778, 773, 813 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 5.2 ( " }, { "type": "equation_inline", "content": "\\frac{p\\sin x + q\\cos x + s}{a\\sin x + b\\cos x + c}dx" }, { "type": "text", "content": " )" } ], "level": 2 }, "bbox": [ 122, 835, 364, 864 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这类积分可以将分子写为 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 倍分母加 " }, { "type": "equation_inline", "content": "B" }, { "type": "text", "content": " 倍分母导数加 " }, { "type": "equation_inline", "content": "C" }, { "type": "text", "content": " 常数，其中常数项用前一个定理进行计算。" } ] }, "bbox": [ 114, 866, 845, 884 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "47" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.9三角根号 (1) " }, { "type": "equation_inline", "content": "\\int \\sin x\\cos x\\sqrt{a^2\\sin^2x + b^2\\cos^2x}\\mathrm{d}x" } ] }, "bbox": [ 58, 82, 519, 110 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)写为 " }, { "type": "equation_inline", "content": "\\frac{1}{2}\\int \\sqrt{(a^2 - b^2)\\sin^2x + b^2}\\mathrm{d}(\\sin^2 x)" }, { "type": "text", "content": " 即可" } ] }, "bbox": [ 87, 109, 475, 139 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.2 对数与指数" } ], "level": 2 }, "bbox": [ 92, 158, 253, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 5.10 几道经典指数积分 (1) " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{e^{x}+e^{-x}}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{\\sqrt{1+e^{2x}}}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int e^{x}\\left(\\frac{1-x}{1+x^{2}}\\right)^{2}dx" } ] }, "bbox": [ 58, 186, 695, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)本质是 " }, { "type": "equation_inline", "content": "e^x" }, { "type": "text", "content": " 的有理函数，故 " }, { "type": "equation_inline", "content": "I = \\int \\frac{\\mathrm{d}(e^x)}{e^{2x} + 1} = \\arctan (e^x) + c" } ] }, "bbox": [ 87, 219, 573, 246 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 同乘 " }, { "type": "equation_inline", "content": "e^{-x}" }, { "type": "text", "content": "，故 " }, { "type": "equation_inline", "content": "I = -\\int \\frac{\\mathrm{d}(e^{-x})}{\\sqrt{e^{-2x} + 1}} = -\\ln |e^{-x} + \\sqrt{e^{-2x} + 1}| + c" } ] }, "bbox": [ 121, 246, 621, 277 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "I=\\int\\frac{e^{x}}{1+x^{2}}dx-\\int\\frac{e^{x}}{(1+x^{2})^{2}}d(x^{2}+1)" }, { "type": "text", "content": " ，第一项实在是没法处理，只能期待第二项，第二项等于 " }, { "type": "equation_inline", "content": "\\int e^{x}d(\\frac{1}{1+x^{2}})" }, { "type": "text", "content": " ，因此用分部积分得到" } ] }, "bbox": [ 90, 275, 934, 317 ] }, { "type": "equation_interline", "content": { "math_content": "\\int {\\frac {e ^ {x}}{1 + x ^ {2}}} \\mathrm{d} x + \\int e ^ {x} \\mathrm{d} (\\frac {1}{1 + x ^ {2}}) = \\frac {e ^ {x}}{1 + x ^ {2}} + c", "math_type": "latex", "image_source": { "path": "images/717d4da27ee349b97027538143d37aedc711834882c820bd1490ef91ad13e838.jpg" } }, "bbox": [ 334, 313, 662, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.11几道经典对数积分 (1) " }, { "type": "equation_inline", "content": "\\int x^n\\ln x\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int \\left(\\frac{\\ln x}{x}\\right)^2\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int \\sqrt{x}\\ln^2 x\\mathrm{d}x" }, { "type": "text", "content": " (4) 特殊： " }, { "type": "equation_inline", "content": "\\int \\frac{1}{1 - x^2}\\ln \\frac{1 + x}{1 - x}\\mathrm{d}x" }, { "type": "text", "content": " 解(1)列表法得到： " }, { "type": "equation_inline", "content": "I = \\frac{x^{n + 1}\\ln x}{n + 1} -\\int \\frac{x^n}{n + 1}\\mathrm{d}x = \\frac{x^{n + 1}}{1 + n} (\\ln x - \\frac{1}{n + 1}) + c" } ] }, "bbox": [ 58, 359, 892, 424 ] }, { "type": "table", "content": { "image_source": { "path": "images/3623663f8cdefaa2ef062ce586721f529a462452fc11c7907f2ac2f7c13542c8.jpg" }, "table_caption": [], "table_footnote": [], "html": "
$ \\ln x $$ \\frac{1}{x} $
$ x^{n} $$ \\frac{1}{n+1}x^{n+1} $
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 423, 431, 574, 483 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 无法直接列表，因为中间项会交叉，因此可以用表格法一步一步列，" } ] }, "bbox": [ 121, 492, 645, 506 ] }, { "type": "equation_interline", "content": { "math_content": "I = - \\frac {\\ln^ {2} x}{x} + \\int \\frac {2 \\ln x}{x ^ {2}} \\mathrm{d} x = - \\frac {\\ln^ {2} x}{x} - \\frac {2 \\ln x}{x} + \\int \\frac {2}{x ^ {2}} \\mathrm{d} x = - \\frac {1}{x} (\\ln^ {2} x + 2 \\ln x + 2) + c", "math_type": "latex", "image_source": { "path": "images/c9bb6bf76f8cc4c836e510beba5c41dc46ba9990fee73729ffea0a1773e396ef.jpg" } }, "bbox": [ 188, 510, 806, 539 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 同理用表格法一步一步列，" } ] }, "bbox": [ 122, 544, 344, 558 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {2}{3} x ^ {\\frac {3}{2}} \\ln^ {2} x - \\frac {4}{3} \\int x ^ {\\frac {1}{2}} \\ln x \\mathrm{d} x = \\frac {2}{3} x ^ {\\frac {3}{2}} \\ln^ {2} x - \\frac {4}{3} \\left(\\frac {2}{3} \\ln x x ^ {\\frac {3}{2}} - \\frac {2}{3} \\int x ^ {\\frac {1}{2}} \\mathrm{d} x\\right)", "math_type": "latex", "image_source": { "path": "images/0a462b8a99397914783f9f2b7aacee9c1ba0625f592cf36e9a89a7e187c93c3d.jpg" } }, "bbox": [ 228, 560, 764, 592 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4)一看就是特殊配凑的，要注意到 " }, { "type": "equation_inline", "content": "\\frac{1}{1 - x^2}\\mathrm{d}x = \\frac{1}{2}\\mathrm{d}\\left(\\ln \\frac{1 + x}{1 - x}\\right)" } ] }, "bbox": [ 121, 595, 589, 626 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.3 分式问题" } ], "level": 2 }, "bbox": [ 92, 645, 233, 662 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.12几个基础型(1) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{ax^2 + b} (a > 0)" }, { "type": "text", "content": " (2)重点： " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{(ax^2 + b)^2}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{(ax^2 + b)^n}" } ] }, "bbox": [ 58, 671, 705, 703 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 重点: " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{ax^2 + bx + c}" }, { "type": "text", "content": " (5) 重点: " }, { "type": "equation_inline", "content": "\\int \\frac{x}{ax^2 + bx + c}\\mathrm{d}x" } ] }, "bbox": [ 122, 701, 532, 731 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)原式 " }, { "type": "equation_inline", "content": "= \\frac{1}{a}\\int \\frac{\\mathrm{d}x}{x^2 + \\frac{b}{a}}" }, { "type": "text", "content": " ，需要分类讨论： " }, { "type": "equation_inline", "content": "b > 0" }, { "type": "text", "content": " 时变为 " }, { "type": "equation_inline", "content": "\\frac{1}{a}\\int \\frac{\\mathrm{d}x}{x^2 + (\\sqrt{\\frac{b}{a}})^2} = \\frac{1}{a}\\cdot \\sqrt{\\frac{a}{b}}\\arctan (\\sqrt{\\frac{a}{b}} x)" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "b < 0" }, { "type": "text", "content": " 时变为 " }, { "type": "equation_inline", "content": "\\frac{1}{a}\\int \\frac{\\mathrm{d}x}{x^2 + (\\sqrt{-\\frac{b}{a}})^2}" }, { "type": "text", "content": " ，后面同理" } ] }, "bbox": [ 87, 728, 905, 804 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 原式 " }, { "type": "equation_inline", "content": "= -\\int \\frac{1}{2ax} \\mathrm{d}\\frac{1}{ax^2 + b} = -\\frac{1}{2ax} \\cdot \\frac{1}{ax^2 + b} + \\int \\frac{1}{ax^2 + b} \\mathrm{d}\\frac{1}{2ax} = -\\frac{1}{2ax} \\cdot \\frac{1}{ax^2 + b} - \\int \\frac{1}{ax^2 + b} \\cdot \\frac{1}{2ax^2} \\mathrm{d}x," }, { "type": "text", "content": " 再对 " }, { "type": "equation_inline", "content": "\\frac{1}{2ax^2(ax^2 + b)} = \\frac{\\frac{1}{b}}{2ax^2} + \\frac{-\\frac{1}{2b}}{ax^2 + b}" }, { "type": "text", "content": " 有理分解，代入后全变为基础型1。" } ] }, "bbox": [ 90, 804, 905, 866 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 同理转化为 " }, { "type": "equation_inline", "content": "\\int \\frac{1}{2ax(1 - n)}\\mathrm{d}\\left(\\frac{1}{(ax^2 + b)^{n - 1}}\\right)" }, { "type": "text", "content": "，再用分部积分去做。" } ] }, "bbox": [ 121, 866, 621, 896 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "48" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 原式 " }, { "type": "equation_inline", "content": "= \\frac{1}{a} \\int \\frac{\\mathrm{d}x}{x^2 + \\frac{b}{a} x + \\frac{c}{a}} = \\frac{1}{a} \\int \\frac{\\mathrm{d}x}{(x + \\frac{b}{2a})^2 + \\frac{c}{a} - \\frac{b^2}{4a^2}}" }, { "type": "text", "content": "，将 " }, { "type": "equation_inline", "content": "\\frac{c}{a} - \\frac{b^2}{4a^2}" }, { "type": "text", "content": " 看作整体，变为 (1)" } ] }, "bbox": [ 121, 80, 791, 116 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 原式 = " }, { "type": "equation_inline", "content": "\\frac{1}{2a}\\int\\frac{2ax}{ax^{2}+bx+c}=\\frac{1}{2a}\\int\\frac{2ax+b}{ax^{2}+bx+c}-\\frac{b}{ax^{2}+bx+c}dx=\\frac{1}{2a}\\left(\\int\\frac{\\mathrm{d}(ax^{2}+bx+c)}{ax^{2}+bx+c}-\\int\\frac{b}{ax^{2}+bx+c}\\mathrm{d}x\\right)" }, { "type": "text", "content": " ，变为 (4)." } ] }, "bbox": [ 90, 114, 976, 159 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 5.13 次方和差 尽量记住下面的分解方法:" } ] }, "bbox": [ 58, 180, 433, 193 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{x^3 + 1}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{x^3 - 1}" } ] }, "bbox": [ 122, 193, 310, 221 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{x^{4}-1}" }, { "type": "text", "content": " (4)重点： " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{x^{4}+1}" }, { "type": "text", "content": " (一定要记住！)" } ] }, "bbox": [ 122, 219, 480, 249 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{x^{6}-1}" }, { "type": "text", "content": " (6) " }, { "type": "equation_inline", "content": "\\int\\frac{dx}{x^{6}+1}" } ] }, "bbox": [ 124, 248, 312, 277 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)写为 " }, { "type": "equation_inline", "content": "\\frac{1}{x^3 + 1} = \\frac{1}{3(x + 1)} + \\frac{-x + 2}{3(x^2 - x + 1)}" }, { "type": "text", "content": "，后者给分母配方即可" } ] }, "bbox": [ 89, 273, 603, 305 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 用立方差即可" } ] }, "bbox": [ 122, 303, 252, 317 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 用 " }, { "type": "equation_inline", "content": "\\frac{1}{x^4 - 1} = \\frac{1}{2}\\left(\\frac{1}{x^2 - 1} - \\frac{1}{x^2 + 1}\\right)" } ] }, "bbox": [ 122, 316, 406, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 构造 " }, { "type": "equation_inline", "content": "M = \\int \\frac{x^2 - 1}{x^4 + 1} \\, \\mathrm{d}x, N = \\int \\frac{x^2 + 1}{x^4 + 1}" }, { "type": "text", "content": "，得到" } ] }, "bbox": [ 122, 346, 485, 376 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} M = \\int \\frac {1 - \\frac {1}{x ^ {2}}}{x ^ {2} + \\frac {1}{x ^ {2}}} = \\int \\frac {\\mathrm{d} (x + \\frac {1}{x})}{(x + \\frac {1}{x}) ^ {2} - 2} \\\\ N = \\int \\frac {x ^ {2} + 1}{x ^ {4} + 1} \\mathrm{d} x = \\int \\frac {1 + \\frac {1}{x ^ {2}}}{x ^ {2} + \\frac {1}{x ^ {2}}} \\mathrm{d} x = \\int \\frac {\\mathrm{d} (x - \\frac {1}{x})}{(x - \\frac {1}{x}) ^ {2} + 2} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/ece6fb1258294e38683cdbd48b1b8b33b795fe588ce1860eb671595bbfdc0ddb.jpg" } }, "bbox": [ 287, 379, 690, 445 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此所求即 " }, { "type": "equation_inline", "content": "\\frac{1}{2} (N - M)" } ] }, "bbox": [ 90, 448, 263, 476 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 用平方差 (不要用立方差): " }, { "type": "equation_inline", "content": "\\frac{1}{x^6 - 1} = \\frac{1}{2}\\left(\\frac{1}{x^3 - 1} + \\frac{1}{x^3 + 1}\\right)" }, { "type": "text", "content": ", 转换为 (1)(2)" } ] }, "bbox": [ 122, 470, 682, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(6) 类似四次方的讨论 + 立方和" } ] }, "bbox": [ 122, 500, 356, 513 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} I = \\frac {1}{2} \\int \\frac {x ^ {4} + 1}{x ^ {6} + 1} \\mathrm{d} x - \\frac {1}{2} \\int \\frac {x ^ {4} - 1}{x ^ {6} + 1} \\mathrm{d} x \\\\ = \\frac {1}{2} \\int \\frac {x ^ {2} + (x ^ {4} - x ^ {2} + 1)}{x ^ {6} + 1} \\mathrm{d} x - \\frac {1}{2} \\int \\frac {(x ^ {2} - 1) (x ^ {2} + 1)}{(x ^ {2} + 1) (x ^ {4} - x ^ {2} + 1)} \\mathrm{d} x \\\\ = \\frac {1}{2} \\int \\frac {x ^ {2}}{x ^ {6} + 1} \\mathrm{d} x + \\frac {1}{2} \\int \\frac {\\mathrm{d} x}{x ^ {2} + 1} - \\frac {1}{2} \\int \\frac {x ^ {2} - 1}{x ^ {4} - x ^ {2} + 1} \\mathrm{d} x \\\\ = \\frac {1}{6} \\int \\frac {\\mathrm{d} (x ^ {3})}{(x ^ {3}) ^ {2} + 1} + \\frac {1}{2} \\arctan x - \\frac {1}{2} \\int \\frac {\\mathrm{d} (x + \\frac {1}{x})}{(x + \\frac {1}{x}) ^ {2} + 3} \\mathrm{d} x \\end{array}", "math_type": "latex", "image_source": { "path": "images/a132ed60747b967df966ecf27606dfbaca181d400087573df1bec87b79164208.jpg" } }, "bbox": [ 267, 517, 729, 649 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.14几个经典题 (1) " }, { "type": "equation_inline", "content": "\\int \\frac{x^2}{(1 - x)^{100}}\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{(x + a)^2(x + b)^2}" }, { "type": "text", "content": "解(1)显然待定系数法有理分解不可行,因此用技巧 " }, { "type": "equation_inline", "content": "x^{2}=[(1-x)-1]^{2}" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "\\frac{x^{2}}{(1-x)^{100}}=\\frac{(1-x)^{2}-2(1-x)+1}{(1-x)^{100}}" }, { "type": "text", "content": " ,做换元即可" } ] }, "bbox": [ 58, 657, 559, 690 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 87, 689, 905, 731 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 这种有理分解比较常见，最好记下来，" } ] }, "bbox": [ 122, 736, 428, 750 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {1}{(x + a) ^ {2} (x + b) ^ {2}}} = {\\frac {1}{(b - a) ^ {2}}} \\left({\\frac {1}{x + a}} - {\\frac {1}{x + b}}\\right) ^ {2} = {\\frac {1}{(b - a) ^ {2}}} \\left[ {\\frac {1}{(x + a) ^ {2}}} + {\\frac {1}{(x + b) ^ {2}}} - {\\frac {2}{(x + a) (x + b)}} \\right]", "math_type": "latex", "image_source": { "path": "images/4009bbed28f0ef1d576a56e70e1805760524cff461e5663265c15e789d8441c2.jpg" } }, "bbox": [ 134, 753, 858, 789 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.4 根式问题" } ], "level": 2 }, "bbox": [ 92, 806, 233, 822 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.15三个基础公式的证明(1) " }, { "type": "equation_inline", "content": "\\int \\sqrt{a^2 - x^2}\\mathrm{d}x" }, { "type": "text", "content": " (2)重点： " }, { "type": "equation_inline", "content": "\\int \\sqrt{x^2 + a^2}\\mathrm{d}x" }, { "type": "text", "content": " (3)重点： " }, { "type": "equation_inline", "content": "\\int \\sqrt{x^2 - a^2}\\mathrm{d}x" }, { "type": "text", "content": " 解(1)设 " }, { "type": "equation_inline", "content": "x = a\\sin t" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "I = \\int a\\cos t\\cdot a\\cos t\\mathrm{d}t = \\frac{a^2}{2}\\int (1 + \\cos 2t)\\mathrm{d}t = \\frac{a^2}{2} (t + \\sin t\\cos t) + C" }, { "type": "text", "content": " 。画一个三角形做辅助，角 " }, { "type": "equation_inline", "content": "t" }, { "type": "text", "content": " 对着 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " ，斜边为 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\sin t = \\frac{x}{a},\\cos t = \\frac{\\sqrt{a^2 - x^2}}{a}" }, { "type": "text", "content": " ，带回得到：" } ] }, "bbox": [ 58, 831, 905, 892 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "49" } ] }, "bbox": [ 489, 953, 507, 964 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 90, 83, 648, 114 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {1}{2} (x \\sqrt {a ^ {2} - x ^ {2}} + a ^ {2} \\arcsin {\\frac {x}{a}}) + C", "math_type": "latex", "image_source": { "path": "images/995fa4b2a205cd0c4f4ebbdc54e75dc5a367ceeb6fbc9139bfb305ca2d9f035c.jpg" } }, "bbox": [ 359, 116, 636, 142 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "x = a\\tan t" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "I = \\int a^2\\sec^3 x\\mathrm{d}t = \\frac{a^2}{2} (\\sec x\\tan x) + \\frac{a^2}{2}\\ln |\\tan x + \\sec x|" }, { "type": "text", "content": " （一定要记住 " }, { "type": "equation_inline", "content": "\\sec x,\\sec^3 x" }, { "type": "text", "content": " 的积分，如果一次方都没记住，就死翘翘了。。），因此得到结果为" } ] }, "bbox": [ 89, 143, 907, 189 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {1}{2} (x \\sqrt {x ^ {2} + a ^ {2}} + a ^ {2} \\ln | x + \\sqrt {x ^ {2} + a ^ {2}} |) + C", "math_type": "latex", "image_source": { "path": "images/95154b05e8bc37172e538f8e4db46c32345ce8ebf8169818e633b18c1c4a82f5.jpg" } }, "bbox": [ 341, 191, 655, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 设 " }, { "type": "equation_inline", "content": "x = a \\sec t" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "I = a^2 \\int \\tan^2 t \\sec t \\, \\mathrm{d}t = a^2 \\int \\sec^3 t - \\sec t \\, \\mathrm{d}t" }, { "type": "text", "content": "，背公式以及画三角形得到：" } ] }, "bbox": [ 122, 219, 835, 250 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {a ^ {2}}{2} (\\sec t \\tan t - \\ln | \\tan t + \\sec t |) + C = \\frac {1}{2} (x \\sqrt {x ^ {2} - a ^ {2}} - a ^ {2} \\ln | x + \\sqrt {x ^ {2} - a ^ {2}} |) + C", "math_type": "latex", "image_source": { "path": "images/7c975952934ec1cc1e26a0f2bfe44dbfaca7c6bd3c0d8276c33fadc8e780e231.jpg" } }, "bbox": [ 179, 252, 816, 282 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.16 " }, { "type": "equation_inline", "content": "(x - a)(b - x)" }, { "type": "text", "content": " 型(1) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{\\sqrt{x(1 - x)}}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{\\sqrt{(x - a)(b - x)}}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int \\sqrt{(x - a)(b - x)}\\mathrm{d}x" } ] }, "bbox": [ 58, 294, 763, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)设 " }, { "type": "equation_inline", "content": "x = \\sin^2 t" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "I = \\int \\frac{2\\sin t\\cos t}{\\sin t\\cos t}\\mathrm{d}t = 2t + C" }, { "type": "text", "content": " ，直接带回得到 " }, { "type": "equation_inline", "content": "I = 2\\arcsin \\sqrt{x} +C" } ] }, "bbox": [ 87, 326, 751, 355 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 令 " }, { "type": "equation_inline", "content": "t = x - a" }, { "type": "text", "content": "，则变为 " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}t}{\\sqrt{t(b - a - t)}}" }, { "type": "text", "content": "，令 " }, { "type": "equation_inline", "content": "t = (b - a)\\sin^2 s" }, { "type": "text", "content": " 即可，故" } ] }, "bbox": [ 121, 355, 663, 385 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int {\\frac {2 (b - a) \\sin s \\cos s \\mathrm{d} s}{(b - a) \\sin s \\cos s}} = 2 s = 2 \\arcsin {\\sqrt {\\frac {t}{(b - a)}}} = 2 \\arcsin {\\sqrt {\\frac {x - a}{b - a}}} + C", "math_type": "latex", "image_source": { "path": "images/f09ef1a0bcf3bf8aff6d79df1144ae464371bff7c1a0fbe4b2e3b5616665342d.jpg" } }, "bbox": [ 213, 391, 781, 425 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 同理得到 " }, { "type": "equation_inline", "content": "I = \\int 2(b - a)^2 \\sin^2 s \\cos^2 s \\, \\mathrm{d}s = \\frac{(b - a)^2}{4} \\int (1 - \\cos 4t) \\, \\mathrm{d}t" }, { "type": "text", "content": "，后续略" } ] }, "bbox": [ 122, 430, 710, 462 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.17其他特别经典的根式积分问题(1) " }, { "type": "equation_inline", "content": "\\int \\left(\\sqrt{\\frac{1 + x}{1 - x}} +\\sqrt{\\frac{1 - x}{1 + x}}\\right)\\mathrm{d}x" } ] }, "bbox": [ 58, 468, 621, 505 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\frac {\\mathrm{d} x}{(\\sqrt {x} + 1) (x + 3)}", "math_type": "latex", "image_source": { "path": "images/d64d8b780d4d65dc50a476a2f8b66afedbc528faff09b3e307754082d24edfb9.jpg" } }, "bbox": [ 147, 504, 287, 535 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{x\\sqrt{x^2 - 2x - 3}}" } ] }, "bbox": [ 124, 533, 285, 569 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {\\mathrm{d} x}{x + \\sqrt {x ^ {2} - x + 1}}", "math_type": "latex", "image_source": { "path": "images/694a9835c4c3a63f75fa7139b6a255940601d22d8ab27635d94d51bb5209d649.jpg" } }, "bbox": [ 147, 568, 302, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)让分母一样： " }, { "type": "equation_inline", "content": "\\frac{1 + x}{\\sqrt{1 - x^2}} +\\frac{1 - x}{\\sqrt{1 - x^2}} = \\frac{2}{\\sqrt{1 - x^2}}" }, { "type": "text", "content": " ，直接转换为了 " }, { "type": "equation_inline", "content": "\\arcsin x" }, { "type": "text", "content": " 的积分" } ] }, "bbox": [ 87, 587, 707, 620 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 令 " }, { "type": "equation_inline", "content": "t = \\sqrt{x}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "I = \\int \\frac{2t}{(t + 1)(t^2 + 3)}\\mathrm{d}t = \\int \\frac{A}{t + 1}\\mathrm{d}t + \\int \\frac{Bt + C}{t^2 + 3}\\mathrm{d}t" }, { "type": "text", "content": "，解出 " }, { "type": "equation_inline", "content": "A = -\\frac{1}{2}, B = \\frac{1}{2}, C = \\frac{3}{2}" }, { "type": "text", "content": "，从而\n " }, { "type": "equation_inline", "content": "I = -\\frac{1}{2}\\ln |t + 1| + \\int \\frac{1}{4}\\ln |t^2 + 3| + \\frac{\\sqrt{3}}{2}\\arctan \\frac{t}{\\sqrt{3}} + C" } ] }, "bbox": [ 90, 615, 902, 690 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3.1) 方法 1: 使用 Euler 变换。设 " }, { "type": "equation_inline", "content": "\\sqrt{x^2 - 2x + 3} = x - t" }, { "type": "text", "content": "，则得到 " }, { "type": "equation_inline", "content": "x = \\frac{t^2 + 3}{2(t - 1)}" }, { "type": "text", "content": "，进而 " }, { "type": "equation_inline", "content": "\\mathrm{d}x = \\frac{t^2 - 2t + 3}{2(t - 1)^2}\\mathrm{d}t" }, { "type": "text", "content": "，此时积分变为" } ] }, "bbox": [ 89, 695, 905, 739 ] }, { "type": "equation_interline", "content": { "math_content": "I = - \\int {\\frac {2}{t ^ {2} + 3}} \\mathrm{d} t = - \\frac {2}{\\sqrt {3}} \\arctan {\\frac {t}{\\sqrt {3}}} + C", "math_type": "latex", "image_source": { "path": "images/fea37fce8ace434ad915e68ec0ea38507b10c91a1fdad385712a4f4a211c5390.jpg" } }, "bbox": [ 342, 736, 653, 768 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3.2) 方法 2: 配方 + 三角函数 (一般还是推荐三角函数)。 " }, { "type": "equation_inline", "content": "I = \\int \\frac{\\mathrm{d}x}{x[(x-1)^{2}-4]^{\\frac{1}{2}}}" }, { "type": "text", "content": " ，做换元 " }, { "type": "equation_inline", "content": "x - 1 = 2 \\sec t" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 90, 776, 905, 820 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int {\\frac {2 \\sec {t} \\tan {t}}{(2 \\sec {t} + 1) 2 \\tan {t}}} \\mathrm{d} t = \\int {\\frac {\\mathrm{d} t}{2 + \\cos {t}}}", "math_type": "latex", "image_source": { "path": "images/991b5d197dd9b98271d4ffc86371db03782096dc70d8d0b0a252b855036906b2.jpg" } }, "bbox": [ 344, 816, 652, 848 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "背公式即 " }, { "type": "equation_inline", "content": "\\frac{2}{\\sqrt{1 - a^2}}\\arctan \\left(\\sqrt{\\frac{1 - a}{1 + a}}\\tan \\frac{x}{2}\\right) + C" }, { "type": "text", "content": " ，代入 " }, { "type": "equation_inline", "content": "a = \\frac{1}{2}" }, { "type": "text", "content": " 。注意带回时需要用半角公式， " }, { "type": "equation_inline", "content": "\\tan \\frac{t}{2} = \\frac{\\sin t}{1 + \\cos t}" }, { "type": "text", "content": " 再画个三角形进行辅助。" } ] }, "bbox": [ 89, 857, 907, 908 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "50" } ] }, "bbox": [ 489, 953, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 可以用 Euler 变换或者配方三角替换。" } ] }, "bbox": [ 122, 85, 423, 99 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.5 配对积分法" } ], "level": 2 }, "bbox": [ 92, 121, 253, 137 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习5.18经典配对积分法 (1) 计算 " }, { "type": "equation_inline", "content": "I_{1} = \\int e^{ax}\\cos bx\\mathrm{d}x,I_{2} = \\int e^{ax}\\sin bx\\mathrm{d}x" }, { "type": "text", "content": " (2) 计算 " }, { "type": "equation_inline", "content": "I_{1} = \\sin (\\ln x)\\mathrm{d}x,I_{2} = \\int \\cos (\\ln x)\\mathrm{d}x" } ] }, "bbox": [ 58, 147, 905, 202 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)用分部积分得到 " }, { "type": "equation_inline", "content": "I_{1} = \\frac{1}{a}\\left(e^{ax}\\cos bx + bI_{2}\\right)" }, { "type": "text", "content": "，同理 " }, { "type": "equation_inline", "content": "I_{2} = \\frac{1}{a}\\left(e^{ax}\\sin x - bI_{1}\\right)" }, { "type": "text", "content": "，列出线性方程组：" } ] }, "bbox": [ 87, 203, 796, 231 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} a I _ {1} - b I _ {2} = e ^ {a x} \\cos b x \\\\ b I _ {1} + a I _ {2} = e ^ {a x} \\sin b x \\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{l} I _ {1} = \\frac {e ^ {a x}}{a ^ {2} + b ^ {2}} (a \\cos b x + b \\sin b x) + C \\\\ I _ {2} = \\frac {e ^ {a x}}{a ^ {2} + b ^ {2}} (a \\sin b x - b \\cos b x) + C \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/76c0c844cfe2f956310f8fe17aefccf0d793c7059d78901ef606ce4f28189eed.jpg" } }, "bbox": [ 238, 234, 742, 285 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 直接用分部积分： " }, { "type": "equation_inline", "content": "\\int \\sin (\\ln x)\\mathrm{d}x = x\\sin (\\ln x) - \\int \\cos (\\ln x)\\mathrm{d}x" }, { "type": "text", "content": " ，同理 " }, { "type": "equation_inline", "content": "\\int \\cos (\\ln x)\\mathrm{d}x = x\\cos (\\ln x) + \\int \\sin (\\ln x)\\mathrm{d}x" } ] }, "bbox": [ 87, 288, 905, 346 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.6 递推问题" } ], "level": 2 }, "bbox": [ 92, 363, 233, 381 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 5.19 几个最常见递推 (1) " }, { "type": "equation_inline", "content": "I_{n}=\\int\\frac{\\sin nx}{\\sin x}dx" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "I_{n}=\\int\\sec^{n}x\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "I_{n}=\\int e^{ax}\\sin^{n}x\\mathrm{d}x" } ] }, "bbox": [ 58, 391, 749, 423 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)首先根据和差公式和积化和差:" } ] }, "bbox": [ 87, 419, 366, 434 ] }, { "type": "equation_interline", "content": { "math_content": "\\sin n x = \\sin [ (n - 1) x + x ] = \\sin (n - 1) x \\cos x + \\cos (n - 1) x \\sin x = \\frac {1}{2} [ \\sin n x + \\sin (n - 2) x ] + \\cos (n - 1) x \\sin x", "math_type": "latex", "image_source": { "path": "images/aa6fddb8af7e86da6727d9001ed93c9f088b6c6e5a0a096f3aa4adee32b37950.jpg" } }, "bbox": [ 92, 436, 900, 466 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此" } ] }, "bbox": [ 92, 466, 129, 480 ] }, { "type": "equation_interline", "content": { "math_content": "I _ {n} = \\frac {1}{2} (I _ {n} + I _ {n - 2}) + \\frac {1}{n - 1} \\sin (n - 1) x \\Rightarrow I _ {n} = \\frac {2}{n - 1} \\sin (n - 1) x + I _ {n - 2}", "math_type": "latex", "image_source": { "path": "images/47a2192ef69a4a1244aa7b2dc4dca2577682427dc25cc3ff0a564f6c5027b21a.jpg" } }, "bbox": [ 228, 476, 766, 506 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 先凑出 " }, { "type": "equation_inline", "content": "\\mathrm{d}\\tan x" }, { "type": "text", "content": " 并使用分部积分:" } ] }, "bbox": [ 122, 513, 384, 527 ] }, { "type": "equation_interline", "content": { "math_content": "\\int \\sec^ {n} x \\mathrm{d} x = \\int \\sec^ {n - 2} x \\mathrm{d} (\\tan x) = \\tan x \\sec^ {n - 2} x - (n - 2) \\int \\tan^ {2} x \\sec^ {n - 2} x \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/34dc357aa4e1147d7e5b450e9687c8e8affad69b5390aef9177a334671884880.jpg" } }, "bbox": [ 201, 530, 794, 560 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "再代入 " }, { "type": "equation_inline", "content": "\\tan^2 x = \\sec^2 x - 1" }, { "type": "text", "content": ":" } ] }, "bbox": [ 92, 562, 294, 577 ] }, { "type": "equation_interline", "content": { "math_content": "I _ {n} = \\tan x \\sec^ {n - 2} x - (n - 2) (I _ {n} - I _ {n - 2}) \\Rightarrow I _ {n} = \\frac {\\tan x \\sec^ {n - 2} x}{n - 1} + \\frac {n - 2}{n - 1} I _ {n - 2}", "math_type": "latex", "image_source": { "path": "images/6e2d760c6fb62fc0303c8f5866c755b3812170414bf5350ade2325127923de88.jpg" } }, "bbox": [ 213, 580, 781, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 " }, { "type": "equation_inline", "content": "\\int^{\\text{(3)}}\\sec x\\mathrm{d}x = \\ln |\\tan x + \\sec x|" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "\\int \\sec^3 x\\mathrm{d}x = \\frac{1}{2}\\tan x\\sec x + \\frac{1}{2}\\ln |\\tan x + \\sec x|" }, { "type": "text", "content": " 在根式中经常用到，要记住，具体证明过程见前面部分。" } ] }, "bbox": [ 55, 613, 905, 665 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "5.3 不定积分表选例" } ] }, "bbox": [ 754, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "51" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第6章 定积分" } ], "level": 2 }, "bbox": [ 410, 87, 603, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.1 定积分理论" } ], "level": 2 }, "bbox": [ 92, 145, 265, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.1.1 可积性" } ], "level": 2 }, "bbox": [ 92, 184, 213, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 6.1 (Riemann 积分)" } ], "level": 2 }, "bbox": [ 122, 217, 307, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上的函数，将 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 划分为 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 等份 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": " ，若 " }, { "type": "equation_inline", "content": "\\lim_{||T||\\to 0}\\sum_{i = 1}^{n}f(\\xi_i)\\Delta x_i = J" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "J" }, { "type": "text", "content": " 为常数，则称 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积，且记 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x = J" } ] }, "bbox": [ 112, 236, 882, 306 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 6.2 (Darboux 上下和)" } ], "level": 2 }, "bbox": [ 122, 330, 322, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在每个区间上取 " }, { "type": "equation_inline", "content": "M_{i} = \\sup_{x\\in \\Delta_{i}}f(x),m_{i} = \\inf_{x\\in \\Delta_{i}}f(x)" }, { "type": "text", "content": " ，记振幅 " }, { "type": "equation_inline", "content": "\\omega_{i} = M_{i} - m_{i}" }, { "type": "text", "content": " ，定义Darboux上下和为" } ] }, "bbox": [ 114, 349, 828, 374 ] }, { "type": "equation_interline", "content": { "math_content": "S (T) = \\sum_ {i = 1} ^ {n} M _ {i} \\Delta x _ {i}, s (T) = \\sum_ {i = 1} ^ {n} m _ {i} \\Delta x _ {i}", "math_type": "latex", "image_source": { "path": "images/d8e74a691279f1c19d7f5102158576a853cac50e16247220fd3b47bcb8680a47.jpg" } }, "bbox": [ 356, 378, 640, 416 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "定义 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 的上下积分为：" } ] }, "bbox": [ 115, 418, 361, 434 ] }, { "type": "equation_interline", "content": { "math_content": "S = \\inf _ {T} S (T), s = \\sup _ {T} s (T)", "math_type": "latex", "image_source": { "path": "images/5e70973eaf4378d58904929107ecc3cf3381e826b2d609ab6da1094dd09be055.jpg" } }, "bbox": [ 398, 436, 596, 458 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.1 (可积函数的性质)" } ], "level": 2 }, "bbox": [ 122, 486, 319, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上可积，则：" } ] }, "bbox": [ 115, 506, 327, 521 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 有界性： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上有界 (反之不正确，有界不一定可积，例如 Dirichlet 函数)" } ] }, "bbox": [ 137, 524, 759, 539 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 反设 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 无界，则存在某个小区间 " }, { "type": "equation_inline", "content": "\\Delta_k" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\Delta_k" }, { "type": "text", "content": " 上 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 无界，假设其他区间都有界，此时 Riemann 和" } ] }, "bbox": [ 89, 558, 853, 575 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\sum_ {i = 1} ^ {n} f \\left(\\xi_ {i}\\right) \\Delta x _ {i} \\right| \\geq \\left| f \\left(\\xi_ {k}\\right) \\Delta x _ {k} \\right| - \\left| \\sum_ {i \\neq k} f \\left(\\xi_ {i}\\right) \\Delta x _ {i} \\right|", "math_type": "latex", "image_source": { "path": "images/ae90a862ee565bea2adcb8e4a67c191ea6b57a1fe86ffb0dc51348264b6c4640.jpg" } }, "bbox": [ 327, 577, 668, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对 " }, { "type": "equation_inline", "content": "\\forall M > 0" }, { "type": "text", "content": " ，无论如何分割，Riemann和都能大于 " }, { "type": "equation_inline", "content": "M" }, { "type": "text", "content": " ，因此无界不可积。" } ] }, "bbox": [ 92, 620, 621, 634 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.2 (可积性充要条件)" } ], "level": 2 }, "bbox": [ 122, 645, 319, 659 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积当且仅当" } ] }, "bbox": [ 114, 665, 315, 680 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 上下积分角度： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 的上积分 " }, { "type": "equation_inline", "content": "s" }, { "type": "text", "content": " 等于下积分 " }, { "type": "equation_inline", "content": "S" } ] }, "bbox": [ 139, 683, 557, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 上下和角度 (最常用！): " }, { "type": "equation_inline", "content": "\\forall \\epsilon" }, { "type": "text", "content": "，存在分割 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "S(T) - s(T) = \\sum_{T} \\omega_i \\Delta x_i < \\epsilon" } ] }, "bbox": [ 139, 699, 717, 728 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 振幅角度：对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\eta" }, { "type": "text", "content": "，总存在分割 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": "，使得振幅 " }, { "type": "equation_inline", "content": "\\omega_{k'} \\geq \\epsilon" }, { "type": "text", "content": " 的那些小区间总长 " }, { "type": "equation_inline", "content": "\\sum \\omega_{k'} < \\eta" } ] }, "bbox": [ 139, 730, 781, 747 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 实变角度： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 几乎处处连续" } ] }, "bbox": [ 139, 747, 442, 765 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.1Dirichlet函数的不可积性证明Dirichlet函数在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 不可积" } ] }, "bbox": [ 58, 803, 591, 818 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于任意分割 T，振幅 " }, { "type": "equation_inline", "content": "\\omega = 1" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "S(T) - s(T) = \\sum_{i=1}^{n} \\Delta x_i = b - a" }, { "type": "text", "content": " ，因此不可积。" } ] }, "bbox": [ 87, 822, 741, 854 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.3 (Riemann 可积函数类)" } ], "level": 2 }, "bbox": [ 122, 87, 357, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 定义于 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": "，常见的有三个Riemann可积函数类：" } ] }, "bbox": [ 114, 107, 522, 122 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "• " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续函数" } ] }, "bbox": [ 139, 126, 267, 141 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 只有有限个间断点的有界函数" } ] }, "bbox": [ 139, 143, 438, 159 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上单调函数" } ] }, "bbox": [ 139, 162, 284, 177 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据一致连续性，每个 " }, { "type": "equation_inline", "content": "\\omega_{i}" }, { "type": "text", "content": " 都可以足够小，因此 " }, { "type": "equation_inline", "content": "\\sum \\omega_{i} \\Delta x_{i}" }, { "type": "text", "content": " 足够小" } ] }, "bbox": [ 90, 196, 636, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 将间断点处的 " }, { "type": "equation_inline", "content": "\\Delta x_{i}" }, { "type": "text", "content": " 设置得足够小，控制住间断点差距" } ] }, "bbox": [ 122, 217, 542, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\omega_{i}=f(x_{i})-f(x_{i-1})" }, { "type": "text", "content": " ，只需要 " }, { "type": "equation_inline", "content": "\\omega_{i}" }, { "type": "text", "content": " 取得足够小" } ] }, "bbox": [ 122, 235, 477, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.2可积性判断(1)证明Riemann函数 " }, { "type": "equation_inline", "content": "R(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 可积，且 " }, { "type": "equation_inline", "content": "\\int_0^1 R(x)\\mathrm{d}x = 0" } ] }, "bbox": [ 58, 263, 697, 296 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 证明下面的 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 上可积:" } ] }, "bbox": [ 122, 294, 394, 310 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = \\left\\{ \\begin{array}{l l} 0, & x = 0 \\\\ \\frac {1}{x} - \\left[ \\frac {1}{x} \\right], & x \\in (0, 1 ] \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/e22eadfcc274ea6e384921183c35d1e134ff4d91a1a9c46b21eb466e0a22ea9c.jpg" } }, "bbox": [ 379, 313, 613, 366 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 只需证 " }, { "type": "equation_inline", "content": "\\sum \\omega_{i} \\Delta x_{i} < \\epsilon" }, { "type": "text", "content": " ，给定 " }, { "type": "equation_inline", "content": "\\forall \\epsilon" }, { "type": "text", "content": " ，满足 " }, { "type": "equation_inline", "content": "\\frac{1}{q} > \\frac{\\epsilon}{2}" }, { "type": "text", "content": " 的只有有限个，且它们的函数值均小于等于 " }, { "type": "equation_inline", "content": "\\frac{1}{2}" }, { "type": "text", "content": " 。把这有限个全部列出 " }, { "type": "equation_inline", "content": "x_{1}, \\cdots, x_{k}" }, { "type": "text", "content": " ，这些点每个最多落入两个小区间中，取 " }, { "type": "equation_inline", "content": "\\|T\\| < \\frac{\\epsilon}{2k}" }, { "type": "text", "content": " 即可，则该部分 " }, { "type": "equation_inline", "content": "\\sum \\omega_{i} \\Delta x_{i} < \\frac{\\epsilon}{2}" }, { "type": "text", "content": " 。另一侧直接 " }, { "type": "equation_inline", "content": "\\sum \\omega_{i} \\Delta x_{i} < \\frac{\\epsilon}{2} \\sum \\Delta x_{i} \\leq \\frac{\\epsilon}{2}" }, { "type": "text", "content": " 就行。" } ] }, "bbox": [ 89, 369, 905, 442 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 振幅小于等于1，不连续点为 " }, { "type": "equation_inline", "content": "0, \\frac{1}{2}, \\frac{1}{3}, \\dots, \\frac{1}{n}, \\dots" }, { "type": "text", "content": "。对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta_1" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "|\\Delta x_i| < \\delta_1" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "(\\epsilon, 1)" }, { "type": "text", "content": " 分割满足 " }, { "type": "equation_inline", "content": "\\sum \\omega_i \\Delta x_i < \\epsilon" }, { "type": "text", "content": "。而 " }, { "type": "equation_inline", "content": "(0, \\epsilon)" }, { "type": "text", "content": " 的任意分割也满足 " }, { "type": "equation_inline", "content": "\\sum \\omega_i \\Delta x_i < \\epsilon" }, { "type": "text", "content": "，因此取 " }, { "type": "equation_inline", "content": "\\delta = \\min\\{\\delta_1, \\epsilon\\}" }, { "type": "text", "content": " 时，整体的分割满足：" } ] }, "bbox": [ 87, 443, 905, 489 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum \\omega_ {i} \\Delta x _ {i} < \\epsilon", "math_type": "latex", "image_source": { "path": "images/777dfee46606a548e07ba4723169097cc1d6d34d936fc1ac542e98c7b426e4e3.jpg" } }, "bbox": [ 443, 492, 552, 512 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.3 可积性相关证明 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 有定义， " }, { "type": "equation_inline", "content": "\\forall \\epsilon" }, { "type": "text", "content": "，存在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上可积函数 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b], |f(x) - g(x)| < \\epsilon" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积" } ] }, "bbox": [ 58, 516, 905, 550 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 由于 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 可以，存在分割 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\sum_{T} \\omega_{i}^{g} \\Delta x_{i} < \\epsilon" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\omega_{i}^{f} \\leq \\omega_{i}^{f - g} + \\omega_{i}^{g}" }, { "type": "text", "content": "，以及 " }, { "type": "equation_inline", "content": "\\omega_{i}^{f - g} \\leq 2\\epsilon" }, { "type": "text", "content": "（因为 " }, { "type": "equation_inline", "content": "g" }, { "type": "text", "content": " 本身可能 " }, { "type": "equation_inline", "content": "\\epsilon" }, { "type": "text", "content": " 震荡， " }, { "type": "equation_inline", "content": "|f(x) - g(x)|" }, { "type": "text", "content": " 又可能 " }, { "type": "equation_inline", "content": "\\epsilon" }, { "type": "text", "content": " 震荡），得到" } ] }, "bbox": [ 87, 551, 905, 596 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {T} \\omega_ {i} ^ {f} \\Delta x _ {i} \\leq \\sum_ {T} \\omega_ {i} ^ {f - g} \\Delta x _ {i} + \\sum_ {T} \\omega_ {i} ^ {g} \\Delta x _ {i} < 2 \\epsilon (b - a) + \\epsilon", "math_type": "latex", "image_source": { "path": "images/bcba7f4db2dff318096a149dde3f7855197033c09ffe91cf8cc9170310820f38.jpg" } }, "bbox": [ 292, 600, 702, 630 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.1.2 Newton-Leibniz 公式" } ], "level": 2 }, "bbox": [ 94, 655, 334, 671 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.4 (Newton-Leibniz 公式)" } ], "level": 2 }, "bbox": [ 121, 687, 356, 702 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积， " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上连续，且除有限个点外， " }, { "type": "equation_inline", "content": "F^{\\prime}(x) = f(x)" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 114, 707, 692, 722 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} f (x) \\mathrm{d} x = F (b) - F (a)", "math_type": "latex", "image_source": { "path": "images/8c6e62e9adb03161484b7020e1142963b77b6392921e6c322738431bb38aa764.jpg" } }, "bbox": [ 400, 726, 596, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 Newton-Leibniz 公式的意义在于将定积分的计算转换为了原函数的计算，不过可积与存在原函数也没有直接关系，具体见下面题目。" } ] }, "bbox": [ 87, 778, 905, 812 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.4 可积与原函数 举例说明可积未必有原函数，有原函数未必可积" } ] }, "bbox": [ 58, 834, 618, 848 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)例如 " }, { "type": "equation_inline", "content": "[-1, 1]" }, { "type": "text", "content": " 上的 " }, { "type": "equation_inline", "content": "\\operatorname{sgn}(x)" }, { "type": "text", "content": "，其可积但无原函数" } ] }, "bbox": [ 89, 852, 463, 866 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.1 定积分理论" } ] }, "bbox": [ 788, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "53" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)例如原函数如下，其在 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 可导，但是 " }, { "type": "equation_inline", "content": "F^{\\prime}(x)" }, { "type": "text", "content": " 无界" } ] }, "bbox": [ 122, 85, 529, 99 ] }, { "type": "equation_interline", "content": { "math_content": "F (x) = \\left\\{ \\begin{array}{l l} 0, & x = 0 \\\\ x ^ {\\frac {3}{2}} \\sin \\frac {1}{x}, & x \\in (0, 1 ] \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/b5a680f015d3be9c42d117822cd3a646e3ad254fe4a8700307a31892e0382777.jpg" } }, "bbox": [ 381, 105, 611, 152 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.5使用Newton-Leibniz公式(1)已知 " }, { "type": "equation_inline", "content": "f(x) = \\left\\{ \\begin{array}{ll}2022, & x = 0\\\\ 2x\\sin \\frac{1}{x} -\\cos \\frac{1}{x}, & x\\in (0,1] \\end{array} \\right." }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在[0,1]可积，求 " }, { "type": "equation_inline", "content": "\\int_0^1 f(x)\\mathrm{d}x" }, { "type": "text", "content": " 解(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在0处为跳跃间断点，只有一处，而[0,1]上 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 有界，故可积。构造 " }, { "type": "equation_inline", "content": "F(x)" } ] }, "bbox": [ 58, 159, 905, 243 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在0处为跳跃间断点，只有一处，而 " }, { "type": "equation_inline", "content": "[0,1]" }, { "type": "text", "content": " 上 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 有界，故可积。构造 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " ：" } ] }, "bbox": [ 87, 233, 737, 249 ] }, { "type": "equation_interline", "content": { "math_content": "F (x) = \\left\\{ \\begin{array}{l l} 0, & x = 0 \\\\ x ^ {2} \\sin \\frac {1}{x}, & x \\in (0, 1 ] \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/6c2d1763f84f905e179fd64171d17416f842b042d79a035d0ab87743d5ebed99.jpg" } }, "bbox": [ 384, 255, 608, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 除了0处外满足 " }, { "type": "equation_inline", "content": "F^{\\prime}(x) = f(x)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\int_0^1 f(x)\\mathrm{d}x = F(1) - F(0) = \\sin 1" } ] }, "bbox": [ 90, 302, 650, 338 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.5 (微积分基本定理)" } ], "level": 2 }, "bbox": [ 122, 354, 321, 369 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积函数，定义 " }, { "type": "equation_inline", "content": "F(x) = \\int_{a}^{x}f(t)\\mathrm{d}t,x\\in [a,b]" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 112, 374, 571, 406 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上的连续函数" } ] }, "bbox": [ 139, 403, 361, 418 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "x_0 \\in [a, b]" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 连续点，则 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 可导，且 " }, { "type": "equation_inline", "content": "F'(x_0) = f(x_0)" } ] }, "bbox": [ 139, 420, 665, 437 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上连续函数，则 " }, { "type": "equation_inline", "content": "F(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上连续可微函数，且 " }, { "type": "equation_inline", "content": "F'(x) = f(x)" } ] }, "bbox": [ 139, 439, 747, 456 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.6 使用微积分基本定理 (1) 重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[A, B]" }, { "type": "text", "content": " 上可积, " }, { "type": "equation_inline", "content": "a, b \\in (A, B)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的两个连续点, 证明" } ] }, "bbox": [ 58, 494, 865, 508 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {h \\rightarrow 0} \\int_ {a} ^ {b} \\frac {f (x + h) - f (x)}{h} \\mathrm{d} x = f (b) - f (a)", "math_type": "latex", "image_source": { "path": "images/23374db42f4bdbbadee64913af3a7802551f55df36eaae8fc8adfd7f291a2e4a.jpg" } }, "bbox": [ 342, 513, 650, 545 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 对左侧做如下化简:" } ] }, "bbox": [ 89, 548, 304, 562 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\int_ {a} ^ {b} \\frac {f (x + h) - f (x)}{h} \\mathrm{d} x = \\frac {1}{h} \\left(\\int_ {a} ^ {b} f (x + h) \\mathrm{d} x - \\int_ {a} ^ {b} f (x) \\mathrm{d} x\\right) \\\\ \\qquad = \\frac {1}{h} \\left(\\int_ {a + h} ^ {b + h} f (x) \\mathrm{d} x - \\int_ {a} ^ {b} f (x) \\mathrm{d} x\\right) \\\\ \\qquad = \\frac {1}{h} \\left(\\int_ {b} ^ {b + h} f (x) \\mathrm{d} x - \\int_ {a} ^ {a + h} f (x) \\mathrm{d} x\\right) \\end{array}", "math_type": "latex", "image_source": { "path": "images/618754dd10cde04b6f58fa0b62e58c728e1ab5cf08f70416d575ce1ca12830a5.jpg" } }, "bbox": [ 272, 565, 721, 682 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "令 " }, { "type": "equation_inline", "content": "F(x) = \\int_{A}^{x}f(t)\\mathrm{d}t" }, { "type": "text", "content": " ，则显然 " }, { "type": "equation_inline", "content": "F^{\\prime}(b) = \\lim_{h\\to 0^{+}}\\frac{F(b + h) - F(b)}{h} = f(b),F^{\\prime}(a) = f(a)" }, { "type": "text", "content": " 同理。" } ] }, "bbox": [ 90, 686, 749, 719 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.1.3 四则运算与复合" } ], "level": 2 }, "bbox": [ 92, 736, 292, 753 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.6 (四则运算的可积性)" } ], "level": 2 }, "bbox": [ 122, 770, 337, 785 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积，则 " }, { "type": "equation_inline", "content": "kf(x), f(x) \\pm g(x), f(x)g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积，但 " }, { "type": "equation_inline", "content": "g(x) \\neq 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\frac{f(x)}{g(x)}" }, { "type": "text", "content": " 也不一定可积" } ] }, "bbox": [ 114, 788, 895, 837 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 数乘、加减显然" } ] }, "bbox": [ 90, 851, 280, 866 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 乘法：根据可积可知有界，设 " }, { "type": "equation_inline", "content": "|f(x)| \\leq M, |g(x)| \\leq N" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\sum_{T} \\omega_{i}^{fg} \\Delta x_{i} < M \\sum_{T} \\omega_{i}^{g} \\Delta x_{i} + N \\sum_{T} \\omega_{i}^{f} \\Delta x_{i}" }, { "type": "text", "content": " 即可" } ] }, "bbox": [ 122, 866, 905, 898 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "54" } ] }, "bbox": [ 489, 954, 509, 964 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.1 定积分理论" } ] }, "bbox": [ 788, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3)除法：例如 " }, { "type": "equation_inline", "content": "f(x) = 1,g(x) = \\left\\{ \\begin{array}{ll}x, & x\\in (0,1]\\\\ 1, & x = 0 \\end{array} \\right." }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\frac{1}{g(x)}" }, { "type": "text", "content": " 因为无界而不可积" } ] }, "bbox": [ 121, 82, 668, 128 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.7除法增加条件可积设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上函数，证明" } ] }, "bbox": [ 58, 136, 517, 152 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 可积，且 " }, { "type": "equation_inline", "content": "|f(x)| \\geq m > 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\frac{1}{f(x)}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积" } ] }, "bbox": [ 121, 154, 563, 180 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续且恒不为零，则 " }, { "type": "equation_inline", "content": "\\frac{1}{f(x)}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积" } ] }, "bbox": [ 121, 181, 557, 208 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 考虑振幅放缩:" } ] }, "bbox": [ 90, 206, 270, 221 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\frac {1}{f (x ^ {\\prime})} - \\frac {1}{f (x ^ {\\prime \\prime})} \\right| = \\left| \\frac {f (x ^ {\\prime \\prime}) - f (x ^ {\\prime})}{f (x ^ {\\prime}) f (x ^ {\\prime \\prime})} \\right| \\leq \\frac {1}{m ^ {2}} \\omega_ {i} ^ {f}", "math_type": "latex", "image_source": { "path": "images/3a24b2d0e09c738f0e494e626af72ec8c3b8773c15ed3189f4613e44e870eada.jpg" } }, "bbox": [ 332, 218, 662, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于连续，能找到最小值，可转换为 (1)" } ] }, "bbox": [ 122, 258, 445, 273 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.7 (复合 (较难，重点！))" } ], "level": 2 }, "bbox": [ 122, 300, 352, 315 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "\\varphi(t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[\\alpha,\\beta]" }, { "type": "text", "content": " 可积，且 " }, { "type": "equation_inline", "content": "a \\leq \\varphi(t) \\leq b, t \\in [\\alpha,\\beta]" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "f(\\varphi(t))" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[\\alpha,\\beta]" }, { "type": "text", "content": " 可积" } ] }, "bbox": [ 110, 316, 902, 341 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续，因此有界且一致连续。设 " }, { "type": "equation_inline", "content": "M = \\max_{x\\in [a,b]}f(x)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, \\forall x', x'' \\in [a,b]" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "|x' - x''| < \\delta" }, { "type": "text", "content": " 就有 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| < \\epsilon" }, { "type": "text", "content": "。由于 " }, { "type": "equation_inline", "content": "\\varphi" }, { "type": "text", "content": " 可积，存在分割 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\sum_{\\omega_i^\\varphi \\geq \\delta} \\Delta t_i < \\epsilon" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 89, 353, 905, 410 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} & {\\sum_ {T} \\omega_ {i} ^ {f \\circ \\varphi} \\Delta t _ {i} = \\sum_ {\\omega_ {i} ^ {\\varphi} \\geq \\delta} \\omega_ {i} ^ {f \\circ \\varphi} \\Delta t _ {i} + \\sum_ {\\omega_ {i} ^ {\\varphi} < \\delta} \\omega_ {i} ^ {f \\circ \\varphi} \\Delta t _ {i}} \\\\ & {\\qquad \\leq 2 M \\sum_ {\\omega_ {i} ^ {\\varphi} \\geq \\delta} \\Delta t _ {i} + \\epsilon \\sum_ {\\omega_ {i} ^ {\\varphi} < \\delta} \\Delta t _ {i}} \\\\ & {\\qquad < 2 M \\epsilon + (\\beta - \\alpha) \\epsilon = \\epsilon^ {\\prime}} \\end{array}", "math_type": "latex", "image_source": { "path": "images/9ad9e4567ba2651aefe28f20b771c53cd08a2692c4918377437d0d1d314c4b06.jpg" } }, "bbox": [ 329, 414, 665, 507 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.1.4 定积分的保号性" } ], "level": 2 }, "bbox": [ 92, 533, 294, 550 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.8 (定积分保号性)" } ], "level": 2 }, "bbox": [ 122, 565, 305, 580 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "f(x)\\geq 0" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x\\geq 0" }, { "type": "text", "content": " ，且 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x = 0" }, { "type": "text", "content": " 当且仅当 " }, { "type": "equation_inline", "content": "f(x)\\equiv 0" } ] }, "bbox": [ 110, 583, 761, 620 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.8等于0的推广(1)重点: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续,且对任意满足 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}g(x)\\mathrm{d}x=0" }, { "type": "text", "content": " 的连续函数 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)g(x)\\mathrm{d}x=0" }, { "type": "text", "content": ",证明: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为常值函数" } ] }, "bbox": [ 60, 649, 927, 695 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点 (ZJU2020): " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可导, " }, { "type": "equation_inline", "content": "f(a) = 0" }, { "type": "text", "content": ", 对 " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x)| \\geq |f'(x)|" }, { "type": "text", "content": ", 证明: " }, { "type": "equation_inline", "content": "f(x) \\equiv 0" }, { "type": "text", "content": " 证明 (1) 取 " }, { "type": "equation_inline", "content": "g(x) = f(x) - \\frac{1}{b - a} \\int_{a}^{b} f(t) \\mathrm{d}t" }, { "type": "text", "content": ", 显然 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} g(x) \\mathrm{d}x = 0" }, { "type": "text", "content": " 满足条件, 此时" } ] }, "bbox": [ 87, 697, 865, 746 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} g ^ {2} (x) \\mathrm{d} x = \\int_ {a} ^ {b} g (x) \\left(f (x) - \\frac {1}{b - a} \\int_ {a} ^ {b} f (x) \\mathrm{d} x\\right) = 0", "math_type": "latex", "image_source": { "path": "images/6e8a0f3a5e4940a9fd1ab904eb1094f7a0f58cc947d22a082cb690f5037b2cf0.jpg" } }, "bbox": [ 295, 747, 702, 784 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 可证明 " }, { "type": "equation_inline", "content": "\\int_{a}^{x} f(t) \\mathrm{d}t = 0, \\forall x \\in [a, b]" }, { "type": "text", "content": "，则考虑 " }, { "type": "equation_inline", "content": "g(x) = \\int_{a}^{x} |f(t)| \\mathrm{d}t" }, { "type": "text", "content": "，有" } ] }, "bbox": [ 122, 788, 631, 820 ] }, { "type": "equation_interline", "content": { "math_content": "g (x) \\geq \\int_ {a} ^ {x} | f ^ {\\prime} (t) | \\mathrm{d} t \\geq \\left| \\int_ {a} ^ {x} f ^ {\\prime} (t) \\mathrm{d} t \\right| = | f (x) | = g ^ {\\prime} (x)", "math_type": "latex", "image_source": { "path": "images/8520f79c39286c1cb1b7988073fbc705d598f10c0d919671a6ad1e9bc28b50a2.jpg" } }, "bbox": [ 305, 822, 687, 854 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此得到 " }, { "type": "equation_inline", "content": "(e^{-x}g(x))' = e^{-x}(g'(x) - g(x))\\leq 0" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "e^{-x}g(x)" }, { "type": "text", "content": " 单调减，而 " }, { "type": "equation_inline", "content": "e^{-a}g(a) = 0" }, { "type": "text", "content": " ，且 " }, { "type": "equation_inline", "content": "e^{-x}g(x)\\geq 0" }, { "type": "text", "content": " ，综上得到 " }, { "type": "equation_inline", "content": "g(x)\\equiv 0" } ] }, "bbox": [ 87, 856, 905, 892 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "55" } ] }, "bbox": [ 489, 954, 507, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.1 定积分理论" } ] }, "bbox": [ 788, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.9 零点问题 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[0, \\pi]" }, { "type": "text", "content": " 上连续函数，且 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi} f(x) \\sin x \\, \\mathrm{d}x = \\int_{0}^{\\pi} f(x) \\cos x \\, \\mathrm{d}x = 0" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, \\pi)" }, { "type": "text", "content": " 上至少有两个零点。\n\n(2) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上连续函数， " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) \\, \\mathrm{d}x = \\int_{a}^{b} x f(x) \\, \\mathrm{d}x = \\cdots = \\int_{a}^{b} x^n f(x) \\, \\mathrm{d}x = 0" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 上至少有 " }, { "type": "equation_inline", "content": "n + 1" }, { "type": "text", "content": " 个零点\n\n证明 (1) 反设无零点，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, \\pi)" }, { "type": "text", "content": " 不变号，而 " }, { "type": "equation_inline", "content": "\\sin x" }, { "type": "text", "content": " 也不变号，因此与 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi} f(x) \\sin x \\, \\mathrm{d}x = 0" }, { "type": "text", "content": " 矛盾，因此至少有一个零点。若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 只有一个零点 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, \\xi), (\\xi, \\pi)" }, { "type": "text", "content": " 内不变号，由于 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi} f(x) \\sin x \\, \\mathrm{d}x = 0" }, { "type": "text", "content": "，因此在上述两个区间符号相反，而 " }, { "type": "equation_inline", "content": "\\sin (x - \\xi)" }, { "type": "text", "content": " 也仅在 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": " 为 0， " }, { "type": "equation_inline", "content": "(0, \\xi), (\\xi, \\pi)" }, { "type": "text", "content": " 符号相反，因此" } ] }, "bbox": [ 58, 82, 905, 240 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {\\pi} f (x) \\sin (x - \\xi) \\mathrm{d} x = \\cos \\xi \\int_ {0} ^ {\\pi} f (x) \\sin x \\mathrm{d} x - \\sin \\xi \\int_ {0} ^ {\\pi} f (x) \\cos x \\mathrm{d} x = 0", "math_type": "latex", "image_source": { "path": "images/bf24f4dc2a99bea030055354180a0c1c06d90ffca57f206386c51ba3ce9d3a6e.jpg" } }, "bbox": [ 228, 242, 768, 273 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这与 " }, { "type": "equation_inline", "content": "f(x)\\sin (x - \\xi)" }, { "type": "text", "content": " 非零矛盾，因此 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 至少有两个零点" } ] }, "bbox": [ 89, 275, 521, 290 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 反设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 上至多 n 个零点，则其最多改变 n 次符号。由于 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) \\, dx = 0" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 至少改变一次符号，一定有能从 n 个零点中找出 k 个零点 " }, { "type": "equation_inline", "content": "x_{1}, x_{2}, \\cdots, x_{k}" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, x_{1}), (x_{1}, x_{2}), \\cdots, (x_{k}, b)" }, { "type": "text", "content": " 上不变号，且在相邻区间变号，而 " }, { "type": "equation_inline", "content": "g(x) = (x - x_{1})(x - x_{2}) \\cdots (x - x_{k})" }, { "type": "text", "content": " 有同样的性质，于是 " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 不变号，但 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x)g(x) \\, dx = 0" }, { "type": "text", "content": " ，这显然矛盾。" } ] }, "bbox": [ 89, 291, 905, 386 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.1.5 积分中值定理" } ], "level": 2 }, "bbox": [ 92, 403, 273, 419 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.9 (积分第一中值定理)" } ], "level": 2 }, "bbox": [ 121, 435, 337, 451 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 不变号，则 " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (a, b)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) g(x) \\mathrm{d}x = f(\\xi) \\int_{a}^{b} g(x) \\mathrm{d}x" } ] }, "bbox": [ 110, 453, 845, 491 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 不妨设 " }, { "type": "equation_inline", "content": "g(x) \\geq 0" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续，设 " }, { "type": "equation_inline", "content": "M, m" }, { "type": "text", "content": " 分别是最大值和最小值，则 " }, { "type": "equation_inline", "content": "mg(x) \\leq f(x)g(x) \\leq Mg(x)" }, { "type": "text", "content": "，因此得到 " }, { "type": "equation_inline", "content": "m\\int_{a}^{b} g(x) \\mathrm{d}x \\leq \\int_{a}^{b} f(x)g(x) \\mathrm{d}x \\leq M\\int_{a}^{b} g(x) \\mathrm{d}x" }, { "type": "text", "content": "，根据介值性可知（具体讨论略去）。" } ] }, "bbox": [ 89, 506, 902, 557 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 6.1 (第一中值定理的推广)" } ], "level": 2 }, "bbox": [ 122, 565, 354, 580 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积 (有上下确界但不一定连续)， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 不变号， " }, { "type": "equation_inline", "content": "M, m" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 的上下确界，则 " }, { "type": "equation_inline", "content": "\\exists \\eta \\in [m, M]" }, { "type": "text", "content": " (不一定能为函数值 " }, { "type": "equation_inline", "content": "f(\\xi)" }, { "type": "text", "content": ") 使得 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) g(x) \\mathrm{d}x = \\eta \\int_{a}^{b} g(x) \\mathrm{d}x" } ] }, "bbox": [ 112, 583, 895, 636 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.10第一中值定理(技巧较强) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是[0,1]连续函数， " }, { "type": "equation_inline", "content": "\\int_0^1 f(x)\\mathrm{d}x = \\int_0^1 xf(x)\\mathrm{d}x" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\int_0^\\xi f(x)\\mathrm{d}x = 0" }, { "type": "text", "content": " 证明考虑 " }, { "type": "equation_inline", "content": "F(x) = \\int_0^x f(t)\\mathrm{d}t" }, { "type": "text", "content": " ，想使用积分第一中值定理可证 " }, { "type": "equation_inline", "content": "\\int_0^1 F(x)\\mathrm{d}x = 0" }, { "type": "text", "content": " ，则可推出 " }, { "type": "equation_inline", "content": "F(\\xi) = 0" }, { "type": "text", "content": " 。考虑 " }, { "type": "equation_inline", "content": "\\int_0^1 F(x)\\mathrm{d}x = \\int_0^1\\mathrm{d}x\\int_0^x f(t)\\mathrm{d}t" }, { "type": "text", "content": " ，交换积分顺序得到 " }, { "type": "equation_inline", "content": "\\int_0^1\\mathrm{d}t\\int_t^1 f(t)\\mathrm{d}x = \\int_0^1 (1 - t)f(t)\\mathrm{d}t = 0" } ] }, "bbox": [ 58, 659, 905, 789 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.10 (积分第二中值定理)" } ], "level": 2 }, "bbox": [ 122, 804, 346, 819 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 单调，则：(记忆方式：非负情况取最大)\n若 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 单增非负： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) g(x) \\mathrm{d}x = g(b) \\int_{\\xi}^{b} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 若 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 单减非负： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) g(x) \\mathrm{d}x = g(a) \\int_{a}^{\\xi} f(x) \\mathrm{d}x" }, { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 单调： " }, { "type": "equation_inline", "content": "\\exists \\xi \\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)g(x)\\mathrm{d}x = g(a)\\int_{a}^{\\xi}f(x)\\mathrm{d}x + g(b)\\int_{\\xi}^{b}f(x)\\mathrm{d}x" } ] }, "bbox": [ 115, 822, 678, 906 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.1 定积分理论" } ] }, "bbox": [ 788, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "56" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 139, 93, 754, 131 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 只证明 (3) 的特殊情况：设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 连续可微且单调" } ] }, "bbox": [ 90, 149, 579, 165 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\int_ {a} ^ {b} f (x) g (x) \\mathrm{d} x = \\int_ {a} ^ {b} g (x) \\mathrm{d} (F (x)) = g (x) F (x) | _ {a} ^ {b} - \\int_ {a} ^ {b} F (x) g ^ {\\prime} (x) \\mathrm{d} x \\\\ \\qquad = g (b) F (b) - g (a) F (a) - F (\\xi) \\int_ {a} ^ {b} g ^ {\\prime} (x) \\mathrm{d} x \\\\ \\qquad = g (a) [ F (\\xi) - F (a) ] + g (b) [ F (b) - F (\\xi) ] \\\\ \\qquad = g (a) \\int_ {a} ^ {\\xi} f (x) \\mathrm{d} x + g (b) \\int_ {\\xi} ^ {b} f (x) \\mathrm{d} x \\end{array}", "math_type": "latex", "image_source": { "path": "images/0855d78728749daec134f15213b3b367a2978297a6428983f6dca93d806a610b.jpg" } }, "bbox": [ 250, 167, 744, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 第二中值定理证明过于繁琐，这里不给出全部，只给出最后一种特殊情况的证明。其最大的应用是证明反常积分的 Abel 和 Dirichlet 判别法。" } ] }, "bbox": [ 87, 303, 905, 337 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.2 三角函数定积分计算" } ], "level": 2 }, "bbox": [ 92, 363, 361, 382 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "6.2.1 " }, { "type": "equation_inline", "content": "[0, \\frac{\\pi}{2}]" }, { "type": "text", "content": " 上三角函数积分公式" } ] }, "bbox": [ 92, 397, 384, 429 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理6.11 " }, { "type": "equation_inline", "content": "\\left[\\frac{0}{2},\\frac{\\pi}{2}\\right]" }, { "type": "text", "content": " 三角函数积分）" } ], "level": 2 }, "bbox": [ 122, 439, 356, 463 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 是连续函数，则 " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2}}f(\\sin x,\\cos x)\\mathrm{d}x = \\int_0^{\\frac{\\pi}{2}}f(\\cos x,\\sin x)\\mathrm{d}x" } ] }, "bbox": [ 114, 469, 621, 504 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 令 " }, { "type": "equation_inline", "content": "x = \\frac{\\pi}{2} - t" }, { "type": "text", "content": "，则左侧为 " }, { "type": "equation_inline", "content": "\\int_{\\frac{\\pi}{2}}^{0} f(\\cos t, \\sin t)(-\\mathrm{d}t) = \\int_{0}^{\\frac{\\pi}{2}} f(\\cos t, \\sin t) \\mathrm{d}t" } ] }, "bbox": [ 89, 513, 640, 549 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.11重要练习结论(1)证明： " }, { "type": "equation_inline", "content": "\\forall k" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2}}\\frac{\\sin^k x}{\\sin^k x + \\cos^k x}\\mathrm{d}x = \\int_0^{\\frac{\\pi}{2}}\\frac{\\cos^k x}{\\sin^k x + \\cos^k x}\\mathrm{d}x = \\int_0^{\\frac{\\pi}{2}}\\frac{x}{1 + \\tan^k x} = \\frac{\\pi}{4}" } ] }, "bbox": [ 60, 552, 892, 587 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2}}\\ln \\tan x\\mathrm{d}x" } ] }, "bbox": [ 122, 583, 297, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 计算 " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\mathrm{d}x}{(1+x^{2})(1+x^{2022})}" } ] }, "bbox": [ 122, 615, 376, 648 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 计算 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty} \\frac{\\ln x}{4 + x^2} \\, \\mathrm{d}x" } ] }, "bbox": [ 124, 646, 309, 677 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据对称性可知 " }, { "type": "equation_inline", "content": "2S = \\int_{0}^{\\frac{\\pi}{2}} \\mathrm{d}x = \\frac{\\pi}{2}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "S = \\frac{\\pi}{4}" } ] }, "bbox": [ 89, 676, 514, 707 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\int_{0}^{\\frac{\\pi}{2}}\\ln \\sin x\\mathrm{d}x - \\int_{0}^{\\frac{\\pi}{2}}\\ln \\cos x\\mathrm{d}x = 0" } ] }, "bbox": [ 122, 706, 405, 739 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 设 " }, { "type": "equation_inline", "content": "x = \\tan t" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2}} \\frac{1}{1 + \\tan^{2022} x} \\mathrm{d}t" }, { "type": "text", "content": "，用 (1) 结论可知为 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{4}" } ] }, "bbox": [ 122, 739, 601, 769 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 令 " }, { "type": "equation_inline", "content": "x = 2 \\tan t" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "I = \\int_{0}^{\\frac{\\pi}{2}} \\frac{\\ln(2 \\tan t)}{4 \\sec^{2} t} 2 \\sec^{2} t \\, \\mathrm{d}t = \\frac{\\pi}{4} \\ln 2 + \\frac{1}{2} \\int_{0}^{\\frac{\\pi}{2}} \\ln(\\tan t) \\, \\mathrm{d}t = \\frac{\\pi}{4} \\ln 2" } ] }, "bbox": [ 122, 769, 779, 801 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.2 三角函数定积分计算" } ] }, "bbox": [ 721, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "57" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.12 (Wallis 推广)" } ], "level": 2 }, "bbox": [ 122, 86, 295, 102 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} & {I (m, n) = \\int_ {0} ^ {\\frac {\\pi}{2}} \\cos^ {m} x \\sin^ {n} x \\mathrm{d} x, \\text {则} I (m, n) = I (n, m), \\text {且}} \\\\ & {\\qquad I (m, n) = \\frac {m - 1}{m + n} I (m - 2, n) = \\frac {n - 1}{m + n} I (m, n - 2) = \\left\\{ \\begin{array}{l l} {\\frac {(m - 1) ! ! (n - 1) ! !}{(m + n) ! !},} & {m, n \\text {不全为偶数}} \\\\ {\\frac {(m - 1) ! ! (n - 1) ! !}{(m + n) ! !} \\frac {\\pi}{2},} & {m, n \\text {均为偶数}} \\end{array} \\right.} \\end{array}", "math_type": "latex", "image_source": { "path": "images/6dc9dc867a2836ee90a1ef39625fd60be8911227e667a4304bd902efb3e706fd.jpg" } }, "bbox": [ 112, 105, 840, 205 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.12基础练习 " }, { "type": "equation_inline", "content": "(1)\\int_{0}^{\\frac{\\pi}{2}}\\cos^{3}x\\sin^{2}x\\mathrm{d}x" } ] }, "bbox": [ 58, 231, 400, 266 ] }, { "type": "equation_interline", "content": { "math_content": "(3) \\int_ {0} ^ {\\frac {\\pi}{2}} \\sin^ {3} \\cos^ {5} x \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/5c48fd3c1a276848c5d32ca6237bdca49a6ec6885b30ca5faffaa245dfc746c7.jpg" } }, "bbox": [ 124, 297, 277, 326 ] }, { "type": "equation_interline", "content": { "math_content": "(4) \\int_ {0} ^ {\\frac {\\pi}{2}} \\sin^ {2} x \\cos^ {4} x \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/9f729f92fc984386dd94deb42fd3f1fa7da05536c8347229c070113b4d6adc95.jpg" } }, "bbox": [ 124, 326, 285, 357 ] }, { "type": "equation_interline", "content": { "math_content": "\\text {解} (1) I (3, 2) = \\frac {2 ! ! \\cdot 1 ! !}{5 ! ! _ {3 \\cdot 2}} = \\frac {2 \\cdot 1}{5 \\cdot 3 \\cdot 1} = \\frac {2}{1 5}", "math_type": "latex", "image_source": { "path": "images/e269c59940102059c49d2be0d61ede686f661715cf869bd9ec31bfdf32d809d6.jpg" } }, "bbox": [ 90, 355, 374, 391 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} (2) I (4, 3) = \\frac {5 \\cdot 2}{7 \\cdot 5 \\cdot 3} = \\frac {2}{3 5} \\\\ (2) I (3, 5) = \\frac {2 \\cdot 4 \\cdot 2}{1} \\end{array}", "math_type": "latex", "image_source": { "path": "images/2906878721c36bb29807ebfc495e39a339e3b61d3685a319aecfffdc42da00db.jpg" } }, "bbox": [ 122, 387, 329, 418 ] }, { "type": "equation_interline", "content": { "math_content": "(3) I (3, 5) = \\frac {2 \\cdot 4 \\cdot 2}{8 \\cdot 6 \\cdot 4 \\cdot 2} = \\frac {1}{2 4}", "math_type": "latex", "image_source": { "path": "images/7da464a3a5f8fc4723dd8a357248e4419fe065915eb441408714ba7cfe23313a.jpg" } }, "bbox": [ 122, 412, 332, 437 ] }, { "type": "equation_interline", "content": { "math_content": "(4) I (2, 4) = \\frac {8 \\cdot 6 \\cdot 4 \\cdot 2}{1 \\cdot 3 \\cdot 1} \\frac {\\pi}{2} = \\frac {2 4}{\\pi}", "math_type": "latex", "image_source": { "path": "images/cb54d188ff7b843d9d5765fc7f520cf541383b7c944cdfdf26fc7b6604f5654e.jpg" } }, "bbox": [ 122, 424, 332, 460 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.2.2 " }, { "type": "equation_inline", "content": "[0, \\pi], [0, 2\\pi]" }, { "type": "text", "content": " 上三角函数积分" } ], "level": 2 }, "bbox": [ 92, 476, 400, 495 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.13 (对称性的积分)" } ], "level": 2 }, "bbox": [ 122, 510, 312, 524 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "(x_0,0)" }, { "type": "text", "content": " 中心对称，则 " }, { "type": "equation_inline", "content": "\\int_{x_0 - a}^{x_0 + a}f(x)\\mathrm{d}x = 0" }, { "type": "text", "content": " 。若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "x = x_0" }, { "type": "text", "content": " 轴对称，则 " }, { "type": "equation_inline", "content": "\\int_{x_0 - a}^{x_0 + a}f(x)\\mathrm{d}x = 2\\int_{x_0 - a}^{x_0}f(x)\\mathrm{d}x = 2\\int_{x_0}^{x_0 + a}f(x)\\mathrm{d}x" } ] }, "bbox": [ 112, 527, 884, 598 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 若 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 都关于 " }, { "type": "equation_inline", "content": "x = x_0" }, { "type": "text", "content": " 轴对称，则 " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 也关于 " }, { "type": "equation_inline", "content": "x = x_0" }, { "type": "text", "content": " 轴对称。 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 轴对称， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 中心对称，则 " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 中心对称。 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 都中心对称，则 " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 轴对称。" } ] }, "bbox": [ 89, 614, 905, 649 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.14 ([0, π] 上 Wallis 公式推广)" } ], "level": 2 }, "bbox": [ 122, 659, 388, 675 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对任意非负整数 " }, { "type": "equation_inline", "content": "m, n" }, { "type": "text", "content": "，有" } ] }, "bbox": [ 115, 680, 307, 695 ] }, { "type": "equation_interline", "content": { "math_content": "J (m, n) = \\int_ {0} ^ {\\pi} \\sin^ {m} x \\cos^ {n} x \\mathrm{d} x = {\\left\\{ \\begin{array}{l l} {0,} & {n \\text {为奇数}} \\\\ {2 I (m, n),} & {n \\text {为偶数}} \\end{array} \\right.}", "math_type": "latex", "image_source": { "path": "images/4fa7f549406f4f59da611e74d88bd033bdf6e7e8d6eabb05c2320dd0b2c8e703.jpg" } }, "bbox": [ 295, 700, 700, 744 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "注意这里 " }, { "type": "equation_inline", "content": "J(m,n)\\neq J(n,m)" } ] }, "bbox": [ 115, 747, 324, 763 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 " }, { "type": "equation_inline", "content": "\\sin^m x\\cos^n x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 为偶关于 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{2}" }, { "type": "text", "content": " 轴对称， " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 为奇时中心对称。" } ] }, "bbox": [ 90, 778, 603, 804 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.15 ( " }, { "type": "equation_inline", "content": "[0, 2\\pi]" }, { "type": "text", "content": " 上 Wallis 公式推广)" } ], "level": 2 }, "bbox": [ 122, 815, 396, 831 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对任意非负 " }, { "type": "equation_inline", "content": "m, n" }, { "type": "text", "content": "，有" } ] }, "bbox": [ 115, 837, 273, 851 ] }, { "type": "equation_interline", "content": { "math_content": "K (m, n) = \\int_ {0} ^ {2 \\pi} \\sin^ {m} x \\cos^ {n} x \\mathrm{d} x = {\\left\\{ \\begin{array}{l l} {0,} & {m, n \\text {不全为偶数}} \\\\ {4 I (m, n),} & {m, n \\text {均为偶数}} \\end{array} \\right.}", "math_type": "latex", "image_source": { "path": "images/2127b6bac085627a24ffc2c128eabb69a9edc5f2e36852d7bf08acc1114868f8.jpg" } }, "bbox": [ 260, 854, 734, 901 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.2 三角函数定积分计算" } ] }, "bbox": [ 721, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "58" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这里 " }, { "type": "equation_inline", "content": "K(m,n) = K(n,m)" } ] }, "bbox": [ 115, 93, 302, 111 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.13(0～π)上的计算(1) " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi}\\cos^{3}x\\sin^{2}xdx(2)\\int_{0}^{\\pi}\\cos^{4}x\\sin^{3}xdx(3)\\int_{0}^{\\pi}\\sin^{3}x\\cos^{5}xdx(4)\\int_{0}^{\\pi}\\sin^{2}x\\cos^{4}xdx" } ] }, "bbox": [ 58, 140, 907, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1) " }, { "type": "equation_inline", "content": "\\cos x" }, { "type": "text", "content": " 是奇数幂，因此为0" } ] }, "bbox": [ 89, 186, 324, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 偶数幂，因此 " }, { "type": "equation_inline", "content": "2I(4,3) = 2\\frac{3 \\cdot 2}{7 \\cdot 5 \\cdot 3} = \\frac{4}{35}" } ] }, "bbox": [ 121, 198, 440, 227 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 奇数，为 0" } ] }, "bbox": [ 122, 223, 235, 237 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 偶数幂，但 " }, { "type": "equation_inline", "content": "m, n" }, { "type": "text", "content": " 均为偶数，因此结果为 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{16}" } ] }, "bbox": [ 122, 237, 470, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.14(0 ~ 2π) 上的计算 (1) " }, { "type": "equation_inline", "content": "\\int_{0}^{2\\pi}\\cos^{3}x\\sin^{2}xdx" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_{0}^{2\\pi}\\cos^{2}x\\sin^{3}xdx" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_{0}^{2\\pi}\\sin^{2}x\\cos^{4}xdx" }, { "type": "text", "content": " 解 (1) 不全为偶，因此为 0 (2)0 (3) 均为偶数，结果为 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{8}" } ] }, "bbox": [ 58, 268, 791, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 周期函数的积分区域可以转换至任意周期区间，因此其他区间也可计算。" } ] }, "bbox": [ 55, 338, 660, 356 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.15非标准区间 (1) " }, { "type": "equation_inline", "content": "\\int_{-\\pi}^{\\pi}\\cos^4 x\\sin^4 x\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_{-\\pi}^{3\\pi}\\sin^5 x\\cos^6 x\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_{-\\pi}^{0}\\cos^3 x\\sin^2 x\\mathrm{d}x" }, { "type": "text", "content": " (4) " }, { "type": "equation_inline", "content": "\\int_{-\\pi}^{0}\\cos^4 x\\sin^3 x\\mathrm{d}x" }, { "type": "text", "content": " (5) " }, { "type": "equation_inline", "content": "\\int_{\\pi}^{2\\pi}\\sin^2 x\\cos^4 x\\mathrm{d}x" }, { "type": "text", "content": " (6) " }, { "type": "equation_inline", "content": "\\int_0^\\pi |\\cos^5 x|\\mathrm{d}x" }, { "type": "text", "content": " 解(1) " }, { "type": "equation_inline", "content": "2J(4,4) = 4I(4,4) = \\frac{3}{64}\\pi" } ] }, "bbox": [ 58, 354, 905, 439 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 先转换到 " }, { "type": "equation_inline", "content": "[0, 4\\pi]" }, { "type": "text", "content": "，再变成 2 倍的 " }, { "type": "equation_inline", "content": "[0, 2\\pi]" }, { "type": "text", "content": "，非全为偶，因此结果为 0" } ] }, "bbox": [ 121, 439, 631, 456 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 偶函数，转换为 " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\cos^3 x\\sin^2 x\\mathrm{d}x = 0" } ] }, "bbox": [ 122, 455, 440, 483 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 奇函数，转换为 " }, { "type": "equation_inline", "content": "-\\int_{0}^{\\pi}\\cos^{4}x\\sin^{2}xdx=-2I(4,3)=-\\frac{4}{35}" } ] }, "bbox": [ 122, 483, 569, 513 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 要看 " }, { "type": "equation_inline", "content": "\\sin" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = \\pi" }, { "type": "text", "content": " 的对称性，其是中心对称，因此等价于 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi} \\sin^{2} x \\cos^{4} x dx = \\frac{\\pi}{16}" } ] }, "bbox": [ 122, 513, 747, 542 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(6) 绝对值肯定轴对称，因此 " }, { "type": "equation_inline", "content": "2I(5,0) = \\frac{16}{15}" } ] }, "bbox": [ 122, 539, 440, 568 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.16其他题目证明： " }, { "type": "equation_inline", "content": "\\int_0^{2\\pi}f(a\\sin x + b\\cos x)\\mathrm{d}x = 2\\int_0^\\pi f(\\sqrt{a^2 + b^2}\\cos x)\\mathrm{d}x" }, { "type": "text", "content": " 证明根据 " }, { "type": "equation_inline", "content": "a\\sin x + b\\cos x = \\sqrt{a^2 + b^2}\\cos (x + \\varphi)" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "\\cos \\varphi = \\frac{b}{\\sqrt{a^2 + b^2}}" }, { "type": "text", "content": " 。因此 " }, { "type": "equation_inline", "content": "\\int_0^{2\\pi}f(\\sqrt{a^2 + b^2}\\cos (x + y))\\mathrm{d}x = \\int_y^{2\\pi +y}f(\\sqrt{a^2 + b^2}\\cos t)\\mathrm{d}t = \\int_0^{2\\pi}f(\\sqrt{a^2 + b^2}\\cos x)\\mathrm{d}x = 2\\int_0^\\pi f(\\sqrt{a^2 + b^2}\\cos x)\\mathrm{d}x" } ] }, "bbox": [ 58, 574, 942, 676 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.2.3 区间再现公式" } ], "level": 2 }, "bbox": [ 94, 708, 273, 725 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.16 (区间再现)" } ], "level": 2 }, "bbox": [ 122, 741, 278, 756 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为连续函数， " }, { "type": "equation_inline", "content": "\\int_0^\\pi xf(\\sin x)\\mathrm{d}x = \\frac{\\pi}{2}\\int_0^\\pi f(\\sin x)\\mathrm{d}x = \\pi \\int_0^{\\frac{\\pi}{2}}f(\\sin x)\\mathrm{d}x" } ] }, "bbox": [ 112, 760, 655, 795 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 令 " }, { "type": "equation_inline", "content": "x = \\pi - t" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "I = \\int_{\\pi}^{0} (\\pi - t) f(\\sin t)(-\\mathrm{d}t) = \\pi \\int_{0}^{\\pi} f(\\sin t) \\mathrm{d}t - \\int_{0}^{\\pi} t f(\\sin t) \\mathrm{d}t" }, { "type": "text", "content": "，因此得到 " }, { "type": "equation_inline", "content": "I = \\frac{\\pi}{2} \\int_{0}^{\\pi} f(\\sin t) \\mathrm{d}t" } ] }, "bbox": [ 89, 804, 905, 868 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.17区间再现公式的应用 (1) " }, { "type": "equation_inline", "content": "\\int_0^\\pi x\\ln \\sin x\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\frac{x\\sin x}{1 + \\cos^2x}\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\frac{x}{1 + \\cos^2x}\\mathrm{d}x" } ] }, "bbox": [ 58, 872, 747, 904 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.2 三角函数定积分计算" } ] }, "bbox": [ 721, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "59" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)变为 " }, { "type": "equation_inline", "content": "I = \\frac{\\pi}{2}\\int_0^\\pi \\ln \\sin x dx = -\\frac{\\pi^2}{2}\\ln 2" } ] }, "bbox": [ 89, 82, 410, 112 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "I=\\frac{\\pi}{2}\\int_{0}^{\\pi}\\frac{\\sin x}{1+\\cos^{2}x}dx=-\\frac{\\pi}{2}\\arctan\\cos x|_{0}^{\\pi}=\\frac{\\pi^{2}}{4}" } ] }, "bbox": [ 122, 112, 514, 142 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {\\pi} \\frac {x}{1 + \\cos^ {2} x} d x = \\frac {\\pi}{2} \\int_ {0} ^ {\\pi} \\frac {1}{1 + \\cos^ {2} x} d x = \\pi \\int_ {0} ^ {\\frac {\\pi}{2}} \\frac {\\sec^ {2} x}{\\sec^ {2} x + 1} d x = \\pi \\int_ {0} ^ {\\frac {\\pi}{2}} \\frac {d (\\tan x)}{\\tan^ {2} x + 2} = \\frac {\\pi^ {2}}{2 \\sqrt {2}}", "math_type": "latex", "image_source": { "path": "images/9069396918c2667c17ca82992a0db3118b9c27ee441a31f89dccc2fddb731edb.jpg" } }, "bbox": [ 122, 141, 825, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4)本题区间比较麻烦，不能用区间再现。可尝试用三角变换： " }, { "type": "equation_inline", "content": "I = \\int_{0}^{\\frac{1}{4}}\\frac{x}{2\\cos^{2}x} dx = \\frac{1}{2}\\int_{0}^{\\frac{1}{4}}xd(\\tan x)" }, { "type": "text", "content": "，用分部积分得到" } ] }, "bbox": [ 90, 175, 905, 217 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {1}{2} x \\tan x \\bigg | _ {0} ^ {\\frac {\\pi}{4}} + \\frac {1}{2} \\ln \\cos x \\bigg | _ {0} ^ {\\frac {\\pi}{4}} = \\frac {\\pi}{8} + \\frac {1}{2} \\ln \\frac {\\sqrt {2}}{2}", "math_type": "latex", "image_source": { "path": "images/8b82e51d162d38d188dfc87edd9143b73918d68ea3661e064881b5c8fce87096.jpg" } }, "bbox": [ 344, 215, 650, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记上面几个虽然并非严格的 " }, { "type": "equation_inline", "content": "xf(\\sin x)" }, { "type": "text", "content": " 形式，但是也可以使用区间再现。" } ] }, "bbox": [ 58, 261, 633, 279 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.2.4 Euler 积分" } ], "level": 2 }, "bbox": [ 92, 300, 245, 317 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.18Euler积分(1)Euler积分: " }, { "type": "equation_inline", "content": "I=\\int_{0}^{\\frac{\\pi}{2}}\\ln\\sin x\\mathrm{d}x=-\\frac{\\pi}{2}\\ln2" }, { "type": "text", "content": " (2)推论: " }, { "type": "equation_inline", "content": "\\int_{0}^{\\frac{\\pi}{2}}\\ln\\cos x\\mathrm{d}x=-\\frac{\\pi}{2}\\ln2,\\int_{0}^{\\frac{\\pi}{2}}\\ln\\tan x\\mathrm{d}x=0,\\int_{0}^{\\pi}\\ln\\sin x\\mathrm{d}x=-\\pi\\ln2" } ] }, "bbox": [ 58, 326, 921, 387 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)该积分是瑕积分, " }, { "type": "equation_inline", "content": "I=\\int_{0}^{\\frac{\\pi}{2}}\\ln\\sin xdx=\\int_{0}^{\\frac{\\pi}{2}}\\ln\\cos xdx=\\frac{1}{2}\\int_{0}^{\\frac{\\pi}{2}}\\ln\\sin x+\\ln\\cos xdx=\\frac{1}{2}\\int_{0}^{\\frac{\\pi}{2}}\\ln\\sin x\\cos xdx" }, { "type": "text", "content": " ,\n根据倍角公式得到" } ] }, "bbox": [ 87, 385, 907, 432 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {1}{2} \\int_ {0} ^ {\\frac {\\pi}{2}} \\ln \\frac {1}{2} \\sin 2 x \\mathrm{d} x = \\frac {1}{2} \\int_ {0} ^ {\\frac {\\pi}{2}} \\ln \\sin 2 x \\mathrm{d} x - \\frac {\\pi}{4} \\ln 2 = \\frac {1}{4} \\int_ {0} ^ {\\pi} \\ln \\sin t \\mathrm{d} t - \\frac {\\pi}{4} \\ln 2 = \\frac {1}{2} I - \\frac {\\pi}{4} \\ln 2", "math_type": "latex", "image_source": { "path": "images/b185cb851fcb86ff5546e7a9c40fcf19998c787fb49b68cf5d6e8b8ee5d94d0f.jpg" } }, "bbox": [ 156, 434, 840, 468 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此得到 " }, { "type": "equation_inline", "content": "I = -\\frac{\\pi}{2}\\ln 2" } ] }, "bbox": [ 92, 470, 255, 495 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记要证明Euler积分收敛，只需证明 " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0^{+}}x\\ln \\sin x = 0" } ] }, "bbox": [ 55, 495, 517, 518 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.19Euler积分的应用 (1) " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2}}x\\cot x\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{\\ln x}{\\sqrt{1 - x^2}}\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{\\arcsin x}{x}\\mathrm{d}x" }, { "type": "text", "content": " (4) " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\frac{x\\sin x}{1 - \\cos x}\\mathrm{d}x" } ] }, "bbox": [ 58, 527, 818, 560 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {\\frac {\\pi}{2}} x \\mathrm{d} (\\ln \\sin x) = x \\ln \\sin x \\big | _ {0} ^ {\\frac {\\pi}{2}} - \\int_ {0} ^ {\\frac {\\pi}{2}} \\ln \\sin x \\mathrm{d} x = \\frac {\\pi}{2} \\ln 2", "math_type": "latex", "image_source": { "path": "images/538d31c2eaf5be806c7d9533980c8ab73defa19fa98fd9130209215cbaaa6dfc.jpg" } }, "bbox": [ 110, 558, 566, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 令 " }, { "type": "equation_inline", "content": "x = \\sin t" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "I = \\int_{0}^{\\frac{\\pi}{2}} \\frac{\\ln \\sin t}{\\cos t} \\cos t \\, \\mathrm{d}t = -\\frac{\\pi}{2} \\ln 2" } ] }, "bbox": [ 121, 590, 532, 621 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 令 " }, { "type": "equation_inline", "content": "x = \\sin t" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\frac{\\pi}{2}} \\frac{t}{\\sin t} \\cos t \\, \\mathrm{d}t = \\int_{0}^{\\frac{\\pi}{2}} t \\cot t \\, \\mathrm{d}t" }, { "type": "text", "content": "，即 (1)" } ] }, "bbox": [ 122, 621, 571, 653 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 注意 " }, { "type": "equation_inline", "content": "\\cot \\frac{x}{2} = \\frac{\\sin x}{1 - \\cos x}" }, { "type": "text", "content": " (或者可以当场对 " }, { "type": "equation_inline", "content": "1 - \\cos x" }, { "type": "text", "content": " 展开)，因此 " }, { "type": "equation_inline", "content": "I = \\int_{0}^{\\pi} x \\cot \\frac{x}{2} \\mathrm{d}x = 4 \\int_{0}^{\\frac{\\pi}{2}} t \\cot(t) \\mathrm{d}t" }, { "type": "text", "content": "，变为 (1)" } ] }, "bbox": [ 90, 652, 902, 697 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.2.5 Dirichlet 积分" } ], "level": 2 }, "bbox": [ 94, 738, 273, 753 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.17 (Dirichlet 积分)" } ], "level": 2 }, "bbox": [ 122, 770, 312, 784 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Dirichlet 积分 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi}\\frac{\\sin\\frac{2n+1}{2}x}{2\\sin\\frac{x}{2}}=\\frac{\\pi}{2}dx" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 114, 790, 384, 822 ] }, { "type": "equation_interline", "content": { "math_content": "\\mathrm{推论} \\colon \\int_ {0} ^ {\\pi} {\\frac {\\sin {\\frac {2 n + 1}{2}} x}{\\sin {\\frac {x}{2}}}} \\mathrm{d} x = \\pi , \\int_ {0} ^ {\\frac {\\pi}{2}} {\\frac {\\sin (2 n + 1) x}{\\sin x}} \\mathrm{d} x = {\\frac {\\pi}{2}}, \\int_ {0} ^ {\\frac {\\pi}{2}} {\\frac {\\sin (2 n) x}{\\sin x}} \\mathrm{d} x = 0, \\int_ {0} ^ {\\pi} {\\frac {\\sin (2 n + 1) x}{\\sin x}} \\mathrm{d} x = \\pi", "math_type": "latex", "image_source": { "path": "images/eb3a9e7aa146fe211082a5b25b0c6fafe061ac22011e84d098574ea3ff048e2e.jpg" } }, "bbox": [ 115, 822, 850, 856 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.2 三角函数定积分计算" } ] }, "bbox": [ 721, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "60" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1)Dirichlet: 关键在于 " }, { "type": "equation_inline", "content": "\\cos x + \\cos 2x + \\cdots + \\cos nx = \\frac{\\sin(n + \\frac{1}{2})x - \\sin\\frac{x}{2}}{2\\sin\\frac{x}{2}} = \\frac{\\sin\\frac{2n+1}{2}x}{2\\sin\\frac{x}{2}} - \\frac{1}{2}" }, { "type": "text", "content": " ，从而得到：\n " }, { "type": "equation_inline", "content": "\\int_{0}^{\\pi}\\frac{\\sin\\frac{2n+1}{2}x}{2\\sin\\frac{x}{2}}dx=\\sum_{k=1}^{n}\\int_{0}^{\\pi}\\cos kxdx+\\frac{\\pi}{2}=\\sum_{k=1}^{n}\\left.\\frac{1}{k}\\sin kx\\right|_{0}^{\\pi}+\\frac{\\pi}{2}=\\frac{\\pi}{2}" } ] }, "bbox": [ 90, 80, 895, 156 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 推论：第一个就是把分母 2 拿走，第二个 (分子奇数情况) 取 " }, { "type": "equation_inline", "content": "t = 2x" }, { "type": "text", "content": " ，第四个用对称性" } ] }, "bbox": [ 121, 160, 774, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 第三个设 " }, { "type": "equation_inline", "content": "I_{n} = \\int_{0}^{\\pi}\\frac{\\sin 2nx}{\\sin x}\\mathrm{d}x," } ] }, "bbox": [ 122, 174, 373, 203 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{c} I _ {n} - I _ {n - 1} = \\int_ {0} ^ {\\pi} \\frac {\\sin 2 n x - \\sin (2 n - 2) x}{\\sin x} d x = \\int_ {0} ^ {\\pi} \\frac {2}{\\sin x} (\\cos \\frac {4 n - 2}{2} x \\cdot \\sin \\frac {2 n - (2 n - 2)}{2} x) \\mathrm{d} x \\\\ = 2 \\int_ {0} ^ {\\pi} \\cos (2 n - 1) x \\mathrm{d} x = \\frac {2}{2 n - 1} \\sin (2 n - 1) x \\bigg | _ {0} ^ {\\pi} = 0 \\end{array}", "math_type": "latex", "image_source": { "path": "images/a8c87a70c4a9ae2751e02a8bd84e0b5d19b4b8981f68511c5f0390ab9f871591.jpg" } }, "bbox": [ 163, 204, 831, 271 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "I_{1} = 0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "I_{n} = 0" } ] }, "bbox": [ 90, 274, 265, 288 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.2.6 其他几个常见三角积分" } ], "level": 2 }, "bbox": [ 92, 310, 354, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.20几个常见三角分数积分(1) " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\frac{dx}{1 + \\sin^2x} dx" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_0^{2\\pi}\\frac{dx}{1 + a\\cos x},(-1 < a < 1)" }, { "type": "text", "content": " 解(1)上下同乘 " }, { "type": "equation_inline", "content": "\\sec^2 x" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "\\int \\frac{\\sec^2x}{\\sec^2x + \\tan^2x} dx = \\int \\frac{d(\\tan x)}{2\\tan^2x + 1} = \\frac{1}{\\sqrt{2}}\\arctan (\\sqrt{2}\\tan x) + c," }, { "type": "text", "content": " 注意 " }, { "type": "equation_inline", "content": "\\tan x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{2}" }, { "type": "text", "content": " 是间断的！根据对称性转换为 " }, { "type": "equation_inline", "content": "I = 2\\int_0^{\\frac{\\pi}{2}}\\frac{dx}{1 + \\sin^2x} = \\sqrt{2}\\arctan (\\sqrt{2}\\tan x)|_0^{\\frac{\\pi}{2}} = \\sqrt{2} (\\frac{\\pi}{2} -0) = 2\\frac{\\pi}{\\sqrt{2}}" } ] }, "bbox": [ 58, 337, 949, 432 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)具体的不定积分计算见不定积分一节,是非常经典的积分。 " }, { "type": "equation_inline", "content": "I=2\\int_{0}^{\\pi}\\frac{dx}{1+a\\cos x}=\\frac{4}{\\sqrt{1-a^{2}}}\\arctan\\sqrt{\\frac{1-a}{1+a}}\\tan\\frac{x}{2}|_{0}^{\\pi}" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "x=\\pi" }, { "type": "text", "content": " 为间断点，使用对称性非常正确！因此结果为 " }, { "type": "equation_inline", "content": "\\frac{2\\pi}{\\sqrt{1-a^{2}}}" } ] }, "bbox": [ 90, 431, 976, 492 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 一定要注意函数的连续性，用对称性转换。因此能用对称性的一定要优先用对称性。上面两个题目也是重要的结论。" } ] }, "bbox": [ 87, 494, 905, 527 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.3 积分 (不) 等式" } ], "level": 2 }, "bbox": [ 92, 573, 292, 592 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.3.1 整体放缩" } ], "level": 2 }, "bbox": [ 92, 612, 233, 629 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.21几个简单积分放缩证明(1) " }, { "type": "equation_inline", "content": "1<\\int_{0}^{\\frac{\\pi}{2}}\\frac{\\sin x}{x}\\mathrm{d}x<\\frac{\\pi}{2}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\frac{2}{9}\\pi^{2}<\\int_{\\frac{\\pi}{6}}^{\\frac{\\pi}{2}}\\frac{2x}{\\sin x}\\mathrm{d}x<\\frac{1}{3}\\pi^{2}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "3\\sqrt{e}<\\int_{e}^{4e}\\frac{\\ln x}{\\sqrt{x}}<6" }, { "type": "text", "content": " 证明(1)根据 " }, { "type": "equation_inline", "content": "f'(x)=\\frac{x-\\tan x}{x^{2}\\sec x}" }, { "type": "text", "content": " 可知函数单减，因此" } ] }, "bbox": [ 58, 638, 902, 701 ] }, { "type": "equation_interline", "content": { "math_content": "f \\left(\\frac {\\pi}{2}\\right) = \\frac {2}{\\pi} \\leq \\frac {\\sin x}{x} < f (0) = 1", "math_type": "latex", "image_source": { "path": "images/4f906acf4835c8772b69b59173d5be0109228d3d8d848b2a08cb764109251913.jpg" } }, "bbox": [ 379, 702, 616, 731 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据导数可知单调增，因此" } ] }, "bbox": [ 122, 731, 356, 746 ] }, { "type": "equation_interline", "content": { "math_content": "f \\left(\\frac {\\pi}{6}\\right) = \\frac {2}{3} \\pi \\leq \\frac {2 x}{\\sin x} \\leq f \\left(\\frac {\\pi}{2}\\right) = \\pi", "math_type": "latex", "image_source": { "path": "images/6b5fc4baa49a6a51cac75c8b1631c60539e2e287a14f9c40a798a8413d1919da.jpg" } }, "bbox": [ 366, 747, 628, 776 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f'(x)=\\frac{2-\\ln x}{2x\\sqrt{x}}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\frac{1}{\\sqrt{e}}=f(e)\\leq f(x)\\leq f(e^{2})=\\frac{2}{e}" } ] }, "bbox": [ 121, 777, 552, 809 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.3.2 同区域单调性放缩" } ], "level": 2 }, "bbox": [ 92, 828, 315, 844 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.22几道经典同区域相减(1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上严格增的连续函数，证明： " }, { "type": "equation_inline", "content": "\\int_{a}^{b}xf(x)\\mathrm{d}x > \\frac{a + b}{2}\\int_{a}^{b}f(x)\\mathrm{d}x" }, { "type": "text", "content": " (2)证明 " }, { "type": "equation_inline", "content": "\\int_0^{\\frac{\\pi}{2}}\\frac{\\cos x}{1 + x^2}\\mathrm{d}x > \\int_0^{\\frac{\\pi}{2}}\\frac{\\sin x}{1 + x^2}\\mathrm{d}x" }, { "type": "text", "content": " (3)证明 " }, { "type": "equation_inline", "content": "\\int_0^{\\sqrt{2\\pi}}\\sin x^2\\mathrm{d}x > 0" } ] }, "bbox": [ 58, 854, 887, 922 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.3 积分 (不) 等式" } ] }, "bbox": [ 768, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "61" } ] }, "bbox": [ 487, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 等价于证明 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}\\left(x - \\frac{a + b}{2}\\right)f(x)\\mathrm{d}x > 0" }, { "type": "text", "content": "，进行有目的性的换元，分为两段，用 " }, { "type": "equation_inline", "content": "t" }, { "type": "text", "content": " 表示 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\frac{a + b}{2}" }, { "type": "text", "content": " 之间的距离：" } ] }, "bbox": [ 90, 82, 904, 128 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\int_ {a} ^ {b} \\left(x - \\frac {a + b}{2}\\right) f (x) \\mathrm{d} x = \\int_ {a} ^ {\\frac {a + b}{2}} \\left(x - \\frac {a + b}{2}\\right) f (x) \\mathrm{d} x + \\int_ {\\frac {a + b}{2}} ^ {b} \\left(x - \\frac {a + b}{2}\\right) f (x) \\mathrm{d} x \\\\ = \\int_ {\\frac {b - a}{2}} ^ {0} (- t) f (\\frac {a + b}{2} - t) (- \\mathrm{d} t) + \\int_ {0} ^ {\\frac {b - a}{2}} t f (t + \\frac {a + b}{2}) \\mathrm{d} t \\\\ = \\int_ {0} ^ {\\frac {b - a}{2}} t \\left[ f (t + \\frac {a + b}{2}) - f (\\frac {a + b}{2} - t) \\right] \\mathrm{d} t > 0 \\end{array}", "math_type": "latex", "image_source": { "path": "images/35c156e78d04f4cf23398e333f14094c1daaa15e04425a4a8c1b289347311dc8.jpg" } }, "bbox": [ 194, 131, 801, 244 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 等价于 " }, { "type": "equation_inline", "content": "\\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin x - \\cos x}{1 + x^2} \\mathrm{d}x < 0" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\sin x - \\cos x = \\sqrt{2} \\sin (x - \\frac{\\pi}{4})" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "\\left(\\frac{\\pi}{4}, 0\\right)" }, { "type": "text", "content": " 中心对称，且 " }, { "type": "equation_inline", "content": "\\frac{1}{1 + x^2}" }, { "type": "text", "content": " 单减。因此令 " }, { "type": "equation_inline", "content": "u = x - \\frac{\\pi}{4}" }, { "type": "text", "content": ":" } ] }, "bbox": [ 89, 248, 904, 300 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} & {\\int_ {0} ^ {\\frac {\\pi}{2}} \\frac {\\sin x - \\cos x}{1 + x ^ {2}} \\mathrm{d} x = \\int_ {0} ^ {\\frac {\\pi}{2}} \\frac {\\sqrt {2} \\sin (x - \\frac {\\pi}{4})}{1 + x ^ {2}} \\mathrm{d} x = \\sqrt {2} \\int_ {- \\frac {\\pi}{4}} ^ {\\frac {\\pi}{4}} \\int_ {- \\frac {\\pi}{4}} ^ {\\frac {\\pi}{4}} \\frac {\\sin u}{1 + (\\frac {\\pi}{4} + u) ^ {2}} \\mathrm{d} u} \\\\ & {\\qquad = \\sqrt {2} \\left(\\int_ {0} ^ {\\frac {\\pi}{4}} \\frac {\\sin x}{1 + (\\frac {\\pi}{4} + x) ^ {2}} \\mathrm{d} x - \\int_ {0} ^ {\\frac {\\pi}{4}} \\frac {\\sin x}{1 + (\\frac {\\pi}{4} - x) ^ {2}} \\mathrm{d} x\\right)} \\\\ & {\\qquad = \\sqrt {2} \\int_ {0} ^ {\\frac {\\pi}{4}} \\left(\\frac {1}{(\\frac {\\pi}{4} + x) ^ {2}} - \\frac {1}{(\\frac {\\pi}{4} - x) ^ {2}}\\right) \\sin x \\mathrm{d} x < 0} \\end{array}", "math_type": "latex", "image_source": { "path": "images/ee74cd2a69cfb0c32ced61e52f9f958d3ea358be9ce9f54966ea88c6cd85ec63.jpg" } }, "bbox": [ 218, 305, 778, 417 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.23一道特殊题(1)证明 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{\\cos x}{\\sqrt{1 - x^2}}\\mathrm{d}x > \\int_0^1\\frac{\\sin x}{\\sqrt{1 - x^2}}\\mathrm{d}x" } ] }, "bbox": [ 58, 425, 564, 460 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.3.3 利用 Newton-Leibniz 公式" } ], "level": 2 }, "bbox": [ 92, 478, 379, 495 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.18 (Newton-Leibniz 放缩)" } ], "level": 2 }, "bbox": [ 121, 510, 364, 525 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上连续可微函数，则对 " }, { "type": "equation_inline", "content": "\\forall x_1,x_2\\in [a,b]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f(x_{2}) = f(x_{1}) + \\int_{x_{1}}^{x_{2}}f^{\\prime}(x)\\mathrm{d}x" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 114, 530, 801, 562 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x _ {2}) | \\leq | f (x _ {1}) | + \\int_ {a} ^ {b} | f ^ {\\prime} (x) | \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/d45cd9d943186df48fc5a23c9b0df981a6096e8ee7599029c5a7dd82169c2811.jpg" } }, "bbox": [ 376, 565, 618, 599 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.24 几道经典题 下面均假设 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在区间上连续可微" } ] }, "bbox": [ 58, 636, 517, 650 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 证明 " }, { "type": "equation_inline", "content": "|f(0)| \\leq \\frac{1}{a} \\int_{0}^{a} |f(x)| \\, \\mathrm{d}x + \\int_{0}^{a} |f'(x)| \\, \\mathrm{d}x" } ] }, "bbox": [ 122, 651, 473, 680 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 经典：证明 " }, { "type": "equation_inline", "content": "|f(x)| \\leq \\left|\\frac{1}{b - a}\\int_{a}^{b} f(x)\\mathrm{d}x\\right| + \\int_{a}^{b}|f'(x)|\\mathrm{d}x" } ] }, "bbox": [ 122, 680, 551, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 证明 " }, { "type": "equation_inline", "content": "|f(x)| \\leq \\int_{0}^{1} (|f(x)| + |f'(x)|) \\, \\mathrm{d}x" } ] }, "bbox": [ 122, 715, 421, 746 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 重点：证明 " }, { "type": "equation_inline", "content": "\\left|f\\left(\\frac{1}{2}\\right)\\right| \\leq \\int_{0}^{1}|f(x)|\\mathrm{d}x + \\frac{1}{2}\\int_{0}^{1}|f'(x)|\\mathrm{d}x" } ] }, "bbox": [ 122, 745, 542, 778 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 用积分第一中值定理可知 " }, { "type": "equation_inline", "content": "\\frac{1}{a} \\int_{0}^{a} |f(x)| \\mathrm{d}x = |f(\\xi)|" }, { "type": "text", "content": "，因此得证" } ] }, "bbox": [ 90, 776, 601, 808 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)(3) 同理" } ] }, "bbox": [ 122, 804, 206, 820 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 设 " }, { "type": "equation_inline", "content": "|f(x_0)| = \\min_{x \\in [0, \\frac{1}{2}]} |f(x)|" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "|f(x_0)| = 2\\int_0^{\\frac{1}{2}} |f(x_0)| \\, \\mathrm{d}x \\leq 2\\int_0^{\\frac{1}{2}} |f(x)| \\, \\mathrm{d}x" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 122, 816, 729, 854 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f \\left(\\frac {1}{2}\\right) \\right| = \\left| f (x _ {0}) + \\int_ {x _ {0}} ^ {\\frac {1}{2}} f ^ {\\prime} (x) \\mathrm{d} x \\right| \\leq 2 \\int_ {0} ^ {\\frac {1}{2}} | f (x) | \\mathrm{d} x + \\int_ {0} ^ {\\frac {1}{2}} | f ^ {\\prime} (x) | \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/8fdba8cd0d9a65ebf6a986cefda5c5f2fe443b083bd05ef424e05f3a602285c4.jpg" } }, "bbox": [ 253, 857, 742, 894 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.3 积分 (不) 等式" } ] }, "bbox": [ 768, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "62" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理取 " }, { "type": "equation_inline", "content": "[\\frac{1}{2}, 1]" }, { "type": "text", "content": " 的部分可得到" } ] }, "bbox": [ 90, 80, 292, 110 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f \\left(\\frac {1}{2}\\right) \\right| \\leq 2 \\int_ {\\frac {1}{2}} ^ {1} | f (x) | \\mathrm{d} x + \\int_ {\\frac {1}{2}} ^ {1} | f ^ {\\prime} (x) | \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/3f34a3e8686c3ba594121fa521826489b4cc0aeaf5692f6312a46bfbc35c991a.jpg" } }, "bbox": [ 344, 105, 650, 142 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此两式相加除以 2 即得到结果。" } ] }, "bbox": [ 90, 149, 339, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.3.4 利用 Cauchy-Schwarz 不等式" } ], "level": 2 }, "bbox": [ 92, 187, 406, 204 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 6.19 (Cauchy-Schwarz 不等式)" } ], "level": 2 }, "bbox": [ 121, 219, 386, 235 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上可积，则" } ] }, "bbox": [ 114, 240, 359, 256 ] }, { "type": "equation_interline", "content": { "math_content": "\\left(\\int_ {a} ^ {b} f (x) g (x) \\mathrm{d} x\\right) ^ {2} \\leq \\int_ {a} ^ {b} f ^ {2} (x) \\mathrm{d} x \\int_ {a} ^ {b} g ^ {2} (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/4d691d178c2514086984633dc50051321f37f6a670af9555ced1ce44123d69b5.jpg" } }, "bbox": [ 329, 259, 670, 299 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论6.2" } ], "level": 2 }, "bbox": [ 124, 324, 188, 338 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " 非负可积，则" } ] }, "bbox": [ 114, 343, 317, 359 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {a} ^ {b} f (x) \\mathrm{d} x \\int_ {a} ^ {b} g (x) \\mathrm{d} x \\geq \\left(\\int_ {a} ^ {b} \\sqrt {f (x) g (x)} \\mathrm{d} x\\right) ^ {2}", "math_type": "latex", "image_source": { "path": "images/e02e3b81f160a0b6b6017c91a1538868e56db918efcf552d83cc25c6234f22e1.jpg" } }, "bbox": [ 324, 362, 670, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.25 一些经典题目 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 [0,1] 连续可微， " }, { "type": "equation_inline", "content": "f(1)=f(0)=1" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\int_{0}^{1}[f'(x)]^{2}\\mathrm{d}x\\geq1" }, { "type": "text", "content": "\n\n证明 (1) 直接根据 Cauchy-Schwarz 不等式得到 " }, { "type": "equation_inline", "content": "\\int_{0}^{1}[f'(x)]^{2}\\mathrm{d}x\\geq\\left[\\int_{0}^{1}f'(x)\\mathrm{d}x\\right]^{2}\\geq\\frac{1}{4}" } ] }, "bbox": [ 58, 436, 771, 500 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6.3.5 利用微分中值定理" } ], "level": 2 }, "bbox": [ 92, 518, 314, 535 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 6.26 几道具体分段题 (1) 一阶导: " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0,2]" }, { "type": "text", "content": " 上可导, " }, { "type": "equation_inline", "content": "f(0)=f(2)=1" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "|f'(x)|\\leq1" }, { "type": "text", "content": " , 证明 " }, { "type": "equation_inline", "content": "1<\\int_{0}^{2}f(x)\\mathrm{d}x<3" }, { "type": "text", "content": " \n(2) 二阶导:" } ] }, "bbox": [ 58, 548, 905, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 左侧：进行分段处理，对于 " }, { "type": "equation_inline", "content": "[0, 1], [1, 2]" }, { "type": "text", "content": " 分别有" } ] }, "bbox": [ 87, 594, 495, 609 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l l} f (x) = f (0) + f ^ {\\prime} (\\xi) x \\geq 1 - x, & x \\in [ 0, 1 ] \\\\ f (x) = f (2) + f ^ {\\prime} (\\eta) (x - 2) \\geq x - 1, & x \\in [ 1, 2 ] \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/939e44834082973eedc9ab0c3cbe40596f42c33497f51de4f190edc79b8fc706.jpg" } }, "bbox": [ 317, 614, 678, 658 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\int_0^2 f(x)\\mathrm{d}x\\geq \\int_0^1 (1 - x)\\mathrm{d}x + \\int_1^2 (x - 1)\\mathrm{d}x = 1" }, { "type": "text", "content": " ，且等号取到时在1处不可导，故严格。同理可证明 " }, { "type": "equation_inline", "content": "f(x)\\geq h(x)" }, { "type": "text", "content": " 其中 " }, { "type": "equation_inline", "content": "h(x)" }, { "type": "text", "content": " 为" } ] }, "bbox": [ 89, 662, 905, 708 ] }, { "type": "equation_interline", "content": { "math_content": "h (x) = \\left\\{ \\begin{array}{l l} 1 + x, & x \\in [ 0, 1 ] \\\\ 3 - x, & x \\in [ 1, 2 ] \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/2d7a51d03d00bda1ba6e312faf840c6a9d65299eae291870fd87567cd2838f0e.jpg" } }, "bbox": [ 396, 708, 598, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\int_0^2 f(x)\\mathrm{d}x\\leq 3" } ] }, "bbox": [ 90, 760, 245, 795 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.27利用一阶导数研究积分 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续可微， " }, { "type": "equation_inline", "content": "M = \\max_{x\\in [a,b]}|f'(x)|" }, { "type": "text", "content": "，证明" } ] }, "bbox": [ 58, 807, 702, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 若 " }, { "type": "equation_inline", "content": "f(a) = 0" }, { "type": "text", "content": " 或 " }, { "type": "equation_inline", "content": "f(b) = 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\left|\\int_{a}^{b} f(x) \\mathrm{d}x\\right| \\leq \\int_{a}^{b} |f(x)| \\mathrm{d}x \\leq \\frac{M}{2}(b - a)^{2}" } ] }, "bbox": [ 122, 831, 660, 866 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 若 " }, { "type": "equation_inline", "content": "f(a) = f(b) = 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\left|\\int_{a}^{b} f(x) \\mathrm{d}x\\right| \\leq \\int_{a}^{b} |f(x)| \\mathrm{d}x \\leq \\frac{M}{4}(b - a)^2" } ] }, "bbox": [ 122, 866, 628, 904 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.3 积分 (不) 等式" } ] }, "bbox": [ 769, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "63" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 若 " }, { "type": "equation_inline", "content": "f\\left(\\frac{a + b}{2}\\right) = 0" } ] }, "bbox": [ 122, 82, 285, 114 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 设 " }, { "type": "equation_inline", "content": "f(a) = 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "f(x) = f'(\\xi)(x - a)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}|f(x)|\\mathrm{d}x \\leq M\\int_{a}^{b}(x - a)\\mathrm{d}x = \\frac{M}{2}(b - a)^2" }, { "type": "text", "content": " (2) 若 " }, { "type": "equation_inline", "content": "f(a) = f(b) = 0" }, { "type": "text", "content": "，则分两段在 " }, { "type": "equation_inline", "content": "a, b" }, { "type": "text", "content": " 展开" } ] }, "bbox": [ 92, 111, 806, 158 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\int_ {a} ^ {\\frac {a + b}{2}} | f (x) | \\mathrm{d} x \\leq \\frac {M}{2} \\cdot \\frac {(b - a) ^ {2}}{4} \\\\ \\int_ {\\frac {a + b}{2}} ^ {b} | f (x) | \\mathrm{d} x \\leq \\frac {M}{2} \\cdot \\frac {(b - a) ^ {2}}{4} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/ed4336db98e04a823a1c437fb02ca97629c9d4e5241be771e8673306e747c938.jpg" } }, "bbox": [ 368, 161, 606, 228 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 若 " }, { "type": "equation_inline", "content": "f\\left(\\frac{a + b}{2}\\right) = 0" }, { "type": "text", "content": "，则分两段在 " }, { "type": "equation_inline", "content": "f\\left(\\frac{a + b}{2}\\right)" }, { "type": "text", "content": " 展开，和 (2) 类似可得到结果" } ] }, "bbox": [ 122, 230, 680, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习6.28利用二阶导数研究积分 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 二阶可微， " }, { "type": "equation_inline", "content": "|f''(x)| \\leq M" }, { "type": "text", "content": "，证明" } ] }, "bbox": [ 58, 275, 662, 291 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 若 " }, { "type": "equation_inline", "content": "f\\left(\\frac{a + b}{2}\\right) = 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\left|\\int_{a}^{b}f(x)\\mathrm{d}x\\right| \\leq \\frac{M}{24}(b - a)^3" } ] }, "bbox": [ 122, 291, 512, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 若 " }, { "type": "equation_inline", "content": "f(a) = f(b) = 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\left|\\int_{a}^{b} f(x) \\, \\mathrm{d}x\\right| \\leq \\frac{M}{12}(b - a)^3" } ] }, "bbox": [ 122, 328, 517, 363 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 在 " }, { "type": "equation_inline", "content": "\\frac{a + b}{2}" }, { "type": "text", "content": " 处展开:" } ] }, "bbox": [ 87, 362, 285, 390 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = f \\left(\\frac {a + b}{2}\\right) + f ^ {\\prime} \\left(\\frac {a + b}{2}\\right) \\left(x - \\frac {a + b}{2}\\right) + \\frac {f ^ {\\prime \\prime} (\\xi)}{2} \\left(x - \\frac {a + b}{2}\\right) ^ {2}", "math_type": "latex", "image_source": { "path": "images/b9186effdc05fdebfa8937c9a6ad50bc4b2e27e2e913b3bbed8bb9610dc5b040.jpg" } }, "bbox": [ 242, 393, 754, 426 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "显然一阶导数项中心对称，故消去，二阶导数项用积分第一中值定理可以得到： " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x=(b-a)f\\left(\\frac{a+b}{2}\\right)+\\frac{f''(\\eta)}{24}(b-a)^{3}" }, { "type": "text", "content": " ，经过放缩即结论。" } ] }, "bbox": [ 90, 430, 904, 489 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "6.3 积分 (不) 等式" } ] }, "bbox": [ 769, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "64" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第7章 反常积分" } ], "level": 2 }, "bbox": [ 396, 87, 618, 111 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "一般的定积分都定义于闭区间中，但很多实际问题中的积分可能区间无穷或者某点不连续，这就需要引出无穷积分和瑕积分的概念。" } ] }, "bbox": [ 87, 140, 905, 174 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.1 反常积分理论" } ], "level": 2 }, "bbox": [ 94, 200, 289, 219 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.1.1 无穷积分与瑕积分概念" } ], "level": 2 }, "bbox": [ 94, 239, 352, 256 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 7.1 (无穷积分与瑕积分)" } ], "level": 2 }, "bbox": [ 122, 272, 337, 287 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 无穷积分：设 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 有定义，若极限 " }, { "type": "equation_inline", "content": "\\lim_{b \\to +\\infty} \\int_{a}^{b} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 存在，则记其为 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": "。" } ] }, "bbox": [ 139, 291, 836, 324 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 瑕积分： " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b]" }, { "type": "text", "content": " 有定义，且 " }, { "type": "equation_inline", "content": "\\lim_{x \\to a^{+}} f(x) = \\infty" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\lim_{\\epsilon \\to 0^{+}} \\int_{a + \\epsilon}^{b} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则记其为 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) \\mathrm{d}x" } ] }, "bbox": [ 139, 323, 875, 359 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 如果 " }, { "type": "equation_inline", "content": "\\lim_{x\\to a^{+}}f(x)\\neq \\infty" }, { "type": "text", "content": " ，则不能称 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的瑕点，比如 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{x}{1 - e^x}\\mathrm{d}x" }, { "type": "text", "content": " ，0不是其瑕点，其就是个定积分。" } ] }, "bbox": [ 87, 369, 895, 401 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.1 (反常积分 Leibniz 公式)" } ], "level": 2 }, "bbox": [ 122, 412, 366, 426 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 无穷积分：设 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 存在，且 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 有原函数 " }, { "type": "equation_inline", "content": "F" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x = F(+\\infty) - F(a)" } ] }, "bbox": [ 139, 432, 803, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 瑕积分：若 " }, { "type": "equation_inline", "content": "F" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b]" }, { "type": "text", "content": " 的一个原函数， " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " 是瑕点，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) \\mathrm{d}x = F(b) - F(a + 0)" } ] }, "bbox": [ 139, 467, 798, 497 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.2 (定义判敛)" } ], "level": 2 }, "bbox": [ 124, 520, 270, 535 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "无穷积分： " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛当且仅当 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists A,\\forall u\\geq A,|\\int_{u}^{+\\infty}f(x)\\mathrm{d}x| < \\epsilon" } ] }, "bbox": [ 139, 539, 710, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 瑕积分： " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛当且仅当 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists \\delta ,\\forall u\\in (a,a + \\delta)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|\\int_{a}^{u}f(x)\\mathrm{d}x| < \\epsilon" } ] }, "bbox": [ 139, 571, 715, 605 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据收敛的定义可知。" } ] }, "bbox": [ 90, 621, 295, 637 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.3 ( " }, { "type": "equation_inline", "content": "\\frac{1}{x^{p}}" }, { "type": "text", "content": " 的反常积分)" } ], "level": 2 }, "bbox": [ 122, 646, 312, 675 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\int_0^1\\frac{\\mathrm{d}x}{x^p}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "p < 1" }, { "type": "text", "content": " 时收敛， " }, { "type": "equation_inline", "content": "p \\geq 1" }, { "type": "text", "content": " 时发散。 " }, { "type": "equation_inline", "content": "\\int_1^{+\\infty}\\frac{\\mathrm{d}x}{x^p}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 时收敛， " }, { "type": "equation_inline", "content": "p \\leq 1" }, { "type": "text", "content": " 时发散" } ] }, "bbox": [ 112, 677, 712, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 只讨论 " }, { "type": "equation_inline", "content": "[1, +\\infty)" }, { "type": "text", "content": " 上的，直接积分：" } ] }, "bbox": [ 89, 731, 376, 747 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {1} ^ {u} \\frac {\\mathrm{d} x}{x ^ {p}} = \\left\\{ \\begin{array}{l l} \\frac {1}{1 - p} (u ^ {1 - p} - 1), & p \\neq 1 \\\\ \\ln u, & p = 1 \\end{array} \\right. \\Rightarrow \\lim _ {u \\to + \\infty} \\int_ {1} ^ {u} \\frac {\\mathrm{d} x}{x ^ {p}} = \\left\\{ \\begin{array}{l l} \\frac {1}{p - 1}, & p > 1 \\\\ + \\infty , & p \\leq 1 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/e8524bd91e7639647bc87a447a5f91d05fc2cd7f9294e351cf6db166fd009f18.jpg" } }, "bbox": [ 226, 751, 766, 803 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.1.2 反常积分比较判别法" } ], "level": 2 }, "bbox": [ 94, 85, 332, 103 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.4 (Cauchy 收敛准则)" } ], "level": 2 }, "bbox": [ 122, 118, 329, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "无穷积分： " }, { "type": "equation_inline", "content": "\\int_{a}^{\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛当且仅当 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists G\\geq a,\\forall u_1,u_2 > G" }, { "type": "text", "content": " ，满足" } ] }, "bbox": [ 139, 137, 662, 171 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {u _ {1}} ^ {u _ {2}} f (x) \\mathrm{d} x \\right| < \\epsilon", "math_type": "latex", "image_source": { "path": "images/3f8bec7493e44e711741073c1ebfff5847e4f2c453f07932894b5af7fdec7b30.jpg" } }, "bbox": [ 450, 172, 588, 204 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 瑕积分： " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 的瑕点为 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " ，则其收敛当且仅当 " }, { "type": "equation_inline", "content": "\\forall \\epsilon ,\\exists \\delta ,\\forall u_1,u_2\\in (a,a + \\delta)" }, { "type": "text", "content": " 满足" } ] }, "bbox": [ 139, 206, 769, 240 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {u _ {1}} ^ {u _ {2}} f (x) \\mathrm{d} x \\right| < \\epsilon", "math_type": "latex", "image_source": { "path": "images/daf72b675eca0a8026ddb483de8c65d16effeae52f4ab5db4cb7ae409ba18c43.jpg" } }, "bbox": [ 450, 241, 586, 274 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 7.2 (绝对收敛与条件收敛)" } ], "level": 2 }, "bbox": [ 122, 298, 352, 313 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}|f(x)|\\mathrm{d}x" }, { "type": "text", "content": " 收敛（ " }, { "type": "equation_inline", "content": "b" }, { "type": "text", "content": " 可以为 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": "），则称 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 绝对收敛；若 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛但不绝对收敛，则称其条件收敛。" } ] }, "bbox": [ 115, 316, 880, 365 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.5 (绝对收敛)" } ], "level": 2 }, "bbox": [ 122, 388, 270, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 无穷积分：若 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}|f(x)|\\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 也收敛，且有下述等式" } ] }, "bbox": [ 139, 406, 715, 442 ] }, { "type": "equation_interline", "content": { "math_content": "| \\int_ {a} ^ {+ \\infty} f (x) \\mathrm{d} x | \\leq \\int_ {a} ^ {+ \\infty} | f (x) | \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/0550af3f693bfe29ad6d88b83ceccd3d2a1e1c982d8a4843b816ac4cf491efd2.jpg" } }, "bbox": [ 396, 443, 640, 476 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 瑕积分：设 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 瑕点为 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b]" }, { "type": "text", "content": " 任一内闭区间上可积，则当 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} |f(x)| \\mathrm{d}x" }, { "type": "text", "content": " 收敛时， " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 也收敛，且" } ] }, "bbox": [ 139, 478, 880, 523 ] }, { "type": "equation_interline", "content": { "math_content": "| \\int_ {a} ^ {b} f (x) \\mathrm{d} x | \\leq \\int_ {a} ^ {b} | f (x) | \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/69df50c1daa0fafa5e534a812b4aff75aad3f20e109a2168b132462bd4acff32.jpg" } }, "bbox": [ 416, 521, 621, 556 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 用 Cauchy 收敛准则转换至有限区间，再根据积分绝对值放缩可得到结论。" } ] }, "bbox": [ 90, 573, 673, 589 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.6 (比较判别法)" } ], "level": 2 }, "bbox": [ 122, 600, 287, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下面考虑 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 非负，则有" } ] }, "bbox": [ 115, 619, 312, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 无穷积分： " }, { "type": "equation_inline", "content": "f, g" }, { "type": "text", "content": " 在任意有限区间可积，且 " }, { "type": "equation_inline", "content": "0 \\leq f(x) \\leq g(x)" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} g(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛时， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 必收敛" } ] }, "bbox": [ 139, 632, 882, 677 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 瑕积分： " }, { "type": "equation_inline", "content": "f, g" }, { "type": "text", "content": " 在内包有限闭区间可积，且 " }, { "type": "equation_inline", "content": "0 \\leq f(x) \\leq g(x)" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} g(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛时， " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 必收敛" } ] }, "bbox": [ 139, 675, 882, 719 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 7.1 (等价无穷小判别)" } ], "level": 2 }, "bbox": [ 122, 741, 321, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 无穷积分： " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在任意有限闭区间可积，且 " }, { "type": "equation_inline", "content": "x \\to \\infty" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "f(x) \\sim \\frac{\\lambda}{x^p}" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 则收敛， " }, { "type": "equation_inline", "content": "p \\leq 1" }, { "type": "text", "content": " 则发散\n- 瑕积分： " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在任意闭区间可积， " }, { "type": "equation_inline", "content": "x \\to a" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "f(x) \\sim \\frac{\\lambda}{(x - a)^p}" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "p < 1" }, { "type": "text", "content": " 则收敛， " }, { "type": "equation_inline", "content": "p \\geq 1" }, { "type": "text", "content": " 则发散" } ] }, "bbox": [ 137, 762, 860, 824 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.1基础训练 判断(1) " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{x\\arctan x}{1+x^{p}}\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{1}{x^{p}+x^{q}}\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{x^{p}}{1+x^{q}}\\mathrm{d}x" }, { "type": "text", "content": " \n解(1)0不是瑕点，只需考虑 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 。若p>0， " }, { "type": "equation_inline", "content": "\\frac{x\\arctan x}{1+x^{p}}\\sim\\frac{\\pi}{2}\\frac{1}{x^{p-1}}" }, { "type": "text", "content": " ，因此要求p>2，此时x不是瑕点。若p=0，显然发散。若p<0， " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 显然发散。" } ] }, "bbox": [ 58, 829, 905, 902 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "7.1 反常积分理论" } ] }, "bbox": [ 771, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "66" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "0, +\\infty" }, { "type": "text", "content": " 均有可能是瑕点。先考虑 " }, { "type": "equation_inline", "content": "p = q" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "\\frac{1}{x^p + x^q} = \\frac{1}{2x^p}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 处要求 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": "，但是 0 处要求 " }, { "type": "equation_inline", "content": "p < 1" }, { "type": "text", "content": "，因此不可能成立。若 " }, { "type": "equation_inline", "content": "p < q" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "x \\to 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\frac{1}{x^p + x^q} \\sim \\frac{1}{x^p}" }, { "type": "text", "content": "，要求 " }, { "type": "equation_inline", "content": "p < 1" }, { "type": "text", "content": "，同理 " }, { "type": "equation_inline", "content": "x \\to \\infty" }, { "type": "text", "content": " 时要求 " }, { "type": "equation_inline", "content": "q > 1" } ] }, "bbox": [ 90, 80, 905, 135 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\frac{x^p}{1 + x^q} = \\frac{1}{x^{-p} + x^{q - p}}" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "-p < 1 < q - p" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "q - p < 1 < -p" }, { "type": "text", "content": " 时收敛，其余情况发散。" } ] }, "bbox": [ 122, 135, 789, 162 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 7.2 (对数、指数比较判别法)" } ], "level": 2 }, "bbox": [ 121, 173, 371, 187 ] }, { "type": "equation_interline", "content": { "math_content": "\\forall \\alpha > 0, \\lim _ {x \\rightarrow 0 ^ {+}} x ^ {\\alpha} \\ln x = 0, \\lim _ {x \\rightarrow \\infty} \\frac {\\ln x}{x ^ {\\alpha}} = 0, \\lim _ {x \\rightarrow + \\infty} x ^ {\\alpha} e ^ {- x} = 0:", "math_type": "latex", "image_source": { "path": "images/3c8e1948e2fd781503519af4e23deeee17a01e4a891f2a9874be979bb9a28a96.jpg" } }, "bbox": [ 136, 196, 568, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "x \\to \\infty" }, { "type": "text", "content": " 时，等价视为 " }, { "type": "equation_inline", "content": "\\ln x \\sim x^{\\epsilon}, e^{x} \\sim x^{+\\infty}" } ] }, "bbox": [ 139, 219, 468, 235 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "x \\to 0" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\ln (1 + x) \\sim x, e^x - 1 \\sim x, \\ln x \\sim \\frac{1}{x^\\epsilon}, e^x \\sim 1" } ] }, "bbox": [ 139, 235, 556, 260 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.2对数、指数比较判别训练 判断收敛性：(1) " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{x}{1 - e^x}\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\ln(1 + x)}{x^p}\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\ln x}{e^x}\\mathrm{d}x" }, { "type": "text", "content": " （重点！）" } ] }, "bbox": [ 58, 288, 907, 332 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)由于0处极限为1，不是瑕点。 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 处 " }, { "type": "equation_inline", "content": "e^x\\sim x^\\infty" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\lim_{x\\to \\infty}x^2\\frac{x}{1 - e^x} = 0" }, { "type": "text", "content": " ，根据比较判别法可知收敛" } ] }, "bbox": [ 87, 334, 863, 355 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)0处等价于 " }, { "type": "equation_inline", "content": "\\frac{1}{x^{p - 1}}" }, { "type": "text", "content": " ，要求 " }, { "type": "equation_inline", "content": "p < 2" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "+\\infty" }, { "type": "text", "content": " 处等价于 " }, { "type": "equation_inline", "content": "\\frac{1}{x^p}" }, { "type": "text", "content": " ，要求 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "1 < p < 2" }, { "type": "text", "content": " 时收敛" } ] }, "bbox": [ 121, 356, 779, 382 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}\\frac{\\ln x\\cdot x^{1 / 2}}{e^x} = 0" }, { "type": "text", "content": " ，因此0处收敛。 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 处 " }, { "type": "equation_inline", "content": "\\lim_{x\\to \\infty}x^2\\frac{\\ln x}{e^x} = 0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 处也收敛。" } ] }, "bbox": [ 122, 382, 724, 412 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.3对数比较判别进阶 判断 " }, { "type": "equation_inline", "content": "\\int_{3}^{+\\infty}\\frac{\\mathrm{d}x}{x^p(\\ln x)^q(\\ln\\ln x)^r}" }, { "type": "text", "content": " 的敛散性解 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 肯定收敛， " }, { "type": "equation_inline", "content": "p < 1" }, { "type": "text", "content": " 肯定发散。若 " }, { "type": "equation_inline", "content": "p = 1" }, { "type": "text", "content": " ，则积分变为 " }, { "type": "equation_inline", "content": "\\int_{3}^{+\\infty}\\frac{\\mathrm{d}(\\ln x)}{(\\ln x)^q(\\ln\\ln x)^r}" }, { "type": "text", "content": "，要求 " }, { "type": "equation_inline", "content": "q > 1" }, { "type": "text", "content": " 。若 " }, { "type": "equation_inline", "content": "p = q = 1" }, { "type": "text", "content": " 则同理要求 " }, { "type": "equation_inline", "content": "r > 1" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 58, 413, 902, 486 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.4变号情况：绝对收敛 " }, { "type": "equation_inline", "content": "+" }, { "type": "text", "content": " 比较判别法 判断敛散性：(1) " }, { "type": "equation_inline", "content": "\\int_{1}^{+\\infty}\\frac{\\cos x\\sin\\frac{1}{x}}{x}\\mathrm{d}x" } ] }, "bbox": [ 58, 483, 680, 514 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\left|\\frac{\\cos x\\sin\\frac{1}{x}}{x}\\right|\\sim\\frac{1}{x^{2}}" }, { "type": "text", "content": " ，因此绝对收敛" } ] }, "bbox": [ 87, 512, 384, 544 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.5三角双瑕点训练重点： " }, { "type": "equation_inline", "content": "|q| < 1" }, { "type": "text", "content": " ，判断 " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\frac{\\sin^{p - 1}x}{|1 + q\\cos x|^p}\\mathrm{d}x" }, { "type": "text", "content": " 的敛散性解0处 " }, { "type": "equation_inline", "content": "\\frac{\\sin^{p - 1}x}{|1 + q\\cos x|^p}\\sim \\frac{x^{p - 1}}{|1 + q|^p}" }, { "type": "text", "content": " ，要求 " }, { "type": "equation_inline", "content": "p - 1 < 1" }, { "type": "text", "content": " 。π处 " }, { "type": "equation_inline", "content": "\\frac{\\sin^{p - 1}x}{|1 + q\\cos x|^p} = \\frac{\\sin^{p - 1}(\\pi -x)}{|1 + q\\cos x|^p}\\sim \\frac{(\\pi -x)^{p - 1}}{|1 + q|^p}" }, { "type": "text", "content": " ，要求 " }, { "type": "equation_inline", "content": "p - 1 < 1" } ] }, "bbox": [ 58, 546, 905, 615 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.1.3 Dirichlet 与 Abel 判别法" } ], "level": 2 }, "bbox": [ 92, 637, 363, 653 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.7 (Dirichlet 与 Abel 判别法)" } ], "level": 2 }, "bbox": [ 122, 670, 379, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- Dirichlet 判别法：若 " }, { "type": "equation_inline", "content": "(1)F(x) = \\int_{a}^{x}f(t)\\mathrm{d}t" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, + \\infty)" }, { "type": "text", "content": " 有界(2) " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, + \\infty)" }, { "type": "text", "content": " 上当 " }, { "type": "equation_inline", "content": "x\\to \\infty" }, { "type": "text", "content": " 时单调趋于0，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)g(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛。" } ] }, "bbox": [ 139, 690, 882, 749 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- Abel 判别法：若 (1) " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛 (2) " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 单调有界，则 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) g(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 139, 747, 766, 784 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.6在 " }, { "type": "equation_inline", "content": "\\sin x" }, { "type": "text", "content": " 反常积分判敛的应用判断以下带三角函数的反常积分的绝对收敛性与条件收敛性：" } ] }, "bbox": [ 58, 819, 816, 834 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "\\int_{1}^{+\\infty}\\frac{\\sin x}{x^p}\\mathrm{d}x,\\int_{1}^{+\\infty}\\frac{\\cos x}{x^p}\\mathrm{d}x" }, { "type": "text", "content": " ： " }, { "type": "equation_inline", "content": "p > 0" }, { "type": "text", "content": " 就收敛， " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 绝对收敛， " }, { "type": "equation_inline", "content": "0 < p\\leq 1" }, { "type": "text", "content": " 条件收敛" } ] }, "bbox": [ 110, 834, 744, 863 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\sin x}{x^{p}}dx:1 1" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\left|\\frac{\\sin x}{x^p}\\right| \\leq \\frac{1}{x^p}" }, { "type": "text", "content": " 因此绝对收敛。 " }, { "type": "equation_inline", "content": "0 < p \\leq 1" }, { "type": "text", "content": " 时根据Dirichlet判别法可知收敛，下证不绝对收敛，根据 " }, { "type": "equation_inline", "content": "|\\sin x| \\geq \\sin^2 x" }, { "type": "text", "content": " ，得到" } ] }, "bbox": [ 89, 205, 905, 244 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\frac {\\sin x}{x ^ {p}} \\right| \\geq \\frac {\\sin^ {2} x}{x ^ {p}} = \\frac {1}{2} \\left(\\frac {1}{x ^ {p}} - \\frac {\\cos 2 x}{x ^ {p}}\\right)", "math_type": "latex", "image_source": { "path": "images/549e30e39ba9d4e2465595247881c92d4285a084e2ab2a16903b0d9daeb744c3.jpg" } }, "bbox": [ 361, 242, 633, 274 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "\\int_{1}^{+\\infty}\\frac{1}{x^p}\\mathrm{d}x" }, { "type": "text", "content": " 不收敛，故 " }, { "type": "equation_inline", "content": "\\int_1^{+\\infty}\\left|\\frac{\\sin x}{x^p}\\right|" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "0 < p\\leq 1" }, { "type": "text", "content": " 时不收敛。 " }, { "type": "equation_inline", "content": "p\\leq 0" }, { "type": "text", "content": " 时，取 " }, { "type": "equation_inline", "content": "u_{1} = 2k\\pi ,u_{2} = 2k\\pi +\\frac{\\pi}{2}" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 90, 281, 875, 315 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {u _ {1}} ^ {u _ {2}} \\frac {\\sin x}{x ^ {p}} \\mathrm{d} x \\right| \\geq \\int_ {u _ {1}} ^ {u _ {2}} \\sin x \\mathrm{d} x = 1", "math_type": "latex", "image_source": { "path": "images/03b20f9ed55ccf0d5d0b3e8234b5bf1159f5373cf4e3d68829fbae28c30fb65d.jpg" } }, "bbox": [ 374, 317, 621, 350 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "显然发散。" } ] }, "bbox": [ 92, 353, 171, 367 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 分两段，这里考虑 " }, { "type": "equation_inline", "content": "\\int_0^1 \\frac{\\sin x}{x^p} \\mathrm{d}x" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "x \\to 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\frac{\\sin x}{x^p} \\sim \\frac{1}{x^{p-1}}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "p < 2" }, { "type": "text", "content": " 时收敛且绝对收敛， " }, { "type": "equation_inline", "content": "p \\geq 2" }, { "type": "text", "content": " 时发散。结合 (1) 得到 " }, { "type": "equation_inline", "content": "1 < p < 2" }, { "type": "text", "content": " 时绝对收敛， " }, { "type": "equation_inline", "content": "0 < p \\leq 1" }, { "type": "text", "content": " 时条件收敛，其余发散" } ] }, "bbox": [ 90, 360, 905, 411 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\frac{\\cos x}{x^{p}}\\sim\\frac{1}{x^{p}}" }, { "type": "text", "content": " ，结合(1)得到0 1" }, { "type": "text", "content": " 时绝对收敛， " }, { "type": "equation_inline", "content": "0 < 2 - p \\leq 1" }, { "type": "text", "content": " 条件收敛。" } ] }, "bbox": [ 87, 432, 905, 525 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) " }, { "type": "equation_inline", "content": "q = 0" }, { "type": "text", "content": " 时显然无法保证 " }, { "type": "equation_inline", "content": "0, +\\infty" }, { "type": "text", "content": " 同时收敛，原因在于0要求 " }, { "type": "equation_inline", "content": "p < 1" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " 要求 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "q > 0" }, { "type": "text", "content": " 时，令 " }, { "type": "equation_inline", "content": "t = x^q" }, { "type": "text", "content": " ，得到" } ] }, "bbox": [ 87, 524, 905, 558 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} x ^ {p} \\sin x ^ {q} \\mathrm{d} x = \\frac {1}{q} \\int_ {0} ^ {+ \\infty} \\frac {\\sin t}{t ^ {1 - \\frac {p + 1}{q}}} \\mathrm{d} t", "math_type": "latex", "image_source": { "path": "images/fcbcbd5383531585af1eeaaca1a0cfffba77d13f9f6f26a4595e9b3478dfe507.jpg" } }, "bbox": [ 357, 555, 638, 589 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "转换为前面的问题。" } ] }, "bbox": [ 90, 596, 236, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin x}{x^p}\\mathrm{d}x,\\int_0^{+\\infty}\\frac{\\cos x}{x^p}\\mathrm{d}x" }, { "type": "text", "content": " 的收敛情况要当作结论背下来。" } ] }, "bbox": [ 55, 607, 571, 640 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 7.7 三角有理式敛散性判别 判断如下几道三角函数有理式的敛散性 (难度较高):" } ] }, "bbox": [ 58, 653, 707, 669 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 判断 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty} \\frac{\\sin x}{2x + 3\\sin x} \\mathrm{d}x" }, { "type": "text", "content": " 的敛散性" } ] }, "bbox": [ 121, 669, 413, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 判断 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} \\frac{\\sin x}{x^p + \\sin x} \\, \\mathrm{d}x" }, { "type": "text", "content": " 的敛散性" } ] }, "bbox": [ 122, 699, 401, 728 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 证明 " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty} \\frac{\\mathrm{d}x}{1 + x^2 \\sin x^2} \\mathrm{d}x" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 122, 728, 386, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)该积分没有瑕点，只需要考虑 " }, { "type": "equation_inline", "content": "\\infty" }, { "type": "text", "content": " ，其敛散性等价于 " }, { "type": "equation_inline", "content": "\\int_2^\\infty \\frac{\\sin x}{2x + 3\\sin x}\\mathrm{d}x" }, { "type": "text", "content": " 。这里必须拆项(不能直接放缩原因在于分母阶数不够)" } ] }, "bbox": [ 87, 757, 905, 801 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {\\sin x}{2 x + 3 \\sin x}} = {\\frac {2 x \\sin x}{2 x (2 x + 3 \\sin x)}} = {\\frac {(2 x + 3 \\sin x) \\sin x - 3 \\sin^ {2} x}{2 x (2 x + 3 \\sin x)}} = {\\frac {\\sin x}{2 x}} - {\\frac {3 \\sin^ {2} x}{2 x (2 x + 3 \\sin x)}}", "math_type": "latex", "image_source": { "path": "images/97f7054baffd189a3d77a822d7c9b6a11cc6972e2adec8b1dc41590aa3f64b81.jpg" } }, "bbox": [ 168, 804, 826, 837 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "考虑前者，显然 " }, { "type": "equation_inline", "content": "\\frac{1}{2x}" }, { "type": "text", "content": " 单调递减趋于0，而" } ] }, "bbox": [ 90, 839, 389, 866 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {2} ^ {u} \\sin \\mathrm{d} x \\right| = | \\cos 2 - \\cos u | \\leq 2", "math_type": "latex", "image_source": { "path": "images/6bd90899e40aadd9d906ecc03e4a8dad37c3be5a4e892976c2dc02bffffff420.jpg" } }, "bbox": [ 376, 866, 620, 901 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "7.1 反常积分理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "68" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 Dirichlet 判别法知前者收敛。后者先进行放缩" } ] }, "bbox": [ 90, 85, 467, 99 ] }, { "type": "equation_interline", "content": { "math_content": "0 \\leq \\frac {3 \\sin^ {2} x}{2 x (2 x + 3 \\sin x)} \\leq \\frac {3}{2 x (2 x - 3)}", "math_type": "latex", "image_source": { "path": "images/cddc3330d2aacdc14f2ea3bc9737451ca59090ffbf54532b2391e1693b2ae877.jpg" } }, "bbox": [ 366, 102, 626, 134 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在无穷处等价于 " }, { "type": "equation_inline", "content": "\\frac{1}{x^2}" }, { "type": "text", "content": "，根据比较原则得到收敛。综上收敛" } ] }, "bbox": [ 90, 135, 505, 164 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.1.4 函数在无穷远处的性质" } ], "level": 2 }, "bbox": [ 92, 181, 354, 198 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "一、若极限存在，则必为0" } ], "level": 2 }, "bbox": [ 90, 210, 295, 225 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.8 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, u]" }, { "type": "text", "content": " 可积， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，若 " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} f(x) = A" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "A = 0" }, { "type": "text", "content": "。证明 若 " }, { "type": "equation_inline", "content": "A \\neq 0" }, { "type": "text", "content": "，不妨设 " }, { "type": "equation_inline", "content": "A > 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\exists M \\geq a, \\forall x \\geq M, f(x) \\geq \\frac{A}{2}" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} \\frac{A}{2} \\mathrm{d}x" }, { "type": "text", "content": " 发散，根据比较判别法可知 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 58, 239, 905, 332 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.9 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 上连续可微，且 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x, \\int_{a}^{+\\infty} f'(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} f(x) = 0" }, { "type": "text", "content": " 证明由于 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f'(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，因此 " }, { "type": "equation_inline", "content": "\\lim_{u \\to +\\infty} \\int_{a}^{u} f'(x) \\mathrm{d}x = \\lim_{u \\to +\\infty} f(u) - f(a)" }, { "type": "text", "content": " 存在，根据极限存在可知极限为0(前一题结论)。" } ] }, "bbox": [ 58, 336, 905, 413 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.10 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 上连续， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则存在递增趋于 " }, { "type": "equation_inline", "content": "+\\infty" }, { "type": "text", "content": " 的数列 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} f(x_{n}) = 0" }, { "type": "text", "content": "。证明根据Cauchy收敛准则，取 " }, { "type": "equation_inline", "content": "\\epsilon_{1} = 1, \\exists A_{1}', A_{1}'' = A_{1}' + 1" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\left| \\int_{A_1'}^{A_1''} f(x) \\mathrm{d}x \\right| < \\epsilon_{1} = 1" }, { "type": "text", "content": "。同理对 " }, { "type": "equation_inline", "content": "\\epsilon_{2} = \\frac{1}{2}, \\exists A_{2}' \\geq A_{1}', A_{2}'' = A_{2}' + 1" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\left| \\int_{A_2'}^{A_2''} f(x) \\mathrm{d}x \\right| < \\epsilon_{2} = \\frac{1}{2}" }, { "type": "text", "content": "。以此类推，根据积分第一中值定理 " }, { "type": "equation_inline", "content": "\\exists x_{i} \\in [A_{i}', A_{i}'']" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(x_{n})| = \\left| \\int_{A_{n}'}^{A_{n}''} f(x) \\mathrm{d}x \\right|" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 单增， " }, { "type": "equation_inline", "content": "|f(x_{n})| < \\frac{1}{n} \\to 0" }, { "type": "text", "content": "。" } ] }, "bbox": [ 58, 424, 905, 562 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "二、极限不一定存在：光满足非负、连续、可导等并不能说明极限存在，甚至不一定有界" } ], "level": 2 }, "bbox": [ 87, 571, 749, 587 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.11举例说明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在任意有限区间 " }, { "type": "equation_inline", "content": "[a, u]" }, { "type": "text", "content": " 非负可积， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 不一定有界。解 " }, { "type": "equation_inline", "content": "f(x) = \\begin{cases} x, & x \\text{为整数} \\\\ 0, & x \\text{不为整数} \\end{cases}" }, { "type": "text", "content": "，如果要求恒正则取 " }, { "type": "equation_inline", "content": "g(x) = f(x) + e^{-x}" } ] }, "bbox": [ 58, 599, 816, 676 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.12举例说明 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 上连续函数， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛，不一定有 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = 0" }, { "type": "text", "content": " 解例如 " }, { "type": "equation_inline", "content": "\\int_1^{+\\infty}\\sin x^2\\mathrm{d}x" }, { "type": "text", "content": " ，令 " }, { "type": "equation_inline", "content": "t = x^{2}" }, { "type": "text", "content": " 以及Dirichlet判别法可知收敛，但是极限不为0。" } ] }, "bbox": [ 58, 678, 789, 739 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.13举例说明 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 恒正的连续可微函数， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 不一定有界解 " }, { "type": "equation_inline", "content": "\\int_1^{+\\infty}\\frac{x}{1 + x^6\\sin^2x}\\mathrm{d}x" } ] }, "bbox": [ 58, 741, 759, 807 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "三、获得极限存在：虽然非负、连续、可导不能推出 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = 0" }, { "type": "text", "content": " ，但一致连续和单调可以推出该性质。" } ], "level": 2 }, "bbox": [ 87, 818, 870, 837 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.14经典老题：一致连续 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 上一致连续， " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty}f(x)\\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = 0" }, { "type": "text", "content": "证明 反设 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x)\\neq 0" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\exists \\epsilon_0" }, { "type": "text", "content": " 以及趋于无穷的 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(x_n)|\\geq \\epsilon_0" }, { "type": "text", "content": " ，对上述 " }, { "type": "equation_inline", "content": "\\epsilon_0" }, { "type": "text", "content": " ，根据一致连续 " }, { "type": "equation_inline", "content": "\\exists \\delta ,\\forall x^{\\prime},x^{\\prime \\prime}\\in" }, { "type": "equation_inline", "content": "[a, + \\infty)" }, { "type": "text", "content": " 只要 " }, { "type": "equation_inline", "content": "|x^{\\prime} - x^{\\prime \\prime}|\\leq \\delta" }, { "type": "text", "content": " ，就有 " }, { "type": "equation_inline", "content": "|f(x^{\\prime}) - f(x^{\\prime \\prime})| < \\frac{\\epsilon_0}{2}" }, { "type": "text", "content": " 。对每个 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " ，当 " }, { "type": "equation_inline", "content": "x\\in [x_n,x_n + \\delta ]" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "|f(x) - f(x_n)| < \\frac{\\epsilon_0}{2}" }, { "type": "text", "content": " 从而 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[x_n,x_n + \\delta ]" }, { "type": "text", "content": " 不变号，且" } ] }, "bbox": [ 58, 846, 828, 879 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "7.1 反常积分理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "69" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 87, 83, 907, 143 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x) | \\geq | f (x _ {n}) | - | f (x) - f (x _ {n}) | > \\frac {\\epsilon_ {0}}{2}", "math_type": "latex", "image_source": { "path": "images/0117f26d443cbccf895f101f9aaba6b31a29cc4767ff8367f29724c886635a62.jpg" } }, "bbox": [ 352, 146, 642, 170 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对 " }, { "type": "equation_inline", "content": "\\forall M > a, \\exists N, x_N > M" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|\\int_{x_N}^{x_N + \\delta} f(x) \\mathrm{d}x| = \\int_{x_N}^{x_N + \\delta} |f(x)| \\mathrm{d}x \\geq \\int_{x_N}^{x_N + \\delta} \\frac{\\epsilon_0}{2} \\mathrm{d}x = \\frac{\\epsilon_0 \\delta}{2}" }, { "type": "text", "content": " 不趋于 0，根据 Cauchy 收敛准则逆否命题可知 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 发散，矛盾。" } ] }, "bbox": [ 89, 170, 905, 239 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.15单调 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 单调，且 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} f(x) = 0" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} x f(x) = 0" }, { "type": "text", "content": "。笔记反之不一定，即使有 " }, { "type": "equation_inline", "content": "\\lim_{x \\to +\\infty} x f(x) = 0" }, { "type": "text", "content": "，也不一定有 " }, { "type": "equation_inline", "content": "\\int_{a}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，例如 " }, { "type": "equation_inline", "content": "f(x) = \\frac{1}{x \\ln x}" } ] }, "bbox": [ 55, 242, 845, 306 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.2 反常积分的计算" } ], "level": 2 }, "bbox": [ 92, 341, 314, 361 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.2.1 两个常用的基础反常积分" } ], "level": 2 }, "bbox": [ 92, 380, 374, 397 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.16计算 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-ax}\\cos bx\\mathrm{d}x,\\int_0^{+\\infty}e^{-ax}\\sin bx\\mathrm{d}x" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "a > 0" }, { "type": "text", "content": " 解直接写不定积分即可(具体计算见配对积分法一节)， " }, { "type": "equation_inline", "content": "I_{1} = \\frac{e^{-ax}}{a^{2} + b^{2}} (a\\cos bx + b\\sin bx)\\bigg|_{0}^{+\\infty} = \\frac{a}{a^2 + b^2}" }, { "type": "text", "content": " ，同理得到 " }, { "type": "equation_inline", "content": "I_{2} = \\frac{b}{a^{2} + b^{2}}" } ] }, "bbox": [ 58, 406, 905, 499 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.17对数配对积分(1)计算 " }, { "type": "equation_inline", "content": "I_{n} = \\int_{0}^{1}(\\ln x)^{n}\\mathrm{d}x" }, { "type": "text", "content": " (2)计算 " }, { "type": "equation_inline", "content": "I(m,n) = \\int_0^1 x^m (\\ln x)^n\\mathrm{d}x" }, { "type": "text", "content": " 解(1)做一次分部积分得到 " }, { "type": "equation_inline", "content": "I_{n} = x(\\ln x)^{n}\\Big|_{0}^{1} - \\int_{0}^{1}n(\\ln x)^{n - 1}\\mathrm{d}x = -nI_{n - 1}" }, { "type": "text", "content": " ，而 " }, { "type": "equation_inline", "content": "I_{1} = \\int_{0}^{1}\\ln x\\mathrm{d}x = x\\ln x\\Big|_{0}^{1} - x\\Big|_{0}^{1} = -1" }, { "type": "text", "content": " ，综上得到 " }, { "type": "equation_inline", "content": "I_{n} = (-1)^{n}n!" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "I(m,n) = -\\frac{n}{m + 1} I(m,n - 1)" }, { "type": "text", "content": " ，结合 " }, { "type": "equation_inline", "content": "I(m,0) = \\frac{1}{m + 1}" }, { "type": "text", "content": " 可得 " }, { "type": "equation_inline", "content": "I(m,n) = \\frac{(-1)^{n}n!}{(m + 1)^{n + 1}}" } ] }, "bbox": [ 58, 505, 905, 615 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.2.2 富如兰尼积分" } ], "level": 2 }, "bbox": [ 94, 633, 273, 650 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.8 (两侧极限存在则无穷积分收敛)" } ], "level": 2 }, "bbox": [ 121, 665, 421, 680 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在任意有限区间可积， " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x) = A,\\lim_{x\\to -\\infty}f(x) = B" }, { "type": "text", "content": " ，给定 " }, { "type": "equation_inline", "content": "a\\in \\mathbb{R}" }, { "type": "text", "content": " ，则下面积分收敛，且值为 " }, { "type": "equation_inline", "content": "a(A - B)" } ] }, "bbox": [ 112, 684, 882, 720 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {- \\infty} ^ {+ \\infty} [ f (a + x) - f (x) ] \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/ed8224fc90c02e136e2940cd42c30aaf88586ecb69d69edc342638af1beece73.jpg" } }, "bbox": [ 401, 719, 593, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据定义， " }, { "type": "equation_inline", "content": "\\int_{-\\infty}^{+\\infty}f(x)\\mathrm{d}x = \\lim_{M\\to +\\infty}\\lim_{m\\to -\\infty}\\int_{m}^{M}[f(a + x) - f(x)]\\mathrm{d}x" }, { "type": "text", "content": " ，对有限区域积分做如下变换：" } ] }, "bbox": [ 90, 763, 845, 797 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {m} ^ {M} [ f (a + x) - f (x) ] \\mathrm{d} x = \\int_ {m + a} ^ {M + a} f (x) \\mathrm{d} x - \\int_ {m} ^ {M} f (x) \\mathrm{d} x = \\int_ {M} ^ {M + a} f (x) \\mathrm{d} x - \\int_ {m} ^ {m + a} f (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/37006b6ef5fe5a6fa9b156ae5d12dc28bd3bd8556a653d6e72985163923c36f7.jpg" } }, "bbox": [ 152, 800, 843, 835 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "代入 " }, { "type": "equation_inline", "content": "A, B" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "I = \\int_{M}^{M + a}[f(x) - A]\\mathrm{d}x - \\int_{m}^{m + a}[f(x) - B]\\mathrm{d}x + a(A - B)" }, { "type": "text", "content": "，根据极限 " }, { "type": "equation_inline", "content": "\\lim_{M\\to +\\infty}[f(x) - A]\\mathrm{d}x = 0" }, { "type": "text", "content": "，同理 " }, { "type": "equation_inline", "content": "\\lim_{N\\to +\\infty}[f(x) - B]\\mathrm{d}x = 0" }, { "type": "text", "content": "，因此取极限得到收敛，且值为 " }, { "type": "equation_inline", "content": "a(A - B)" } ] }, "bbox": [ 90, 837, 905, 890 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "7.2 反常积分的计算" } ] }, "bbox": [ 754, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "70" } ] }, "bbox": [ 489, 953, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.9 (富如兰尼积分)" } ], "level": 2 }, "bbox": [ 122, 86, 304, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 连续函数， " }, { "type": "equation_inline", "content": "a, b > 0" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 115, 105, 411, 121 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. 若 " }, { "type": "equation_inline", "content": "\\lim_{x\\to +\\infty}f(x)" }, { "type": "text", "content": " 存在，则" } ] }, "bbox": [ 136, 124, 334, 146 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} \\frac {f (a x) - f (b x)}{x} \\mathrm{d} x = [ f (0) - f (+ \\infty) ] \\ln \\frac {b}{a}", "math_type": "latex", "image_source": { "path": "images/d48bd21a726af5c5ccf97bc2fa44781241953e14646e56dbef749faba8286825.jpg" } }, "bbox": [ 347, 149, 690, 181 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. 若 " }, { "type": "equation_inline", "content": "\\exists A > 0" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\int_{A}^{+\\infty} \\frac{f(x)}{x} \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则" } ] }, "bbox": [ 134, 184, 462, 216 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} \\frac {f (a x) - f (b x)}{x} \\mathrm{d} x = f (0) \\ln {\\frac {b}{a}}", "math_type": "latex", "image_source": { "path": "images/cd1fa5002a9f540b7c9ef34591fb3374a5427e337dfa3202e7c0c6af5dab5d54.jpg" } }, "bbox": [ 389, 219, 650, 252 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 下面对 " }, { "type": "equation_inline", "content": "I = \\int_{m}^{M}\\frac{f(ax) - f(bx)}{x}\\mathrm{d}x" }, { "type": "text", "content": " 进行化简：" } ] }, "bbox": [ 90, 263, 460, 297 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {m} ^ {M} \\frac {f (a x)}{x} \\mathrm{d} x - \\int_ {m} ^ {M} \\frac {f (b x)}{x} \\mathrm{d} x = \\int_ {a m} ^ {a M} \\frac {f (x)}{x} \\mathrm{d} x - \\int_ {b m} ^ {b M} \\frac {f (x)}{x} \\mathrm{d} x = \\int_ {a m} ^ {b m} \\frac {f (x)}{x} \\mathrm{d} x - \\int_ {a M} ^ {b M} \\frac {f (x)}{x} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/2bb4c70eaba969b87c6f9af653c546caeb4e75aef091beb303946b48deba25a7.jpg" } }, "bbox": [ 136, 298, 858, 332 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 根据积分第一中值定理得到" } ] }, "bbox": [ 122, 335, 354, 349 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\int_ {a m} ^ {b m} \\frac {f (x)}{x} \\mathrm{d} x = f (\\xi) \\int_ {a \\eta} ^ {b m} \\frac {1}{x} \\mathrm{d} x = f (\\xi) \\ln \\frac {b}{a} \\\\ \\int_ {a M} ^ {b M} \\frac {f (x)}{x} \\mathrm{d} x = f (\\eta) \\int_ {a M} ^ {b M} \\frac {1}{x} \\mathrm{d} x = f (\\eta) \\ln \\frac {b}{a} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/392dee4e24ef0f48e9bf5b664873cd26f8b689ef6d613551f6beb5842ff7fb3b.jpg" } }, "bbox": [ 324, 353, 655, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "m \\to 0^{+}, M \\to +\\infty" }, { "type": "text", "content": " 时可代入极限，因此结果为 " }, { "type": "equation_inline", "content": "[f(0) - f(+\\infty)] \\ln \\frac{b}{a}" } ] }, "bbox": [ 92, 419, 615, 445 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于 " }, { "type": "equation_inline", "content": "\\int_{A}^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，因此第二项 " }, { "type": "equation_inline", "content": "\\lim_{M \\to +\\infty} \\frac{f(x)}{x} \\mathrm{d}x = 0" }, { "type": "text", "content": "，同时第一项用积分中值定理取极限得到 " }, { "type": "equation_inline", "content": "f(0) \\ln \\frac{b}{a}" } ] }, "bbox": [ 89, 443, 905, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 7.18 富如兰尼积分的应用 计算以下反常积分" } ] }, "bbox": [ 60, 517, 460, 531 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\arctan ax - \\arctan bx}{x}\\mathrm{d}x" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{e^{-ax} - e^{-bx}}{x}\\mathrm{d}x" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\cos ax - \\cos bx}{x}\\mathrm{d}x" } ] }, "bbox": [ 122, 529, 744, 561 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\arctan 0 = 0, \\arctan (+\\infty) = \\frac{\\pi}{2}" }, { "type": "text", "content": "，因此积分为 " }, { "type": "equation_inline", "content": "-\\frac{\\pi}{2} \\ln \\frac{b}{a}" } ] }, "bbox": [ 90, 560, 522, 588 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "e^{0}=1,e^{-\\infty}=0" }, { "type": "text", "content": " ，因此积分为 " }, { "type": "equation_inline", "content": "\\ln\\frac{b}{a}" } ] }, "bbox": [ 122, 586, 401, 613 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\cos 0 = 1, \\int_{A}^{+\\infty} \\frac{\\cos x}{x} \\mathrm{d}x" }, { "type": "text", "content": " 收敛，因此积分为 " }, { "type": "equation_inline", "content": "\\ln \\frac{b}{a}" } ] }, "bbox": [ 122, 612, 495, 642 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7.2.3 Dirichlet 积分" } ], "level": 2 }, "bbox": [ 95, 675, 273, 690 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "引理7.1" } ], "level": 2 }, "bbox": [ 124, 706, 189, 721 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "p > 0" }, { "type": "text", "content": " ，则含参积分 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-px}\\frac{\\sin bx - \\sin ax}{x}\\mathrm{d}x = \\arctan \\frac{b}{p} -\\arctan \\frac{a}{p}" } ] }, "bbox": [ 115, 725, 655, 758 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 首先 " }, { "type": "equation_inline", "content": "\\frac{\\sin bx - \\sin ax}{x} = \\int_{a}^{b} \\cos xy \\, dy" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 89, 771, 443, 803 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {+ \\infty} \\mathrm{d} x \\int_ {a} ^ {b} e ^ {- p x} \\cos x y \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/8c977f15010dc4193b639dcdfe36b159e7d9600a5409aa040ebbe0ad9aeadded.jpg" } }, "bbox": [ 384, 806, 613, 839 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 " }, { "type": "equation_inline", "content": "|e^{-px}\\cos xy|\\leq e^{-px}" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-px}\\mathrm{d}x" }, { "type": "text", "content": " 绝对收敛，因此 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-px}\\cos xy\\mathrm{d}x,y\\in [a,b]" }, { "type": "text", "content": " 一致收敛，交换积" } ] }, "bbox": [ 92, 840, 905, 875 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "7.2 反常积分的计算" } ] }, "bbox": [ 754, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "71" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "分顺序得到" } ] }, "bbox": [ 90, 85, 179, 99 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {a} ^ {b} \\mathrm{d} y \\int_ {0} ^ {+ \\infty} e ^ {- p x} \\cos x y \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/02539852d5a2e6e2e557bdf9d3a39669fa38f9a85f24295f2578f5a86c1e5379.jpg" } }, "bbox": [ 384, 101, 613, 135 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据表格法得到:" } ] }, "bbox": [ 90, 137, 220, 151 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} e ^ {- p x} \\cos x y \\mathrm{d} x = - \\frac {1}{p} e ^ {- p x} \\cos x y \\bigg | _ {0} ^ {+ \\infty} + \\frac {y}{p ^ {2}} e ^ {- p x} \\sin x y \\bigg | _ {0} ^ {+ \\infty} - \\int_ {0} ^ {+ \\infty} \\frac {y ^ {2}}{p ^ {2}} e ^ {- p x} \\cos x y \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/ba18d204fdfba5219998efa19e619b496add28fd375e28f29ff5404ed6d33ec0.jpg" } }, "bbox": [ 178, 153, 818, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此得到 " }, { "type": "equation_inline", "content": "\\frac{y^2 + p^2}{p^2}\\int_0^{+\\infty}\\cos xy\\mathrm{d}y = \\frac{1}{p}" } ] }, "bbox": [ 90, 191, 368, 224 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {a} ^ {b} {\\frac {p}{p ^ {2} + y ^ {2}}} \\mathrm{d} y = \\arctan {\\frac {b}{p}} - \\arctan {\\frac {a}{p}}", "math_type": "latex", "image_source": { "path": "images/e206c14b4f2aee70e02ba97c9ff32a7c6d984b4aa2f5ed999ae0b679fa480154.jpg" } }, "bbox": [ 344, 225, 650, 260 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 7.10 (Dirichlet 积分)" } ], "level": 2 }, "bbox": [ 122, 274, 312, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "积分 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin x}{x}\\mathrm{d}x = \\frac{\\pi}{2}" }, { "type": "text", "content": " ，进一步地， " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin px}{x}\\mathrm{d}x = \\frac{\\pi}{2}\\mathrm{sgn}(p)" } ] }, "bbox": [ 114, 294, 584, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 令引理中 " }, { "type": "equation_inline", "content": "a = 0" }, { "type": "text", "content": "，将 " }, { "type": "equation_inline", "content": "p" }, { "type": "text", "content": " 看成参数， " }, { "type": "equation_inline", "content": "G(p) = \\int_{0}^{+\\infty} e^{-px} \\frac{\\sin bx}{x} \\mathrm{d}x = \\arctan \\frac{b}{p}" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty} \\frac{\\sin bx}{x} \\mathrm{d}x" }, { "type": "text", "content": " 条件收敛， " }, { "type": "equation_inline", "content": "e^{-px}" }, { "type": "text", "content": " 在固定 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 时关于 " }, { "type": "equation_inline", "content": "p" }, { "type": "text", "content": " 单调，且一致有界，根据 Abel 判别法可知 " }, { "type": "equation_inline", "content": "G(p)" }, { "type": "text", "content": " 在 [0,1] 上一致收敛，从而" } ] }, "bbox": [ 89, 337, 907, 382 ] }, { "type": "equation_interline", "content": { "math_content": "G (0) = \\lim _ {p \\to 0 ^ {+}} G (p) = \\lim _ {p \\to 0 ^ {+}} \\arctan {\\frac {b}{p}} = \\frac {\\pi}{2} \\mathrm{sgn} (b)", "math_type": "latex", "image_source": { "path": "images/bb9f3e1f557dc102d0bc237c181b11f0f26733bcb8a5d8229c829ecd4f9b7933.jpg" } }, "bbox": [ 327, 387, 668, 416 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "特别地， " }, { "type": "equation_inline", "content": "b = 1" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin x}{x}\\mathrm{d}x = \\frac{\\pi}{2}" } ] }, "bbox": [ 90, 420, 379, 451 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习7.19Dirichlet积分的应用 计算以下反常积分：(1)重点： " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin x^2}{x}\\mathrm{d}x" }, { "type": "text", "content": " (2)重点： " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin^2x}{x^2}\\mathrm{d}x" } ] }, "bbox": [ 58, 455, 838, 488 ] }, { "type": "equation_interline", "content": { "math_content": "(3) \\int_ {0} ^ {+ \\infty} \\frac {\\sin^ {4} x}{x ^ {2}} \\mathrm{d} x (4) \\int_ {0} ^ {+ \\infty} \\frac {\\sin^ {4} x}{x ^ {4}} \\mathrm{d} x (5) \\int_ {0} ^ {+ \\infty} \\frac {x - \\sin x}{x ^ {3}} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/f11728396c1aae96351800d74c46ab66225b281a789c058e7491aa87c5eed60b.jpg" } }, "bbox": [ 122, 486, 542, 518 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)做 " }, { "type": "equation_inline", "content": "t = x^2" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "\\frac{1}{2}\\int_0^{+\\infty}\\frac{\\sin t}{t}\\mathrm{d}t = \\frac{\\pi}{4}" } ] }, "bbox": [ 89, 519, 406, 546 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\sin^{2}x}{x^{2}}\\mathrm{d}x=-\\int_{0}^{+\\infty}\\sin^{2}x\\mathrm{d}\\left(\\frac{1}{x}\\right)" }, { "type": "text", "content": " ，用分部积分得到" } ] }, "bbox": [ 122, 546, 564, 580 ] }, { "type": "equation_interline", "content": { "math_content": "I = - \\frac {\\sin^ {2} x}{x} \\bigg | _ {0} ^ {+ \\infty} + \\int_ {0} ^ {+ \\infty} \\frac {\\sin 2 x}{2 x} \\mathrm{d} 2 x = \\frac {\\pi}{2}", "math_type": "latex", "image_source": { "path": "images/580824b8ebbe3fca1c554aad45f376c1abb47620fb010ce34d051a1341bb073d.jpg" } }, "bbox": [ 347, 583, 648, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 先分部积分得到" } ] }, "bbox": [ 122, 619, 270, 633 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} \\frac {\\sin^ {4} x}{x ^ {2}} \\mathrm{d} x = - \\int_ {0} ^ {+ \\infty} \\sin^ {4} x \\mathrm{d} \\left(\\frac {1}{x}\\right) = \\int_ {0} ^ {+ \\infty} \\frac {4 \\sin^ {3} x \\cos x}{x} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/1debc7c943749df9d15474e858965a89d8019bf19cd85778cff9700e2d68ad4f.jpg" } }, "bbox": [ 265, 636, 731, 669 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 " }, { "type": "equation_inline", "content": "4\\sin^3 x\\cos x = \\sin 2x(1 - \\cos 2x) = \\sin 2x - \\frac{1}{2}\\sin 4x" }, { "type": "text", "content": " 得到" } ] }, "bbox": [ 90, 674, 549, 699 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {+ \\infty} \\frac {\\sin 2 x}{2 x} \\mathrm{d} (2 x) - \\frac {1}{2} \\int_ {0} ^ {+ \\infty} \\frac {\\sin 4 x}{4 x} \\mathrm{d} (4 x) = \\frac {\\pi}{4}", "math_type": "latex", "image_source": { "path": "images/9bfb9cf858e53ff27fb3d7d818397b2ce9b913699af56fa5a33553fd5f8571a1.jpg" } }, "bbox": [ 312, 702, 682, 734 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "7.2 反常积分的计算" } ] }, "bbox": [ 754, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "72" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第8章 数项级数" } ], "level": 2 }, "bbox": [ 396, 87, 618, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.1 数项级数基本理论" } ], "level": 2 }, "bbox": [ 92, 145, 337, 164 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 8.1 (数项级数收敛)" } ], "level": 2 }, "bbox": [ 122, 186, 302, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "给定数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": "，定义 " }, { "type": "equation_inline", "content": "S_n = \\sum_{k=1}^{n} u_n" }, { "type": "text", "content": " 为部分和函数，若 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} S_n = S" }, { "type": "text", "content": " 收敛，则称 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛，否则称数项级数发散。" } ] }, "bbox": [ 114, 205, 882, 258 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.1 (数项级数收敛的各种条件)" } ], "level": 2 }, "bbox": [ 122, 281, 388, 296 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "给定数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": "，其收敛的条件：" } ] }, "bbox": [ 114, 300, 398, 338 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 必要条件 (一定能推出): " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}u_n = 0" } ] }, "bbox": [ 139, 337, 433, 354 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 充要条件 (Cauchy 准则): " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall m, m + p > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|u_{m+1} + \\cdots + u_{m+p}| < \\epsilon" } ] }, "bbox": [ 139, 355, 721, 372 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "笔记 一般证明发散要么不满足必要条件，要么用 Cauchy 准则逆否命题" } ], "level": 2 }, "bbox": [ 87, 390, 611, 406 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.1Cauchy 收敛准则 (1) 证明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{1}{n}" }, { "type": "text", "content": " 发散 (2) 用 Cauchy 收敛准则证明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}" }, { "type": "text", "content": " 收敛\n\n证明 (1) " }, { "type": "equation_inline", "content": "\\exists\\epsilon_{0}=0,\\forall N,m=N+1,m+p=2N" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\frac{1}{m+1}+\\frac{1}{m+2}+\\cdots+\\frac{1}{2m}\\geq\\frac{1}{2}=\\epsilon_{0}" }, { "type": "text", "content": " ，根据 Cauchy 逆否命题可证 (也可以用积分判别法证明)\n\n(2) 用 " }, { "type": "equation_inline", "content": "\\frac{1}{n^{2}}<\\frac{1}{n(n-1)}" }, { "type": "text", "content": " ，直接求和或者用 Cauchy 收敛准则都行" } ] }, "bbox": [ 58, 418, 905, 521 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.2 (子列问题)" } ], "level": 2 }, "bbox": [ 122, 542, 270, 557 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} u_n = 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "S_n = \\sum_{k=1}^{n} u_k" }, { "type": "text", "content": " 有一个子列 " }, { "type": "equation_inline", "content": "S_{np}" }, { "type": "text", "content": " ( " }, { "type": "equation_inline", "content": "p" }, { "type": "text", "content": " 为固定整数) 收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 112, 558, 899, 604 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.2 交错子列 证明 " }, { "type": "equation_inline", "content": "1 + \\frac{1}{2} + (\\frac{1}{3} - 1) + \\frac{1}{4} + \\frac{1}{5} + (\\frac{1}{6} - \\frac{1}{2}) + \\cdots = \\ln 3" } ] }, "bbox": [ 60, 631, 621, 659 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "S_{3n}=\\sum_{k=1}^{3n}u_{k}=(1+\\frac{1}{2}+\\cdots+\\frac{1}{3n})-(1+\\frac{1}{2}+\\cdots+\\frac{1}{n})" }, { "type": "text", "content": " ，配合 Euler 常数可知 " }, { "type": "equation_inline", "content": "S_{3n}=\\ln3n+\\gamma-\\ln n-\\gamma=\\ln3" } ] }, "bbox": [ 65, 653, 902, 699 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.3 (去除有限项)" } ], "level": 2 }, "bbox": [ 122, 711, 285, 726 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "去掉、增加、改变级数的有限项并不改变数项级数的收敛性。" } ] }, "bbox": [ 117, 731, 564, 746 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.2 正项级数" } ], "level": 2 }, "bbox": [ 92, 87, 240, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.2.1 比较原则" } ], "level": 2 }, "bbox": [ 92, 126, 233, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.4 (比较原则)" } ], "level": 2 }, "bbox": [ 122, 160, 270, 174 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n,\\sum_{n = 1}^{\\infty}v_n" }, { "type": "text", "content": " 是两个正项级数，" } ] }, "bbox": [ 112, 179, 347, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 非极限形式：若 " }, { "type": "equation_inline", "content": "\\exists N > 0" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\forall n > N" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "u_{n} \\leq v_{n}" }, { "type": "text", "content": "，则根据 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} v_{n}" }, { "type": "text", "content": " 收敛推出 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_{n}" }, { "type": "text", "content": " 收敛；根据 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_{n}" }, { "type": "text", "content": " 发散，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} v_{n}" }, { "type": "text", "content": " 也发散" } ] }, "bbox": [ 137, 214, 882, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 极限形式： " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{u_n}{v_n} = \\lambda" }, { "type": "text", "content": " 。当 " }, { "type": "equation_inline", "content": "0 < \\lambda < \\infty" }, { "type": "text", "content": " ，同敛散。 " }, { "type": "equation_inline", "content": "\\lambda = 0" }, { "type": "text", "content": " 则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}v_n" }, { "type": "text", "content": " 敛推 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 敛。 " }, { "type": "equation_inline", "content": "\\lambda = \\infty" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}v_n" }, { "type": "text", "content": " 发散推 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}v_n" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 139, 286, 887, 365 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 比较原则极限形式的 " }, { "type": "equation_inline", "content": "0 < \\lambda < \\infty" }, { "type": "text", "content": " 情况即可以使用等价无穷小。但在一些特殊情况下也需要用 " }, { "type": "equation_inline", "content": "\\lambda = 0, \\lambda = \\infty" }, { "type": "text", "content": " 的形式。" } ] }, "bbox": [ 87, 379, 905, 413 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "例题8.1常用比较对象 " }, { "type": "equation_inline", "content": "\\sum_{n = 0}^{\\infty}\\frac{1}{n^p}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 时收敛，在 " }, { "type": "equation_inline", "content": "p\\leq 1" }, { "type": "text", "content": " 时发散证明用积分判别法" } ] }, "bbox": [ 87, 426, 569, 478 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.3使用等价无穷小 判断敛散性：(1) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{1}{\\sqrt{n^2 + 1}}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} 2^n \\sin \\frac{\\pi}{3^n}" } ] }, "bbox": [ 58, 491, 611, 530 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (\\sqrt[n]{2} - 1)" }, { "type": "text", "content": " (4) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (a^{\\frac{1}{n}} + a^{-\\frac{1}{n}} - 2)(a > 1)" } ] }, "bbox": [ 121, 527, 566, 565 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\frac{1}{\\sqrt{n^2 + 1}} \\sim \\frac{1}{n}" }, { "type": "text", "content": "，发散" } ] }, "bbox": [ 90, 562, 287, 593 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)和等比级数比，等价于 " }, { "type": "equation_inline", "content": "(\\frac{2}{3})^n\\pi" }, { "type": "text", "content": " ，收敛" } ] }, "bbox": [ 122, 592, 421, 618 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 根据 " }, { "type": "equation_inline", "content": "2^{\\frac{1}{n}} - 1 \\sim \\frac{1}{n} \\ln 2" }, { "type": "text", "content": "，发散 (注意这里用的是 " }, { "type": "equation_inline", "content": "a^x \\sim x \\ln a - 1" }, { "type": "text", "content": ")" } ] }, "bbox": [ 122, 618, 600, 643 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 根据 " }, { "type": "equation_inline", "content": "a^{\\frac{1}{n}} + a^{-\\frac{1}{n}} - 2 = (a^{\\frac{1}{2n}} - a^{-\\frac{1}{2n}})^2 = a^{-\\frac{1}{n}}(a^{\\frac{1}{n}} - 1)^2 \\sim \\frac{\\ln^2 a}{n^2}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 122, 642, 640, 673 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.4含对数与指数(1)重点： " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{1}{3^{\\ln n}}" }, { "type": "text", "content": " (2)重点： " }, { "type": "equation_inline", "content": "\\sum_{n = 2}^{\\infty}\\frac{1}{(\\ln n)^{\\ln n}}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\sum_{n = 3}^{\\infty}\\frac{1}{(\\ln\\ln n)^{\\ln n}}" }, { "type": "text", "content": " (4)特殊： " }, { "type": "equation_inline", "content": "\\sum_{n = 2}^{\\infty}\\frac{1}{(\\ln n)^n}" } ] }, "bbox": [ 58, 678, 867, 718 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{1}{n^{2n \\sin \\frac{1}{n}}}" }, { "type": "text", "content": " (6) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{\\ln(n+1)}{e^n}" }, { "type": "text", "content": " (7) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (e^{\\frac{1}{n^2}} - \\cos \\frac{\\pi}{n})" } ] }, "bbox": [ 117, 715, 618, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 对数需要用到恒等式 " }, { "type": "equation_inline", "content": "a^{\\ln b} = b^{\\ln a}" }, { "type": "text", "content": " (两侧取对数即可证明)" } ] }, "bbox": [ 90, 752, 515, 766 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\frac{1}{3^{\\ln n}} = \\frac{1}{n^{\\ln 3}}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 122, 766, 278, 791 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\frac{1}{(\\ln n)^{\\ln n}}=\\frac{1}{n^{\\ln(\\ln n)}}<\\frac{1}{n^{2}}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 122, 790, 376, 820 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\frac{1}{(\\ln \\ln n)^{\\ln n}} = \\frac{1}{n^{\\ln \\ln \\ln n}} < \\frac{1}{n^2}" }, { "type": "text", "content": "，收敛" } ] }, "bbox": [ 122, 819, 413, 847 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "\\frac{1}{(\\ln n)^n} < \\frac{1}{2^n}" }, { "type": "text", "content": "，收敛" } ] }, "bbox": [ 124, 846, 297, 875 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) " }, { "type": "equation_inline", "content": "\\frac{1}{n^{2n\\sin\\frac{1}{n}}} < \\frac{1}{n^{\\frac{3}{2}}}" }, { "type": "text", "content": "，因此收敛。" } ] }, "bbox": [ 122, 875, 351, 903 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.2 正项级数" } ] }, "bbox": [ 805, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "74" } ] }, "bbox": [ 489, 953, 509, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(6) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}n^2\\frac{\\ln(n + 1)}{e^n}\\rightarrow 0" } ] }, "bbox": [ 122, 82, 310, 111 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(7)Taylor 展开: " }, { "type": "equation_inline", "content": "e^{\\frac{1}{n^2}} - \\cos \\frac{\\pi}{n} = 1 + \\frac{1}{n^2} - 1 + \\frac{\\pi^2}{2n^2} + o\\left(\\frac{1}{n^2}\\right) = \\left(1 + \\frac{\\pi^2}{2}\\right)\\frac{1}{n^2}" }, { "type": "text", "content": " 收敛笔记要灵活使用比较判别法。有时用等价无穷小，有时直接放缩，有时乘上 " }, { "type": "equation_inline", "content": "n^2" }, { "type": "text", "content": " (一般在分母极大)，有时Taylor展开" } ] }, "bbox": [ 87, 108, 907, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.5 等价无穷小的进一步应用 " }, { "type": "equation_inline", "content": "(1)a_{n}>0" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "(1+a_{1})(1+a_{2})\\cdots(1+a_{n})" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}a_{n}" }, { "type": "text", "content": " 敛散性相同\n\n " }, { "type": "equation_inline", "content": "(2)a_{n}=(1-\\frac{p\\ln n}{n})^{n}" }, { "type": "text", "content": " ，讨论 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}b_{n}" }, { "type": "text", "content": " 的敛散性" } ] }, "bbox": [ 58, 190, 833, 260 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[-1, 1]" }, { "type": "text", "content": " 二阶连续可微，且 " }, { "type": "equation_inline", "content": "\\lim_{x \\to 0} \\frac{f(x)}{x} = 0" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} f\\left(\\frac{1}{n}\\right)" }, { "type": "text", "content": " 绝对收敛证明 (1) 取对数即 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\ln(1 + a_n)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_n = 0" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\frac{\\ln(1 + a_n)}{a_n} = 1" }, { "type": "text", "content": "，故同敛散。若 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_n \\neq 0" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n" }, { "type": "text", "content": " 发散，显然 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\ln(1 + a_n) \\neq 0" }, { "type": "text", "content": "，故也发散。" } ] }, "bbox": [ 90, 260, 905, 351 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "a_{n}=e^{n\\ln(1-\\frac{p\\ln n}{n})}=e^{n[-\\frac{p\\ln n}{n}+o(\\frac{1}{n})]}=e^{-p\\ln n}\\sim\\frac{1}{p^{p}}" }, { "type": "text", "content": " ，当p>1时收敛， " }, { "type": "equation_inline", "content": "p\\leq1" }, { "type": "text", "content": " 时发散。" } ] }, "bbox": [ 122, 351, 771, 376 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 根据 " }, { "type": "equation_inline", "content": "\\lim_{x\\to0}\\frac{f(x)}{x}=0" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "f'(0)=0" }, { "type": "text", "content": " ，根据 Lagrange 余项得到 " }, { "type": "equation_inline", "content": "f(\\frac{1}{n})=f'(\\xi)\\frac{1}{n^{2}}" }, { "type": "text", "content": " ，因此收敛" } ] }, "bbox": [ 122, 376, 791, 406 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.5 (比式比较判别法)" } ], "level": 2 }, "bbox": [ 122, 425, 319, 439 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n, \\sum_{n=1}^{\\infty} v_n" }, { "type": "text", "content": " 为正项级数， " }, { "type": "equation_inline", "content": "\\exists N, \\forall n \\geq N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\frac{u_{n+1}}{u_n} \\leq \\frac{v_{n+1}}{v_n}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} v_n" }, { "type": "text", "content": " 收敛可推出 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 发散可推出 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} v_n" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 112, 442, 899, 523 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 " }, { "type": "equation_inline", "content": "\\frac{u_{n}}{u_{N}} = \\frac{u_{N+1}}{u_{N}} \\frac{u_{N+2}}{u_{N+1}} \\cdots \\frac{u_{n}}{u_{n-1}} \\leq \\frac{v_{N+1}}{v_{N}} \\frac{v_{N+2}}{v_{N+1}} \\cdots \\frac{v_{n}}{v_{n+1}} = \\frac{v_{n}}{v_{N}}" }, { "type": "text", "content": " ，由此推出 " }, { "type": "equation_inline", "content": "u_{n} \\leq \\frac{u_{N}}{v_{N}} v_{n}" }, { "type": "text", "content": " 只相差一个常数，因此根据一般的比较判别法可得到结论。" } ] }, "bbox": [ 89, 533, 905, 575 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.6比式比较判别法基本应用 " }, { "type": "equation_inline", "content": "a_{n} > 0,0 < \\alpha \\leq 1,\\beta >1" }, { "type": "text", "content": " ，证明：(1)若 " }, { "type": "equation_inline", "content": "\\frac{a_{n + 1}}{a_n}\\geq (1 - \\frac{1}{n})^\\alpha" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 发散(2)若 " }, { "type": "equation_inline", "content": "\\frac{a_{n + 1}}{a_n}\\leq (1 - \\frac{1}{n})^\\beta" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 60, 590, 905, 659 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据 " }, { "type": "equation_inline", "content": "\\frac{a_{n+1}}{a_n} \\geq \\left(\\frac{n-1}{n}\\right)^\\alpha = \\frac{\\frac{1}{n^\\alpha}}{\\frac{1}{(n-1)^\\alpha}}" }, { "type": "text", "content": "，因此根据结论发散 (2) 同理" } ] }, "bbox": [ 89, 659, 613, 697 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.2.2 比式判别法与根式判别法" } ], "level": 2 }, "bbox": [ 94, 715, 374, 733 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "前面比较判别法常用的比较对象是 " }, { "type": "equation_inline", "content": "\\frac{1}{n^p}" }, { "type": "text", "content": "，而根式判别法与比式判别法的比较对象是等比级数。" } ] }, "bbox": [ 122, 741, 800, 772 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.6 (比式判别法与根式判别法)" } ], "level": 2 }, "bbox": [ 122, 781, 388, 795 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 是正项级数，则" } ] }, "bbox": [ 114, 801, 289, 838 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 比式判别法：若 " }, { "type": "equation_inline", "content": "\\varlimsup_{n\\to \\infty}\\frac{u_{n + 1}}{u_n} = q < 1" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛，若 " }, { "type": "equation_inline", "content": "\\varliminf_{n\\to \\infty}\\frac{u_{n + 1}}{u_n} >1" }, { "type": "text", "content": " 则发散。" } ] }, "bbox": [ 139, 837, 752, 875 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.2 正项级数" } ] }, "bbox": [ 806, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "75" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 根式判别法：若 " }, { "type": "equation_inline", "content": "\\varlimsup_{n\\to \\infty}\\sqrt[n]{u_n} = l < 1" }, { "type": "text", "content": " 则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛， " }, { "type": "equation_inline", "content": "\\varlimsup_{n\\to \\infty}\\sqrt[n]{u_n} >1" }, { "type": "text", "content": " 则发散。(双上极限)" } ] }, "bbox": [ 139, 93, 796, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 若 " }, { "type": "equation_inline", "content": "\\varlimsup_{n\\to \\infty}\\frac{u_{n + 1}}{u_n} = q < 1" }, { "type": "text", "content": "，则对 " }, { "type": "equation_inline", "content": "\\forall r\\in (q,1)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists N,\\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\frac{u_{n + 1}}{u_n} < r = \\frac{r^{n + 1}}{r^n}" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}r^n" }, { "type": "text", "content": " 收敛，根据比式比较判别法可知。另一侧同理。" } ] }, "bbox": [ 89, 146, 905, 197 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2.1) 若 " }, { "type": "equation_inline", "content": "\\varlimsup_{n\\to \\infty}\\sqrt[n]{u_n} = l < 1" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\forall r\\in (l,1)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists N,\\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\sqrt[n]{u_n} < r" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "u_{n} < r^{n}" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}r^n" }, { "type": "text", "content": " 收敛，因此 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛。" } ] }, "bbox": [ 87, 199, 905, 267 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2.2) 另一侧若 " }, { "type": "equation_inline", "content": "\\varlimsup_{n\\to \\infty}\\sqrt[n]{u_n} = l > 1" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\forall r\\in (1,l)" }, { "type": "text", "content": "，存在子列 " }, { "type": "equation_inline", "content": "\\sqrt[n]{u_{n_k}}\\geq r" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "u_{n_k}\\geq r^{n_k}\\to \\infty" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 发散。" } ] }, "bbox": [ 89, 268, 905, 315 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记理论上根式判别法证明发散只需要证上极限, 比比式条件更弱, 因此更好用。而且由于 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{u_{n+1}}{u_n}\\leq\\lim_{n\\to\\infty}\\sqrt[n]{u_n}\\leq\\overline{\\lim}_{n\\to\\infty}\\sqrt[n]{u_n}\\leq\\overline{\\lim}_{n\\to\\infty}\\frac{u_{n+1}}{u_n}" }, { "type": "text", "content": " 可知能用比式的一定能用根式, 一定要记住 " }, { "type": "equation_inline", "content": "\\sqrt[n]{n!}\\sim\\frac{n}{e}" } ] }, "bbox": [ 87, 317, 929, 368 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.7使用根式判别法用根式判别法法证明(1) " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{n^2}{(2 + \\frac{1}{n})^n}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{n!}{n^n}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{n!3^n}{n^n}" } ] }, "bbox": [ 58, 374, 715, 411 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{(n!)^{2}}{(2n)!}" }, { "type": "text", "content": " (5) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{(2n-1)!!}{n!}" }, { "type": "text", "content": " (6) 讨论 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^{n} n!}{n^{n}}" }, { "type": "text", "content": " 的收敛性, " }, { "type": "equation_inline", "content": "x \\geq 0" } ] }, "bbox": [ 121, 409, 712, 447 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)极限为 " }, { "type": "equation_inline", "content": "\\frac{1}{2}" }, { "type": "text", "content": ", 收敛" } ] }, "bbox": [ 90, 445, 258, 472 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 极限为 " }, { "type": "equation_inline", "content": "\\frac{n / e}{n} = \\frac{1}{e}" }, { "type": "text", "content": ", 收敛" } ] }, "bbox": [ 122, 470, 322, 499 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 极限为 " }, { "type": "equation_inline", "content": "\\frac{3}{e} > 1" }, { "type": "text", "content": " ，发散" } ] }, "bbox": [ 122, 497, 302, 524 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 极限为 " }, { "type": "equation_inline", "content": "\\sqrt[n]{\\frac{(n!)^2}{(2n)!}} = \\frac{\\left(\\frac{n}{e}\\right)^2}{\\sqrt[2n]{\\left(\\frac{2n}{e}\\right)^2}} \\sim \\frac{1}{4}" } ] }, "bbox": [ 122, 523, 394, 563 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{(2n-1)!!}{n!}=\\sum_{n=1}^{\\infty}\\frac{(2n)!}{n!(2n)!!}=\\sum_{n=1}^{\\infty}\\frac{(2n)!}{n!2^{n}n!}" }, { "type": "text", "content": " ，因此根据根式判别法等价于2>1。" } ] }, "bbox": [ 122, 564, 729, 602 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(7) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{\\frac{x^n n!}{n^n}} = \\frac{x}{e}, x\\in (0,e)" }, { "type": "text", "content": " 收敛， " }, { "type": "equation_inline", "content": "x\\in (e, + \\infty)" }, { "type": "text", "content": " 发散。当 " }, { "type": "equation_inline", "content": "x = e" }, { "type": "text", "content": " 则 " }, { "type": "equation_inline", "content": "\\frac{a_{n + 1}}{a_n} = \\frac{e}{(1 + \\frac{1}{n})^n} >1" }, { "type": "text", "content": " ，极限不趋于0。" } ] }, "bbox": [ 87, 605, 905, 646 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.8仅能用根式判别法(1)证明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{1}{2^{n+(-1)^n}}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 58, 661, 497, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 讨论 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^n}{1 + x^{2n}}" }, { "type": "text", "content": " 的敛散性" } ] }, "bbox": [ 122, 697, 349, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{\\frac{1}{2^{n + (-1)^n}}} = \\frac{1}{2} < 1" }, { "type": "text", "content": " 因此收敛" } ] }, "bbox": [ 89, 734, 418, 764 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 分母 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{1 + x^{2n}} = \\max \\{1,x^2\\}" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 122, 766, 455, 787 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} \\sqrt [ n ]{u _ {n}} = \\frac {x}{\\max \\{1 , x ^ {2} \\}} = \\left\\{ \\begin{array}{l l} < 1, & x \\neq 1 \\\\ = 1, & x = 1 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/2b035384f266510b6caeae2371f022fbf3531b54b84e297751318c91fa9aedf5.jpg" } }, "bbox": [ 339, 791, 653, 835 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "x \\neq 1" }, { "type": "text", "content": " 时收敛， " }, { "type": "equation_inline", "content": "x = 1" }, { "type": "text", "content": " 时发散。" } ] }, "bbox": [ 90, 840, 347, 854 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.9 根式判别法上下极限 (1) 判断 " }, { "type": "equation_inline", "content": "b + c + b^{2} + c^{2} + \\cdots + b^{n} + c^{n} + \\cdots" }, { "type": "text", "content": " 的敛散性，这里 0 < b < c < 1 解 (1) " }, { "type": "equation_inline", "content": "\\sqrt[n]{u_{n}}" }, { "type": "text", "content": " 在偶数项为 " }, { "type": "equation_inline", "content": "\\sqrt{c}" }, { "type": "text", "content": " ，奇数项为 " }, { "type": "equation_inline", "content": "\\sqrt{b}" }, { "type": "text", "content": " ，因此上极限为 " }, { "type": "equation_inline", "content": "\\sqrt{c}" }, { "type": "text", "content": " ，故收敛" } ] }, "bbox": [ 58, 876, 863, 910 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.2 正项级数" } ] }, "bbox": [ 805, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "76" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.2.3 积分判别法和拉贝判别法" } ], "level": 2 }, "bbox": [ 92, 85, 374, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.7 (积分判别法与拉贝判别法)" } ], "level": 2 }, "bbox": [ 122, 118, 389, 134 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛最本质的充要条件为部分和函数 " }, { "type": "equation_inline", "content": "S_{n}" }, { "type": "text", "content": " 有界，其推论为" } ] }, "bbox": [ 114, 137, 584, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 拉贝判别法(常用于 " }, { "type": "equation_inline", "content": "\\frac{a_{n+1}}{a_n} \\to 1" }, { "type": "text", "content": "): " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} n(1 - \\frac{u_{n+1}}{u_n}) = r > 1" }, { "type": "text", "content": ", 则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛。若 " }, { "type": "equation_inline", "content": "r < 1" }, { "type": "text", "content": " 则发散, " }, { "type": "equation_inline", "content": "r = 1" }, { "type": "text", "content": " 需要具体判定\n- 积分判别法: " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[1, +\\infty)" }, { "type": "text", "content": " 上非负减函数, 则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} f(n)" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\int_1^{+\\infty} f(x) \\mathrm{d}x" }, { "type": "text", "content": " 同时收敛与发散" } ] }, "bbox": [ 139, 175, 890, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 拉贝判别法:" } ] }, "bbox": [ 90, 279, 253, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "收敛情况：根据条件， " }, { "type": "equation_inline", "content": "\\exists N, \\forall n > N" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "n\\left(1 - \\frac{u_{n+1}}{u_n}\\right) > r" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "\\frac{u_{n+1}}{u_n} < 1 - \\frac{r}{n}" }, { "type": "text", "content": "，根据伯努利不等式：" } ] }, "bbox": [ 122, 292, 868, 323 ] }, { "type": "equation_interline", "content": { "math_content": "(1 + x) ^ {r} \\geq 1 + r x, x > - 1, r > 1 \\Rightarrow \\left(1 - \\frac {1}{n}\\right) ^ {r} \\geq 1 - \\frac {r}{n}", "math_type": "latex", "image_source": { "path": "images/366acf88d4870ae7b0bbfa9f8887ca89fb8ef2d2c7c4085db8ad3d51b105ffa0.jpg" } }, "bbox": [ 294, 326, 702, 357 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\frac{u_{n + 1}}{u_n}\\leq \\left(1 - \\frac{1}{n}\\right)^r = \\left(\\frac{n - 1}{n}\\right)^r = \\frac{\\frac{1}{n^r}}{\\frac{1}{(n - 1)^r}}" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{1}{n^r}" }, { "type": "text", "content": " 收敛，因此 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 90, 360, 721, 398 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "发散情况： " }, { "type": "equation_inline", "content": "\\frac{u_{n + 1}}{u_n}\\geq 1 - \\frac{1}{n} = \\frac{n - 1}{n} = \\frac{\\frac{1}{n}}{\\frac{1}{n - 1}}" }, { "type": "text", "content": " ，而 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{1}{n}" }, { "type": "text", "content": " 发散，因此发散。" } ] }, "bbox": [ 122, 398, 648, 434 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "笔记 拉贝判别法更多地用于比式和根式判别法都失效的情况下。" } ], "level": 2 }, "bbox": [ 55, 436, 559, 453 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.10 拉贝判别法 判断敛散性 (1) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{n^{n}}{e^{n}n!}" } ] }, "bbox": [ 58, 468, 428, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 显然根式判别法失效。因此设 " }, { "type": "equation_inline", "content": "u_{n} = \\frac{n^{n}}{e^{n}n!}" }, { "type": "text", "content": "，使用拉贝判别法以及 " }, { "type": "equation_inline", "content": "(1 + \\frac{1}{n})^{n} = e - \\frac{e}{2n}" }, { "type": "text", "content": " (用 " }, { "type": "equation_inline", "content": "(1 + x)^{\\frac{1}{x}}" }, { "type": "text", "content": " 在 0 处 Taylor 展开) 得到" } ] }, "bbox": [ 87, 500, 904, 543 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} n \\left(1 - \\frac {u _ {n + 1}}{u _ {n}}\\right) = \\lim _ {n \\to \\infty} n \\left[ 1 - \\frac {(1 + \\frac {1}{n}) ^ {n}}{e} \\right] = \\lim _ {n \\to \\infty} n \\left[ \\frac {1}{2 n} + o (\\frac {1}{n}) \\right] = \\frac {1}{2} < 1", "math_type": "latex", "image_source": { "path": "images/0cc960332937c82d4360d9485af051a1dcc5dcfd3130232488a6b5fd87cdd050.jpg" } }, "bbox": [ 211, 546, 784, 580 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据拉贝判别法可知发散。" } ] }, "bbox": [ 90, 583, 289, 596 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.2.4 放缩证明敛散性" } ], "level": 2 }, "bbox": [ 92, 619, 292, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.11两个简单放缩 " }, { "type": "equation_inline", "content": "(1)a_{n}\\geq 0" }, { "type": "text", "content": " ，讨论 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\sqrt{a_na_{n + 1}}" }, { "type": "text", "content": " 收敛性的关系" } ] }, "bbox": [ 58, 645, 668, 682 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 设 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n^2, \\sum_{n=1}^{\\infty} b_n^2" }, { "type": "text", "content": " 收敛，证明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n b_n, \\sum_{n=1}^{\\infty} (a_n + b_n)^2" }, { "type": "text", "content": " 均收敛" } ] }, "bbox": [ 122, 680, 589, 720 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "(1)\\sqrt{a_n + b_n} \\leq \\frac{1}{2}(a_n + b_n)" }, { "type": "text", "content": "，根据比较定理可知" } ] }, "bbox": [ 90, 719, 482, 745 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据 Cauchy 不等式， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{N}a_{n}b_{n}\\leq\\left(\\sum_{n=1}^{N}a_{n}^{2}\\right)\\left(\\sum_{n=1}^{N}b_{n}^{2}\\right)" }, { "type": "text", "content": " ，因此收敛。 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{N}(a_{n}+b_{n})^{2}=\\sum_{n=1}^{N}a_{n}^{2}+2a_{n}b_{n}+b_{n}^{2}" }, { "type": "text", "content": " 即可" } ] }, "bbox": [ 90, 741, 905, 793 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.3 任意项级数" } ] }, "bbox": [ 788, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "77" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/6a2f4b6cc6f1a88a4b7ff3a2c0646790a46f3c0d06e9f32f3b45405572f20b4a.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 751, 78, 770 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.3 任意项级数" } ], "level": 2 }, "bbox": [ 92, 87, 265, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.3.1 交错项级数" } ], "level": 2 }, "bbox": [ 92, 127, 253, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.8 (Leibniz 判别法)" } ], "level": 2 }, "bbox": [ 122, 160, 312, 174 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单减趋于0，那么交替项级数 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}(-1)^{n - 1}a_{n}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 112, 179, 524, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单减趋于 0, " }, { "type": "equation_inline", "content": "S_{n} = \\sum_{k=1}^{n} (-1)^{k-1} a_{k}" }, { "type": "text", "content": ", 下面考虑奇偶部分和:" } ] }, "bbox": [ 87, 229, 594, 266 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} S _ {2 n} = (a _ {1} - a _ {2}) + (a _ {3} - a _ {4}) + \\dots + (a _ {2 n - 1} - a _ {2 n}) \\\\ S _ {2 n + 1} = (a _ {1}) - (a _ {2} - a _ {3}) - (a _ {4} - a _ {5}) - \\dots - (a _ {2 n} - a _ {2 n + 1}) \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/2fbd3520abf0e69288e2a25f269822886cc5db3e01ec9208f48b94489a1ca6c3.jpg" } }, "bbox": [ 265, 273, 712, 316 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "显然 " }, { "type": "equation_inline", "content": "S_{2n}" }, { "type": "text", "content": " 单增， " }, { "type": "equation_inline", "content": "S_{2n+1}" }, { "type": "text", "content": " 单减， " }, { "type": "equation_inline", "content": "a_{1}-a_{2}\\leq S_{2n}\\leq S_{2n}+a_{2n+1}=S_{2n+1}\\leq a_{1}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "S_{2n}, S_{2n+1}" }, { "type": "text", "content": " 均有界，根据单调有界定理可知收敛，而奇偶子列均收敛说明 " }, { "type": "equation_inline", "content": "S_{n}" }, { "type": "text", "content": " 收敛。" } ] }, "bbox": [ 90, 321, 902, 354 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.12 讨论下面级数的条件收敛与绝对收敛性:" } ] }, "bbox": [ 58, 376, 458, 388 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{(-1)^n}{n^{p+\\frac{1}{n}}}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\sum_{n=2}^{\\infty} \\frac{(-1)^n}{n^p \\ln^q n}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^n \\sin \\frac{x}{n}" } ] }, "bbox": [ 122, 390, 522, 425 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^{n} \\frac{\\arctan n}{n^{x}}" }, { "type": "text", "content": " (5) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\sin(n\\pi + \\frac{1}{n^{p}})" }, { "type": "text", "content": " (6) 重点: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\sin(\\pi \\sqrt{n^{2} + p^{2}})" } ] }, "bbox": [ 122, 425, 704, 460 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\frac{1}{n^{p + \\frac{1}{n}}} \\sim \\frac{1}{n^p}" }, { "type": "text", "content": " 因此 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 时绝对收敛。条件收敛性只需要看是否单调，考虑 " }, { "type": "equation_inline", "content": "1 \\geq p > 0" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f(x) = x^{p + \\frac{1}{x}}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f'(x) = px^{p - 1}x^{\\frac{1}{x}} + x^p x^{\\frac{1}{x}}\\frac{1 - \\ln x}{x^2} = x^{p - 1}x^{\\frac{1}{x}}(p + \\frac{1 - \\ln x}{x}) > 0" }, { "type": "text", "content": " （ " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 足够大时），因此整体单减趋于 0。 " }, { "type": "equation_inline", "content": "p \\leq 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\frac{(-1)^n}{n^{p + \\frac{1}{n}}} \\neq 0" }, { "type": "text", "content": " ，因此发散。" } ] }, "bbox": [ 89, 460, 905, 539 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 加绝对值后可以用积分判别法，在 " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 时绝对收敛， " }, { "type": "equation_inline", "content": "p = 1, q > 1" }, { "type": "text", "content": " 绝对收敛。 " }, { "type": "equation_inline", "content": "p > 0" }, { "type": "text", "content": "，构造 " }, { "type": "equation_inline", "content": "g(x) = x^p \\ln^q x" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "g'(x) = x^{p-1} \\ln^{q-1} x (p \\ln x + q) > 0" }, { "type": "text", "content": "（ " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 足够大），从而 " }, { "type": "equation_inline", "content": "p = 1, q \\leq 1" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "0 < p < 1" }, { "type": "text", "content": " 条件收敛 " }, { "type": "equation_inline", "content": "p = 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "q > 0" }, { "type": "text", "content": " 条件收敛， " }, { "type": "equation_inline", "content": "q \\leq 0" }, { "type": "text", "content": " 发散。 " }, { "type": "equation_inline", "content": "p < 0" }, { "type": "text", "content": " 发散。" } ] }, "bbox": [ 89, 539, 905, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 则绝对收敛。 " }, { "type": "equation_inline", "content": "x \\neq 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|(-1)^n \\sin \\frac{x}{n} | \\sim \\frac{|x|}{n}" }, { "type": "text", "content": " 发散。而 " }, { "type": "equation_inline", "content": "\\sin \\frac{x}{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 足够大时单调，因此条件收敛" } ] }, "bbox": [ 122, 590, 868, 619 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 加绝对值时等价于 " }, { "type": "equation_inline", "content": "\\frac{1}{n^x}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "x > 1" }, { "type": "text", "content": " 时绝对收敛。不加绝对值时， " }, { "type": "equation_inline", "content": "x < 0" }, { "type": "text", "content": " 一定发散， " }, { "type": "equation_inline", "content": "x > 0" }, { "type": "text", "content": " 时，取 " }, { "type": "equation_inline", "content": "g(y) = \\frac{\\arctan y}{y^x}" }, { "type": "text", "content": "，求导发现小于 0，因此收敛" } ] }, "bbox": [ 90, 618, 905, 669 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 等于 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^{n} \\sin \\frac{1}{n^{p}}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 绝对收敛, " }, { "type": "equation_inline", "content": "0 < p \\leq 1" }, { "type": "text", "content": " 条件收敛, " }, { "type": "equation_inline", "content": "p \\leq 0" }, { "type": "text", "content": " 发散。" } ] }, "bbox": [ 121, 668, 692, 708 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(6) 等于 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^n \\sin (\\pi \\sqrt{n^2 + p^2} - n\\pi) = \\sum_{n=1}^{\\infty} (-1)^n \\sin \\frac{p^2\\pi}{\\sqrt{n^2 + p^2} + n}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "p = 0" }, { "type": "text", "content": " 时绝对收敛， " }, { "type": "equation_inline", "content": "p \\neq 0" }, { "type": "text", "content": " 条件收敛。" } ] }, "bbox": [ 90, 703, 902, 752 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 单调性常常使用求导辅助。" } ] }, "bbox": [ 85, 757, 324, 771 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.3 任意项级数" } ] }, "bbox": [ 788, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "78" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.3.2 Dirichlet 与 Abel 判别法" } ], "level": 2 }, "bbox": [ 92, 85, 363, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.9 (Dirichlet 判别法)" } ], "level": 2 }, "bbox": [ 122, 118, 321, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{k}, b_{k}" }, { "type": "text", "content": " 为两个数列，且满足(1) " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{k} a_{i}" }, { "type": "text", "content": " 有界(2) " }, { "type": "equation_inline", "content": "b_{k}" }, { "type": "text", "content": " 单调趋于0，则 " }, { "type": "equation_inline", "content": "\\sum_{k=1}^{\\infty} a_{k} b_{k}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 114, 139, 677, 179 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.10 (Abel 判别法)" } ], "level": 2 }, "bbox": [ 122, 203, 299, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a_{k}, b_{k}" }, { "type": "text", "content": " 满足 (1) " }, { "type": "equation_inline", "content": "\\sum_{i=1}^{\\infty} a_{i}" }, { "type": "text", "content": " 收敛 (2) " }, { "type": "equation_inline", "content": "b_{k}" }, { "type": "text", "content": " 单调有界，则 " }, { "type": "equation_inline", "content": "\\sum_{k=1}^{\\infty} a_{k} b_{k}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 112, 222, 546, 263 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "笔记 Leibniz 判别法本质只是 Dirichlet 和 Abel 判别法的一个特例" } ], "level": 2 }, "bbox": [ 87, 281, 564, 296 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.13常用三角结论(1)重点： " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单减趋于0， " }, { "type": "equation_inline", "content": "x\\neq 2k\\pi" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n\\cos nx,\\sum_{n = 1}^{\\infty}a_n\\sin nx" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 60, 294, 825, 332 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点： " }, { "type": "equation_inline", "content": "x \\neq k\\pi" }, { "type": "text", "content": "，讨论 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{\\cos nx}{n^{p}}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{\\sin nx}{n^{p}}" }, { "type": "text", "content": " 的条件收敛和绝对收敛性" } ] }, "bbox": [ 122, 329, 658, 366 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点：讨论 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^n \\frac{\\sin n}{n}" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^n \\frac{\\sin^2 n}{n}" }, { "type": "text", "content": " 的条件收敛与绝对收敛性证明 (1) 根据 " }, { "type": "equation_inline", "content": "|\\sum_{k=1}^{n}\\cos kx| = |\\frac{\\sin(n + \\frac{1}{2})x - \\sin\\frac{x}{2}}{2\\sin\\frac{x}{2}}| \\leq \\frac{1}{|\\sin\\frac{x}{2}|}" }, { "type": "text", "content": "，同理 " }, { "type": "equation_inline", "content": "|\\sum_{k=1}^{\\infty}\\sin kx| = |\\frac{\\cos\\frac{x}{2} - \\cos(n + \\frac{1}{2})x}{2\\sin\\frac{x}{2}}| \\leq \\frac{1}{|\\sin\\frac{x}{2}|}" }, { "type": "text", "content": "，根据 Dirichlet 判别法可知。" } ] }, "bbox": [ 124, 366, 675, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 90, 400, 905, 466 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "p > 1" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\left|\\frac{\\cos nx}{n^p}\\right| \\leq \\frac{1}{n^p}" }, { "type": "text", "content": "，因此绝对收敛。 " }, { "type": "equation_inline", "content": "p > 0" }, { "type": "text", "content": " 时根据Dirichlet可知条件收敛， " }, { "type": "equation_inline", "content": "p \\leq 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\frac{\\cos nx}{n^p} \\neq 0" }, { "type": "text", "content": " 发散。" } ] }, "bbox": [ 90, 468, 905, 507 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下面证明 " }, { "type": "equation_inline", "content": "0 < p \\leq 1" }, { "type": "text", "content": " 时不绝对收敛：" } ] }, "bbox": [ 124, 508, 381, 524 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\frac {\\cos n x}{n ^ {p}} \\right| \\geq \\left| \\frac {\\cos^ {2} n x}{n ^ {p}} \\right| \\geq \\left| \\frac {1 - \\sin^ {2} x}{n ^ {p}} \\right|", "math_type": "latex", "image_source": { "path": "images/b2c859129cb0551451e4ae0199c53b9de93122c46b4010433c45b80564eee0df.jpg" } }, "bbox": [ 366, 527, 628, 558 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{1}{n^p}" }, { "type": "text", "content": " 发散，因此发散。" } ] }, "bbox": [ 90, 562, 295, 599 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 这种双交错要把 " }, { "type": "equation_inline", "content": "(-1)^n" }, { "type": "text", "content": " 吸收进三角。 " }, { "type": "equation_inline", "content": "\\sin(x + n\\pi) = (-1)^n \\sin x, \\cos(n + n\\pi) = (-1)^n \\cos x" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^n \\frac{\\sin n}{n} = \\sum_{n=1}^{\\infty} \\frac{\\sin(1 + \\pi)n}{n}" }, { "type": "text", "content": " 条件收敛 (根据 (1) 结论)。第二个转换为 " }, { "type": "equation_inline", "content": "\\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{(-1)^n}{n} - \\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{\\cos(2 + \\pi)n}{n}" }, { "type": "text", "content": " 条件收敛。绝对值发散的原因是 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{\\sin^2 n}{n} = \\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{1}{n} - \\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{\\cos 2n}{n}" } ] }, "bbox": [ 87, 602, 942, 708 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.14具体例子讨论 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{\\cos 3n}{n} (1 + \\frac{1}{n})^n" }, { "type": "text", "content": " 的收敛性" } ] }, "bbox": [ 60, 709, 492, 745 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{\\cos 3n}{n}" }, { "type": "text", "content": " 根据前面结论收敛， " }, { "type": "equation_inline", "content": "(1 + \\frac{1}{n})^n \\to e" }, { "type": "text", "content": " 单调有界，因此由 Abel 可知收敛" } ] }, "bbox": [ 87, 745, 692, 781 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.4 数项级数理论" } ], "level": 2 }, "bbox": [ 92, 804, 289, 822 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "8.4.1 级数的加括号" } ] }, "bbox": [ 92, 843, 273, 860 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "级数的加括号本质是数列收敛及其子列收敛的关系" } ] }, "bbox": [ 122, 871, 497, 887 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.4 数项级数理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "79" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.11 (加括号收敛)" } ], "level": 2 }, "bbox": [ 122, 86, 295, 101 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 中任意加括号得到级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} v_n" }, { "type": "text", "content": " 也收敛，且 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n = \\sum_{n=1}^{\\infty} v_n" }, { "type": "text", "content": " (即加括号既不改变收敛性，又不改变和)" } ] }, "bbox": [ 112, 105, 882, 156 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 显然 " }, { "type": "equation_inline", "content": "S_{n} = \\sum_{k = 1}^{n}u_{k}" }, { "type": "text", "content": " 收敛，从而 " }, { "type": "equation_inline", "content": "S_{k}^{\\prime} = \\sum_{k = 1}^{m}v_{k}" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "S_{n}" }, { "type": "text", "content": " 子列，从而收敛且极限相同。" } ] }, "bbox": [ 90, 168, 697, 206 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 若级数发散，则加括号得到的新级数可能收敛，例如 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (-1)^n" }, { "type": "text", "content": "。若级数加括号以后发散，则原级数一定收敛。" } ] }, "bbox": [ 87, 206, 905, 254 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.12 (加括号收敛反推级数收敛)" } ], "level": 2 }, "bbox": [ 121, 263, 396, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 加括号后 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}v_{n}" }, { "type": "text", "content": " 收敛，" } ] }, "bbox": [ 117, 282, 352, 322 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}u_n = 0" }, { "type": "text", "content": " 且每个括号里项个数小于固定值 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 139, 319, 648, 357 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若每个括号里符号相同，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 139, 355, 452, 393 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.4.2 级数的重排" } ], "level": 2 }, "bbox": [ 94, 431, 253, 448 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 8.2 (重排)" } ], "level": 2 }, "bbox": [ 122, 463, 236, 478 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "考虑级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[n] \\to [n]" }, { "type": "text", "content": " 的映射, 则称" } ] }, "bbox": [ 115, 486, 460, 521 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {n = 1} ^ {\\infty} v _ {n} = \\sum_ {n = 1} ^ {\\infty} u _ {f (n)}", "math_type": "latex", "image_source": { "path": "images/b9eefb1365ce8aae322a26e21b214d713901e1bd8300f5f98871e186c4fbe080.jpg" } }, "bbox": [ 428, 524, 566, 563 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "为 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 的一个重排" } ] }, "bbox": [ 115, 564, 277, 602 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.13 (正项级数的重排)" } ], "level": 2 }, "bbox": [ 122, 627, 329, 642 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "正项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 收敛，则其任意重排 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_{f(n)}" }, { "type": "text", "content": " 也收敛，且" } ] }, "bbox": [ 114, 646, 552, 684 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {n = 1} ^ {\\infty} u _ {n} = \\sum_ {n = 1} ^ {\\infty} u _ {f (n)}", "math_type": "latex", "image_source": { "path": "images/4ce79440d2888d729272d78af0cdd1381c1a0d460ee3f7155e2fd3392ff1d52e.jpg" } }, "bbox": [ 426, 687, 566, 725 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 记 " }, { "type": "equation_inline", "content": "S_{n}=\\sum_{k=1}^{n}u_{k},S_{n}^{\\prime}=\\sum_{k=1}^{n}u_{f(k)}" }, { "type": "text", "content": " ，记 " }, { "type": "equation_inline", "content": "M=\\max\\{f(1),\\cdots,f(n)\\}" }, { "type": "text", "content": " ，则显然 " }, { "type": "equation_inline", "content": "S_{n}^{\\prime}\\leq S_{M}" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}S_{n}" }, { "type": "text", "content": " 存在可知 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}S_{n}^{\\prime}" }, { "type": "text", "content": " 存在，且" } ] }, "bbox": [ 90, 738, 905, 793 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} S _ {n} ^ {\\prime} \\leq \\lim _ {n \\to \\infty} S _ {n} ^ {\\prime}", "math_type": "latex", "image_source": { "path": "images/3efaf3056bb524b877966428c8b79fd7265a927c17400910c41f6e527886a955.jpg" } }, "bbox": [ 426, 791, 569, 813 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "反之也可以将 " }, { "type": "equation_inline", "content": "S_{n}" }, { "type": "text", "content": " 看作 " }, { "type": "equation_inline", "content": "S_{n}^{\\prime}" }, { "type": "text", "content": " 的重排，因此也满足 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}S_n\\leq \\lim_{n\\to \\infty}S_n'" } ] }, "bbox": [ 124, 819, 610, 840 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.4 数项级数理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "80" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.14 (一般项级数的重排)" } ], "level": 2 }, "bbox": [ 122, 86, 346, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是重排映射，若 " }, { "type": "equation_inline", "content": "\\exists M\\in \\mathbb{Z}^{+}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(n) - n|\\leq M" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 收敛当且仅当 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_{f(n)}" }, { "type": "text", "content": " 收敛，且" } ] }, "bbox": [ 114, 105, 811, 145 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {n = 1} ^ {\\infty} u _ {n} = \\sum_ {n = 1} ^ {\\infty} u _ {f (n)}", "math_type": "latex", "image_source": { "path": "images/9ff57284801dad0b181bb52f7ffebe1bdd6ef42968cd66458b6d0b40dc227018.jpg" } }, "bbox": [ 426, 146, 566, 185 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.15 (绝对收敛级数的重排)" } ], "level": 2 }, "bbox": [ 124, 209, 361, 224 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n" }, { "type": "text", "content": " 绝对收敛，则其任意重排 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_{f(n)}" }, { "type": "text", "content": " 绝对收敛，且" } ] }, "bbox": [ 115, 229, 552, 267 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {n = 1} ^ {\\infty} u _ {n} = \\sum_ {n = 1} ^ {\\infty} u _ {f (n)}", "math_type": "latex", "image_source": { "path": "images/02ad00276a7a9699dcf3cc839efbb4e712319d0df2de764ba1f0c9a3655a6f2d.jpg" } }, "bbox": [ 426, 269, 566, 307 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 正负部分开即可" } ] }, "bbox": [ 92, 326, 253, 341 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.16 (Riemann 定理：条件收敛级数的重排)" } ], "level": 2 }, "bbox": [ 122, 351, 482, 367 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 条件收敛，则对 " }, { "type": "equation_inline", "content": "\\forall S \\in \\mathbb{R} \\cup \\{\\pm \\infty\\}" }, { "type": "text", "content": "，存在重排 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} v_n" }, { "type": "text", "content": " 收敛到 " }, { "type": "equation_inline", "content": "S" } ] }, "bbox": [ 114, 372, 704, 411 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.4.3 正部与负部" } ], "level": 2 }, "bbox": [ 94, 448, 253, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于任意数列 " }, { "type": "equation_inline", "content": "u_{n}" }, { "type": "text", "content": "，考虑非负数列 " }, { "type": "equation_inline", "content": "p_n = \\frac{|u_n| + u_n}{2}, q_n = \\frac{|u_n| - u_n}{2}" }, { "type": "text", "content": "，分别为 " }, { "type": "equation_inline", "content": "u_{n}" }, { "type": "text", "content": " 的正部与负部，下面研究它们之间的关系。" } ] }, "bbox": [ 90, 474, 905, 516 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.17 (正负部)" } ], "level": 2 }, "bbox": [ 124, 525, 260, 539 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 绝对收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} p_n, \\sum_{n=1}^{\\infty} q_n" }, { "type": "text", "content": " 均收敛。若 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 条件收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{k=1}^{n} p_k, \\sum_{k=1}^{n} q_k" }, { "type": "text", "content": " 发散到正无穷，且 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^{n} p_k}{\\sum_{k=1}^{n} q_k} = 1" } ] }, "bbox": [ 110, 542, 887, 648 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "0 \\leq p_{n}, q_{n} \\leq |u_{n}|" }, { "type": "text", "content": "，因此根据比较原则可发现 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} p_{n}, q_{n}" }, { "type": "text", "content": " 都收敛，且极限可拆开" } ] }, "bbox": [ 90, 656, 726, 690 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n" }, { "type": "text", "content": " 条件收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (p_n - q_n)" }, { "type": "text", "content": " 收敛，但 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (p_n + q_n) = +\\infty" }, { "type": "text", "content": "，假设 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} p_n" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} q_n" }, { "type": "text", "content": " 收敛，这与 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} (p_n + q_n)" }, { "type": "text", "content": " 发散矛盾。" } ] }, "bbox": [ 90, 689, 904, 765 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.15 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单减， " }, { "type": "equation_inline", "content": "a_{n} \\geq 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_{n}" }, { "type": "text", "content": " 发散，证明 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\frac{a_{2} + a_{4} + \\cdots + a_{2n}}{a_{1} + a_{3} + \\cdots + a_{2n-1}} = 1" } ] }, "bbox": [ 60, 764, 670, 797 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 极限小于等于 1 显然。而 " }, { "type": "equation_inline", "content": "\\frac{a_{2}+\\cdots+a_{2n}}{a_{1}+\\cdots+a_{2n-1}}\\geq\\frac{a_{3}+a_{5}+\\cdots+a_{2n-1}}{a_{1}+a_{3}+\\cdots+a_{2n-1}}=1-\\frac{a_{1}}{a_{1}+\\cdots+a_{2n-1}}" }, { "type": "text", "content": " ，根据单减以及发散可知 " }, { "type": "equation_inline", "content": "2(a_{1}+a_{3}+\\cdots+a_{2n-1})\\geq a_{1}+a_{2}+\\cdots+a_{2n}\\rightarrow\\infty" }, { "type": "text", "content": " ，因此大于等于 1。" } ] }, "bbox": [ 80, 797, 907, 843 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.4 数项级数理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "81" } ] }, "bbox": [ 489, 954, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.4.4 级数的乘积" } ], "level": 2 }, "bbox": [ 92, 85, 253, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "8.4.5 级数的相互控制" } ], "level": 2 }, "bbox": [ 94, 115, 292, 131 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 8.18 (级数相互控制)" } ], "level": 2 }, "bbox": [ 122, 148, 312, 162 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "a_{n}\\leq c_{n}\\leq b_{n}" }, { "type": "text", "content": " 则" } ] }, "bbox": [ 115, 168, 257, 183 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 数列 " }, { "type": "equation_inline", "content": "a_{n}, b_{n}" }, { "type": "text", "content": " 收敛不能得出 " }, { "type": "equation_inline", "content": "c_{n}" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 139, 186, 405, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n, \\sum_{n=1}^{\\infty} b_n" }, { "type": "text", "content": " 收敛可得出 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} c_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 139, 200, 473, 237 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 例如: " }, { "type": "equation_inline", "content": "a_{n} = -2 - \\frac{1}{n}, b_{n} = 2 + \\frac{1}{n}, c_{n} = (-1)^{n}" } ] }, "bbox": [ 89, 250, 487, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 此时 " }, { "type": "equation_inline", "content": "0 \\leq c_{n} - a_{n} \\leq b_{n} - a_{n}" }, { "type": "text", "content": ", 显然 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} b_{n} - a_{n}" }, { "type": "text", "content": " 收敛, 根据比较原则得到 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} c_{n} - a_{n}" }, { "type": "text", "content": " 收敛, 因此得到 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} c_{n}" }, { "type": "text", "content": " 收敛。" } ] }, "bbox": [ 89, 275, 905, 349 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 8.1 (控制是相互的)" } ], "level": 2 }, "bbox": [ 122, 360, 302, 375 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "a_{n} \\leq b_{n} \\leq a_{n + 1}" }, { "type": "text", "content": "，则显然推出 " }, { "type": "equation_inline", "content": "b_{n - 1} \\leq a_{n} \\leq b_{n}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty} a_{n}, \\sum_{n = 1}^{\\infty} b_{n}" }, { "type": "text", "content": " 同敛散。" } ] }, "bbox": [ 112, 379, 687, 418 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习8.16相互控制练习(1)重点： " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 每一项都非零， " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = a\\neq 0" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}|a_{n + 1} - a_n|,\\sum_{n = 1}^{\\infty}\\left|\\frac{1}{a_{n + 1}} -\\frac{1}{a_n}\\right|" }, { "type": "text", "content": " 敛散性相同" } ] }, "bbox": [ 58, 448, 902, 497 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 是单调减的正数列，证明 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\sum_{m = 1}^{\\infty}2^{m}a_{2^{m}}" }, { "type": "text", "content": " 同敛散" } ] }, "bbox": [ 122, 495, 566, 533 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据极限保号性， " }, { "type": "equation_inline", "content": "\\exists N, \\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\frac{1}{2} a \\leq a_n \\leq \\frac{3}{2} a" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\left|\\frac{1}{a_{n+1}} - \\frac{1}{a_n}\\right| = \\left|\\frac{a_n - a_{n+1}}{a_na_{n+1}}\\right|" }, { "type": "text", "content": " 因此" } ] }, "bbox": [ 89, 531, 793, 563 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {4}{9 a ^ {2}} | a _ {n + 1} - a _ {n} | \\leq \\frac {| a _ {n + 1} - a _ {n} |}{| a _ {n} a _ {n + 1} |} \\leq \\frac {4}{a ^ {2}} | a _ {n + 1} - a _ {n} |", "math_type": "latex", "image_source": { "path": "images/4a81a91077861b4955e59561828e4ec8f516b414d2a32689b66d08e2dc168d31.jpg" } }, "bbox": [ 322, 565, 673, 596 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "从而相互控制，同敛散" } ] }, "bbox": [ 90, 600, 263, 614 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单调减可得" } ] }, "bbox": [ 122, 618, 295, 633 ] }, { "type": "equation_interline", "content": { "math_content": "2 ^ {k - 1} a _ {2 ^ {k}} \\leq a _ {2 ^ {k - 1} + 1} + a _ {2 ^ {k - 1} + 2} + \\dots + a _ {2 ^ {k}} \\leq 2 ^ {k - 1} a _ {2 ^ {k - 1}}", "math_type": "latex", "image_source": { "path": "images/91c3fd77ff3840c0d6e5eac426eec32d047f17de4db15724a979733869af6b71.jpg" } }, "bbox": [ 299, 640, 694, 657 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对 " }, { "type": "equation_inline", "content": "k" }, { "type": "text", "content": " 求和得到" } ] }, "bbox": [ 90, 664, 198, 678 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {1}{2} \\sum_ {k = 1} ^ {n} 2 ^ {k} a _ {2 ^ {k}} \\leq a _ {1} + a _ {2} + \\dots + a _ {2 ^ {n}} \\leq a _ {1} + \\sum_ {k = 0} ^ {n - 1} 2 ^ {k} a _ {2 ^ {k}}", "math_type": "latex", "image_source": { "path": "images/dcfd543461aecbc8ce60468960803f5fdf711502abd4cdb8f8ba5bd354b3ac9c.jpg" } }, "bbox": [ 309, 677, 685, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这说明两者部分和等价，因此同敛散。" } ] }, "bbox": [ 90, 724, 371, 738 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 8.17 比较原则对于一般项级数不成立的反例 (1) " }, { "type": "equation_inline", "content": "a_{n}, b_{n}" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{a_{n}}{b_{n}}=l>0" }, { "type": "text", "content": " ，此时显然 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}|a_{n}|,\\sum_{n=1}^{\\infty}|b_{n}|" }, { "type": "text", "content": " 敛散性相同，但举例说明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}a_{n},\\sum_{n=1}^{\\infty}b_{n}" }, { "type": "text", "content": " 敛散性可以不同。" } ] }, "bbox": [ 60, 755, 904, 825 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{b_n} = 0" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}b_n" }, { "type": "text", "content": " 绝对收敛可推出 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 绝对收敛，举例 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}b_n" }, { "type": "text", "content": " 收敛推不出 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 122, 822, 855, 863 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3)举例说明 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n" }, { "type": "text", "content": " 收敛， " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n^2" }, { "type": "text", "content": " 不一定收敛" } ] }, "bbox": [ 122, 860, 460, 898 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.4 数项级数理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "82" } ] }, "bbox": [ 489, 953, 507, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 例如 " }, { "type": "equation_inline", "content": "a_{n} = \\frac{(-1)^{n}}{\\sqrt{n}}, b_{n} = \\frac{(-1)^{n}}{\\sqrt{n}} + \\frac{1}{n}" }, { "type": "text", "content": ", 显然 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{a_n}{b_n} = 1" }, { "type": "text", "content": ", 但 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛, " }, { "type": "equation_inline", "content": "b_{n}" }, { "type": "text", "content": " 发散" } ] }, "bbox": [ 90, 80, 702, 112 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 比如 " }, { "type": "equation_inline", "content": "a_{n} = \\frac{1}{n}, b_{n} = \\frac{(-1)^{n}}{\\sqrt{n}}" } ] }, "bbox": [ 122, 112, 337, 141 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 例如 " }, { "type": "equation_inline", "content": "a_{n} = \\frac{(-1)^{n}}{\\sqrt{n}}" } ] }, "bbox": [ 124, 141, 277, 171 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "8.4 数项级数理论" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "83" } ] }, "bbox": [ 489, 954, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第9章 函数列与函数项级数" } ], "level": 1 }, "bbox": [ 324, 87, 690, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.1 函数列的一致收敛" } ], "level": 2 }, "bbox": [ 92, 145, 337, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.1.1 一致收敛的概念" } ], "level": 2 }, "bbox": [ 92, 185, 292, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 9.1 (一致收敛)" } ], "level": 2 }, "bbox": [ 119, 216, 270, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上的函数列，若对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall x, \\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_{n}(x) - f(x)| < \\epsilon" }, { "type": "text", "content": " ，则称 " }, { "type": "equation_inline", "content": "f_{n}" }, { "type": "text", "content": " 一致收敛至 " }, { "type": "equation_inline", "content": "f" } ] }, "bbox": [ 112, 235, 902, 261 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记一致收敛与点态收敛的区别：一致收敛的 N 不受 x 影响，因此当 n > N 时，所有 x 都会在 " }, { "type": "equation_inline", "content": "f(x) \\pm \\epsilon" }, { "type": "text", "content": " 带中。" } ] }, "bbox": [ 87, 272, 905, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.1定义说明不一致收敛 " }, { "type": "equation_inline", "content": "(1)\\frac{x}{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0,a]" }, { "type": "text", "content": " 一致收敛到0，但在 " }, { "type": "equation_inline", "content": "(0, + \\infty)" }, { "type": "text", "content": " 不一致收敛(2)说明 " }, { "type": "equation_inline", "content": "x^{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 上收敛但不一致收敛证明 " }, { "type": "equation_inline", "content": "(1)\\exists \\epsilon_0 = \\frac{1}{2},\\forall N,\\exists n_0 = N + 1,\\exists x_0 = n_0" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|\\frac{x_0}{n_0} -0| = 1 > \\epsilon_0" }, { "type": "text", "content": " ，因此不一致收敛" } ] }, "bbox": [ 58, 307, 905, 369 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 点态收敛显然。不一致收敛的原因在 1 处，因此考虑 " }, { "type": "equation_inline", "content": "x_{n} = 1 - \\frac{1}{n}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}|x_n^n -0| = \\lim_{n\\to \\infty}(1 - \\frac{1}{n})^n = e^{-1}\\neq 0" }, { "type": "text", "content": " 笔记 " }, { "type": "equation_inline", "content": "\\frac{x}{n}, x^n" }, { "type": "text", "content": " 是极度重要的不一致收敛反例，在很多反例里都会用到它" } ] }, "bbox": [ 58, 367, 905, 434 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.1 (Cauchy 收敛准则)" } ], "level": 2 }, "bbox": [ 122, 453, 329, 468 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上的函数列， " }, { "type": "equation_inline", "content": "f_{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛于 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 的充要条件为 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall n, m > N" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "\\sup_{x \\in I} |f_{m}(x) - f_{n}(x)| < \\epsilon" } ] }, "bbox": [ 102, 470, 900, 507 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 Cauchy 判别法一般用于证明其他判别法，本身对于证明具体一致收敛用处较少 (因为具体函数列的极限一般容易算出)" } ] }, "bbox": [ 60, 516, 905, 551 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 9.2 (内闭一致收敛)" } ], "level": 2 }, "bbox": [ 122, 561, 304, 576 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若函数列 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在区间 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 任意一个闭区间都一致收敛，则称 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上内闭一致收敛。" } ] }, "bbox": [ 110, 580, 900, 606 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 内闭一致收敛的意义在于大部分不一致收敛都是在区间端点出现了问题，因此内闭一致收敛反应了区间整体的性质，不受个别端点影响。" } ] }, "bbox": [ 87, 617, 904, 651 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.1.2 一致收敛的基本判别法" } ], "level": 2 }, "bbox": [ 92, 673, 352, 689 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.2 (Weiestrass 判别法)" } ], "level": 2 }, "bbox": [ 122, 705, 334, 720 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若存在 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}a_n = 0" }, { "type": "text", "content": " 的正项数列 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " ，使得 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上有 " }, { "type": "equation_inline", "content": "|f(x) - f_n(x)|\\leq a_n" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" } ] }, "bbox": [ 112, 722, 900, 756 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根本原因是 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛的 N 与 x 无关，用 Cauchy 收敛准则易证。" } ] }, "bbox": [ 90, 766, 591, 781 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.2M判别法的使用(1)重点：证明 " }, { "type": "equation_inline", "content": "\\sqrt{x^2 + \\frac{1}{n^2}}" }, { "type": "text", "content": " 在任意区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛" } ] }, "bbox": [ 58, 794, 660, 827 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点： " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上的连续函数， " }, { "type": "equation_inline", "content": "f_{0}(x) = f(x)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "f_{n+1}(x) = \\int_{a}^{x} f_{n}(t) \\, dt" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 一致收敛\n\n证明 (1) " }, { "type": "equation_inline", "content": "\\left|\\sqrt{x^{2} + \\frac{1}{n^{2}}} - |x|\\right| = \\frac{\\frac{1}{n^{2}}}{\\sqrt{x^{2} + \\frac{1}{n^{2}}} + |x|} \\leq \\frac{1}{n} \\to 0" } ] }, "bbox": [ 87, 828, 900, 898 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于连续，因此 " }, { "type": "equation_inline", "content": "|f(x)| \\leq M" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "|f_1(x)| \\leq \\int_0^x |f_1(t)|\\mathrm{d}t \\leq M(x - a)" }, { "type": "text", "content": " (暂时不要进一步放缩)，同理得到 " }, { "type": "equation_inline", "content": "|f_2(x)| \\leq \\int_a^x M(t - a)\\mathrm{d}t = \\frac{(x - a)^2}{2!} M" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "|f_n(x)| \\leq \\frac{(x - a)^n}{n!} M \\leq \\frac{(b - a)^n}{n!} M \\to 0" }, { "type": "text", "content": " (最后再放缩)" } ] }, "bbox": [ 90, 82, 905, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.3 (确界判别法)" } ], "level": 2 }, "bbox": [ 121, 161, 287, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上函数列，则" } ] }, "bbox": [ 114, 181, 277, 195 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的充要条件为 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sup_{x\\in I}|f_n(x) - f(x)| = 0" }, { "type": "text", "content": "。" } ] }, "bbox": [ 139, 198, 642, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 不一致收敛于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的充要条件是存在 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "f_{n}(x_{n}) - f(x_{n})" }, { "type": "text", "content": " 不收敛到0(极度重要)" } ] }, "bbox": [ 139, 222, 831, 237 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 左推右：对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall x, \\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x) - f(x)| < \\epsilon" }, { "type": "text", "content": "，根据上确界定义可知 " }, { "type": "equation_inline", "content": "\\sup_{\\epsilon \\in I} |f_n(x) - f(x)| \\leq \\epsilon" } ] }, "bbox": [ 89, 256, 853, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "右推左： " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall n > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\sup_{x \\in I} |f_n(x) - f(x)| < \\epsilon" }, { "type": "text", "content": "，而对 " }, { "type": "equation_inline", "content": "\\forall x \\in I" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x) - f(x)| \\leq \\sup_{x \\in I} |f_n(x) - f(x)|" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\forall x, |f_n(x) - f(x)| < \\epsilon" }, { "type": "text", "content": "，即一致收敛。" } ] }, "bbox": [ 90, 279, 902, 318 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 一般一致收敛用 Weiestrass 判别法判定，不一致收敛用确界判别法第二条判定" } ] }, "bbox": [ 87, 321, 697, 335 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.1.3 一致收敛判别练习" } ], "level": 2 }, "bbox": [ 92, 357, 312, 374 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 9.3 三角相关 求下列函数列的收敛区间、极限函数与一致收敛区间：" } ] }, "bbox": [ 58, 386, 626, 400 ] }, { "type": "equation_interline", "content": { "math_content": "(1) f _ {n} (x) = \\frac {\\sin x}{n} (2) f _ {n} (x) = \\sin \\frac {x}{n} (3) f _ {n} (x) = \\frac {x}{n} \\sin \\frac {x}{n} (4) f _ {n} (x) = n \\sin \\frac {x}{n}", "math_type": "latex", "image_source": { "path": "images/4ba19d13920eeb029adf5ad713f962d43e7246cd589c6ccb30a56861aaaaf881.jpg" } }, "bbox": [ 122, 401, 648, 426 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)根据 " }, { "type": "equation_inline", "content": "|f_n(x)|\\leq \\frac{1}{n}\\to 0" }, { "type": "text", "content": " ，显然" } ] }, "bbox": [ 89, 426, 347, 450 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 先求极限函数， " }, { "type": "equation_inline", "content": "\\forall x \\in (-\\infty, +\\infty)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} f_n(x) = 0" }, { "type": "text", "content": "，因此点态收敛到 0。取 " }, { "type": "equation_inline", "content": "x_n = n" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\sin \\frac{x_n}{n} = \\sin 1" }, { "type": "text", "content": " 不趋于 0，因此在 " }, { "type": "equation_inline", "content": "(- \\infty, +\\infty)" }, { "type": "text", "content": " 不一致收敛。但是任意 " }, { "type": "equation_inline", "content": "[a, b] \\subseteq (-\\infty, +\\infty)" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "|x| \\leq M" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "|\\sin \\frac{x}{n}| \\leq |\\frac{x}{n}| \\leq \\frac{M}{n} \\to 0" }, { "type": "text", "content": "，因此内闭一致收敛" } ] }, "bbox": [ 87, 450, 905, 510 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 点态收敛到 " }, { "type": "equation_inline", "content": "f(x) = 0" }, { "type": "text", "content": "，取 " }, { "type": "equation_inline", "content": "x_{n} = n" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}f_n(x_n) = \\sin 1\\neq 0" }, { "type": "text", "content": "。取 " }, { "type": "equation_inline", "content": "[a,b]\\subseteq (-\\infty , + \\infty)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "|f_n(x) - 0|\\leq |\\frac{x}{n} |\\rightarrow 0" }, { "type": "text", "content": " 因此内闭一致收敛。" } ] }, "bbox": [ 90, 514, 905, 558 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 极限函数为 1，考虑 " }, { "type": "equation_inline", "content": "x_{n} = n" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}|f_n(x_n) - x_n| = |n\\sin 1 - n| \\neq 0" }, { "type": "text", "content": "，因此在 " }, { "type": "equation_inline", "content": "(- \\infty, + \\infty)" }, { "type": "text", "content": " 不一致收敛。取 " }, { "type": "equation_inline", "content": "[a,b] \\subseteq (-\\infty, + \\infty)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "|f_n(x) - x| \\leq |n(\\sin \\frac{x}{n} - \\frac{x}{n})| \\leq n(\\frac{x}{n})^2 \\leq \\frac{M^2}{n} \\to 0" }, { "type": "text", "content": "，中间的不等式使用了 Taylor 展开，因此内闭一致收敛。" } ] }, "bbox": [ 90, 556, 907, 617 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 9.4 分式相关 判断在 " }, { "type": "equation_inline", "content": "[0, +\\infty)" }, { "type": "text", "content": " 的一致收敛性" } ] }, "bbox": [ 58, 638, 450, 652 ] }, { "type": "equation_interline", "content": { "math_content": "(1) f _ {n} (x) = \\frac {x}{1 + n ^ {2} x ^ {2}} (2) f _ {n} (x) = \\frac {x}{1 + n x} (3) f _ {n} (x) = \\frac {1}{1 + n x} (4) f _ {n} (x) = \\frac {n x}{1 + n ^ {2} x ^ {2}}", "math_type": "latex", "image_source": { "path": "images/e8a7301e6219ba6a9b6e14de6ebac2e5b8d9404a659f355182eae63c13b99d28.jpg" } }, "bbox": [ 122, 652, 709, 680 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 " }, { "type": "equation_inline", "content": "(1)f_{n}(x)\\leq \\frac{x}{2nx} = \\frac{1}{2n} (x\\neq 0)\\to 0" }, { "type": "text", "content": " ，而 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "f_{n}(0) = 0" }, { "type": "text", "content": " ，因此一致收敛" } ] }, "bbox": [ 87, 680, 653, 705 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 显然， " }, { "type": "equation_inline", "content": "x \\neq 0" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "f_{n}(x) \\leq \\frac{1}{n} \\to 0" }, { "type": "text", "content": " ，一致收敛" } ] }, "bbox": [ 121, 703, 502, 731 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 先考虑极限函数 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 时, 极限函数为 " }, { "type": "equation_inline", "content": "1, x \\neq 0" }, { "type": "text", "content": " 时, 极限函数为 0。考虑 " }, { "type": "equation_inline", "content": "(0, +\\infty)" }, { "type": "text", "content": ", 取 " }, { "type": "equation_inline", "content": "x_{n} = \\frac{1}{n}, f_{n}(x) - f(x) = \\frac{1}{2} \\neq 0" }, { "type": "text", "content": ", 因此不一致收敛。在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f_{n}(x) - 0 = \\frac{1}{1 + nx} \\leq \\frac{1}{1 + na} \\to 0" }, { "type": "text", "content": " 一致。" } ] }, "bbox": [ 87, 730, 904, 780 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 极限函数为 0。取 " }, { "type": "equation_inline", "content": "x_{n} = \\frac{1}{n}" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "f_{n}(x) - 0 = \\frac{1}{2}" }, { "type": "text", "content": " 不趋于 0，因此不一致收敛。在 " }, { "type": "equation_inline", "content": "[a, +\\infty)" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "f_{n}(x) \\leq \\frac{nx}{n^{2}x^{2}} \\leq \\frac{1}{nx} \\leq \\frac{1}{na} \\to 0" } ] }, "bbox": [ 90, 782, 905, 833 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 9.5 导数与积分一致收敛 (1) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 R 上有连续导函数， " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 为单调趋于 " }, { "type": "equation_inline", "content": "+\\infty" }, { "type": "text", "content": " 的正数列，证明 " }, { "type": "equation_inline", "content": "f_{n}(x) = a_{n}\\left[f\\left(x + \\frac{1}{a_{n}}\\right) - f(x)\\right]" }, { "type": "text", "content": " 在任意有限闭区间 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "f'(x)" } ] }, "bbox": [ 58, 846, 905, 892 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "85" } ] }, "bbox": [ 489, 954, 507, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.1 函数列的一致收敛" } ] }, "bbox": [ 737, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 特例：证明 " }, { "type": "equation_inline", "content": "f_{n}(x) = n\\left(\\sqrt{x + \\frac{1}{n}} - \\sqrt{x}\\right)" }, { "type": "text", "content": " 在任意有限闭区间 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "\\frac{1}{2\\sqrt{x}}" } ] }, "bbox": [ 121, 82, 766, 118 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上连续函数，记 " }, { "type": "equation_inline", "content": "f_{n}(x) = \\frac{1}{n}\\sum_{k = 1}^{n}f\\left(x + \\frac{k}{n}\\right)" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在任意有限区间 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上一致收敛于 " }, { "type": "equation_inline", "content": "F(x) = \\int_0^1 f(x + t)\\mathrm{d}t" } ] }, "bbox": [ 90, 120, 905, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 特例：证明 " }, { "type": "equation_inline", "content": "f_{n}(x) = \\sum_{k=1}^{n} \\frac{1}{n} \\cos \\left( x + \\frac{k}{n} \\right)" }, { "type": "text", "content": " 在任意有限区域 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "\\sin (x + 1) - \\sin x" }, { "type": "text", "content": " 证明 (1) 给定 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": "，根据 Lagrange 中值定理， " }, { "type": "equation_inline", "content": "f_{n}(x) = f'(x + \\xi_{n})" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "\\xi_{n} < \\frac{1}{a_{n}}" }, { "type": "text", "content": "。由于 " }, { "type": "equation_inline", "content": "f'(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b + 1]" }, { "type": "text", "content": " 连续，因此一致连续， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta \\in (0, 1)" }, { "type": "text", "content": " 当 " }, { "type": "equation_inline", "content": "|x_{1} - x_{2}| < \\delta" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "|f'(x_{1}) - f'(x_{2})| < \\epsilon" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\exists N, \\forall n > N, \\xi_{n} < \\frac{1}{a_{n}} < \\delta" }, { "type": "text", "content": "，此时" } ] }, "bbox": [ 90, 186, 905, 275 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f _ {n} (x) - f ^ {\\prime} (x) \\right| = \\left| f ^ {\\prime} \\left(x _ {n} + \\xi_ {n}\\right) - f ^ {\\prime} (x) \\right| < \\epsilon", "math_type": "latex", "image_source": { "path": "images/112610bf6e7a7af241b8190886c92a2f95a75795d163228460c21748c2ae49f7.jpg" } }, "bbox": [ 339, 279, 657, 296 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此一致收敛" } ] }, "bbox": [ 90, 303, 196, 317 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 即 (1) 的特例" } ] }, "bbox": [ 122, 321, 248, 335 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "F(x)=\\sum_{k=1}^{n}\\int_{\\frac{k-1}{n}}^{\\frac{k}{n}}f(x+t)\\mathrm{d}t" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 122, 334, 396, 373 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f _ {n} (x) - F (x) \\right| = \\left| \\sum_ {k = 1} ^ {n} \\int_ {\\frac {k - 1}{n}} ^ {\\frac {k}{n}} f \\left(x + \\frac {k}{n}\\right) - \\sum_ {k = 1} ^ {n} \\int_ {\\frac {k - 1}{n}} ^ {\\frac {k}{n}} f (x + t) \\mathrm{d} t \\right| \\leq \\sum_ {k = 1} ^ {n} \\int_ {\\frac {k - 1}{n}} ^ {\\frac {k}{n}} \\left| f \\left(x + \\frac {k}{n}\\right) - f (x + t) \\right| \\mathrm{d} t", "math_type": "latex", "image_source": { "path": "images/2cbe5763425e9f08b17273e61ade19394418d9d34d747c0ad4297fa8539b8d7f.jpg" } }, "bbox": [ 132, 375, 863, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "显然 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b + 1]" }, { "type": "text", "content": " 一致连续， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, |x_1 - x_2| < \\delta" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "|f(x') - f(x'')| < \\epsilon" }, { "type": "text", "content": "。而 " }, { "type": "equation_inline", "content": "\\exists N, \\forall n > N, \\frac{1}{n} < \\delta" }, { "type": "text", "content": "，对于 " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b], t \\in \\left[\\frac{k - 1}{n}, \\frac{k}{n}\\right]" }, { "type": "text", "content": "，有" } ] }, "bbox": [ 89, 420, 905, 473 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\left(x + \\frac {k}{n}\\right) - (x + t) \\right| = \\left| \\frac {k}{n} - t \\right| \\leq \\frac {1}{n} < \\delta", "math_type": "latex", "image_source": { "path": "images/0128f62dc66f31d634cfe06de0673919c7416a1fa5a15da26ebc19933a8fba5f.jpg" } }, "bbox": [ 349, 470, 648, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "综上得到" } ] }, "bbox": [ 90, 511, 163, 526 ] }, { "type": "equation_interline", "content": { "math_content": "\\forall x \\in [ a, b ], | f _ {n} (x) - F (x) | < \\sum_ {k = 1} ^ {n} \\int_ {\\frac {k - 1}{n}} ^ {\\frac {k}{n}} \\epsilon \\mathrm{d} t = \\epsilon", "math_type": "latex", "image_source": { "path": "images/b5a794f1ac4a16bfaa89a210653770117f9add77fe424154ccaff4f149ceb38a.jpg" } }, "bbox": [ 329, 523, 665, 562 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.1.4 四则运算、复合极限的一致收敛性" } ], "level": 2 }, "bbox": [ 92, 590, 453, 607 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 9.3 (一致有界)" } ], "level": 2 }, "bbox": [ 122, 621, 270, 637 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\exists M, \\forall n, \\forall x \\in I" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x)| \\leq M" } ] }, "bbox": [ 114, 643, 344, 659 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "引理 9.1 (极限的有界性)" } ], "level": 2 }, "bbox": [ 124, 683, 304, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在区间 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上点态收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 115, 703, 423, 718 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若一致收敛，且每个 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 有界，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 有界， " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致有界" } ] }, "bbox": [ 139, 722, 631, 737 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若一致收敛，且 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 有界，则 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 除去至多有限项外一致有界" } ] }, "bbox": [ 139, 740, 635, 755 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致有界，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 有界" } ] }, "bbox": [ 139, 758, 401, 774 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 设 " }, { "type": "equation_inline", "content": "|f_n(x)| \\leq M_n" }, { "type": "text", "content": "。根据一致收敛，取 " }, { "type": "equation_inline", "content": "\\epsilon = 1, \\exists N, \\forall n > N, \\forall x \\in I" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x) - f(x)| \\leq 1" }, { "type": "text", "content": "。因此得到 " }, { "type": "equation_inline", "content": "|f(x)| \\leq M_N + 1 := M" }, { "type": "text", "content": "。而 " }, { "type": "equation_inline", "content": "n \\geq N" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\forall x \\in I" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x)| \\leq |f(x)| + 1 = M + 1" }, { "type": "text", "content": "，前面有限项取最大即可，因此一致有界" } ] }, "bbox": [ 89, 794, 905, 845 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 包含于 (1) 的推导过程中" } ] }, "bbox": [ 122, 850, 331, 864 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 若 " }, { "type": "equation_inline", "content": "\\exists M, \\forall n, |f_n(x)| \\leq M" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} f_n(x) = f(x)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\forall x \\in I, \\exists \\epsilon_0 = 1, \\exists N, n \\geq N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f(x) - f_n(x)| \\leq 1" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "|f(x)| \\leq |f_n(x)| + 1 \\leq M + 1" }, { "type": "text", "content": "，因此有界。" } ] }, "bbox": [ 90, 868, 905, 903 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.1 函数列的一致收敛" } ] }, "bbox": [ 737, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "86" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.4 (四则运算)" } ], "level": 2 }, "bbox": [ 122, 108, 270, 123 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}, g_{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上分别一致收敛到 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 115, 128, 443, 143 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 数乘： " }, { "type": "equation_inline", "content": "kf_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "kf(x)" } ] }, "bbox": [ 139, 147, 396, 161 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 加法： " }, { "type": "equation_inline", "content": "f_{n}(x) + g_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x) + g(x)" } ] }, "bbox": [ 139, 165, 489, 180 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 乘法：若每个 " }, { "type": "equation_inline", "content": "f_{n}(x), g_{n}(x)" }, { "type": "text", "content": " 均有界，则 " }, { "type": "equation_inline", "content": "f_{n}(x)g_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)g(x)" } ] }, "bbox": [ 139, 183, 682, 199 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 乘法推论：若 " }, { "type": "equation_inline", "content": "f_{n}(x), g_{n}(x)" }, { "type": "text", "content": " 为连续函数列且 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 为闭区间，则 " }, { "type": "equation_inline", "content": "f_{n}(x)g_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)g(x)" } ] }, "bbox": [ 139, 202, 835, 217 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 除法：若 " }, { "type": "equation_inline", "content": "\\exists M" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|f(x)| \\geq M" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\exists N, n > N" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "f_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上无零点，且 " }, { "type": "equation_inline", "content": "\\frac{1}{f_n(x)}" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "\\frac{1}{f(x)}" } ] }, "bbox": [ 139, 217, 900, 249 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "证明 (1,2) 显然" } ], "level": 2 }, "bbox": [ 90, 265, 208, 279 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 用 " }, { "type": "equation_inline", "content": "|f_n(x)g_n(x) - f(x)g(x)| \\leq |f_n(x)||g_n(x) - g(x)| + |g(x)||f_n(x) - f(x)|" }, { "type": "text", "content": " (4) 取 " }, { "type": "equation_inline", "content": "\\epsilon = \\frac{M}{2}, \\exists N, \\forall n > N, \\forall x" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x) - f(x)| \\leq \\frac{M}{2}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "|f_n(x)| \\geq |f(x)| - \\frac{M}{2} \\geq \\frac{M}{2}" }, { "type": "text", "content": "，因此无零点。第二个结论用 " }, { "type": "equation_inline", "content": "|\\frac{1}{f_n(x)} - \\frac{1}{f(x)}| = |\\frac{f_n(x) - f(x)}{f_n(x)f(x)}| \\leq \\frac{|f_n(x) - f(x)|}{M \\cdot \\frac{M}{2}} < \\epsilon" } ] }, "bbox": [ 89, 282, 904, 356 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 9.6 反例 (1) 举例说明 " }, { "type": "equation_inline", "content": "f_{n}(x), g_{n}(x)" }, { "type": "text", "content": " 分别一致收敛到 " }, { "type": "equation_inline", "content": "f(x), g(x)" }, { "type": "text", "content": " ，但 " }, { "type": "equation_inline", "content": "f_{n}(x)g_{n}(x)" }, { "type": "text", "content": " 不一致收敛到 " }, { "type": "equation_inline", "content": "f(x)g(x)" }, { "type": "text", "content": " 解 (1) 根据定理函数列要无界，取 " }, { "type": "equation_inline", "content": "f_{n}(x)=\\frac{1}{n}, g_{n}(x)=\\frac{1}{x}" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致收敛到 0， " }, { "type": "equation_inline", "content": "g_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "\\frac{1}{x}" }, { "type": "text", "content": " ，但 " }, { "type": "equation_inline", "content": "f_{n}(x)g_{n}(x)=\\frac{1}{nx}" }, { "type": "text", "content": " 不一致收敛到 0，因为取 " }, { "type": "equation_inline", "content": "x_{n}=\\frac{1}{n}" }, { "type": "text", "content": " 即可。" } ] }, "bbox": [ 58, 366, 905, 435 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.5 (复合)" } ], "level": 2 }, "bbox": [ 121, 454, 238, 468 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，且每个 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 均有界， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "\\mathbb{R}" }, { "type": "text", "content": " 上连续函数，则 " }, { "type": "equation_inline", "content": "g(f_{n}(x))" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "g(f(x))" } ] }, "bbox": [ 110, 472, 890, 510 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据一致收敛以及个个有界得到 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致有界，不妨设对 " }, { "type": "equation_inline", "content": "\\forall n, \\forall x \\in I" }, { "type": "text", "content": " 有" } ] }, "bbox": [ 90, 527, 670, 542 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f _ {n} (x) \\right| \\leq M", "math_type": "latex", "image_source": { "path": "images/a71682950283413cc04e2c85d2eb910dfc26927e3e52ff8ab1f666ac9419318d.jpg" } }, "bbox": [ 450, 551, 546, 565 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[-M, M]" }, { "type": "text", "content": " 上一致连续， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, |x_1 - x_2| < \\delta" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "|g(x_1) - g(x_2)| < \\epsilon" }, { "type": "text", "content": "，对上述 " }, { "type": "equation_inline", "content": "\\delta" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f_n(x)" }, { "type": "text", "content": " 一致收敛， " }, { "type": "equation_inline", "content": "\\exists N, \\forall n > N" }, { "type": "text", "content": " 有" } ] }, "bbox": [ 90, 573, 905, 607 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f _ {n} (x) - f (x) \\right| < \\delta , \\forall x \\in I", "math_type": "latex", "image_source": { "path": "images/1f40fc24b83c4982d3cfd45323977a9ab1e190a86ba7c66e6bc9fdc9c4bec43c.jpg" } }, "bbox": [ 401, 611, 596, 626 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "从而 " }, { "type": "equation_inline", "content": "n > N" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\forall x\\in I" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|g(f_n(x)) - g(f(x))| < \\epsilon" } ] }, "bbox": [ 90, 638, 484, 653 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.2 函数列极限的性质" } ], "level": 2 }, "bbox": [ 92, 680, 337, 699 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.2.1 极限函数的连续性" } ], "level": 2 }, "bbox": [ 92, 719, 312, 736 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.6 (极限顺序交换)" } ], "level": 2 }, "bbox": [ 121, 752, 304, 766 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a,x_0)\\cup (x_0,b)" }, { "type": "text", "content": " 上一致收敛于 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\forall n" }, { "type": "text", "content": "，极限 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}f_n(x) = a_n" }, { "type": "text", "content": " 存在，则" } ] }, "bbox": [ 114, 772, 737, 790 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\to \\infty} \\lim _ {x \\to x _ {0}} f _ {n} (x) = \\lim _ {x \\to x _ {0}} \\lim _ {n \\to \\infty} f _ {n} (x)", "math_type": "latex", "image_source": { "path": "images/e703c03842d0da3813cdea9fd179644f221e355245f9f925e1a572c1f8673f20.jpg" } }, "bbox": [ 368, 795, 628, 816 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，则对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall m, n > N, \\forall x \\in (a, b)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_{n}(x) - f_{m}(x)| < \\epsilon" }, { "type": "text", "content": "，让 " }, { "type": "equation_inline", "content": "x \\to x_{0}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "|a_{n} - a_{m}| < \\epsilon" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 收敛，设 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_{n} = A" }, { "type": "text", "content": "。下证右侧 " }, { "type": "equation_inline", "content": "\\lim_{x \\to x_{0}} f(x) = A" }, { "type": "text", "content": "，根据" } ] }, "bbox": [ 89, 835, 905, 873 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x) - A | \\leq | f (x) - f _ {n} (x) | + | f _ {n} (x) - a _ {n} | + | a _ {n} - A | < 3 \\epsilon", "math_type": "latex", "image_source": { "path": "images/ace05e059d1a8890325ad77575942cddfc29710f596dbaa2db27a11c1e8bab20.jpg" } }, "bbox": [ 280, 879, 714, 894 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "第一个要求 N 足够大，第二个要求 " }, { "type": "equation_inline", "content": "\\delta" }, { "type": "text", "content": " 足够小，第三个要求 N 足够大。" } ] }, "bbox": [ 90, 902, 603, 916 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.2 函数列极限的性质" } ] }, "bbox": [ 737, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "87" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 9.1 (局部连续性)" } ], "level": 2 }, "bbox": [ 122, 86, 287, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 邻域 " }, { "type": "equation_inline", "content": "U(x_0,\\delta)" }, { "type": "text", "content": " 内一致收敛， " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处连续，则极限函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 也在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处连续" } ] }, "bbox": [ 114, 105, 816, 123 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 " }, { "type": "equation_inline", "content": "\\lim_{x\\to x_0}f(x) = \\lim_{x\\to x_0}\\lim_{n\\to \\infty}f_n(x) = \\lim_{n\\to \\infty}\\lim_{x\\to x_0}f_n(x) = \\lim_{n\\to \\infty}f_n(x_0) = f(x_0)" } ] }, "bbox": [ 90, 141, 692, 164 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 9.2 (整体连续性)" } ], "level": 2 }, "bbox": [ 121, 175, 287, 190 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若连续函数列在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上内闭一致收敛于 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上连续" } ] }, "bbox": [ 115, 195, 539, 209 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 原因在于单点的连续性仅与小邻域有关(甚至可以只有半边)，因此内闭的一致收敛即可推出极限函数的连续" } ] }, "bbox": [ 89, 230, 905, 262 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "例题 9.1 例如 " }, { "type": "equation_inline", "content": "x^{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 内闭一致收敛，因此极限函数 0 在 " }, { "type": "equation_inline", "content": "(0,1)" }, { "type": "text", "content": " 连续。" } ] }, "bbox": [ 90, 267, 621, 282 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.7 (一致连续性)" } ], "level": 2 }, "bbox": [ 122, 309, 287, 323 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 中一致收敛，每一项都一致连续，则极限函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上也一致连续" } ] }, "bbox": [ 115, 328, 729, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 思路： " }, { "type": "equation_inline", "content": "|f(x_{1})-f(x_{2})|\\leq|f(x_{1})-f_{n}(x_{1})|+|f_{n}(x_{1})-f_{n}(x_{2})|+|f_{n}(x_{2})-f(x_{2})|<3\\epsilon" }, { "type": "text", "content": " ，中间一项用 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 的一致连续性，左右两项用一致收敛性。" } ] }, "bbox": [ 89, 363, 905, 397 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.7函数列连续性练习 " }, { "type": "equation_inline", "content": "(1)f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，数列 " }, { "type": "equation_inline", "content": "x_{n} \\in [a,b]" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} x_{n} = x_{0}" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 处连续，证明 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} f_{n}(x_{n}) = f(x_{0})" } ] }, "bbox": [ 58, 418, 905, 454 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 收敛到连续 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的充要条件为任意满足 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}x_n = x_0" }, { "type": "text", "content": " 的数列，都有 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}f_n(x_n) = f(x_0)" } ] }, "bbox": [ 90, 456, 905, 492 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 注意与定理的区别，这里有双重 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": "。思路为 " }, { "type": "equation_inline", "content": "|f_n(x_n) - f(x_0)| \\leq |f_n(x_n) - f(x_n)| + |f(x_n) - f(x_0)|" }, { "type": "text", "content": "，前者用一致收敛，后者用连续性。" } ] }, "bbox": [ 89, 493, 902, 526 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据一致收敛性， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N_1, \\forall n > N_1, \\forall x \\in [a, b]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_n(x) - f(x)| < \\epsilon" }, { "type": "text", "content": "，根据 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 连续以及 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} x_n = x_0" }, { "type": "text", "content": " 可知 " }, { "type": "equation_inline", "content": "\\exists N_2, \\forall n > N_2, |f(x_n) - f(x)| < \\epsilon" }, { "type": "text", "content": "，取 " }, { "type": "equation_inline", "content": "N = \\max\\{N_1, N_2\\}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\forall n > N" }, { "type": "text", "content": " 有" } ] }, "bbox": [ 92, 530, 905, 565 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f _ {n} \\left(x _ {n}\\right) - f \\left(x _ {0}\\right) \\right| \\leq \\left| f _ {n} \\left(x _ {n}\\right) - f \\left(x _ {n}\\right) \\right| + \\left| f \\left(x _ {n}\\right) - f \\left(x _ {0}\\right) \\right| < 2 \\epsilon", "math_type": "latex", "image_source": { "path": "images/75e7fc053183a1f4df11b043bd94fb836c12eabf2b6de7d1d33a8f1038f7f999.jpg" } }, "bbox": [ 278, 571, 717, 588 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 左推右：思路： " }, { "type": "equation_inline", "content": "|f_{n}(x_{n}) - f(x_{0})| \\leq |f_{n}(x_{n}) - f(x_{n})| + |f(x_{n}) - f(x_{0})|" }, { "type": "text", "content": " ，注意不要写成 " }, { "type": "equation_inline", "content": "|f_{n}(x_{n}) - f_{n}(x_{0})| + |f_{n}(x_{0}) - f(x_{0})|" }, { "type": "text", "content": " ！原因在于 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 不一定连续。" } ] }, "bbox": [ 89, 595, 902, 629 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "右推左：反设不一致收敛，则 " }, { "type": "equation_inline", "content": "\\exists \\epsilon_0, \\forall N, \\exists n > N, \\exists x_n \\in [a, b]" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|f_n(x_n) - f(x_n)| \\geq \\epsilon_0" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "x_n" }, { "type": "text", "content": " 有界，因此存在收敛子列，不妨就设 " }, { "type": "equation_inline", "content": "x_n \\to x_0 \\in [a, b]" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} |f_n(x_n) - f(x_n)| = \\lim_{n \\to \\infty} |f_n(x_n) - f(x_0)| \\geq \\epsilon_0" }, { "type": "text", "content": "，与所给条件矛盾。" } ] }, "bbox": [ 89, 632, 905, 683 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 反设法下不一致收敛常用的讨论即存在 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\left|f_n(x_n) - f(x_n)\\right| \\geq \\epsilon_0" } ] }, "bbox": [ 87, 687, 653, 703 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.2.2 极限函数的可积性" } ], "level": 2 }, "bbox": [ 92, 724, 312, 740 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.8 (可积性)" } ], "level": 2 }, "bbox": [ 122, 757, 253, 772 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 均在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上可积，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积，且" } ] }, "bbox": [ 114, 776, 744, 793 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {n \\rightarrow \\infty} \\int_ {a} ^ {b} f _ {n} (x) \\mathrm{d} x = \\int_ {a} ^ {b} f (x) \\mathrm{d} x = \\int_ {a} ^ {b} \\lim _ {n \\rightarrow \\infty} f _ {n} (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/875d79fbbefc55fb0ed36397b423ec1988377c3bbfb64e5716621be0f897bfc6.jpg" } }, "bbox": [ 310, 795, 685, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 先说明可积: " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\forall x, |f(x) - f_N(x)| < \\frac{\\epsilon}{b - a}" }, { "type": "text", "content": ", 做划分 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": ", 取 " }, { "type": "equation_inline", "content": "x_1, x_2" }, { "type": "text", "content": " 于同一区间, 考虑振幅:" } ] }, "bbox": [ 90, 845, 887, 869 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x _ {1}) - f (x _ {2}) | \\leq | f (x _ {1}) - f _ {N} (x _ {1}) | + | f _ {N} (x _ {1}) - f _ {N} (x _ {2}) | + | f _ {N} (x _ {2}) - f (x _ {2}) | \\leq \\frac {\\epsilon}{b - a} + \\omega_ {i} ^ {f _ {N}} + \\frac {\\epsilon}{b - a}", "math_type": "latex", "image_source": { "path": "images/f58343cadb67b442fab7c782b738e4a06a212830b3d62e9efe14758970aae3f0.jpg" } }, "bbox": [ 132, 872, 860, 898 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "88" } ] }, "bbox": [ 489, 953, 507, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.2 函数列极限的性质" } ] }, "bbox": [ 737, 55, 904, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 的振幅 " }, { "type": "equation_inline", "content": "\\omega_{i}^{f}\\leq \\frac{2\\epsilon}{b - a} +\\omega_{i}^{f_{N}}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 90, 80, 408, 111 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {T} \\omega_ {i} ^ {f} \\Delta x _ {i} \\leq \\sum_ {T} \\frac {2 \\epsilon}{b - a} \\Delta x _ {i} + \\sum_ {T} \\omega_ {i} ^ {f _ {N}} \\Delta x _ {i} < 3 \\epsilon", "math_type": "latex", "image_source": { "path": "images/9c4a6f65751381ec61d4fd25d4151cfa728e0a625074320560a6977d35d2f5c0.jpg" } }, "bbox": [ 324, 114, 672, 146 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 求积分值：考虑 " }, { "type": "equation_inline", "content": "|\\int_{a}^{b} f_{n}(x) \\mathrm{d}x - \\int_{a}^{b} f(x) \\mathrm{d}x| \\leq \\int_{a}^{b} |f_{n}(x) - f(x)| \\mathrm{d}x < \\int_{a}^{b} \\frac{\\epsilon}{b - a} \\mathrm{d}x = \\epsilon" } ] }, "bbox": [ 122, 149, 778, 184 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 该定理在 ZJU2020 中出现过。" } ] }, "bbox": [ 87, 186, 344, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.8不可积反例举例连续函数列 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 收敛到连续函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，但无 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\int_a^b f_n(x)\\mathrm{d}x = \\int_a^b f(x)\\mathrm{d}x" }, { "type": "text", "content": " 解 " }, { "type": "equation_inline", "content": "f_{n}(x) = nx^{n}(1 - x^{n}),x\\in [0,1]" }, { "type": "text", "content": "，极限函数为 " }, { "type": "equation_inline", "content": "f(x) = 0" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "\\int_0^1 f(x)\\mathrm{d}x = 0" }, { "type": "text", "content": "，但是 " }, { "type": "equation_inline", "content": "\\int_0^1 f_n(x)\\mathrm{d}x = \\frac{n}{n + 1} -" }, { "type": "equation_inline", "content": "\\frac{n}{2n + 1} = \\frac{1}{2}" }, { "type": "text", "content": "。" } ] }, "bbox": [ 58, 216, 902, 306 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.9可积性的应用 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 每项都在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续可微， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续函数，若存在 " }, { "type": "equation_inline", "content": "M > 0" }, { "type": "text", "content": "，使得 " }, { "type": "equation_inline", "content": "\\forall n \\in \\mathbb{N}^{+}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x', x'' \\in [a,b]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_{n}(x') - f_{n}(x'')| \\leq \\frac{M}{n} |x' - x''|" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\int_{a}^{b} f_{n}'(x) g(x) \\, \\mathrm{d}x = 0" }, { "type": "text", "content": " 证明根据条件得到 " }, { "type": "equation_inline", "content": "|\\frac{f_n(x) - f_n(y)}{x - y}| \\leq \\frac{M}{n}" }, { "type": "text", "content": "，令 " }, { "type": "equation_inline", "content": "y \\to x" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "|f_n'(x)| \\leq \\frac{M}{n} \\to 0" }, { "type": "text", "content": "。由于 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 有界，因此 " }, { "type": "equation_inline", "content": "f_{n}'(x) g(x)" }, { "type": "text", "content": " 一致收敛到0，根据一致收敛性质可将极限和积分互换位置，从而结论成立。" } ] }, "bbox": [ 58, 318, 907, 409 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 9.3 (可积性推广)" } ], "level": 2 }, "bbox": [ 122, 435, 287, 449 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 均可积，则 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 可积，且对任意可积 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "x_0 \\in [a,b]" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\int_{x_0}^{x} f_n(t) g(t) \\mathrm{d}t" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "\\int_{x_0}^{x} f(t) g(t) \\mathrm{d}t" } ] }, "bbox": [ 112, 453, 894, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 考虑 " }, { "type": "equation_inline", "content": "\\left|\\int_{x_{0}}^{x}f_{n}(t)g(t)\\mathrm{d}t-\\int_{x_{0}}^{x}f(t)g(t)\\mathrm{d}t\\right|\\leq\\left|\\int_{x_{0}}^{x}|f_{n}(t)-f(t)||g(t)|\\mathrm{d}t\\right|\\leq M\\int_{a}^{b}|f_{n}(x)-f(x)|\\mathrm{d}x<\\epsilon" }, { "type": "text", "content": " \n\n笔记 上述推论常用的特例： " }, { "type": "equation_inline", "content": "\\int_{x_{0}}^{x}f_{n}(t)\\mathrm{d}t" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "\\int_{x_{0}}^{x}f(t)\\mathrm{d}t" } ] }, "bbox": [ 60, 512, 845, 577 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.2.3 极限函数的可微性" } ], "level": 2 }, "bbox": [ 92, 595, 312, 613 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.9 (可微性)" } ], "level": 2 }, "bbox": [ 122, 627, 253, 643 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 的每项在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 有连续导函数， " }, { "type": "equation_inline", "content": "f_{n}^{\\prime}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\exists x_0\\in [a,b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f_{n}(x_{0})" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 一致收敛于一个连续可微函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "f^{\\prime}(x) = g(x)" }, { "type": "text", "content": "，即" } ] }, "bbox": [ 112, 646, 880, 683 ] }, { "type": "equation_interline", "content": { "math_content": "(\\lim _ {n \\to \\infty} f _ {n} (x)) ^ {\\prime} = \\lim _ {n \\to \\infty} f _ {n} ^ {\\prime} (x)", "math_type": "latex", "image_source": { "path": "images/c2418b85f95b03aecab269274aab1b523b6e1e597a1e920b00ffbb2c7751ca00.jpg" } }, "bbox": [ 396, 689, 600, 711 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 极限函数存在：根据 Newton-Leibniz 公式可知 " }, { "type": "equation_inline", "content": "f_{n}(x) = f_{n}(x_{0}) + \\int_{x_{0}}^{x} f_{n}'(t) \\mathrm{d}t" }, { "type": "text", "content": "，根据条件 " }, { "type": "equation_inline", "content": "f_{n}(x_{0})" }, { "type": "text", "content": " 收敛且与 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 无关。根据前面可积性推论可知 " }, { "type": "equation_inline", "content": "\\int_{x_{0}}^{x} f_{n}'(t) \\mathrm{d}t" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "\\int_{x_{0}}^{x} g(t) \\mathrm{d}t" }, { "type": "text", "content": "，根据一致收敛的四则运算可知 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 本身也一致收敛至 " }, { "type": "equation_inline", "content": "f(x) = \\lim_{n \\to \\infty} f_{n}(x_{0}) + \\int_{x_{0}}^{x} g(t) \\mathrm{d}t" } ] }, "bbox": [ 89, 725, 905, 813 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 导数：由于 " }, { "type": "equation_inline", "content": "f_n'(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 连续，从而 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 可导且 " }, { "type": "equation_inline", "content": "f'(x) = g(x)" }, { "type": "text", "content": "。" } ] }, "bbox": [ 97, 813, 705, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 定理中 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 存在 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 这个收敛点的条件必不可少，这为使用 Leibniz 公式奠定了基础。在实际使用中，往往可以直接看出 " }, { "type": "equation_inline", "content": "f_{n}(x), f_{n}^{\\prime}(x)" }, { "type": "text", "content": " 都一致收敛，则可直接得出 " }, { "type": "equation_inline", "content": "(\\lim_{n\\to\\infty}f_{n}(x))^{\\prime}=\\lim_{n\\to\\infty}f_{n}^{\\prime}(x)" }, { "type": "text", "content": " 的结论，不用管 " }, { "type": "equation_inline", "content": "x_{0}" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 87, 829, 905, 871 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 9.10 具体计算 (1) 已知 " }, { "type": "equation_inline", "content": "u_{n}(x)=\\frac{\\cos nx}{n\\sqrt{n}}" }, { "type": "text", "content": " ，判定其在 " }, { "type": "equation_inline", "content": "(0,2\\pi)" }, { "type": "text", "content": " 上是否逐项可导，若可导则求出 " }, { "type": "equation_inline", "content": "S'(x)" } ] }, "bbox": [ 58, 881, 831, 912 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.2 函数列极限的性质" } ] }, "bbox": [ 737, 54, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "89" } ] }, "bbox": [ 489, 953, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)考虑 " }, { "type": "equation_inline", "content": "u_{n}^{\\prime}(x) = -\\frac{\\sin nx}{\\sqrt{n}}" }, { "type": "text", "content": " ，根据Dirichlet判别法可知一致收敛，因此逐项可导， " }, { "type": "equation_inline", "content": "S^{\\prime}(x) = -\\sum_{n = 1}^{\\infty}\\frac{\\sin nx}{\\sqrt{n}}" } ] }, "bbox": [ 89, 83, 855, 118 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.2.4 连续 + 单调：推出一致收敛" } ], "level": 2 }, "bbox": [ 92, 139, 394, 155 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.10 (关于 x 单调)" } ], "level": 2 }, "bbox": [ 121, 171, 297, 186 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "函数列 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 收敛到连续函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，且每个 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 单调增，则 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" } ] }, "bbox": [ 112, 191, 889, 229 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明" } ] }, "bbox": [ 90, 246, 131, 260 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.11 (Dini 定理：关于 n 单调)" } ], "level": 2 }, "bbox": [ 121, 287, 381, 303 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "连续函数列 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 收敛于连续函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，若固定 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": "，每个 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 单调，则 " }, { "type": "equation_inline", "content": "f_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "f(x)" } ] }, "bbox": [ 112, 307, 899, 346 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 先证明在小区间成立：固定 " }, { "type": "equation_inline", "content": "x_0 \\in [a, b]" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f_n(x_0) - f(x_0)" }, { "type": "text", "content": " 单调趋于 0， " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_N(x_0) - f(x_0)| < \\epsilon" }, { "type": "text", "content": "。固定 " }, { "type": "equation_inline", "content": "N" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "f_N(x) - f(x)" }, { "type": "text", "content": " 连续，因此根据保号性 " }, { "type": "equation_inline", "content": "\\exists \\delta, \\forall x \\in U(x_0, \\delta)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|f_N(x) - f(x)| < \\epsilon" }, { "type": "text", "content": "。重新根据单调性得到 " }, { "type": "equation_inline", "content": "\\forall n > N, \\forall x \\in U(x_0, \\delta)" }, { "type": "text", "content": " 有" } ] }, "bbox": [ 87, 362, 905, 414 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f _ {n} (x) - f (x) \\right| \\leq \\left| f _ {N} (x) - f (x) \\right| < \\epsilon", "math_type": "latex", "image_source": { "path": "images/b68015877d1a8cafe79a1f378a88fa4b5503812f1e803f101a27b3c5a604b22a.jpg" } }, "bbox": [ 364, 418, 633, 434 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 再证明在整个区间成立：显然区间集 " }, { "type": "equation_inline", "content": "\\{U(x_{0},\\delta):x_{0}\\in[a,b]\\}" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 的开覆盖，根据有限覆盖定理得到存在有限个点 " }, { "type": "equation_inline", "content": "x_{1},\\cdots,x_{k}" }, { "type": "text", "content": " 的邻域覆盖 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " ，每个对应着 " }, { "type": "equation_inline", "content": "N_{i},\\delta_{i}" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 90, 445, 905, 480 ] }, { "type": "equation_interline", "content": { "math_content": "\\forall n > N _ {i}, \\forall x \\in U (x _ {i}, \\delta_ {i}), | f _ {n} (x) - f (x) | < \\epsilon", "math_type": "latex", "image_source": { "path": "images/96a5997d6a245d69757eb249023393587d59ffa6b2e8b6d87783af3f63677661.jpg" } }, "bbox": [ 339, 487, 655, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此取 " }, { "type": "equation_inline", "content": "N = \\max\\{N_{1}, \\cdots, N_{k}\\}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\forall n > N" }, { "type": "text", "content": " 时，对 " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b]" }, { "type": "text", "content": " ，总能找到 " }, { "type": "equation_inline", "content": "x_{i}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "x \\in U(x_{i}, \\delta_{i})" }, { "type": "text", "content": " ，且 " }, { "type": "equation_inline", "content": "|f_{n}(x) - f(x)| < \\epsilon" }, { "type": "text", "content": " \n笔记 ZJU2020 默写了 Dini 定理的证明" } ] }, "bbox": [ 87, 508, 902, 543 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.3 函数项级数的一致收敛性" } ], "level": 2 }, "bbox": [ 92, 570, 408, 589 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.3.1 一致收敛的基本性质与判别" } ], "level": 2 }, "bbox": [ 92, 609, 393, 626 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.12 (一致收敛性质)" } ], "level": 2 }, "bbox": [ 122, 642, 312, 657 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}u_{n}(x)" }, { "type": "text", "content": " 在 D 上一致收敛，则 " }, { "type": "equation_inline", "content": "u_{n}(x)" }, { "type": "text", "content": " 在 D 上一致收敛到 0。" } ] }, "bbox": [ 112, 664, 559, 700 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.11收敛但不一致收敛的级数说明 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{n}{1 + n^3x}" }, { "type": "text", "content": " 在(0,1)收敛但不一致收敛" } ] }, "bbox": [ 58, 730, 685, 766 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 收敛: 根据 " }, { "type": "equation_inline", "content": "\\frac{n}{1 + n^3 x} \\sim \\frac{1}{n^2 x}" }, { "type": "text", "content": ", 若 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 固定, 则作为数项级数收敛, 因此函数项级数收敛" } ] }, "bbox": [ 87, 766, 781, 794 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 不一致收敛: " }, { "type": "equation_inline", "content": "x_{n} = \\frac{1}{n^{3}}" }, { "type": "text", "content": ", 则 " }, { "type": "equation_inline", "content": "\\frac{n}{1 + n^{3}x_{n}} = \\frac{n}{2}" }, { "type": "text", "content": " 不趋于 0, 因此不一致收敛" } ] }, "bbox": [ 122, 794, 660, 820 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.13 (一致收敛充要条件)" } ], "level": 2 }, "bbox": [ 122, 840, 344, 854 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上一致收敛到 " }, { "type": "equation_inline", "content": "S(x)" }, { "type": "text", "content": " 的充要条件为：" } ] }, "bbox": [ 114, 860, 484, 898 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.3 函数项级数的一致收敛性" } ] }, "bbox": [ 687, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "90" } ] }, "bbox": [ 487, 953, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- Cauchy 收敛准则： " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N, \\forall n > m > N, \\forall x \\in D" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|\\sum_{k=m+1}^{n} u_k(x)| < \\epsilon" } ] }, "bbox": [ 139, 95, 665, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 确界： " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sup_{x\\in D}|\\sum_{k = n + 1}^{\\infty}u_k(x)| = 0" } ] }, "bbox": [ 139, 131, 411, 171 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.12 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上函数列 " }, { "type": "equation_inline", "content": "u_{n}(x)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "u_{i}(x)u_{j}(x) = 0, i \\neq j, \\forall x \\in I" }, { "type": "text", "content": "，若存在正数列 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|u_{n}(x)| \\leq a_{n}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} a_{n} = 0" }, { "type": "text", "content": "，证明： " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_{n}(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 一致收敛" } ] }, "bbox": [ 60, 209, 905, 266 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据条件 " }, { "type": "equation_inline", "content": "\\forall x \\in I, u_{1}(x), \\cdots, u_{n}(x), \\cdots" }, { "type": "text", "content": " 中至多只有一个非零，因此根据确界条件 " }, { "type": "equation_inline", "content": "\\sup_{x \\in I} |\\sum_{k=n+1}^{\\infty} u_{k}(x)| \\leq \\sup_{n} \\{a_{n+1}, a_{n+2}, \\cdots\\} \\to 0" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 87, 267, 905, 325 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.14 (M 判别法)" } ], "level": 2 }, "bbox": [ 122, 343, 282, 357 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上的函数项级数，若存在收敛的数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} M_n" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "\\forall x \\in D, |u_n(x)| \\leq M_n" }, { "type": "text", "content": "，则得到 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上一致收敛" } ] }, "bbox": [ 112, 361, 895, 441 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.13 讨论一致收敛性：(1) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\arctan \\frac{2x}{x^2 + n^3}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(- \\infty, + \\infty)" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^{2n}}{1 + x^{2n+1}}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x \\geq 0" } ] }, "bbox": [ 58, 468, 754, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "|\\arctan \\frac{2x}{x^2 + n^3} |\\leq |\\frac{2x}{x^2 + n^3} |\\leq \\frac{|2x|}{2|x|n^{3/2}} = \\frac{1}{n^{3/2}}" }, { "type": "text", "content": " ，因此一致收敛" } ] }, "bbox": [ 89, 502, 606, 533 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "x \\geq 1" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\frac{x^{2n}}{1 + x^{2n+1}} = \\frac{1}{x}" }, { "type": "text", "content": "，发散。 " }, { "type": "equation_inline", "content": "0 \\leq 1 < 1" }, { "type": "text", "content": " 时根据 " }, { "type": "equation_inline", "content": "\\frac{x^{2n}}{1 + x^{2n+1}} \\leq x^{2n}" }, { "type": "text", "content": " 点态收敛，但是感觉出 1 处会导致一致收敛出问题，因此取 " }, { "type": "equation_inline", "content": "x_n = 1 - \\frac{1}{n}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\frac{x_n^{2n}}{1 + x_n^{2n+1}} \\to \\frac{e^{-2}}{1 + e^{-2}} \\neq 0" }, { "type": "text", "content": "，因此不一致收敛。" } ] }, "bbox": [ 90, 535, 904, 593 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.14积分格式 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 上连续函数， " }, { "type": "equation_inline", "content": "f_0(x) = f(x)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "f_{n + 1}(x) = \\int_a^x f_n(t)\\mathrm{d}t" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}f_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 一致收敛。证明同前面， " }, { "type": "equation_inline", "content": "|f_n(x)|\\leq \\frac{(b - a)^n}{n!} M" }, { "type": "text", "content": "，根据根式或者比式判别法易知。" } ] }, "bbox": [ 58, 600, 905, 675 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.3.2 Dirichlet 与 Abel 判别法" } ], "level": 2 }, "bbox": [ 92, 692, 363, 708 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.15 (Dirichlet 与 Abel 判别法)" } ], "level": 2 }, "bbox": [ 122, 725, 388, 739 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上的函数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x) v_n(x)" }, { "type": "text", "content": "，若满足下面任意一条，则一致收敛\n- Dirichlet: (1) " }, { "type": "equation_inline", "content": "\\sum_{k=1}^{n} u_k(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致有界 (2) " }, { "type": "equation_inline", "content": "v_n(x)" }, { "type": "text", "content": " 固定 " }, { "type": "equation_inline", "content": "x \\in I" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 单调，且在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛到 0\n- Abel: (1) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 一致收敛 (2) " }, { "type": "equation_inline", "content": "v_n(x)" }, { "type": "text", "content": " 固定 " }, { "type": "equation_inline", "content": "x \\in I" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 单调，且在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致有界" } ] }, "bbox": [ 115, 741, 873, 859 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 单调都是关于 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " ，一致都是关于 " }, { "type": "equation_inline", "content": "x" } ] }, "bbox": [ 87, 873, 381, 888 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.3 函数项级数的一致收敛性" } ] }, "bbox": [ 687, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "91" } ] }, "bbox": [ 487, 953, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习9.15经典一致收敛题(1) " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单调收敛于0，证明： " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_n\\cos nx" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[\\alpha ,2\\pi -\\alpha ]" }, { "type": "text", "content": " 上一致收敛证明(1)首先判断一致有界性：" } ] }, "bbox": [ 60, 83, 781, 133 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\sum_ {k = 1} ^ {n} \\cos k x \\right| = \\left| \\frac {\\sin \\left(n + \\frac {1}{2}\\right) x}{2 \\sin \\frac {x}{2}} - \\frac {1}{2} \\right| \\leq \\frac {1}{2 \\left| \\sin \\frac {x}{2} \\right|} + \\frac {1}{2} \\leq \\frac {1}{2 \\sin \\frac {\\alpha}{2}} + 1", "math_type": "latex", "image_source": { "path": "images/ca4b05edeca5c49341b0dd329898bcc4bf315b1fba6a5d90ff6de45d6e565d2b.jpg" } }, "bbox": [ 263, 135, 734, 174 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\sum \\cos nx" }, { "type": "text", "content": " 一致有界，而 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 固定 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 时单调趋于0，因此根据Dirichlet判别法可知一致收敛。" } ] }, "bbox": [ 90, 179, 793, 198 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "9.4 函数项级数极限函数的性质" } ], "level": 2 }, "bbox": [ 92, 222, 433, 242 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.16 (端点法)" } ], "level": 2 }, "bbox": [ 122, 263, 260, 279 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 一致收敛，且每个 " }, { "type": "equation_inline", "content": "u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b)" }, { "type": "text", "content": " 连续，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b)" }, { "type": "text", "content": " 一致收敛，特别的， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(a)" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 112, 281, 885, 332 ] }, { "type": "image", "content": { "image_source": { "path": "images/c161fa1ac200508a16f4bc9d4e777e4dbc7aa39cf920488e81a4031f73dfdf45.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 346, 78, 365 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 开区间一致收敛的函数列往往能延拓到闭区间，因此题目中开区间的函数列往往是不一致收敛的 (除非无定义)。" } ] }, "bbox": [ 90, 351, 905, 385 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.17 (逐项单向极限)" } ], "level": 2 }, "bbox": [ 122, 395, 312, 410 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(a, b)" }, { "type": "text", "content": " 一致收敛， " }, { "type": "equation_inline", "content": "\\lim_{x \\to a^+} u_n(x) = a_n, \\sum_{n=1}^{\\infty} a_n" }, { "type": "text", "content": " 收敛，则" } ] }, "bbox": [ 112, 416, 586, 453 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\to a ^ {+}} \\sum_ {n = 1} ^ {\\infty} u _ {n} (x) = \\sum_ {n = 1} ^ {\\infty} \\lim _ {x \\to a ^ {+}} u _ {n} (x) = \\sum_ {n = 1} ^ {\\infty} a _ {n}", "math_type": "latex", "image_source": { "path": "images/4f0f015928cfbfabda5449e589d61b1ecc2778c94898902e3561067281711c88.jpg" } }, "bbox": [ 339, 456, 655, 495 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.18 (连续性)" } ], "level": 2 }, "bbox": [ 122, 519, 260, 533 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "U(x_0, \\delta)" }, { "type": "text", "content": " 内一致收敛到 " }, { "type": "equation_inline", "content": "S(x)" }, { "type": "text", "content": "，且每个 " }, { "type": "equation_inline", "content": "u_n(x)" }, { "type": "text", "content": " 均在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处连续，则 " }, { "type": "equation_inline", "content": "S(x)" }, { "type": "text", "content": " 也在 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处连续，即" } ] }, "bbox": [ 114, 539, 860, 577 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {n = 1} ^ {\\infty} (\\lim _ {x \\to x _ {0}} u _ {n} (x)) = \\lim _ {x \\to x _ {0}} \\sum_ {n = 1} ^ {\\infty} u _ {n} (x)", "math_type": "latex", "image_source": { "path": "images/c94e867a2ab27a4f77b9bf8d39a68542b63ef274e9b71b36a82f79fddf84a719.jpg" } }, "bbox": [ 369, 577, 623, 618 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.19 (逐项积分)" } ], "level": 2 }, "bbox": [ 122, 640, 278, 656 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "S(x)" }, { "type": "text", "content": "，且每个 " }, { "type": "equation_inline", "content": "u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积，则 " }, { "type": "equation_inline", "content": "S(x)" }, { "type": "text", "content": " 也在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 可积，且" } ] }, "bbox": [ 115, 664, 818, 700 ] }, { "type": "equation_interline", "content": { "math_content": "\\sum_ {n = 1} ^ {\\infty} \\int_ {a} ^ {b} u _ {n} (x) \\mathrm{d} x = \\int_ {a} ^ {b} \\sum_ {n = 1} ^ {\\infty} u _ {n} (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/7d5cea134553f352428b1e560edff851a62705a19b356ee0431a92a95d37d4ff.jpg" } }, "bbox": [ 368, 701, 628, 739 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 9.20 (逐项可微)" } ], "level": 2 }, "bbox": [ 124, 764, 278, 778 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 的每个 " }, { "type": "equation_inline", "content": "u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 有连续偏导数， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n'(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 一致收敛到 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": "，若存在 " }, { "type": "equation_inline", "content": "x_0 \\in [a, b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x_0)" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 上一致收敛，和函数 " }, { "type": "equation_inline", "content": "S(x)" }, { "type": "text", "content": " 可导， " }, { "type": "equation_inline", "content": "S'(x) = g(x)" }, { "type": "text", "content": "，即" } ] }, "bbox": [ 112, 784, 882, 859 ] }, { "type": "equation_interline", "content": { "math_content": "(\\sum_ {n = 1} ^ {\\infty} u _ {n} (x)) ^ {\\prime} = \\sum_ {n = 1} ^ {\\infty} u _ {n} ^ {\\prime}", "math_type": "latex", "image_source": { "path": "images/87f87ac163b9970170123c30c966445733b65b11126eb57d6f7c74a85896871a.jpg" } }, "bbox": [ 416, 860, 579, 898 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "92" } ] }, "bbox": [ 487, 953, 507, 964 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "9.4 函数项级数极限函数的性质" } ] }, "bbox": [ 670, 55, 904, 68 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 10 章 幂级数" } ], "level": 2 }, "bbox": [ 403, 87, 611, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.1 幂级数概念及其收敛区间" } ], "level": 2 }, "bbox": [ 95, 145, 421, 164 ] }, { "type": "table", "content": { "image_source": { "path": "images/af8e5fa60607714eeceb6dc12551446113d24a755c9b9830a30904d5de5ed540.jpg" }, "table_caption": [], "table_footnote": [], "html": "
定义10.1(幂级数)
$\\sum_{n=0}^{\\infty}a_n(x-x_0)^n$ 称为幂级数,但一般只研究 $\\sum_{n=0}^{\\infty}a_nx^n$
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 110, 190, 905, 250 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.1.1 收敛区间与收敛半径" } ], "level": 2 }, "bbox": [ 95, 282, 342, 299 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.1 (Abel 定理)" } ], "level": 2 }, "bbox": [ 122, 315, 282, 329 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若幂级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n x^n" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0 \\neq 0" }, { "type": "text", "content": " 收敛，则对 " }, { "type": "equation_inline", "content": "\\forall x, |x| < |x_0|" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n x^n" }, { "type": "text", "content": " 绝对收敛；若 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 处发散，则 " }, { "type": "equation_inline", "content": "\\forall x, |x| > |x_0|" }, { "type": "text", "content": " 均发散。" } ] }, "bbox": [ 112, 332, 885, 388 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 若 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x_0^n" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "|a_n x^n| = |a_n x_0^n| \\cdot |\\frac{x}{x_0}|^n \\leq M r^n" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "r = |\\frac{x}{x_0}| \\in [0,1)" }, { "type": "text", "content": "。另一侧同理" } ] }, "bbox": [ 90, 398, 794, 435 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 10.2 (收敛半径)" } ], "level": 2 }, "bbox": [ 122, 444, 278, 458 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据Abel定理可知 " }, { "type": "equation_inline", "content": "\\exists R\\in \\mathbb{R}" }, { "type": "text", "content": " ，使得 " }, { "type": "equation_inline", "content": "|x| < R" }, { "type": "text", "content": " 时幂级数绝对收敛(若 " }, { "type": "equation_inline", "content": "R = 0" }, { "type": "text", "content": " 则仅在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 处收敛)，对一切 " }, { "type": "equation_inline", "content": "|x| > R" }, { "type": "text", "content": " 均发散，称 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": " 为幂级数收敛半径， " }, { "type": "equation_inline", "content": "(-R,R)" }, { "type": "text", "content": " 为收敛区间。" } ] }, "bbox": [ 114, 464, 880, 499 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.2 (收敛半径的计算)" } ], "level": 2 }, "bbox": [ 122, 524, 329, 539 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "幂级数收敛半径 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": " (可以为 " }, { "type": "equation_inline", "content": "0, \\infty" }, { "type": "text", "content": ") 的计算有以下几种方式：" } ] }, "bbox": [ 115, 544, 532, 558 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 根式 (柯西-阿达马): " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{|a_n|} = \\frac{1}{R}" }, { "type": "text", "content": " (可以为 " }, { "type": "equation_inline", "content": "0,\\infty" }, { "type": "text", "content": " )" } ] }, "bbox": [ 139, 558, 539, 582 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 比式 (一般用得少): 若 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\frac{|a_{n + 1}|}{|a_n|} = \\frac{1}{R}" }, { "type": "text", "content": ", 也能推出 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{a_n} = \\frac{1}{R}" } ] }, "bbox": [ 139, 582, 657, 613 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根式：只需要考虑 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty}|a_n x^n|" }, { "type": "text", "content": "，用数项级数的根式判别法为 " }, { "type": "equation_inline", "content": "\\varlimsup_{n \\to \\infty} \\sqrt[n]{|a_n x^n|} = \\varliminf_{n \\to \\infty} \\sqrt[n]{|a_n|} \\cdot |x|" }, { "type": "text", "content": "，因此若记 " }, { "type": "equation_inline", "content": "\\varlimsup_{n \\to \\infty} \\sqrt[n]{|a_n|} = \\rho" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "|x| < \\frac{1}{\\rho}" }, { "type": "text", "content": " 则收敛， " }, { "type": "equation_inline", "content": "\\frac{1}{\\rho}" }, { "type": "text", "content": " 为收敛半径。" } ] }, "bbox": [ 89, 625, 902, 690 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习10.1计算收敛半径 计算收敛半径与收敛域(1) " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{x^n}{n^2}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{x^n}{n}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{(2n - 1)!!}{(2n)!!} x^n" } ] }, "bbox": [ 60, 695, 734, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{\\frac{1}{n^2}} = 1" }, { "type": "text", "content": " ，收敛半径1，收敛域[-1,1]" } ] }, "bbox": [ 89, 731, 462, 764 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{\\frac{1}{n}} = 1" }, { "type": "text", "content": " ，收敛半径1，收敛域[-1,1)" } ] }, "bbox": [ 122, 765, 467, 793 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "a_{n}=\\frac{(2n-1)!!}{(2n)!!}" }, { "type": "text", "content": " ，用Wallis+根式或者比式 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\frac{a_{n+1}}{a_{n}}=\\lim_{n\\to\\infty}\\frac{2n+1}{2n+2}=1" }, { "type": "text", "content": " 得到收敛半径为1。而 " }, { "type": "equation_inline", "content": "\\frac{(2n-1)!!}{(2n)!!}\\sim\\frac{1}{\\sqrt{n\\pi}}" }, { "type": "text", "content": " ，因此发散。 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{(2n-1)!!}{(2n)!!}(-1)^{n}" }, { "type": "text", "content": " 根据Leibniz可知收敛。故收敛域为 " }, { "type": "equation_inline", "content": "[-1,1)" } ] }, "bbox": [ 92, 790, 915, 857 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 上述题目也说明在逐项求导后，端点处收敛域可能发生变换" } ] }, "bbox": [ 87, 862, 569, 877 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 10.2 常规格式幂级数 (1) 求收敛域: " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}(1+\\frac{1}{2}+\\cdots+\\frac{1}{n})x^{n}\\sum_{n=1}^{\\infty}\\frac{x^{n}}{1+\\frac{1}{2}+\\cdots+\\frac{1}{n}}" } ] }, "bbox": [ 60, 82, 719, 118 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "0 < a < b" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^n}{a^n + b^n}, \\sum_{n=1}^{\\infty} (a^n + b^n)x^n" }, { "type": "text", "content": " 收敛域" } ] }, "bbox": [ 122, 118, 517, 155 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^{2n}}{n-3^{2n}}, \\sum_{n=1}^{\\infty} \\frac{x^{n^2}}{2^n}" }, { "type": "text", "content": " 收敛域" } ] }, "bbox": [ 122, 154, 356, 193 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{3^{n}+(-2)^{n}}{n}(x+1)^{n}" } ] }, "bbox": [ 122, 192, 324, 225 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 要么根据 Euler 常数得到 " }, { "type": "equation_inline", "content": "\\sqrt[n]{1+\\frac{1}{2}+\\cdots+\\frac{1}{n}}\\sim\\sqrt[n]{\\ln n}=1" }, { "type": "text", "content": " ，要么用 " }, { "type": "equation_inline", "content": "1\\leq1+\\frac{1}{2}+\\cdots+\\frac{1}{n}\\leq n" }, { "type": "text", "content": " 得到收敛半径为 R=1，显然收敛域为 " }, { "type": "equation_inline", "content": "(-1,1)" }, { "type": "text", "content": " 第二个同理得到收敛域为 " }, { "type": "equation_inline", "content": "[-1,1]" } ] }, "bbox": [ 89, 225, 905, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 第一个 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{\\frac{1}{a^n + b^n}} = \\frac{1}{b}" }, { "type": "text", "content": "，收敛半径为 " }, { "type": "equation_inline", "content": "b" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x = \\pm b" }, { "type": "text", "content": " 都震荡发散。第二个 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}\\sqrt[n]{a^n + b^n} = b" }, { "type": "text", "content": "，收敛半径 " }, { "type": "equation_inline", "content": "b" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x = \\pm b" }, { "type": "text", "content": " 震荡发散" } ] }, "bbox": [ 89, 273, 905, 313 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 由于不是 " }, { "type": "equation_inline", "content": "x^n" }, { "type": "text", "content": " 因此开 " }, { "type": "equation_inline", "content": "2n" }, { "type": "text", "content": " 次根号， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\sqrt[2n]{|\\frac{1}{n - 3^{2n}}|} = \\frac{1}{3}" }, { "type": "text", "content": "，半径为 " }, { "type": "equation_inline", "content": "3" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x = \\pm 3" }, { "type": "text", "content": " 时通项不趋于 0，因此发散。 " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\sqrt[n^2]{\\frac{1}{2^n}} = 1" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "x = \\pm 1" }, { "type": "text", "content": " 均收敛" } ] }, "bbox": [ 90, 315, 907, 374 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 将 " }, { "type": "equation_inline", "content": "x + 1" }, { "type": "text", "content": " 视为 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 即可， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} \\sqrt[n]{\\frac{3^n + (-2)^n}{n}} = 3" }, { "type": "text", "content": "，收敛半径为 " }, { "type": "equation_inline", "content": "\\frac{1}{3}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "y = \\frac{1}{3}" }, { "type": "text", "content": " 时为 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\left(\\frac{1}{n} + \\frac{1}{n} \\left(-\\frac{2}{3}\\right)^n\\right)" }, { "type": "text", "content": " 前者发散，后者收敛，因此整体发散。 " }, { "type": "equation_inline", "content": "y = -\\frac{1}{3}" }, { "type": "text", "content": " 时为 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\left(\\frac{(-1)^n}{n} + \\frac{1}{n} \\left(\\frac{2}{3}\\right)^n\\right)" }, { "type": "text", "content": " 收敛，因此收敛域为 " }, { "type": "equation_inline", "content": "[- \\frac{1}{3} - 1, \\frac{1}{3} - 1)" }, { "type": "text", "content": " 即 " }, { "type": "equation_inline", "content": "x \\in [-\\frac{4}{3}, -\\frac{2}{3})" } ] }, "bbox": [ 87, 373, 905, 473 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习10.3非常规格式幂级数(1) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{(x^{2}+x+1)^{n}}{3n}" }, { "type": "text", "content": " 的收敛域(2) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{1}{2^{n}}\\left(\\frac{x-1}{2x+1}\\right)^{n^{2}}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{1^{n}+2^{n}+\\cdots+2022^{n}}{n^{2022}}\\left(\\frac{1-x}{1+x}\\right)^{n}" }, { "type": "text", "content": " 解(1)设 " }, { "type": "equation_inline", "content": "y=x^{2}+x+1" }, { "type": "text", "content": " ，考虑 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\sqrt[n]{\\frac{1}{3n}}=1" }, { "type": "text", "content": " ，R=1，而y=1发散，y=-1收敛，计算 " }, { "type": "equation_inline", "content": "x^{2}+x+1\\in[-1,1)" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "x^{2}+x\\in[-2,0)" }, { "type": "text", "content": " ，注意最终得到 " }, { "type": "equation_inline", "content": "x\\in(-1,0)" }, { "type": "text", "content": " 两侧都是开的！" } ] }, "bbox": [ 60, 479, 976, 565 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 考虑 " }, { "type": "equation_inline", "content": "y = \\frac{x - 1}{2x + 1}" }, { "type": "text", "content": "，计算得到 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 的收敛域为 " }, { "type": "equation_inline", "content": "[-1, 1]" }, { "type": "text", "content": "，计算 " }, { "type": "equation_inline", "content": "-1 \\leq \\frac{x - 1}{2x + 1} \\leq 1" }, { "type": "text", "content": "，直接讨论分母容易出错，因此这里看成两个不等式。 " }, { "type": "equation_inline", "content": "\\frac{x - 1}{2x + 1} + 1 = \\frac{3x}{2x + 1} \\geq 0" }, { "type": "text", "content": " 以及 " }, { "type": "equation_inline", "content": "\\frac{x - 1}{2x + 1} - 1 = \\frac{-x - 2}{2x + 1} \\leq 0" }, { "type": "text", "content": "，第一个要求 " }, { "type": "equation_inline", "content": "x \\geq 0" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "x < -\\frac{1}{2}" }, { "type": "text", "content": "，第二个要求 " }, { "type": "equation_inline", "content": "x \\leq -2" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "x \\geq -\\frac{1}{2}" }, { "type": "text", "content": "，求交得到 " }, { "type": "equation_inline", "content": "x \\in (-\\infty, -2] \\cup [0, +\\infty)" }, { "type": "text", "content": "\n\n(3) " }, { "type": "equation_inline", "content": "y = \\frac{1 - x}{1 + x}" }, { "type": "text", "content": " 得到收敛半径为 " }, { "type": "equation_inline", "content": "\\frac{1}{2022}" }, { "type": "text", "content": " 且端点收敛，考虑 " }, { "type": "equation_inline", "content": "-\\frac{1}{2022} \\leq \\frac{1 - x}{1 + x} \\leq \\frac{1}{2022}" }, { "type": "text", "content": "，分为两部分，得到 " }, { "type": "equation_inline", "content": "\\frac{2023 - 2021x}{1 + x} \\geq 0" }, { "type": "text", "content": " 以及 " }, { "type": "equation_inline", "content": "\\frac{2021 - 2023x}{1 + x} \\leq 0" }, { "type": "text", "content": "，前者解出 " }, { "type": "equation_inline", "content": "-1 < x \\leq \\frac{2023}{2021}" }, { "type": "text", "content": "，后者解出 " }, { "type": "equation_inline", "content": "x < -1" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "x \\geq \\frac{2021}{2023}" }, { "type": "text", "content": "，得出结果为 " }, { "type": "equation_inline", "content": "x \\in [\\frac{2021}{2023}, \\frac{2023}{2021}]" } ] }, "bbox": [ 87, 564, 907, 721 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.1.2 幂级数的一致收敛性" } ], "level": 2 }, "bbox": [ 95, 739, 342, 756 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下面考虑幂级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} a_n x^n" }, { "type": "text", "content": " 的一致收敛性，假设其收敛半径为 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": "，收敛域在端点处需要讨论，下面开始分析。" } ] }, "bbox": [ 124, 765, 905, 804 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.3 (Abel 第二定理)" } ], "level": 2 }, "bbox": [ 122, 815, 315, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若幂级数收敛半径为 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}a_nx^n" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\langle -R,R\\rangle" }, { "type": "text", "content": " 内闭一致收敛(端点未确定)" } ] }, "bbox": [ 115, 835, 658, 875 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "10.1 幂级数概念及其收敛区间" } ] }, "bbox": [ 680, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "94" } ] }, "bbox": [ 489, 954, 509, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x_0^n" }, { "type": "text", "content": " 收敛，(1) 若 " }, { "type": "equation_inline", "content": "x_0 > 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\forall x \\in [0, x_0]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x^n = \\sum_{n=0}^{\\infty} a_n x_0^n \\left(\\frac{x}{x_0}\\right)^n" }, { "type": "text", "content": "，根据 Abel 判别法可证。(2) 若 " }, { "type": "equation_inline", "content": "x_0 < 0" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\forall x \\in [x_0, 0]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x^n = \\sum_{n=0}^{\\infty} a_n x_0^n \\left(\\frac{x}{x_0}\\right)^n" }, { "type": "text", "content": "，根据 Abel 可证。进而任选闭区间都一致收敛。" } ] }, "bbox": [ 89, 80, 905, 156 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.4 (连续性)" } ], "level": 2 }, "bbox": [ 122, 166, 260, 180 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "幂级数的和函数 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x^n" }, { "type": "text", "content": " 在收敛域 " }, { "type": "equation_inline", "content": "\\langle -R, R \\rangle" }, { "type": "text", "content": " 是连续的。" } ] }, "bbox": [ 115, 187, 519, 224 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.5 (区间不变性)" } ], "level": 2 }, "bbox": [ 122, 249, 295, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "幂级数 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x^n" }, { "type": "text", "content": " 的收敛半径为 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": "，则其逐项求导和逐项积分后得到的幂级数的收敛半径仍为 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": "，不过端点处的一致收敛情况可能改变！" } ] }, "bbox": [ 114, 269, 884, 319 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 考虑 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} a_n x^n" }, { "type": "text", "content": " 的逐项求导与逐项积分, 得到 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} na_n" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} \\frac{a_n}{n+1} x^{n+1}" }, { "type": "text", "content": ", 根据 " }, { "type": "equation_inline", "content": "\\varlimsup_{n \\to \\infty} \\sqrt[n]{|a_n|} = \\varliminf_{n \\to \\infty} \\sqrt[n-1]{|na_n|} = \\varlimsup_{n \\to \\infty} \\sqrt[n-1]{|\\frac{a_n}{n+1}|}" }, { "type": "text", "content": " 可知收敛半径相同" } ] }, "bbox": [ 89, 329, 912, 397 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 端点处一致收敛性不保证，例如 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^n}{n}" }, { "type": "text", "content": " 收敛域为 " }, { "type": "equation_inline", "content": "[-1, 1)" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} x^{n-1}" }, { "type": "text", "content": " 收敛域为 " }, { "type": "equation_inline", "content": "(-1, 1)" } ] }, "bbox": [ 87, 395, 773, 436 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.6 (求导与求积)" } ], "level": 2 }, "bbox": [ 122, 445, 295, 458 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "(-R, R)" }, { "type": "text", "content": " 上 " }, { "type": "equation_inline", "content": "f(x) = \\sum_{n=0}^{\\infty} a_n x^n" }, { "type": "text", "content": "，则对 " }, { "type": "equation_inline", "content": "\\forall x \\in (-R, R)" }, { "type": "text", "content": " 有：" } ] }, "bbox": [ 115, 464, 527, 502 ] }, { "type": "equation_interline", "content": { "math_content": "f ^ {\\prime} (x) = \\sum_ {n = 1} ^ {\\infty} n a _ {n} x ^ {n - 1}, \\int_ {0} ^ {x} f (t) \\mathrm{d} t = \\sum_ {n = 0} ^ {\\infty} \\frac {a _ {n}}{n + 1} x ^ {n + 1}", "math_type": "latex", "image_source": { "path": "images/b42a12d95b89977daf73fdd23044181d69f43a3b350fa7f7e7a640bba8313bda.jpg" } }, "bbox": [ 312, 504, 678, 542 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 幂级数的逐项求导和逐项积分要注意其只能保证在收敛区间成立，但不保证在收敛域上成立！" } ] }, "bbox": [ 87, 562, 811, 577 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.1.3 缺项幂级数的收敛区域" } ], "level": 2 }, "bbox": [ 95, 600, 364, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习10.4缺项幂级数的收敛范围(1)求 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}n^{n^2}x^{n^3}" }, { "type": "text", "content": " 的收敛范围" } ] }, "bbox": [ 58, 627, 554, 664 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解令 " }, { "type": "equation_inline", "content": "a_{k} = \\left\\{ \\begin{array}{ll}n^{n^{2}}, & k = n^{3}\\\\ 0, & k\\neq n^{3} \\end{array} \\right." }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}n^{n^2}x^{n^3} = \\sum_{k = 1}^{\\infty}a_kx^k" }, { "type": "text", "content": " ，此时" } ] }, "bbox": [ 87, 665, 541, 707 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {1}{R} = \\varlimsup_ {k \\to \\infty} \\sqrt [ k ]{| a _ {k} |} = \\lim _ {n \\to \\infty} \\sqrt [ n ^ {3} ]{n ^ {n ^ {2}}} = \\lim _ {n \\to \\infty} (n ^ {n ^ {2}}) ^ {\\frac {1}{n ^ {3}}} = 1", "math_type": "latex", "image_source": { "path": "images/39993da7e00efa13bc578b689a1d759fa3b7db10fa91c8a5e7be47df5986069b.jpg" } }, "bbox": [ 307, 712, 687, 741 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.2 幂级数展开式求和" } ], "level": 2 }, "bbox": [ 95, 762, 347, 781 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.7 (必备幂级数展开)" } ], "level": 2 }, "bbox": [ 122, 802, 329, 816 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "以下的幂级数展开会经常使用，基本上是分母较为简单的 Taylor 展开式" } ] }, "bbox": [ 115, 822, 640, 837 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "\\frac{1}{1 - x} = \\sum_{n=0}^{\\infty} x^n" } ] }, "bbox": [ 136, 837, 275, 873 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "10.2 幂级数展开式求和" } ] }, "bbox": [ 732, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "95" } ] }, "bbox": [ 489, 954, 507, 964 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. " }, { "type": "equation_inline", "content": "e^x = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}" } ] }, "bbox": [ 136, 96, 252, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3. " }, { "type": "equation_inline", "content": "\\ln (1 + x) = \\sum_{n = 1}^{\\infty}(-1)^{n - 1}\\frac{x^n}{n}" } ] }, "bbox": [ 136, 133, 357, 167 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "4. " }, { "type": "equation_inline", "content": "\\arctan x = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{2n+1}" } ] }, "bbox": [ 134, 168, 359, 205 ] }, { "type": "image", "content": { "image_source": { "path": "images/f2fe059e28c6b6c8debade1d0b18e4ac23151c56b894b7eeec42bd44c39050ed.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 219, 78, 237 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 具体用哪个主要看 " }, { "type": "equation_inline", "content": "(-1)" }, { "type": "text", "content": " 的指数次和分母的关系。" } ] }, "bbox": [ 87, 225, 485, 240 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.2.1 等比级数" } ], "level": 2 }, "bbox": [ 95, 262, 243, 278 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.8 (等比级数)" } ], "level": 2 }, "bbox": [ 122, 294, 278, 309 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据 " }, { "type": "equation_inline", "content": "f(x) = \\sum_{n=0}^{\\infty} x^n = \\frac{1}{1 - x}" }, { "type": "text", "content": " 可以推出：" } ] }, "bbox": [ 115, 315, 403, 351 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} nx^{n} = \\frac{x}{(1 - x)^{2}}" }, { "type": "text", "content": "，方法为对 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty} x^{n}" }, { "type": "text", "content": " 求导，再同乘 " }, { "type": "equation_inline", "content": "x" } ] }, "bbox": [ 136, 350, 559, 388 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} n^{2} x^{n} = \\frac{x(1 + x)}{(1 - x)^{3}}" }, { "type": "text", "content": "，方法为对 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} nx^{n}" }, { "type": "text", "content": " 求导，再乘 " }, { "type": "equation_inline", "content": "x" } ] }, "bbox": [ 134, 387, 561, 424 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习10.5等比级数 (1) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{n^2 + 3n}{2^n}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{x^n}{n(n+1)}" } ] }, "bbox": [ 60, 436, 467, 473 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)这里展示正推, 进行公式推导:" } ] }, "bbox": [ 89, 470, 366, 486 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} f _ {1} (x) = \\sum_ {n = 0} ^ {\\infty} x ^ {n} = \\frac {1}{1 - x} \\Rightarrow f _ {1} ^ {\\prime} (x) = \\sum_ {n = 1} ^ {\\infty} n x ^ {n - 1} = \\frac {1}{(1 - x) ^ {2}} \\\\ f _ {2} (x) = \\sum_ {n = 1} ^ {\\infty} n x ^ {n} = \\frac {x}{(1 - x) ^ {2}} \\Rightarrow f _ {2} ^ {\\prime} (x) = \\sum_ {n = 1} ^ {\\infty} n ^ {2} x ^ {n - 1} = \\frac {1 + x}{(1 - x) ^ {3}} \\\\ f _ {3} (x) = \\sum_ {n = 1} ^ {\\infty} n ^ {2} x ^ {n} = \\frac {x (1 + x)}{(1 - x) ^ {3}} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/b5e68623767b858b1f0b3301f1e0e507c170fc5afb33293b20f8e3d9a118dda2.jpg" } }, "bbox": [ 252, 489, 726, 595 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "I = f_{3}\\left(\\frac{1}{2}\\right) + 3f_{2}\\left(\\frac{1}{2}\\right) = 6 + 6 = 12" } ] }, "bbox": [ 92, 598, 376, 621 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 这里展示反推:" } ] }, "bbox": [ 122, 623, 260, 637 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} f _ {1} (x) = \\sum_ {n = 1} ^ {\\infty} \\frac {x ^ {n}}{n (n + 1)} \\\\ f _ {2} (x) = \\sum_ {n = 1} ^ {\\infty} \\frac {x ^ {n + 1}}{n (n + 1)} \\Rightarrow f _ {2} ^ {\\prime} (x) = \\sum_ {n = 1} ^ {\\infty} \\frac {x ^ {n}}{n} \\Rightarrow f _ {2} ^ {\\prime \\prime} (x) = \\sum_ {n = 1} ^ {\\infty} x ^ {n - 1} = \\sum_ {n = 0} ^ {\\infty} x ^ {n} = \\frac {1}{1 - x} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/3b5ae9c7df3c3b113a178586feebfe82a89948deb99ab2719bdd764cc60527a8.jpg" } }, "bbox": [ 189, 642, 786, 714 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此推出 " }, { "type": "equation_inline", "content": "f_{2}^{\\prime}(x) = -\\ln (1 - x),f_{2}(x) = (1 - x)\\ln (1 - x) + x" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "f_{1}(x) = \\frac{1 - x}{x}\\ln (1 - x) + 1" } ] }, "bbox": [ 90, 717, 781, 744 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.2.2 " }, { "type": "equation_inline", "content": "\\arctan x" }, { "type": "text", "content": " 级数" } ], "level": 2 }, "bbox": [ 95, 762, 277, 777 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 10.6arctan x (1) 重点：求 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{n(2n+1)}" } ] }, "bbox": [ 58, 788, 425, 826 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "10.2 幂级数展开式求和" } ] }, "bbox": [ 732, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "96" } ] }, "bbox": [ 489, 954, 509, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)相差较大，选择裂项： " }, { "type": "equation_inline", "content": "\\sum_{n = 1}^{\\infty}\\frac{(-1)^{n - 1}}{n(2n + 1)} = 2\\sum_{n = 1}^{\\infty}\\left(\\frac{(-1)^{n - 1}}{2n} -\\frac{(-1)^{n - 1}}{2n + 1}\\right)" }, { "type": "text", "content": " ，两者均收敛，且" } ] }, "bbox": [ 89, 80, 781, 120 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\ln (1 + x) = \\sum_ {n = 1} ^ {\\infty} \\frac {(- 1) ^ {n - 1} x ^ {n}}{n} \\\\ \\arctan x = \\sum_ {n = 0} ^ {\\infty} \\frac {(- 1) ^ {n} x ^ {2 n + 1}}{2 n + 1} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/dd0b45f59bb18020a8a4e47a74d1732b9c54ea578f1272a2da374dcdb9b21603.jpg" } }, "bbox": [ 379, 123, 598, 193 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此结果为 " }, { "type": "equation_inline", "content": "\\ln 2 + \\frac{\\pi}{2} - 2" } ] }, "bbox": [ 90, 196, 273, 223 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.2.3 " }, { "type": "equation_inline", "content": "e^x" }, { "type": "text", "content": " 级数" } ], "level": 2 }, "bbox": [ 95, 241, 225, 256 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 " }, { "type": "equation_inline", "content": "10.7e^{x}" }, { "type": "text", "content": " (1) 计算 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}\\left[1+\\frac{1}{2!}+\\frac{1}{4!}+\\cdots+\\frac{1}{(2n)!}\\right]" } ] }, "bbox": [ 60, 267, 485, 298 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 由于 " }, { "type": "equation_inline", "content": "e^x = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "e^{-x} = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^n}{n!}" }, { "type": "text", "content": "，故" } ] }, "bbox": [ 89, 299, 497, 335 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {e ^ {x} + e ^ {- x}}{2} = \\sum_ {n = 0} ^ {\\infty} \\frac {x ^ {2 n}}{(2 n) !}", "math_type": "latex", "image_source": { "path": "images/ed1d927192353ceb342c948af202d0c6cd48f31c4e14979fe128c6b8e34c4ba5.jpg" } }, "bbox": [ 416, 338, 578, 375 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "取 " }, { "type": "equation_inline", "content": "x = 1" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "\\sum_{n=0}^{\\infty}\\frac{1}{(2n)!} = \\frac{e + e^{-1}}{2}" } ] }, "bbox": [ 90, 378, 359, 414 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.3 幂级数求和式展开" } ], "level": 2 }, "bbox": [ 95, 439, 347, 457 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.3.1 基本理论与直接展开" } ], "level": 2 }, "bbox": [ 95, 476, 342, 494 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 10.9 (幂级数展开)" } ], "level": 2 }, "bbox": [ 122, 510, 295, 524 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0 - R, x_0 + R)" }, { "type": "text", "content": " 上存在任意阶导数，且 " }, { "type": "equation_inline", "content": "\\exists M, N > 0" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "n > N" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\forall x \\in (x_0 - R, x_0 + R)" }, { "type": "text", "content": "，均有 " }, { "type": "equation_inline", "content": "|f^{(n)}(x_0)| \\leq M" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 112, 529, 880, 564 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = \\sum_ {n = 0} ^ {\\infty} \\frac {f ^ {(n)} (x _ {0})}{n !} (x - x _ {0}) ^ {n}, \\forall x \\in (x _ {0} - R, x _ {0} + R)", "math_type": "latex", "image_source": { "path": "images/f667b7c11ad38411d79dfc40058f6be32d06f43f9aafa54fe8050f8432347a25.jpg" } }, "bbox": [ 300, 567, 692, 604 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习10.8幂级数直接展开(1) " }, { "type": "equation_inline", "content": "\\frac{ax + b}{cx + d}" }, { "type": "text", "content": " 型： " }, { "type": "equation_inline", "content": "f(x) = \\frac{x - 3}{x + 1}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 1" }, { "type": "text", "content": " 处的幂级数展开" } ] }, "bbox": [ 58, 634, 684, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f(x)=\\frac{1}{x(x+1)}" }, { "type": "text", "content": " 在 x=1 处幂级数展开" } ] }, "bbox": [ 122, 663, 438, 695 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f(x)=\\frac{1}{1-3x+2x^{2}}" }, { "type": "text", "content": " 在 x=0 处展开" } ] }, "bbox": [ 122, 694, 421, 721 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "f(x) = \\frac{x}{1 + x - 2x^2}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 处展开" } ] }, "bbox": [ 122, 719, 413, 741 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)首先 " }, { "type": "equation_inline", "content": "f(x)=1-\\frac{4}{1+x}" }, { "type": "text", "content": " ，设t=x-1，则 " }, { "type": "equation_inline", "content": "f(x)=g(t)=1-\\frac{4}{2+t}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "f(x)=1-2\\cdot\\frac{1}{1+\\frac{t}{2}}=1-2\\sum_{n=0}^{\\infty}\\left(\\frac{-t}{2}\\right)^{n}" }, { "type": "text", "content": " ，\n因此得到" } ] }, "bbox": [ 90, 741, 905, 789 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = 1 - \\sum_ {n = 0} ^ {\\infty} \\left(\\frac {- t}{2}\\right) ^ {n}", "math_type": "latex", "image_source": { "path": "images/b30e89a58e2833872fe7f8283ef6a18aa48b48db9f06a2139d4d8b77365bfa60.jpg" } }, "bbox": [ 410, 787, 584, 825 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 设 " }, { "type": "equation_inline", "content": "t = x - 1" }, { "type": "text", "content": "，可以得到 " }, { "type": "equation_inline", "content": "f(x) = \\sum_{n=0}^{\\infty} (-1)^n \\left(1 - \\frac{1}{2^{n+1}}\\right)(x - 1)^n" } ] }, "bbox": [ 122, 832, 611, 870 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "10.3 幂级数求和式展开" } ] }, "bbox": [ 732, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "97" } ] }, "bbox": [ 487, 953, 507, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "10.3.2 求导与求积" } ], "level": 2 }, "bbox": [ 95, 85, 263, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习10.9几个经典的积分与求导法(1)计算 " }, { "type": "equation_inline", "content": "f(x) = \\frac{1}{(1 + x)^2}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 处的幂级数展开" } ] }, "bbox": [ 58, 111, 726, 141 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "f(x) = \\arcsin x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 处的幂级数展开" } ] }, "bbox": [ 122, 141, 490, 156 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 计算 " }, { "type": "equation_inline", "content": "f(x) = \\int_{0}^{x} \\frac{\\sin t}{t} t" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 处的幂级数展开" } ] }, "bbox": [ 122, 155, 500, 184 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)积分得到 " }, { "type": "equation_inline", "content": "\\int_0^x f(t)\\mathrm{d}t = -\\frac{1}{1 + t}\\bigg|_0^x = 1 - \\frac{1}{1 + x} = 1 - \\sum_{n = 0}^{\\infty}(-x)^n" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "f(x) = \\left(\\int_0^x f(t)\\mathrm{d}t\\right)' = \\sum_{n = 1}^{\\infty}(-1)^{n + 1}nx^{n - 1}" } ] }, "bbox": [ 89, 181, 919, 219 ] }, { "type": "equation_interline", "content": { "math_content": "(2) f ^ {\\prime} (x) = \\frac {1}{\\sqrt {1 - x ^ {2}}}, \\text { 而 }", "math_type": "latex", "image_source": { "path": "images/cfbee71c6604c3c2290f4097a758d90cfdefe8af84e0ed9a3547bde4a750a3c6.jpg" } }, "bbox": [ 122, 219, 307, 249 ] }, { "type": "equation_interline", "content": { "math_content": "(1 - x ^ {2}) ^ {- \\frac {1}{2}} = 1 + \\sum_ {n = 1} ^ {\\infty} \\frac {(- \\frac {1}{2}) (- \\frac {3}{2}) \\cdots (- \\frac {1}{2} - n + 1)}{n !} (- x ^ {2}) ^ {n} = 1 + \\sum_ {n = 1} ^ {\\infty} \\frac {(2 n - 1) ! !}{(2 n) ! !} x ^ {2 n}", "math_type": "latex", "image_source": { "path": "images/f16c3e265febd5012786f9f964c91f3990ca34e6b68fad738aa1812a54421d24.jpg" } }, "bbox": [ 200, 250, 794, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "f(x) = f(0) + \\int_0^x f'(t)\\mathrm{d}t = x + \\sum_{n = 1}^{\\infty}\\frac{(2n - 1)!!}{(2n + 1)(2n)!!} x^{2n + 1}" } ] }, "bbox": [ 90, 293, 546, 329 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 10.10arctan 经典题目 求幂级数展开:" } ] }, "bbox": [ 58, 338, 405, 353 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "f(x)=\\arctan x" } ] }, "bbox": [ 122, 357, 265, 373 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重点 (ZJU2021): " }, { "type": "equation_inline", "content": "g(x) = \\arctan \\frac{1 - kx}{1 + kx}" } ] }, "bbox": [ 122, 370, 435, 398 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 重点: " }, { "type": "equation_inline", "content": "h(x) = \\arctan \\frac{2x}{1 - x^2}" }, { "type": "text", "content": ", 求导结果很简单" } ] }, "bbox": [ 122, 397, 492, 425 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) " }, { "type": "equation_inline", "content": "f'(x)=\\frac{1}{1+x^{2}}=\\sum_{n=0}^{\\infty}(-x^{2})^{n}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 90, 426, 396, 460 ] }, { "type": "equation_interline", "content": { "math_content": "f (x) = f (0) + \\sum_ {n = 0} ^ {\\infty} \\int_ {0} ^ {x} (- 1) ^ {n} t ^ {2 n} \\mathrm{d} t = \\sum_ {n = 0} ^ {\\infty} (- 1) ^ {n} \\frac {x ^ {2 n + 1}}{2 n + 1}", "math_type": "latex", "image_source": { "path": "images/70de4ca34d98111ed5ecc89f316130c05e351994bda6841330340c03218acbff.jpg" } }, "bbox": [ 302, 464, 692, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\arctan \\frac{1 - kx}{1 + kx} = \\arctan 1 - \\arctan kx" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 122, 502, 468, 533 ] }, { "type": "equation_interline", "content": { "math_content": "g (x) = \\frac {\\pi}{4} - \\sum_ {n = 0} ^ {\\infty} \\frac {(- 1) ^ {n} k ^ {2 n + 1}}{2 n + 1} x ^ {2 n + 1}", "math_type": "latex", "image_source": { "path": "images/faea163fa7da825d53b86bb280f08399e63ec790b9a16b496c0aad66317296bc.jpg" } }, "bbox": [ 369, 536, 625, 573 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "h'(x)=\\frac{2}{1+x^{2}}" }, { "type": "text", "content": " (别怕求导！)，因此 " }, { "type": "equation_inline", "content": "h'(x)=2\\sum_{n=0}^{\\infty}(-x^{2})^{n}" }, { "type": "text", "content": " ，积分得到" } ] }, "bbox": [ 122, 576, 638, 613 ] }, { "type": "equation_interline", "content": { "math_content": "h (x) = 2 \\sum_ {n = 0} ^ {\\infty} \\frac {(- 1) ^ {n}}{2 n + 1} x ^ {2 n + 1}", "math_type": "latex", "image_source": { "path": "images/9fd3797d8f03ffab96bcdcd8b8f1ddc8897d3e36023bc7db9f1eb79d69bd9bc4.jpg" } }, "bbox": [ 400, 615, 593, 653 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "10.3 幂级数求和式展开" } ] }, "bbox": [ 732, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "98" } ] }, "bbox": [ 487, 953, 507, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 11 章 傅里叶级数" } ], "level": 2 }, "bbox": [ 374, 87, 640, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "11.1 以 " }, { "type": "equation_inline", "content": "2\\pi" }, { "type": "text", "content": " 为周期的傅里叶级数" } ], "level": 2 }, "bbox": [ 95, 145, 436, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 11.1 (2π 周期傅里叶展开)" } ], "level": 2 }, "bbox": [ 121, 185, 352, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "2\\pi" }, { "type": "text", "content": " 周期函数 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": "，定义其傅里叶展开为 " }, { "type": "equation_inline", "content": "f(x) \\sim \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} (a_n \\cos nx + b_n \\sin nx)" }, { "type": "text", "content": "，其中" } ] }, "bbox": [ 114, 206, 789, 243 ] }, { "type": "equation_interline", "content": { "math_content": "\\int a _ {n} = \\frac {1}{\\pi} \\int_ {- \\pi} ^ {\\pi} f (x) \\cos n x \\mathrm{d} x \\quad n = 0, 1, 2, \\dots", "math_type": "latex", "image_source": { "path": "images/7a073e64ff50788907380424c8cb0e61fd6495b673656f89ca5d3dda7c5ebdbf.jpg" } }, "bbox": [ 337, 246, 658, 278 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{b _ {n} = \\frac {1}{\\pi} \\int_ {- \\pi} ^ {\\pi} f (x) \\sin n x \\mathrm{d} x \\quad n = 1, 2, \\dots \\right.", "math_type": "latex", "image_source": { "path": "images/a2ba5e2ff50e4037f9bd8172227cba138cf5706f3ad8ca138e2fd9ec4636ea7a.jpg" } }, "bbox": [ 339, 277, 645, 307 ] }, { "type": "image", "content": { "image_source": { "path": "images/94b85b028233997a3678393537cecdea4088e96528b29e922edcbe8d2e7f4793.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 319, 78, 340 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 计算时一定要注意第一项是 " }, { "type": "equation_inline", "content": "\\frac{a_0}{2}" }, { "type": "text", "content": "!" } ] }, "bbox": [ 60, 322, 369, 348 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 11.1 (收敛定理)" } ], "level": 2 }, "bbox": [ 122, 357, 278, 373 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若以 " }, { "type": "equation_inline", "content": "2\\pi" }, { "type": "text", "content": " 为周期的函数 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[- \\pi, \\pi]" }, { "type": "text", "content": " 上按段光滑，则对 " }, { "type": "equation_inline", "content": "\\forall x \\in [-\\pi, \\pi]" }, { "type": "text", "content": "，傅里叶级数收敛于 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 左右极限的算术平均值。" } ] }, "bbox": [ 114, 376, 882, 412 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 11.1 (连续特殊情况)" } ], "level": 2 }, "bbox": [ 122, 435, 312, 450 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 是以 " }, { "type": "equation_inline", "content": "2\\pi" }, { "type": "text", "content": " 为周期的连续函数，且在 " }, { "type": "equation_inline", "content": "[- \\pi, \\pi]" }, { "type": "text", "content": " 上按段光滑，则 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 的傅里叶级数在 " }, { "type": "equation_inline", "content": "(- \\infty, +\\infty)" }, { "type": "text", "content": " 上收敛于 " }, { "type": "equation_inline", "content": "f" } ] }, "bbox": [ 114, 453, 902, 476 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习11.1几个经典展开(1)求 " }, { "type": "equation_inline", "content": "f(x) = x^{2}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(- \\pi, \\pi)" }, { "type": "text", "content": " 上的傅里叶级数，并求 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}" } ] }, "bbox": [ 58, 502, 695, 538 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)计算得到 " }, { "type": "equation_inline", "content": "a_0 = \\frac{2}{3}\\pi^2" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "a_{n} = \\frac{1}{\\pi}\\int_{-\\pi}^{\\pi}x^{2}\\cos nx\\mathrm{d}x = (-1)^{n}\\frac{4}{n^{2}}" }, { "type": "text", "content": " ，因此得到" } ] }, "bbox": [ 87, 536, 642, 568 ] }, { "type": "equation_interline", "content": { "math_content": "x ^ {2} = \\frac {1}{3} \\pi^ {2} + \\sum_ {n = 1} ^ {\\infty} (- 1) ^ {n} \\frac {4}{n ^ {2}}", "math_type": "latex", "image_source": { "path": "images/a97ec404c8bd51115843982bda92c9f0910011765782168edfe19acfc1af8d03.jpg" } }, "bbox": [ 403, 571, 591, 608 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "令 " }, { "type": "equation_inline", "content": "x = \\pi" }, { "type": "text", "content": " 得到" } ] }, "bbox": [ 90, 611, 196, 626 ] }, { "type": "equation_interline", "content": { "math_content": "\\pi^ {2} = \\frac {1}{3} \\pi^ {2} + \\sum_ {n = 1} ^ {\\infty} \\frac {4}{n ^ {2}} \\Rightarrow \\sum_ {n = 1} ^ {\\infty} \\frac {1}{n ^ {2}} = \\frac {\\pi^ {2}}{6}", "math_type": "latex", "image_source": { "path": "images/a60b982df7c0e06183b647bd1e8096c04e40672805aaac3d4bad6cc158ceeb17.jpg" } }, "bbox": [ 363, 623, 631, 661 ] }, { "type": "image", "content": { "image_source": { "path": "images/d54523c5e53334e427bb9e9822136c9ad2e44ba8407c1ddcc6e3a92d9c8e6f2d.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 57, 668, 80, 688 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 注意 " }, { "type": "equation_inline", "content": "\\cos n\\pi = (-1)^n" }, { "type": "text", "content": "。" } ] }, "bbox": [ 87, 673, 289, 689 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "11.2 以 " }, { "type": "equation_inline", "content": "2l" }, { "type": "text", "content": " 为周期的傅里叶级数" } ], "level": 2 }, "bbox": [ 95, 734, 428, 753 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 11.2 (2l 周期的傅里叶级数)" } ], "level": 2 }, "bbox": [ 122, 775, 364, 790 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 以 " }, { "type": "equation_inline", "content": "2l" }, { "type": "text", "content": " 为周期，则其傅里叶级数为 " }, { "type": "equation_inline", "content": "f(x) \\sim \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} (a_n \\cos \\frac{n\\pi x}{l} + b_n \\sin \\frac{n\\pi x}{l})" }, { "type": "text", "content": "，其中" } ] }, "bbox": [ 114, 795, 761, 831 ] }, { "type": "equation_interline", "content": { "math_content": "\\int a _ {n} = \\frac {1}{l} \\int_ {- l} ^ {l} f (x) \\cos \\frac {n \\pi x}{l} \\mathrm{d} x, \\quad n = 0, 1, 2, \\dots", "math_type": "latex", "image_source": { "path": "images/4d3e315f61b6dfc273b22d86127b8a5f09cd2dcc897d411e495493afceb8a4e1.jpg" } }, "bbox": [ 331, 835, 665, 869 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{b _ {n} = \\frac {1}{l} \\int_ {- l} ^ {l} f (x) \\sin {\\frac {n \\pi x}{l}} \\mathrm{d} x, n = 1, 2, \\dots \\right.", "math_type": "latex", "image_source": { "path": "images/4451f8b08984566ae84e0be3ab66422808e166f9be0f6e070037d0b7bfed2aed.jpg" } }, "bbox": [ 332, 869, 650, 900 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "11.3 Parseval 等式" } ], "level": 2 }, "bbox": [ 95, 87, 297, 107 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "定理 11.2 (Parseval 等式)" } ] }, "bbox": [ 122, 129, 310, 143 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {1}{\\pi} \\int_ {- \\pi} ^ {\\pi} [ f (x) ] ^ {2} \\mathrm{d} x = \\frac {a _ {0} ^ {2}}{2} + \\sum_ {n = 1} ^ {\\infty} (a _ {n} ^ {2} + b _ {n} ^ {2})", "math_type": "latex", "image_source": { "path": "images/99828ad576401e15649731aadbd145fd185cd62e4d1992441b7e1c30667c729c.jpg" } }, "bbox": [ 115, 149, 391, 185 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习11.2计算数项级数(1)对 " }, { "type": "equation_inline", "content": "f(x) = x^{2}" }, { "type": "text", "content": " 展开，并证明 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\frac{1}{n^4} = \\frac{\\pi^4}{90}" } ] }, "bbox": [ 60, 200, 593, 236 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "11.3 Parseval 等式" } ] }, "bbox": [ 771, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "100" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/cc6eab2313c8b054f7a9294640558a696f8d7f28aec6df75223441dd840f7835.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 291, 78, 310 ] }, { "type": "image", "content": { "image_source": { "path": "images/ef8b8195cbadf4e004099909588a2b27bf2e5ea56b2eedc28d61fe25a11d8d12.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 682, 78, 701 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 12 章 多元极限与连续" } ], "level": 2 }, "bbox": [ 344, 87, 668, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "12.1 累次极限与重极限" } ], "level": 2 }, "bbox": [ 95, 145, 349, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "12.1.1 基本概念" } ], "level": 2 }, "bbox": [ 95, 184, 243, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 12.1 (重极限)" } ], "level": 2 }, "bbox": [ 121, 217, 262, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a \\in \\mathbb{R}^n" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "f: \\mathbb{R}^n \\to \\mathbb{R}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " 的去心邻域中有定义, 若 " }, { "type": "equation_inline", "content": "\\exists A, \\forall \\epsilon, \\exists \\delta" }, { "type": "text", "content": " 当 " }, { "type": "equation_inline", "content": "\\mathbf{x} \\in \\mathbb{R}^n" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|\\mathbf{x} - a| < \\delta" }, { "type": "text", "content": " 时, 有 " }, { "type": "equation_inline", "content": "|f(\\mathbf{x}) - A| < \\epsilon" }, { "type": "text", "content": ", 则称 " }, { "type": "equation_inline", "content": "\\lim_{\\mathbf{x} \\to a} f(\\mathbf{x}) = A" } ] }, "bbox": [ 112, 235, 882, 274 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 计算重极限的方法:" } ] }, "bbox": [ 58, 293, 275, 310 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 放缩 + 迫敛：例如 " }, { "type": "equation_inline", "content": "\\left|(x + y)\\sin \\frac{1}{x}\\sin \\frac{1}{y}\\right| \\leq |x + y| = 0" } ] }, "bbox": [ 114, 307, 522, 336 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 极坐标变换：例如 " }, { "type": "equation_inline", "content": "\\left|\\frac{x^3 + y^3}{x^2 + y^2}\\right| = \\left|\\frac{r^3\\cos^3\\theta + r^3\\sin^3\\theta}{r^2}\\right| = r|\\cos^3\\theta +\\sin^3\\theta |\\leq 2r\\to 0" } ] }, "bbox": [ 115, 336, 737, 370 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 证明重极限不存在的常用方法:" } ] }, "bbox": [ 87, 372, 359, 388 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 两个累次极限存在但不相等" } ] }, "bbox": [ 115, 391, 339, 406 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 找特殊的趋近方式，如 " }, { "type": "equation_inline", "content": "y = kx" }, { "type": "text", "content": "，使得获得的重极限不同" } ] }, "bbox": [ 115, 410, 544, 425 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习12.1重极限的计算(1)已知 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 如下，计算 " }, { "type": "equation_inline", "content": "\\lim_{(x,y)\\to (0,0)}f(x,y)" } ] }, "bbox": [ 58, 447, 600, 469 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = \\left\\{ \\begin{array}{l l} x y \\frac {x ^ {2} - y ^ {2}}{x ^ {2} + y ^ {2}}, & (x, y) \\neq 0 \\\\ 0, & (x, y) = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/a0b2f3401fbf40ae677cd84156cd3727e20fc8075a3eed8a5c8cf24cbe2be035.jpg" } }, "bbox": [ 366, 474, 630, 525 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 设 " }, { "type": "equation_inline", "content": "x = r\\cos \\theta, y = r\\sin \\theta" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 89, 530, 352, 545 ] }, { "type": "equation_interline", "content": { "math_content": "| f (x, y) | = \\left| \\frac {r ^ {2} \\sin \\theta \\cos \\theta (r ^ {2} \\cos^ {2} \\theta - r ^ {2} \\sin^ {2} \\theta)}{r ^ {2}} \\right| = \\frac {1}{4} r ^ {2} | \\sin 4 \\theta | \\leq \\frac {1}{4} r ^ {2} \\rightarrow 0", "math_type": "latex", "image_source": { "path": "images/664cd9f0060de56c04b4fe52f2a1dac57e12ee2ac035ea9a09039e01dff5cf82.jpg" } }, "bbox": [ 233, 548, 761, 580 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此极限为0" } ] }, "bbox": [ 92, 583, 194, 598 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 12.2 (累次极限)" } ], "level": 2 }, "bbox": [ 122, 624, 278, 639 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_0,y_0" }, { "type": "text", "content": " 的某去心邻域中有定义，则称 " }, { "type": "equation_inline", "content": "\\lim_{y\\to y_0}\\lim_{x\\to x_0}f(x,y),\\lim_{x\\to x_0}\\lim_{y\\to y_0}f(x,y)" }, { "type": "text", "content": " 称为累次极限。" } ] }, "bbox": [ 114, 644, 848, 669 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 重极限是多个分量同时趋近于 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0" }, { "type": "text", "content": "，而累次极限相当于依次取极限的极限。" } ] }, "bbox": [ 87, 686, 665, 702 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 12.1 (重极限与累次极限的关系)" } ], "level": 2 }, "bbox": [ 121, 712, 396, 727 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若重极限 " }, { "type": "equation_inline", "content": "\\lim_{(x,y)\\to (x_0,y_0)}f(x,y) = A" }, { "type": "text", "content": " 存在，且累次极限存在(累次极限可能不存在)，则累次极限的值为 " }, { "type": "equation_inline", "content": "A" } ] }, "bbox": [ 114, 732, 870, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists \\delta, |\\mathbf{x} - \\mathbf{x}_0| < \\delta" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "|f(x, y) - A| < \\epsilon" }, { "type": "text", "content": "，两侧取极限 " }, { "type": "equation_inline", "content": "x \\to x_0" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "|\\lim_{x \\to x_0} f(x, y) - A| < \\epsilon" }, { "type": "text", "content": "，当 " }, { "type": "equation_inline", "content": "|y - y_0| < \\delta" }, { "type": "text", "content": " 时显然结论成立。" } ] }, "bbox": [ 90, 776, 905, 812 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 12.2 (重极限的任意性)" } ], "level": 2 }, "bbox": [ 122, 821, 329, 835 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "(x,y)" }, { "type": "text", "content": " 沿着任意路径趋于 " }, { "type": "equation_inline", "content": "(x_0,y_0)" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "f(x,y)\\to A" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 115, 841, 534, 857 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {(x, y) \\to (x _ {0}, y _ {0})} f (x, y) = A", "math_type": "latex", "image_source": { "path": "images/33bf9a9dc4803a7bad82c6f2e8dd9258b84bdc156040ddd89859a16dd181b29b.jpg" } }, "bbox": [ 410, 865, 586, 888 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 反设 " }, { "type": "equation_inline", "content": "\\lim_{(x,y)\\to(x_{0},y_{0})}f(x,y)\\neq A" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\exists\\epsilon_{0}" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "\\{x_{n}\\}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\lim_{n\\to\\infty}x_{n}=(x_{0},y_{0})" }, { "type": "text", "content": " ，但 " }, { "type": "equation_inline", "content": "|f(x_{n})-A|\\geq\\epsilon_{0}" }, { "type": "text", "content": " ，将 " }, { "type": "equation_inline", "content": "x_{1},x_{2},\\cdots,x_{n},\\cdots" }, { "type": "text", "content": " 连接成折线，折线趋近于 " }, { "type": "equation_inline", "content": "(x_{0},y_{0})" }, { "type": "text", "content": " ，但 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 不趋于 A，这与已知矛盾。" } ] }, "bbox": [ 87, 84, 921, 123 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 12.2 常用反例 讨论 " }, { "type": "equation_inline", "content": "(0,0)" }, { "type": "text", "content": " 处的重极限与累次极限" } ] }, "bbox": [ 58, 143, 490, 158 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 重点: " }, { "type": "equation_inline", "content": "f(x, y) = \\frac{xy}{x^2 + y^2}" } ] }, "bbox": [ 121, 160, 332, 185 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{x-y}{x+y}" } ] }, "bbox": [ 122, 185, 267, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f(x,y)=x\\sin\\frac{1}{y}" } ] }, "bbox": [ 122, 212, 268, 236 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) " }, { "type": "equation_inline", "content": "f(x,y)=x\\sin\\frac{1}{y}+y\\sin\\frac{1}{x}" } ] }, "bbox": [ 122, 234, 339, 266 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 重点: " }, { "type": "equation_inline", "content": "f(x, y) = \\begin{cases} 1, & 0 < y < x^2, -\\infty < x < +\\infty \\\\ 0, & \\text{其他} \\end{cases}" } ] }, "bbox": [ 121, 267, 521, 310 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)累次都是0。重极限若选择 " }, { "type": "equation_inline", "content": "y = kx" }, { "type": "text", "content": "，则得到 " }, { "type": "equation_inline", "content": "f(x, y) = \\frac{kx^2}{(k^2 + 1)x^2} = \\frac{k}{k^2 + 1}" }, { "type": "text", "content": "，重极限不存在。" } ] }, "bbox": [ 87, 309, 811, 341 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 累次为 " }, { "type": "equation_inline", "content": "\\pm 1" }, { "type": "text", "content": "。重极限考虑 " }, { "type": "equation_inline", "content": "y = kx" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "f(x, y) = \\frac{x - kx}{x + kx} = \\frac{1 - k}{1 + k}" }, { "type": "text", "content": "，不存在" } ] }, "bbox": [ 121, 341, 678, 369 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 先 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 极限为 0，先 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 不存在。重极限 " }, { "type": "equation_inline", "content": "x \\sin \\frac{1}{y} < |x| \\to 0" }, { "type": "text", "content": "，因此重极限为 0" } ] }, "bbox": [ 121, 370, 747, 388 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 累次极限不存在。而 " }, { "type": "equation_inline", "content": "|f(x, y)| \\leq |x| + |y| \\to 0" }, { "type": "text", "content": "，重极限为 0。" } ] }, "bbox": [ 121, 394, 584, 409 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 画出图像，发现沿着直线逼近都是 0，沿着抛物线 " }, { "type": "equation_inline", "content": "y = kx^{2} (0 < k < 1)" }, { "type": "text", "content": " 时趋于 1，因此极限不存在。" } ] }, "bbox": [ 121, 412, 870, 426 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 12.3 反例应用 上述反例可作为以下问题的解答:" } ] }, "bbox": [ 58, 449, 484, 463 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. 累次极限存在但不相等: " }, { "type": "equation_inline", "content": "\\frac{x - y}{x + y}" } ] }, "bbox": [ 110, 464, 366, 489 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. 累次极限存在且相等，但重极限不存在： " }, { "type": "equation_inline", "content": "\\frac{xy}{x^2 + y^2}" } ] }, "bbox": [ 107, 489, 499, 513 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3. 累次极限不存在，但重极限存在： " }, { "type": "equation_inline", "content": "x \\sin \\frac{1}{y} + y \\sin \\frac{1}{x}" } ] }, "bbox": [ 109, 513, 507, 542 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "4. 重极限与双累次极限都不存在: " }, { "type": "equation_inline", "content": "\\frac{1}{x} + \\frac{1}{y}" } ] }, "bbox": [ 109, 539, 421, 570 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "12.1.2 重极限的计算" } ], "level": 2 }, "bbox": [ 94, 602, 284, 619 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 12.4 多元分式 讨论 (0,0) 处的重极限: (1) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{xy}{x+y}" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{xy}{x^{2}+y^{2}}" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{x^{2}y^{2}}{(x^{2}+y^{2})^{2}}" }, { "type": "text", "content": " \n(4) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{x^{6}y^{8}}{(x^{2}+y^{4})^{5}}" }, { "type": "text", "content": " (5) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{x^{2}y^{2}}{x^{2}y^{2}+(x-y)^{2}}" }, { "type": "text", "content": " (6) " }, { "type": "equation_inline", "content": "f(x,y)=\\frac{x^{2}y^{2}}{x^{3}+y^{3}}" } ] }, "bbox": [ 58, 640, 902, 703 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 不齐次, 一方面将分子次数升高, " }, { "type": "equation_inline", "content": "y = x" }, { "type": "text", "content": " 得到极限为 0 。另一方面常数消去公因式, 将分母变简单, 例如 " }, { "type": "equation_inline", "content": "y = x^{2} - x" }, { "type": "text", "content": " 化简为 " }, { "type": "equation_inline", "content": "x - 1 \\rightarrow -1" }, { "type": "text", "content": ", 因此重极限不存在" } ] }, "bbox": [ 87, 701, 905, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 齐次， " }, { "type": "equation_inline", "content": "y = kx" }, { "type": "text", "content": "，不存在" } ] }, "bbox": [ 122, 738, 322, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 齐次， " }, { "type": "equation_inline", "content": "y = kx" }, { "type": "text", "content": "，不存在" } ] }, "bbox": [ 122, 757, 322, 771 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 尝试将分母变齐次， " }, { "type": "equation_inline", "content": "x = ky^{2}" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "\\frac{k^{6}y^{20}}{(k^{2} + 1)^{5}y^{20}}" }, { "type": "text", "content": " ，不存在" } ] }, "bbox": [ 121, 769, 578, 799 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(5) 尝试化简 " }, { "type": "equation_inline", "content": "x = y" }, { "type": "text", "content": " 得到 1，其他想不到办法，尝试取零， " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 得到极限为 0。不存在" } ] }, "bbox": [ 121, 800, 757, 814 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(6) 分子次数高： " }, { "type": "equation_inline", "content": "y = x" }, { "type": "text", "content": " 显然极限为 0。另一方面化简分母，设 " }, { "type": "equation_inline", "content": "y^{3} = x^{4} - x^{3}" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "\\frac{x^2(x^4x^3)^{\\frac{2}{3}}}{x^4}" }, { "type": "text", "content": "，极限不是 0。" } ] }, "bbox": [ 121, 814, 907, 843 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 12.5 指数形式 (1) " }, { "type": "equation_inline", "content": "\\lim_{(x,y)\\to(0^{+},0^{+})}x^{y}" } ] }, "bbox": [ 58, 859, 371, 881 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)用累次极限看出一个0, 一个1, 显然重极限不存在" } ] }, "bbox": [ 87, 881, 517, 896 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "12.2 多元函数连续性" } ] }, "bbox": [ 747, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "102" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "12.2 多元函数连续性" } ], "level": 2 }, "bbox": [ 95, 87, 324, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "12.2.1 多元函数连续性" } ], "level": 2 }, "bbox": [ 95, 127, 304, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 12.3 (多元函数连续)" } ], "level": 2 }, "bbox": [ 122, 159, 312, 174 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0,y_0)" }, { "type": "text", "content": " 处满足 " }, { "type": "equation_inline", "content": "\\lim_{(x,y)\\to (x_0,y_0)}f(x,y) = f(x_0,y_0)" }, { "type": "text", "content": "，则称 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0,y_0)" }, { "type": "text", "content": " 处连续。" } ] }, "bbox": [ 100, 175, 798, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "例题 12.1 关于 x, y 连续，不多元连续 举例说明 " }, { "type": "equation_inline", "content": "f(x, y)" }, { "type": "text", "content": " 关于 x, y 均连续，但不多元连续。" } ] }, "bbox": [ 89, 223, 744, 240 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = \\left\\{ \\begin{array}{l l} \\frac {x y}{x ^ {2} + y ^ {2}}, & (x, y) \\neq (0, 0) \\\\ 0, & (x, y) = (0, 0) \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/d95a8d198e98cc9d51dc4004069b7133f4c3f629631843a7e6ca79a336865665.jpg" } }, "bbox": [ 361, 255, 633, 306 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "x,y" }, { "type": "text", "content": " 均连续，但是 " }, { "type": "equation_inline", "content": "\\lim_{(x,y)\\to (0,0)}f(x,y)" }, { "type": "text", "content": " 不存在。" } ] }, "bbox": [ 89, 313, 505, 338 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 12.6 加强条件 若 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 D 中关于 x,y 连续，且增加任意条件可证明 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 连续：" } ] }, "bbox": [ 58, 351, 754, 367 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. 一致连续条件： " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 的连续是关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 一致的" } ] }, "bbox": [ 110, 370, 438, 384 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. Lip 条件： " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 关于 x 满足 Lip 条件，则 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 连续" } ] }, "bbox": [ 109, 388, 544, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3. 单调条件：关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 或 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 单调" } ] }, "bbox": [ 110, 407, 342, 420 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1)(2) 用 " }, { "type": "equation_inline", "content": "|f(x,y) - f(x_0,y_0)| \\leq |f(x,y) - f(x,y_0)| + |f(x,y_0) - f(x_0,y_0)| < 2\\epsilon" }, { "type": "text", "content": " 即可，注意第一个绝对值放缩要求 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 的连续关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 一致！" } ] }, "bbox": [ 89, 424, 904, 458 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 设关于 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 单调增。考虑一点 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": "，用 " }, { "type": "equation_inline", "content": "(x, y)" }, { "type": "text", "content": " 逼近，设 " }, { "type": "equation_inline", "content": "y_0 - \\delta_1 < y < y_0 + \\delta_1" }, { "type": "text", "content": "，首先根据 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 的单调性可知 " }, { "type": "equation_inline", "content": "f(x, y) \\leq f(x, y_0 + \\delta_1)" }, { "type": "text", "content": " 根据 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 的连续性可知 " }, { "type": "equation_inline", "content": "f(x, y_0 + \\delta_1) \\leq f(x, y_0) + \\epsilon" }, { "type": "text", "content": "。进一步地根据 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 的连续性可知" } ] }, "bbox": [ 89, 462, 905, 497 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) \\leq f (x, y _ {0}) + \\epsilon \\leq f (x _ {0}, y _ {0}) + 2 \\epsilon", "math_type": "latex", "image_source": { "path": "images/8062ab9e7ea1d82a58f7ebde66e538dd6b6d79c98f50f355ca1dd6fdfd91b7f2.jpg" } }, "bbox": [ 357, 505, 640, 520 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理可证得 " }, { "type": "equation_inline", "content": "f(x,y)\\geq f(x_0,y_0) - 2\\epsilon" }, { "type": "text", "content": " ，综上可证 " }, { "type": "equation_inline", "content": "\\left|f(x,y) - f(x_0,y_0)\\right|\\leq 2\\epsilon" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 90, 527, 630, 543 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "12.2.2 多元函数的一致连续性" } ], "level": 2 }, "bbox": [ 95, 564, 363, 580 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "引理 12.1 (多元三角不等式)" } ], "level": 2 }, "bbox": [ 122, 596, 329, 612 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对 " }, { "type": "equation_inline", "content": "\\forall \\mathbf{x},\\mathbf{y}\\in \\mathbb{R}^n" }, { "type": "text", "content": " ，有 " }, { "type": "equation_inline", "content": "||\\mathbf{x} - |\\mathbf{y}||\\leq |\\mathbf{x} - \\mathbf{y}|" } ] }, "bbox": [ 112, 615, 902, 640 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 " }, { "type": "equation_inline", "content": "|\\mathbf{x}| = |(\\mathbf{x} - \\mathbf{y}) + \\mathbf{y}| \\leq |\\mathbf{x} - \\mathbf{y}| + |\\mathbf{y}|" }, { "type": "text", "content": "，因此移项得到 " }, { "type": "equation_inline", "content": "|\\mathbf{x}| - |\\mathbf{y}| \\leq |\\mathbf{x} - \\mathbf{y}|" }, { "type": "text", "content": "。同理可得 " }, { "type": "equation_inline", "content": "|\\mathbf{y}| - |\\mathbf{x}| \\leq |\\mathbf{x} - \\mathbf{y}|" } ] }, "bbox": [ 90, 652, 862, 669 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 12.7 多元一致连续 (1) " }, { "type": "equation_inline", "content": "f(\\mathbf{x}) = |\\mathbf{x}|" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(\\mathbf{x})" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}^n" }, { "type": "text", "content": " 上一致连续 (2) " }, { "type": "equation_inline", "content": "f(x, y) = \\sin(xy)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}^2" }, { "type": "text", "content": " 的一致连续性 (3) " }, { "type": "equation_inline", "content": "f(x, y) = \\frac{1}{1 - xy}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, 1) \\times (0, 1)" }, { "type": "text", "content": " 的一致连续性\n\n证明 (1) " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\forall \\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^n" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "|\\mathbf{x} - \\mathbf{y}| < \\epsilon" }, { "type": "text", "content": "，则根据三角不等式可知 " }, { "type": "equation_inline", "content": "|f(\\mathbf{x}) - f(\\mathbf{y})| = ||\\mathbf{x}| - |\\mathbf{y}|| \\leq |\\mathbf{x} - \\mathbf{y}| < \\epsilon" }, { "type": "text", "content": "，因此取 " }, { "type": "equation_inline", "content": "\\delta = \\epsilon" }, { "type": "text", "content": " 即可。\n\n(2) 选 " }, { "type": "equation_inline", "content": "a_n = (\\sqrt{2n\\pi}, \\sqrt{2n\\pi}), b_n = (\\sqrt{2n\\pi + \\frac{\\pi}{2}}, \\sqrt{2n\\pi + \\frac{\\pi}{2}})" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "|b_n - a_n| \\to 0" }, { "type": "text", "content": "，但 " }, { "type": "equation_inline", "content": "|f(b_n) - f(a_n)| \\neq 0" }, { "type": "text", "content": "，因此不一致连续。" } ] }, "bbox": [ 60, 688, 905, 801 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 由于是在 " }, { "type": "equation_inline", "content": "(1,1)" }, { "type": "text", "content": " 处出问题，可以考虑 " }, { "type": "equation_inline", "content": "a_{n} = (1 - \\frac{1}{n}, 1 - \\frac{1}{n}), b_{n} = (1 - \\frac{1}{n + 1}, 1 - \\frac{1}{n + 1})" }, { "type": "text", "content": "，显然 " }, { "type": "equation_inline", "content": "|a_{n} - b_{n}| \\to 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "|f(a_{n}) - f(b_{n})| = \\left|\\frac{n^{2} - 3n^{2} + o(n^{2})}{4n^{2} - 1}\\right| \\to \\frac{1}{2}" }, { "type": "text", "content": "，因此不一致连续。" } ] }, "bbox": [ 89, 803, 907, 860 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "12.2 多元函数连续性" } ] }, "bbox": [ 747, 54, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "103" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 12.3 (多元 Lip 连续与一致连续)" } ], "level": 2 }, "bbox": [ 122, 87, 394, 101 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 是二元函数，若对 " }, { "type": "equation_inline", "content": "\\forall (x_1,y),(x_2,y),(x,y_1),(x,y_2)\\in \\mathbb{R}^2" }, { "type": "text", "content": " 有" } ] }, "bbox": [ 114, 108, 589, 123 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| f \\left(x _ {1}, y\\right) - f \\left(x _ {2}, y\\right) \\right| \\leq L \\left| x _ {1} - x _ {2} \\right|, \\quad \\left| f \\left(x, y _ {1}\\right) - f \\left(x, y _ {2}\\right) \\right| \\leq L \\left| y _ {1} - y _ {2} \\right|", "math_type": "latex", "image_source": { "path": "images/00d74dfbfac7f5ef17d7e1a1f66162a069edd15f6109ad9b1303ab460313f7b9.jpg" } }, "bbox": [ 243, 131, 752, 147 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "则 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbb{R}^2" }, { "type": "text", "content": " 上一致连续" } ] }, "bbox": [ 115, 153, 324, 168 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 考虑 " }, { "type": "equation_inline", "content": "\\left|f(x',y')-f(x'',y'')\\right|\\leq\\left|f(x',y')-f(x',y'')\\right|+\\left|f(x',y'')-f(x'',y'')\\right|" }, { "type": "text", "content": " ，用 Lip 条件放缩即可。" } ] }, "bbox": [ 89, 189, 853, 205 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "12.2 多元函数连续性" } ] }, "bbox": [ 749, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "104" } ] }, "bbox": [ 485, 954, 512, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 13 章 多元微分学" } ], "level": 1 }, "bbox": [ 374, 87, 640, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1 多变量导数与微分" } ], "level": 2 }, "bbox": [ 95, 145, 349, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1.1 全微分与偏导数" } ], "level": 2 }, "bbox": [ 95, 184, 302, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.1 (偏导数)" } ], "level": 2 }, "bbox": [ 121, 217, 263, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0\\in D" }, { "type": "text", "content": " 的偏导数定义为 " }, { "type": "equation_inline", "content": "\\frac{\\partial f}{\\partial x_i} := \\lim_{t\\to 0}\\frac{f(\\mathbf{x}_0 + t\\mathbf{e}_i) - f(\\mathbf{x}_0)}{t}" } ] }, "bbox": [ 112, 235, 563, 267 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 计算偏导 " }, { "type": "equation_inline", "content": "f_{x}(x_{0},y_{0})" }, { "type": "text", "content": " 时，预先将 " }, { "type": "equation_inline", "content": "y=y_{0}" }, { "type": "text", "content": " 代入，得到 " }, { "type": "equation_inline", "content": "f(x,y_{0})" }, { "type": "text", "content": " ，再对 x 求导更加方便。" } ] }, "bbox": [ 55, 284, 737, 303 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.2 (可微)" } ], "level": 2 }, "bbox": [ 121, 313, 247, 328 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "z = f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0(x_0,y_0)" }, { "type": "text", "content": " 邻域 " }, { "type": "equation_inline", "content": "U(P_0)" }, { "type": "text", "content": " 中有定义，且对于 " }, { "type": "equation_inline", "content": "U(P_0)" }, { "type": "text", "content": " 中任意点 " }, { "type": "equation_inline", "content": "(x,y)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists A, B" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 112, 332, 803, 349 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) - f \\left(x _ {0}, y _ {0}\\right) = A \\left(x - x _ {0}\\right) + B \\left(y - y _ {0}\\right) + o \\left(\\sqrt {\\left(x - x _ {0}\\right) ^ {2} + \\left(y - y _ {0}\\right) ^ {2}}\\right)", "math_type": "latex", "image_source": { "path": "images/6159eb6a17f2b4d5b116fcb03f1884089111b3d1eba82636d22b893f5b8b30b9.jpg" } }, "bbox": [ 228, 356, 766, 372 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "则称 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0" }, { "type": "text", "content": " 可微。可知这里 " }, { "type": "equation_inline", "content": "A = \\frac{\\partial f}{\\partial x}, B = \\frac{\\partial f}{\\partial y}" }, { "type": "text", "content": "，因此记 " }, { "type": "equation_inline", "content": "\\mathrm{d}f(x_0,y_0) = \\frac{\\partial f}{\\partial x}(x_0,y_0)\\mathrm{d}x + \\frac{\\partial f}{\\partial y}(x_0,y_0)\\mathrm{d}y" } ] }, "bbox": [ 112, 373, 887, 407 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记可微定义中的范数可以更改，例如 " }, { "type": "equation_inline", "content": "\\Delta z = A\\Delta x + B\\Delta y + \\alpha \\Delta x + \\beta \\Delta y" }, { "type": "text", "content": " ，在 " }, { "type": "equation_inline", "content": "(\\Delta x,\\Delta y)\\to (0,0)" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\alpha ,\\beta \\rightarrow 0" }, { "type": "text", "content": " 该条件也等价于可微。" } ] }, "bbox": [ 87, 423, 905, 458 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 13.1 (偏导与可微)" } ], "level": 2 }, "bbox": [ 122, 467, 295, 483 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 可微，则 " }, { "type": "equation_inline", "content": "f_x(x_0, y_0), f_y(x_0, y_0)" }, { "type": "text", "content": " 一定存在" } ] }, "bbox": [ 115, 487, 522, 504 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "笔记 判断是否可微分两步：" } ], "level": 2 }, "bbox": [ 89, 523, 292, 538 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. 看看 " }, { "type": "equation_inline", "content": "f_{x}, f_{y}" }, { "type": "text", "content": " 是否都存在，不存在则不可微" } ] }, "bbox": [ 110, 542, 438, 556 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. 计算 " }, { "type": "equation_inline", "content": "\\frac{f(x + \\Delta x, y + \\Delta y) - f(x, y) - f_x \\Delta x - f_y \\Delta y}{\\sqrt{(\\Delta x)^2 + (\\Delta y)^2}}" }, { "type": "text", "content": " 是否极限为 0，是则可微" } ] }, "bbox": [ 107, 556, 689, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.1计算全微分(1)求 " }, { "type": "equation_inline", "content": "u = xe^{yz} + e^{-z} + y" }, { "type": "text", "content": " 在原点处的全微分" } ] }, "bbox": [ 58, 600, 581, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)原点处 " }, { "type": "equation_inline", "content": "u_{x} = 1, u_{y} = 1, u_{z} = -1" }, { "type": "text", "content": "，因此原点处 " }, { "type": "equation_inline", "content": "\\mathrm{d}u = \\mathrm{d}x + \\mathrm{d}y - \\mathrm{d}z" } ] }, "bbox": [ 87, 618, 611, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.2判断可微性 判断 " }, { "type": "equation_inline", "content": "f(x,y) = \\left\\{ \\begin{array}{ll}\\frac{xy}{\\sqrt{x^2 + y^2}}, & (x,y)\\neq (0,0)\\\\ 0, & (x,y) = (0,0) \\end{array} \\right." }, { "type": "text", "content": " 在原点的可微性" } ] }, "bbox": [ 58, 648, 700, 699 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 首先计算偏导，计算 x 偏导时将 y 固定为 0(而不是固定为一个很小的值！)" } ] }, "bbox": [ 87, 701, 663, 715 ] }, { "type": "equation_interline", "content": { "math_content": "f _ {x} (0, 0) = \\frac {f (\\Delta x , 0) - f (0 , 0)}{\\Delta x} = 0", "math_type": "latex", "image_source": { "path": "images/5906b2709859ece98dd15b8605809d386c74885c91ef5bb513ea0c51289f32a7.jpg" } }, "bbox": [ 373, 718, 623, 747 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理可知 " }, { "type": "equation_inline", "content": "f_{y}(0,0) = 0" }, { "type": "text", "content": " ，再考虑极限" } ] }, "bbox": [ 90, 747, 351, 763 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {\\rho \\rightarrow 0} \\frac {f (\\Delta x , \\Delta y) - f (0 , 0) - 0 \\Delta x - 0 \\Delta y}{\\sqrt {\\Delta x ^ {2} + \\Delta y ^ {2}}} = \\lim _ {\\rho \\rightarrow 0} \\frac {\\Delta x \\Delta y}{\\Delta x ^ {2} + \\Delta y ^ {2}}", "math_type": "latex", "image_source": { "path": "images/b3d6dde54e9ee877e2979fa14a07590b7c254beb54736845be64b68ada2da59c.jpg" } }, "bbox": [ 282, 766, 710, 799 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "极限不存在，因此不可微。" } ] }, "bbox": [ 90, 803, 289, 818 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1.2 可微性的证明" } ], "level": 2 }, "bbox": [ 94, 840, 284, 857 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.3判断可微性(1)讨论 " }, { "type": "equation_inline", "content": "f(x,y) = \\left\\{ \\begin{array}{ll}y\\ln (x^2 +y^2), & (x,y)\\neq (0,0)\\\\ 0, & (x,y) = (0,0) \\end{array} \\right." }, { "type": "text", "content": " 在原点的可微性" } ] }, "bbox": [ 58, 866, 742, 914 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\alpha > 0" }, { "type": "text", "content": "，讨论 " }, { "type": "equation_inline", "content": "f(x, y) = |xy|^{\\alpha}" }, { "type": "text", "content": " 在原点的可微性" } ] }, "bbox": [ 122, 85, 475, 99 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\alpha > 0" }, { "type": "text", "content": "，讨论 " }, { "type": "equation_inline", "content": "f(x, y) = \\begin{cases} \\frac{|x|^{\\alpha}y}{x^2 + y^2}, & (x, y) \\neq (0, 0) \\\\ 0, & (x, y) = (0, 0) \\end{cases}" }, { "type": "text", "content": " 在原点处的可微性" } ] }, "bbox": [ 122, 99, 650, 148 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)先看看偏导数， " }, { "type": "equation_inline", "content": "f_{x}(0,0)=\\lim_{x\\to0}\\frac{f(x,0)-f(0,0)}{x}=\\frac{0}{x}=0,\\quad f_{y}(0,0)=\\lim_{y\\to0}\\frac{y\\ln(y^{2})-0}{y}=-\\infty" }, { "type": "text", "content": " ，因此不可微" } ] }, "bbox": [ 90, 149, 905, 181 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 先看看偏导数 " }, { "type": "equation_inline", "content": "f_{x}(0,0) = \\lim_{x\\to 0}\\frac{0}{x} = 0,\\quad f_{y}(0,0) = \\lim_{y\\to 0}\\frac{0}{y} = 0" }, { "type": "text", "content": "，考虑：" } ] }, "bbox": [ 121, 180, 633, 206 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {f (x , y) - f (0 , 0) - f _ {x} (0 , 0) x - f _ {y} (0 , 0) y}{\\sqrt {x ^ {2} + y ^ {2}}} = \\frac {| x y | ^ {\\alpha}}{\\sqrt {x ^ {2} + y ^ {2}}}", "math_type": "latex", "image_source": { "path": "images/ab635d16851924f21908b16159888a2d34b6abeeb60fdcd820d5ade614249c23.jpg" } }, "bbox": [ 304, 209, 692, 243 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "考察上述分式是否趋于0，用极坐标变换即可得出若 " }, { "type": "equation_inline", "content": "\\alpha >\\frac{1}{2}" }, { "type": "text", "content": " ，则可微，若 " }, { "type": "equation_inline", "content": "0 < \\alpha \\leq \\frac{1}{2}" }, { "type": "text", "content": " ，则不可微" } ] }, "bbox": [ 90, 244, 791, 273 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "f_{x}(0,0) = f_{y}(0,0) = 0" }, { "type": "text", "content": " ，而 " }, { "type": "equation_inline", "content": "\\frac{f(x,y)}{\\sqrt{x^2 + y^2}} = \\frac{|x|^{\\alpha}y}{(x^2 + y^2)^{\\frac{3}{2}}}" }, { "type": "text", "content": " ，使用极坐标变换得到 " }, { "type": "equation_inline", "content": "r^{\\alpha -2}|\\cos \\theta |^{\\alpha}\\sin \\theta" }, { "type": "text", "content": " 。因此 " }, { "type": "equation_inline", "content": "\\alpha >2" }, { "type": "text", "content": " 时可微， " }, { "type": "equation_inline", "content": "0 < \\alpha \\leq 2" }, { "type": "text", "content": " 时不可微（例如取 " }, { "type": "equation_inline", "content": "\\theta = \\frac{\\pi}{4})" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 90, 271, 907, 328 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1.3 梯度与方向导数" } ], "level": 2 }, "bbox": [ 95, 346, 302, 363 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.3 (梯度)" } ], "level": 2 }, "bbox": [ 122, 378, 245, 393 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "f: R^{n} \\rightarrow R" }, { "type": "text", "content": " ，定义梯度 " }, { "type": "equation_inline", "content": "\\nabla f := [\\frac{\\partial f}{\\partial x_{1}}, \\frac{\\partial f}{\\partial x_{2}}, \\cdots, \\frac{\\partial f}{\\partial x_{n}}]^{T}" } ] }, "bbox": [ 114, 395, 542, 428 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.4 (方向导数)" } ], "level": 2 }, "bbox": [ 122, 451, 278, 467 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "开集 " }, { "type": "equation_inline", "content": "D \\subset \\mathbb{R}^n, f: D \\to \\mathbb{R}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "\\mathbf{u} \\in \\mathbb{R}^n" }, { "type": "text", "content": " 为单位向量, " }, { "type": "equation_inline", "content": "\\mathbf{x}_0 \\in D" }, { "type": "text", "content": ", 定义 " }, { "type": "equation_inline", "content": "\\frac{\\partial f}{\\partial \\mathbf{u}} := \\lim_{t \\to 0} \\frac{f(\\mathbf{x}_0 + t\\mathbf{u}) - f(\\mathbf{x}_0)}{t}" } ] }, "bbox": [ 112, 470, 801, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 要计算方向导数，则可取方向为 " }, { "type": "equation_inline", "content": "(\\cos\\alpha,\\cos\\beta)" }, { "type": "text", "content": " ，且 " }, { "type": "equation_inline", "content": "\\cos^{2}\\alpha+\\cos^{2}\\beta=1" }, { "type": "text", "content": " 。原因在于 " }, { "type": "equation_inline", "content": "\\frac{(x-x_{0},y-y_{0})}{|(x-x_{0},y-y_{0})|}=\\frac{1}{r}(x-x_{0},y-y_{0})=(\\cos\\alpha,\\cos\\beta)" } ] }, "bbox": [ 87, 516, 904, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.4 常用反例 (1) 证明 " }, { "type": "equation_inline", "content": "f(x,y)=\\sqrt[3]{x^{2}+y^{2}}" }, { "type": "text", "content": " 在原点连续，但不存在任意方向导数和偏导数" } ] }, "bbox": [ 58, 585, 786, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 考虑方向 " }, { "type": "equation_inline", "content": "l = (\\cos \\alpha, \\cos \\beta)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "f_{l}(0,0) = \\lim_{r\\to 0^{+}}\\frac{f(r\\cos\\alpha,r\\cos\\beta) - f(0,0)}{r} = \\lim_{r\\to 0^{+}}\\frac{1}{r^{1/3}} = \\infty" } ] }, "bbox": [ 85, 601, 831, 630 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.1 (可微函数方向导数的梯度表达)" } ], "level": 2 }, "bbox": [ 122, 648, 428, 663 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0(x_0,y_0)" }, { "type": "text", "content": " 可微，则 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0" }, { "type": "text", "content": " 沿任一方向 " }, { "type": "equation_inline", "content": "l" }, { "type": "text", "content": " 的方向导数存在，且" } ] }, "bbox": [ 115, 668, 707, 684 ] }, { "type": "equation_interline", "content": { "math_content": "f _ {l} (P _ {0}) = f _ {x} (P _ {0}) \\cos \\alpha + f _ {y} (P _ {0}) \\cos \\beta = \\nabla f \\cdot \\mathbf {n}", "math_type": "latex", "image_source": { "path": "images/415470166ea193a726121048c9cd9e96dd20fc98a78c180cb10fe50c1f6ec834.jpg" } }, "bbox": [ 329, 692, 665, 707 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这里 " }, { "type": "equation_inline", "content": "\\mathbf{n} = (\\cos \\alpha, \\cos \\beta)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "l" }, { "type": "text", "content": " 方向的单位向量" } ] }, "bbox": [ 114, 715, 435, 728 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 考虑方向导数 " }, { "type": "equation_inline", "content": "f_{l}(P_{0}) = \\lim_{r\\to 0^{+}}\\frac{f(x_0 + r\\cos\\alpha, y_0 + r\\cos\\beta) - f(x_0, y_0)}{r}" }, { "type": "text", "content": "，可化简为：" } ] }, "bbox": [ 90, 743, 726, 772 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {r \\rightarrow 0 ^ {+}} \\frac {f _ {x} (x _ {0} , y _ {0}) r \\cos \\alpha + f _ {y} (x _ {0} , y _ {0}) r \\cos \\beta + o (r)}{r} = f _ {x} (x _ {0}, y _ {0}) \\cos \\alpha + f _ {y} (x _ {0}, y _ {0}) \\cos \\beta", "math_type": "latex", "image_source": { "path": "images/65b797dc9ef7cfecdac7f19022aff8781be03bf16a8aafb605509b09cde364bf.jpg" } }, "bbox": [ 191, 775, 803, 804 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记设 " }, { "type": "equation_inline", "content": "\\theta" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\nabla f" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "l" }, { "type": "text", "content": " 的夹角，则 " }, { "type": "equation_inline", "content": "f_{l}(\\mathbf{x}_{0}) = |\\nabla f||\\mathbf{n}|\\cos \\theta = |\\nabla f|\\cos \\theta" }, { "type": "text", "content": " 。可知梯度方向是增长最快的方向。" } ] }, "bbox": [ 87, 812, 840, 828 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.5 梯度与方向导数计算 (1) 计算 u = xyz 在 " }, { "type": "equation_inline", "content": "A(5,1,2)" }, { "type": "text", "content": " 朝向 " }, { "type": "equation_inline", "content": "B(9,4,14)" }, { "type": "text", "content": " 的方向导数" } ] }, "bbox": [ 58, 850, 742, 865 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "u(x,y,z)=xy^{2}z^{3}" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "u(x,y,z)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(1,1,1)" }, { "type": "text", "content": " 处函数值增长最快的方向解(1)可微函数具体计算方向导数时一般用 " }, { "type": "equation_inline", "content": "\\nabla f\\cdot \\mathbf{n}" }, { "type": "text", "content": " 计算。 " }, { "type": "equation_inline", "content": "\\overrightarrow{AB} = (4,3,12)" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "\\mathbf{n} = \\frac{1}{13} (4,3,12)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 处梯度为(2,10,5)，因此 " }, { "type": "equation_inline", "content": "\\nabla f\\cdot \\mathbf{n} = \\frac{1}{13} (8 + 30 + 60) = \\frac{98}{13}" } ] }, "bbox": [ 122, 869, 647, 884 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.1 多变量导数与微分" } ] }, "bbox": [ 731, 54, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "106" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 89, 85, 905, 131 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 即梯度方向， " }, { "type": "equation_inline", "content": "u_{x}=y^{2}z^{3},u_{y}=2xyz^{3},u_{z}=3xy^{2}z^{2}" }, { "type": "text", "content": " ，因此梯度为 " }, { "type": "equation_inline", "content": "(1,2,3)" } ] }, "bbox": [ 122, 131, 673, 147 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.6方向导数的应用已知 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 可微，若 " }, { "type": "equation_inline", "content": "l_{1},l_{2}" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\mathbb{R}^2" }, { "type": "text", "content": " 上一组线性无关的向量，若 " }, { "type": "equation_inline", "content": "f_{l_1}(x,y) = f_{l_2}(x,y)\\equiv 0" }, { "type": "text", "content": "，证明 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 为常值函数。" } ] }, "bbox": [ 60, 168, 905, 203 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 不妨设 " }, { "type": "equation_inline", "content": "n_{l_1} = (\\cos \\alpha, \\sin \\alpha), n_{l_2} = (\\cos \\beta, \\sin \\beta)" }, { "type": "text", "content": "，此时根据条件" } ] }, "bbox": [ 87, 205, 586, 221 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} f _ {l _ {1}} = f _ {x} \\cos \\alpha + f _ {y} \\sin \\alpha \\equiv 0 \\\\ f _ {l _ {2}} = f _ {x} \\cos \\beta + f _ {y} \\sin \\beta \\equiv 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/c7fd9a9b11d1706660dcf5f26f9d6f4a35208c84cdcc010c11f12c3fc14f61de.jpg" } }, "bbox": [ 378, 225, 601, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据线性无关条件可解出 " }, { "type": "equation_inline", "content": "f_{x} \\equiv f_{y} \\equiv 0" }, { "type": "text", "content": "，因此为常值函数。" } ] }, "bbox": [ 90, 274, 512, 288 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1.4 可微、偏导、方向导数、连续关系总结" } ], "level": 2 }, "bbox": [ 95, 311, 504, 328 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.2 (可微的性质)" } ], "level": 2 }, "bbox": [ 121, 344, 295, 357 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0" }, { "type": "text", "content": " 可微，则" } ] }, "bbox": [ 115, 363, 265, 378 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0" }, { "type": "text", "content": " 处连续" } ] }, "bbox": [ 139, 382, 270, 397 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0" }, { "type": "text", "content": " 处存在偏导数和任意方向导数" } ] }, "bbox": [ 139, 400, 436, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 根据定义显然 " }, { "type": "equation_inline", "content": "||\\mathbf{h}|| \\to 0" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "f(\\mathbf{x} + \\mathbf{h}) - f(\\mathbf{x}) \\to 0" }, { "type": "text", "content": "，因此连续" } ] }, "bbox": [ 89, 436, 603, 450 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.3 (可微充分条件)" } ], "level": 2 }, "bbox": [ 121, 462, 312, 476 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 偏导数 " }, { "type": "equation_inline", "content": "f_{x},f_{y}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0,y_0)" }, { "type": "text", "content": " 连续，则 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0,y_0)" }, { "type": "text", "content": " 可微" } ] }, "bbox": [ 112, 482, 608, 498 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 首先写为 " }, { "type": "equation_inline", "content": "\\Delta z = [f(x_0 + \\Delta x, y_0 + \\Delta y) - f(x_0, y_0 + \\Delta y)] + [f(x_0, y_0 + \\Delta y) - f(x_0, y_0)]" }, { "type": "text", "content": "，根据Lagrange中值定理得到" } ] }, "bbox": [ 90, 517, 905, 551 ] }, { "type": "equation_interline", "content": { "math_content": "\\Delta z = f _ {x} (x _ {0} + \\theta_ {1} \\Delta x, y _ {0} + \\Delta y) \\Delta x + f _ {y} (x _ {0}, y _ {0} + \\theta_ {2} \\Delta y) \\Delta y", "math_type": "latex", "image_source": { "path": "images/de22813bafe25e097f2617ae3735f03cd48e9710ad4317a380d17d008bf12fba.jpg" } }, "bbox": [ 289, 556, 707, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据偏导连续， " }, { "type": "equation_inline", "content": "\\exists \\alpha, \\beta" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(\\Delta x, \\Delta y) \\to (0, 0)" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "\\alpha, \\beta \\to 0" }, { "type": "text", "content": "，满足" } ] }, "bbox": [ 90, 583, 544, 598 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} f _ {x} (x _ {0} + \\theta_ {1} \\Delta x, y _ {0} + \\Delta y) = f _ {x} (x _ {0}, y _ {0}) + \\alpha \\\\ f _ {y} (x _ {0}, y _ {0} + \\Delta y) = f _ {y} (x _ {0}, y _ {0}) + \\beta \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/46b794c2ed009bb217f82398709cddfe447989f536888ffd1f47c5fc6abdaa6d.jpg" } }, "bbox": [ 332, 602, 648, 645 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到 " }, { "type": "equation_inline", "content": "\\Delta z = f_{x}(x_{0}, y_{0})\\Delta x + f_{y}(x_{0}, y_{0})\\Delta y + \\alpha\\Delta x + \\beta\\Delta y" }, { "type": "text", "content": " ，即可微。" } ] }, "bbox": [ 90, 651, 568, 667 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 若 " }, { "type": "equation_inline", "content": "f_{x}, f_{y}" }, { "type": "text", "content": " 连续，则一般称 " }, { "type": "equation_inline", "content": "f(x, y)" }, { "type": "text", "content": " 连续可微。" } ] }, "bbox": [ 90, 671, 448, 686 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.4 (偏导与方向导的性质)" } ], "level": 2 }, "bbox": [ 121, 697, 361, 712 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "偏导与方向导数相关性质：" } ] }, "bbox": [ 115, 717, 312, 731 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 偏导数在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 邻域有界，则 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 连续" } ] }, "bbox": [ 139, 736, 557, 751 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 的偏导数存在或任意方向导数存在并不能推出连续或可微" } ] }, "bbox": [ 139, 753, 594, 769 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 首先 " }, { "type": "equation_inline", "content": "f(x, y) - f(x_0, y_0) = f(x, y) - f(x_0, y) + f(x_0, y) - f(x_0, y_0)" }, { "type": "text", "content": "，根据 Lagrange 中值定理得到" } ] }, "bbox": [ 90, 789, 862, 804 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) - f \\left(x _ {0}, y\\right) = f _ {x} (\\xi , y) \\left(x - x _ {0}\\right), \\quad f \\left(x _ {0}, y\\right) - f \\left(x _ {0}, y _ {0}\\right) = f _ {y} \\left(x _ {0}, \\eta\\right) \\left(y - y _ {0}\\right)", "math_type": "latex", "image_source": { "path": "images/4cc1add5b00c491dc6efa8169669d851f44c96610395712ecdbb0fa98c47750c.jpg" } }, "bbox": [ 208, 813, 784, 828 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据偏导有界条件，因此 " }, { "type": "equation_inline", "content": "|f(x,y) - f(x_0,y_0)|\\to 0" }, { "type": "text", "content": " ，因此连续。" } ] }, "bbox": [ 90, 835, 552, 851 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 均见下面的重要反例" } ] }, "bbox": [ 122, 854, 305, 869 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.1 多变量导数与微分" } ] }, "bbox": [ 731, 55, 902, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "107" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.7 方向导数、偏导数存在，但不可微，不连续例子证明下面的 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在原点不连续，但是 " }, { "type": "equation_inline", "content": "f_{x}(0,0)=f_{y}(0,0)=0" }, { "type": "text", "content": " ，任何方向 l 满足 " }, { "type": "equation_inline", "content": "f_{l}(0,0)=0" }, { "type": "text", "content": " ，" } ] }, "bbox": [ 58, 84, 905, 118 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = \\left\\{ \\begin{array}{l l} 1, & 0 < y < x ^ {2}, - \\infty < x < \\infty \\\\ 0, & \\text {其他} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/437843484d9e6138525e7adfe2c98a92f8c25858e0159ea8d2cd40e930e9b901.jpg" } }, "bbox": [ 339, 123, 653, 167 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 不连续的原因是可以选择 0 趋近，也可以选择 1 趋近。但方向导数都为 0 的原因是 " }, { "type": "equation_inline", "content": "y = x^2" }, { "type": "text", "content": " 在原点处切线是水平的，无论原点处切线取什么，都会经过 0 区域。" } ] }, "bbox": [ 87, 172, 905, 206 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1.5 复合求导：链式法则" } ], "level": 2 }, "bbox": [ 94, 247, 342, 263 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.5 (链式法则)" } ], "level": 2 }, "bbox": [ 122, 279, 278, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "x = \\varphi(x, t), y = \\psi(s, t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(s, t) \\in D" }, { "type": "text", "content": " 可微， " }, { "type": "equation_inline", "content": "z = f(x, y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x, y) = (\\varphi(s, t), \\varphi(s, t))" }, { "type": "text", "content": " 可微，则复合函数 " }, { "type": "equation_inline", "content": "z = f(\\varphi(s, t), \\psi(s, t))" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(s, t)" }, { "type": "text", "content": " 可微，且" } ] }, "bbox": [ 112, 298, 880, 332 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\frac {\\partial z}{\\partial s} \\bigg | _ {(s, t)} = \\frac {\\partial z}{\\partial x} \\bigg | _ {(x, y)} \\frac {\\partial x}{\\partial s} \\bigg | _ {(s, t)} + \\frac {\\partial z}{\\partial y} \\bigg | _ {(x, y)} \\frac {\\partial y}{\\partial s} \\bigg | _ {(s, t)} \\\\ \\frac {\\partial z}{\\partial t} \\bigg | _ {(s, t)} = \\frac {\\partial z}{\\partial x} \\bigg | _ {(x, y)} \\frac {\\partial x}{\\partial t} \\bigg | _ {(s, t)} + \\frac {\\partial z}{\\partial y} \\bigg | _ {(x, y)} \\frac {\\partial y}{\\partial t} \\bigg | _ {(s, t)} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/6d9f12195bc525a3994eae416f9f04d7c9c90ac90054ac030b2996198f22ec29.jpg" } }, "bbox": [ 317, 335, 662, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.8 链式法则练习 (1) 设 " }, { "type": "equation_inline", "content": "z = uv + \\sin t" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "u = e^{t}, v = \\cos t" }, { "type": "text", "content": " ，计算 " }, { "type": "equation_inline", "content": "\\frac{dz}{dt}" } ] }, "bbox": [ 58, 432, 665, 461 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "x^{x}" }, { "type": "text", "content": " 的导数" } ] }, "bbox": [ 122, 458, 265, 472 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)展开后如下:" } ] }, "bbox": [ 89, 476, 231, 489 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} & {\\frac {\\mathrm{d} z}{\\mathrm{d} t} = \\frac {\\partial z}{\\partial u} \\frac {\\mathrm{d} u}{\\mathrm{d} t} + \\frac {\\partial z}{\\partial v} \\frac {\\mathrm{d} v}{\\mathrm{d} t} + \\frac {\\partial z}{\\partial t}} \\\\ & {\\qquad = v e ^ {t} + u (- \\sin t) + \\cos t = e ^ {t} (\\cos t - \\sin t) + \\cos t} \\end{array}", "math_type": "latex", "image_source": { "path": "images/42d7de21cc84dc4c63ed1779a7e5405989f94d555843ca57a043f38d5c12c3db.jpg" } }, "bbox": [ 299, 492, 697, 541 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 可以两边取对数隐函数求导。或者令 " }, { "type": "equation_inline", "content": "y = u^v" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\frac{\\mathrm{d}y}{\\mathrm{d}x} = u^v \\ln u + v(u^{v-1}) = x^x (1 + \\ln x)" } ] }, "bbox": [ 122, 548, 789, 574 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.9高阶偏导(1)设 " }, { "type": "equation_inline", "content": "z = f\\left(x,\\frac{x}{y}\\right)" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "\\frac{\\partial^2z}{\\partial x^2},\\frac{\\partial^2z}{\\partial x\\partial y}" } ] }, "bbox": [ 58, 581, 509, 612 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 把每个位置当作一层参数即可，注意偏导的参数仍保持一致" } ] }, "bbox": [ 89, 612, 552, 626 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\frac{\\partial z}{\\partial x} = f_{1}(x, \\frac{x}{y}) + f_{2}(x, \\frac{x}{y})\\frac{1}{y}" }, { "type": "text", "content": "，因此" } ] }, "bbox": [ 122, 626, 398, 653 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {\\partial^ {2} z}{\\partial x ^ {2}} = f _ {1 1} + f _ {1 2} \\frac {1}{y} + \\left[ f _ {2 1} + f _ {2 2} \\frac {1}{y} \\right] \\frac {1}{y} = f _ {1 1} + \\frac {2}{y} f _ {1 2} + \\frac {1}{y ^ {2}} f _ {2 2}", "math_type": "latex", "image_source": { "path": "images/16f2e50a5916d6191b657c92a1aefa820ad8fa6bc58fad56da12e33fe24b3cb9.jpg" } }, "bbox": [ 282, 656, 714, 689 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.1.6 混合偏导" } ], "level": 2 }, "bbox": [ 95, 707, 243, 724 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.6 (混合偏导定理)" } ], "level": 2 }, "bbox": [ 122, 740, 312, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f_{xy}(x,y), f_{yx}(x,y)" }, { "type": "text", "content": " 均在 " }, { "type": "equation_inline", "content": "x_0, y_0" }, { "type": "text", "content": " 邻域连续，则" } ] }, "bbox": [ 115, 760, 462, 775 ] }, { "type": "equation_interline", "content": { "math_content": "f _ {x y} (x _ {0}, y _ {0}) = f _ {y x} (x _ {0}, y _ {0})", "math_type": "latex", "image_source": { "path": "images/e176547d9e4f40bf34c308c6c040e3aa8d545687f500c756c6e20d4a8ec985c8.jpg" } }, "bbox": [ 406, 784, 589, 799 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 定义 " }, { "type": "equation_inline", "content": "F(\\Delta x, \\Delta y) = f(x_0 + \\Delta x, y_0 + \\Delta y) - f(x_0 + \\Delta x, y_0) - f(x_0, y_0 + \\Delta y) + f(x_0, y_0)" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 90, 819, 801, 834 ] }, { "type": "equation_interline", "content": { "math_content": "F (\\Delta x, \\Delta y) = \\left\\{ \\begin{array}{l} [ f _ {x} (x _ {0} + \\theta_ {1} \\Delta x, y _ {0} + \\Delta y) - f _ {x} (x _ {0} + \\theta_ {1} \\Delta x, y _ {0}) ] \\Delta x \\\\ [ f _ {y} (x _ {0} + \\Delta x, y _ {0} + \\theta_ {2} \\Delta y) - f _ {y} (x _ {0}, y _ {0} + \\theta_ {2} \\Delta y) ] \\Delta y \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/77df0da7d59b6dbfde3b036b4c91bcd7cda5894d65a3bc96ea9d11148b0ba8b0.jpg" } }, "bbox": [ 248, 840, 729, 884 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.1 多变量导数与微分" } ] }, "bbox": [ 731, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "108" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "再根据 Lagrange 中值定理:" } ] }, "bbox": [ 90, 85, 292, 99 ] }, { "type": "equation_interline", "content": { "math_content": "F (\\Delta x, \\Delta y) = \\left\\{ \\begin{array}{l} f _ {x y} (x _ {0} + \\theta_ {1} \\Delta x, y _ {0} + \\theta_ {3} \\Delta y) \\Delta x \\Delta y \\\\ f _ {y x} (x _ {0} + \\theta_ {4} \\Delta x, y _ {0} + \\theta_ {2} \\Delta y) \\Delta x \\Delta y \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/00a4dc448f73f3d72b3d8e0e197b9ac522bc77d840083c40e433c35e5299d540.jpg" } }, "bbox": [ 304, 104, 673, 148 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\frac{F(\\Delta x, \\Delta y)}{\\Delta x \\Delta y} = f_{xy}(x_0 + \\theta_1 \\Delta x, y_0 + \\theta_3 \\Delta y) = f_{yx}(x_0 + \\theta_4 \\Delta x, y_0 + \\theta_2 \\Delta y)" }, { "type": "text", "content": "，令 " }, { "type": "equation_inline", "content": "\\Delta x, \\Delta y \\to 0" }, { "type": "text", "content": " 即可得到结果。" } ] }, "bbox": [ 90, 152, 890, 184 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.10 混合偏导不相同的例子 对于下面的函数， " }, { "type": "equation_inline", "content": "f_{xy}(0,0) \\neq f_{yx}(0,0)" }, { "type": "text", "content": " :" } ] }, "bbox": [ 58, 197, 633, 212 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = \\left\\{ \\begin{array}{l l} x y \\frac {x ^ {2} - y ^ {2}}{x ^ {2} + y ^ {2}} & (x, y) \\neq (0, 0) \\\\ 0, & (x, y) = (0, 0) \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/7ecf2fc571649adc7e4cb0c890dee256839c617264e23bff7c94d2d9a7a5bb31.jpg" } }, "bbox": [ 354, 215, 638, 267 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 直接计算可以得到 " }, { "type": "equation_inline", "content": "f_{xy}(0,0) = -1, f_{yx}(0,0) = 1" } ] }, "bbox": [ 87, 272, 460, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.11混合偏导相关练习(1)设 " }, { "type": "equation_inline", "content": "f_{x}, f_{y}, f_{yx}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 邻域存在， " }, { "type": "equation_inline", "content": "f_{yx}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 连续，证明 " }, { "type": "equation_inline", "content": "f_{xy}(x_0, y_0)" }, { "type": "text", "content": " 也存在，且 " }, { "type": "equation_inline", "content": "f_{xy}(x_0, y_0) = f_{yx}(x_0, y_0)" }, { "type": "text", "content": " (2)设 " }, { "type": "equation_inline", "content": "f_{x}, f_{y}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 某邻域存在且在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 可微，证明： " }, { "type": "equation_inline", "content": "f_{xy}(x_0, y_0) = f_{yx}(x_0, y_0)" } ] }, "bbox": [ 58, 307, 905, 360 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.2 多元中值定理与泰勒展开" } ], "level": 2 }, "bbox": [ 95, 388, 421, 407 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.2.1 多元中值定理" } ], "level": 2 }, "bbox": [ 95, 426, 282, 443 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.5 (凸区域)" } ], "level": 2 }, "bbox": [ 122, 458, 260, 474 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "区域 " }, { "type": "equation_inline", "content": "D \\subset \\mathbb{R}^2" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\forall P_1(x_1, y_1), P_2(x_2, y_2) \\in D" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "\\forall \\lambda \\in [0, 1]" }, { "type": "text", "content": " 满足" } ] }, "bbox": [ 115, 480, 581, 495 ] }, { "type": "equation_interline", "content": { "math_content": "\\left(x _ {1} + \\lambda \\left(x _ {2} - x _ {1}\\right), y _ {1} + \\lambda \\left(y _ {2} - y _ {1}\\right)\\right) \\in D", "math_type": "latex", "image_source": { "path": "images/c2d06c9b1c7bb0c45b5db6474986eec6b530c2be2c7d53a83c452f6e547e0ccd.jpg" } }, "bbox": [ 354, 504, 642, 518 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即任意两点的连线都含于 D，则称 D 为凸区域。" } ] }, "bbox": [ 115, 527, 470, 541 ] }, { "type": "image", "content": { "image_source": { "path": "images/2512492e270f8e5160ea87da7c79597fffe8f0ac2bd4813aa99fd215e988f5ce.jpg" }, "content": "", "image_caption": [ { "type": "text", "content": "凸区域" } ], "image_footnote": [] }, "bbox": [ 324, 592, 453, 663 ] }, { "type": "image", "content": { "image_source": { "path": "images/5ee26be4f33a285c0c1af4e019ea2e8a584421a961c78fd1c940bb689030f299.jpg" }, "content": "", "image_caption": [ { "type": "text", "content": "非凸区域" }, { "type": "text", "content": "图 13.1: 凸区域" } ], "image_footnote": [] }, "bbox": [ 484, 574, 655, 661 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.7 (多元中值定理)" } ], "level": 2 }, "bbox": [ 122, 734, 312, 749 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "二元函数 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在凸开域 " }, { "type": "equation_inline", "content": "D\\subset \\mathbb{R}^2" }, { "type": "text", "content": " 连续，在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 内点均可微，则对 " }, { "type": "equation_inline", "content": "\\forall P(a,b),Q(a + h,b + k)\\in D,\\exists \\theta \\in (0,1)" }, { "type": "text", "content": " 满足" } ] }, "bbox": [ 114, 753, 880, 787 ] }, { "type": "equation_interline", "content": { "math_content": "f (a + h, b + k) - f (a, b) = f _ {x} (a + \\theta h, b + \\theta k) h + f _ {y} (a + \\theta h, b + \\theta k) k", "math_type": "latex", "image_source": { "path": "images/01d2066e290a4244825640a256d8222f95f377b62e66a7a84e0a6029f3635e79.jpg" } }, "bbox": [ 248, 791, 749, 807 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 令 " }, { "type": "equation_inline", "content": "\\Phi(t) = f(a + th, b + tk)" }, { "type": "text", "content": "，根据 Lagrange 中值定理得到 " }, { "type": "equation_inline", "content": "\\exists \\theta \\in (0,1)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\Phi(1) - \\Phi(0) = \\Phi'( \\theta)" }, { "type": "text", "content": "，而" } ] }, "bbox": [ 90, 828, 855, 843 ] }, { "type": "equation_interline", "content": { "math_content": "\\Phi^ {\\prime} (\\theta) = f _ {x} (a + \\theta h, b + \\theta k) h + f _ {y} (a + \\theta h, b + \\theta k) k", "math_type": "latex", "image_source": { "path": "images/448c525df6dee3df0d164db0578aa013f02e916794a47ca21cd904dda0dbf84a.jpg" } }, "bbox": [ 315, 851, 682, 866 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 是凸区域，因此 " }, { "type": "equation_inline", "content": "(a + \\theta h, b + \\theta k) \\in D" }, { "type": "text", "content": " ，从而结论成立。" } ] }, "bbox": [ 92, 873, 547, 889 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.2 多元中值定理与泰勒展开" } ] }, "bbox": [ 680, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "109" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.2.2 多元 Taylor 展开" } ], "level": 2 }, "bbox": [ 95, 85, 309, 103 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.8 (多元 Taylor 展开)" } ], "level": 2 }, "bbox": [ 122, 118, 334, 134 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "多元泰勒 " }, { "type": "equation_inline", "content": "f(x + \\Delta x, y + \\Delta y)" }, { "type": "text", "content": " 展开的公式如下：" } ] }, "bbox": [ 115, 139, 457, 154 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) + \\sum_ {l = 1} ^ {n} \\frac {1}{l !} \\left(\\binom {l} {0} \\frac {\\partial^ {l} f}{\\partial y ^ {l}} (x, y) \\Delta y ^ {l} + \\binom {l} {1} \\frac {\\partial^ {l} f}{\\partial x \\partial y ^ {l - 1}} (x, y) \\Delta x \\Delta y ^ {l - 1} + \\dots + \\binom {l} {l} \\frac {\\partial^ {l} f}{\\partial x ^ {l}} (x, y) \\Delta x ^ {l}\\right) + R (\\Delta x, \\Delta y)", "math_type": "latex", "image_source": { "path": "images/d0ab86cb77073f092778f657b079b03c544fee9000f8609c5208c9a3b54b0e1f.jpg" } }, "bbox": [ 114, 156, 882, 196 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "\\Phi(t)=f(x_{0}+th,y_{0}+tk)" }, { "type": "text", "content": " ，用一元 Taylor 公式证明。" } ] }, "bbox": [ 90, 215, 541, 231 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.3 隐函数定理" } ], "level": 2 }, "bbox": [ 95, 256, 277, 277 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.3.1 隐函数存在定理与可微定理" } ], "level": 2 }, "bbox": [ 94, 297, 403, 313 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "隐函数研究的问题是给定一个二维曲线表达式 " }, { "type": "equation_inline", "content": "F(x,y)=0" }, { "type": "text", "content": " ，希望研究 " }, { "type": "equation_inline", "content": "y=y(x)" }, { "type": "text", "content": " 的存在性，并计算 " }, { "type": "equation_inline", "content": "y'(x)" }, { "type": "text", "content": " ，而计算 " }, { "type": "equation_inline", "content": "y'(x)" }, { "type": "text", "content": " 其实不需要得到 " }, { "type": "equation_inline", "content": "y(x)" }, { "type": "text", "content": " 的具体表达式，因此隐函数求导大幅度简化了求导过程，因此作为多元微分学的一部分。" } ] }, "bbox": [ 89, 325, 905, 376 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.6 (隐函数)" } ], "level": 2 }, "bbox": [ 122, 386, 262, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "E \\subset \\mathbb{R}^2" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "F: E \\to \\mathbb{R}" }, { "type": "text", "content": " ，对于方程 " }, { "type": "equation_inline", "content": "F(x, y) = 0" }, { "type": "text", "content": " ，若存在集合 " }, { "type": "equation_inline", "content": "I, J \\subset \\mathbb{R}" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\forall x \\in I" }, { "type": "text", "content": " 存在唯一 " }, { "type": "equation_inline", "content": "y \\in \\mathbb{R}" }, { "type": "text", "content": " ，满足 " }, { "type": "equation_inline", "content": "F(x, y) = 0" }, { "type": "text", "content": " ，则称 " }, { "type": "equation_inline", "content": "F(x, y) = 0" }, { "type": "text", "content": " 定义了隐函数。" } ] }, "bbox": [ 114, 406, 880, 442 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.9 (隐函数存在唯一性定理)" } ], "level": 2 }, "bbox": [ 122, 464, 379, 481 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "F(x,y)" }, { "type": "text", "content": " 满足" } ], "level": 2 }, "bbox": [ 115, 486, 231, 500 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "F" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0(x_0, y_0)" }, { "type": "text", "content": " 为内点的某一区域 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 存在 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 连续偏导" } ] }, "bbox": [ 134, 504, 566, 519 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. " }, { "type": "equation_inline", "content": "F(x_0,y_0) = 0" }, { "type": "text", "content": " ，且 " }, { "type": "equation_inline", "content": "F_{y}(x_{0},y_{0})\\neq 0" }, { "type": "text", "content": " 则：" } ] }, "bbox": [ 115, 523, 401, 555 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. " }, { "type": "equation_inline", "content": "F(x,y) = 0" }, { "type": "text", "content": " 唯一定义了 " }, { "type": "equation_inline", "content": "\\forall x\\in U(x_0,\\delta)" }, { "type": "text", "content": " 上的连续函数 " }, { "type": "equation_inline", "content": "y = f(x)" } ] }, "bbox": [ 136, 558, 608, 575 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "U(x_0,\\delta)" }, { "type": "text", "content": " 上连续可微，且 " }, { "type": "equation_inline", "content": "f^{\\prime}(x) = -\\frac{F_x(x,y)}{F_y(x,y)}" }, { "type": "text", "content": " (这里 " }, { "type": "equation_inline", "content": "F_{x}" }, { "type": "text", "content": " 求导时把 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 当作常数而非函数！）" } ] }, "bbox": [ 134, 575, 847, 605 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3. 特别地，对于 " }, { "type": "equation_inline", "content": "F(x_{1},\\cdots,x_{n},y)=0,\\quad y=f(x_{1},\\cdots,x_{n})" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "f_{x_{i}}(x_{1},\\cdots,x_{n})=-\\frac{F_{x_{i}}}{F_{y}}" } ] }, "bbox": [ 134, 604, 791, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (2) " }, { "type": "equation_inline", "content": "F(x,y)=0" }, { "type": "text", "content": " 两侧对 x 求导得到 " }, { "type": "equation_inline", "content": "F_{x}(x,y)+F_{y}(x,y)y'=0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "y'=-\\frac{F_{x}(x,y)}{F_{y}(x,y)}" } ] }, "bbox": [ 90, 648, 729, 677 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 如果只希望得到 " }, { "type": "equation_inline", "content": "f(x)" }, { "type": "text", "content": " 连续，则条件可减弱为 " }, { "type": "equation_inline", "content": "F(x, y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0(x_0, y_0)" }, { "type": "text", "content": " 某邻域连续， " }, { "type": "equation_inline", "content": "F(x_0, y_0) = 0" }, { "type": "text", "content": "，固定 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "F(x, y)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 严格单调。" } ] }, "bbox": [ 90, 681, 904, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 若要求高阶导，则可以对 " }, { "type": "equation_inline", "content": "y' = -\\frac{F_{x}}{F_{y}}" }, { "type": "text", "content": " 两边求导，再将 " }, { "type": "equation_inline", "content": "y'(x)" }, { "type": "text", "content": " 表达式带进去，或者实际计算时可以 " }, { "type": "equation_inline", "content": "F(x, y) = 0" }, { "type": "text", "content": " 直接两边求两次导。" } ] }, "bbox": [ 87, 715, 905, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.12隐函数求导 " }, { "type": "equation_inline", "content": "(1)y = y(x)" }, { "type": "text", "content": " 由 " }, { "type": "equation_inline", "content": "\\ln \\sqrt{x^2 + y^2} = \\arctan \\frac{y}{x}" }, { "type": "text", "content": " 确定，求 " }, { "type": "equation_inline", "content": "\\frac{\\mathrm{dy}}{\\mathrm{dx}}" } ] }, "bbox": [ 58, 772, 645, 801 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "z = z(x,y)" }, { "type": "text", "content": " 由 " }, { "type": "equation_inline", "content": "F(x - y,y - z) = 0" }, { "type": "text", "content": " 所确定的隐函数，求 " }, { "type": "equation_inline", "content": "z_{x},z_{y},z_{xy}" } ] }, "bbox": [ 122, 797, 623, 814 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "F(x,y,z)=0" }, { "type": "text", "content": " 可以确定 " }, { "type": "equation_inline", "content": "z=z(x,y),y=y(z,x),x=x(y,z)" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\frac{\\partial x}{\\partial y}\\cdot\\frac{\\partial y}{\\partial z}\\cdot\\frac{\\partial z}{\\partial x}=-1" } ] }, "bbox": [ 122, 813, 778, 841 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)可以用隐函数求导公式，此时 " }, { "type": "equation_inline", "content": "F" }, { "type": "text", "content": " 对 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 求导时，将它们视为彼此无关的：" } ] }, "bbox": [ 89, 841, 673, 854 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} F _ {x} = \\frac {x}{x ^ {2} + y ^ {2}} - \\frac {- \\frac {y}{x ^ {2}}}{1 + \\left(\\frac {y}{x}\\right) ^ {2}} = \\frac {x + y}{x ^ {2} + y ^ {2}} \\\\ F _ {y} = \\frac {y - x}{x ^ {2} + y ^ {2}} \\end{array} \\right. \\Rightarrow \\frac {\\mathrm{d} y}{\\mathrm{d} x} = \\frac {x + y}{x - y}", "math_type": "latex", "image_source": { "path": "images/abab86a2f576b0101390efeb1fb1f37bd28a5a68a40dca27f76500610d47953c.jpg" } }, "bbox": [ 287, 858, 709, 920 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.3 隐函数定理" } ] }, "bbox": [ 781, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "110" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "或者直接两侧求导，此时视 " }, { "type": "equation_inline", "content": "y = y(x)" }, { "type": "text", "content": " 与 x 有关：" } ] }, "bbox": [ 90, 85, 443, 99 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {x}{x ^ {2} + y ^ {2}}} + {\\frac {y}{x ^ {2} + y ^ {2}}} \\cdot y ^ {\\prime} + {\\frac {y}{x ^ {2} + y ^ {2}}} + {\\frac {x}{x ^ {2} + y ^ {2}}} \\cdot y ^ {\\prime} = 0", "math_type": "latex", "image_source": { "path": "images/578040eae8e031c6e3db87c3d694362ec20a6bf09ebf66c1d7cc52268ceac1b9.jpg" } }, "bbox": [ 307, 103, 690, 131 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 比较难用隐函数求导公式，只能两边求导。视 " }, { "type": "equation_inline", "content": "z = z(x, y)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 彼此无关，则两侧分别对 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 求导得到：" } ] }, "bbox": [ 122, 134, 907, 149 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} F _ {1} + F _ {2} (- z _ {x}) = 0 \\Rightarrow z _ {x} = \\frac {F _ {1}}{F _ {2}} \\\\ - F _ {1} + F _ {2} (1 - z _ {y}) = 0 \\Rightarrow z _ {y} = 1 - \\frac {F _ {1}}{F _ {2}} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/9c89b63cbaa900df98898b2d9aa3a6761ed407eaf7c5f08c0f3c02e32d9ad7ab.jpg" } }, "bbox": [ 342, 152, 636, 206 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "要求 " }, { "type": "equation_inline", "content": "z_{xy}" }, { "type": "text", "content": "，可以对 " }, { "type": "equation_inline", "content": "F_{1}(x - y, y - z) - z_{x} F_{2}(x - y, y - z) = 0" }, { "type": "text", "content": " 两侧对 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 求导，得到" } ] }, "bbox": [ 90, 209, 682, 225 ] }, { "type": "equation_interline", "content": { "math_content": "\\left(- F _ {1 1} + F _ {1 2} \\left(1 - z _ {y}\\right)\\right) - \\left(F _ {2 1} (- 1) + F _ {2 2} \\left(1 - z _ {y}\\right)\\right) z _ {x} - F _ {2} z _ {x y} = 0 \\Rightarrow z _ {x y} = \\frac {2 F _ {1} F _ {2} F _ {1 2} - F _ {1 1} F _ {2} ^ {2} - F _ {2 2} F _ {1} ^ {2}}{F _ {2} ^ {3}}", "math_type": "latex", "image_source": { "path": "images/666c461c5e5e68f0581e30a60a9fa4a49ed051a115af5f37b82935826af696d8.jpg" } }, "bbox": [ 122, 227, 868, 259 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 注意这里的符号含义，不能简简单单乘起来变成 1。例如 " }, { "type": "equation_inline", "content": "\\frac{\\partial x}{\\partial y}" }, { "type": "text", "content": " 的含义为 " }, { "type": "equation_inline", "content": "\\frac{\\partial x(y, z)}{y}" }, { "type": "text", "content": "，因此根据隐函数定理即" } ] }, "bbox": [ 122, 262, 905, 294 ] }, { "type": "equation_interline", "content": { "math_content": "- \\frac {F _ {y}}{F _ {x}} \\cdot - \\frac {F _ {z}}{F _ {y}} \\cdot - \\frac {F _ {x}}{F _ {z}} = - 1", "math_type": "latex", "image_source": { "path": "images/afd0937ca17be6f8d5c1679ee49fd4f736474ebbd03a68ef8ccb53980cc553e8.jpg" } }, "bbox": [ 418, 294, 589, 326 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.3.2 隐函数组存在定理与微分定理" } ], "level": 2 }, "bbox": [ 95, 346, 425, 363 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.10 (隐函数组定理)" } ], "level": 2 }, "bbox": [ 122, 379, 319, 393 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "F(x,y,u,v),G(x,y,u,v)" }, { "type": "text", "content": " 满足：" } ] }, "bbox": [ 115, 399, 364, 413 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 在 " }, { "type": "equation_inline", "content": "P_0(x_0, y_0, u_0, v_0)" }, { "type": "text", "content": " 邻域中对所有变量有连续偏导" } ] }, "bbox": [ 139, 417, 524, 431 ] }, { "type": "equation_interline", "content": { "math_content": "F (P _ {0}) = G (P _ {0}) = 0, J = \\left. \\frac {\\partial (F , G)}{\\partial (u , v)} \\right| _ {P _ {0}} \\neq 0", "math_type": "latex", "image_source": { "path": "images/ecd78e867c88c08eb9a2223123ee891dead8e9c43840c1f42cdbf0fff18c95b6.jpg" } }, "bbox": [ 139, 431, 467, 463 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "则方程组" } ] }, "bbox": [ 115, 462, 188, 475 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} F (x, y, u, v) = 0 \\\\ G (x, y, u, v) = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/1c82c47fc4e156d6c9e2a111f6b761bc13d333ce85ba4270434001aa2b8b804c.jpg" } }, "bbox": [ 423, 475, 559, 519 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在 " }, { "type": "equation_inline", "content": "P_0" }, { "type": "text", "content": " 邻域中唯一确定了两个二元连续隐函数 " }, { "type": "equation_inline", "content": "u = f(x,y), v = g(x,y)" }, { "type": "text", "content": "，设 " }, { "type": "equation_inline", "content": "J = \\frac{\\partial(F,G)}{\\partial(u,v)}" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 112, 529, 781, 558 ] }, { "type": "equation_interline", "content": { "math_content": "u _ {x} = - \\frac {1}{J} \\frac {\\partial (F , G)}{\\partial (x , v)}, u _ {y} = - \\frac {1}{J} \\frac {\\partial (F , G)}{\\partial (y , v)}, v _ {x} = - \\frac {1}{J} \\frac {\\partial (F , G)}{\\partial (u , x)}, v _ {y} = - \\frac {1}{J} \\frac {\\partial (F , G)}{\\partial (u , y)}", "math_type": "latex", "image_source": { "path": "images/51dda4f57d07cfc73f1365112ed039d2618ecc8c09db4b4af470211a38e26be5.jpg" } }, "bbox": [ 231, 561, 763, 593 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "记忆口诀为求什么就换什么，例如求 " }, { "type": "equation_inline", "content": "u_{x}" }, { "type": "text", "content": " 就把 " }, { "type": "equation_inline", "content": "\\frac{\\partial(F,G)}{\\partial(u,v)}" }, { "type": "text", "content": " 中的 " }, { "type": "equation_inline", "content": "u" }, { "type": "text", "content": " 换成 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "\\frac{\\partial(F,G)}{\\partial(x,v)}" } ] }, "bbox": [ 114, 595, 727, 629 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 求导结论使用了 Cramer 法则" } ] }, "bbox": [ 89, 646, 349, 661 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 一般而言有几个方程就有几个因变量，其余的为自变量。" } ] }, "bbox": [ 55, 664, 542, 680 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.13 容易分不清的题 (1) 错过：已知 " }, { "type": "equation_inline", "content": "x = \\varphi(t), y = \\phi(x)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\varphi'(t) = e^{t} + 1, \\frac{\\mathrm{d}^{3}y}{\\mathrm{d}x^{3}} = e^{t} + 2t" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "\\frac{d^{4}y}{dx^{4}}" }, { "type": "text", "content": " 解 (1) 直接求：" } ] }, "bbox": [ 58, 677, 838, 719 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {\\mathrm{d} ^ {4} y}{\\mathrm{d} x ^ {4}}} = {\\frac {\\left({\\frac {\\mathrm{d} ^ {3} y}{\\mathrm{d} x ^ {3}}}\\right) / \\mathrm{d} t}{\\mathrm{d} x / \\mathrm{d} t}} = {\\frac {e ^ {t} + 2}{e ^ {t} + 1}}", "math_type": "latex", "image_source": { "path": "images/3ab67cd5711d91e4050aa7185c17a09f3c72450e9f3feccec970d344ae355477.jpg" } }, "bbox": [ 396, 718, 601, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.14 隐函数组求导 (1) 证明" } ] }, "bbox": [ 58, 780, 342, 795 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} x ^ {2} + y ^ {2} + u ^ {2} - v ^ {2} = 2 \\\\ x + y + u + v = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/e70853c73f23c506ee8588c4b12b5b33103274f26fb86c61f8afe12f076a7b00.jpg" } }, "bbox": [ 400, 793, 579, 839 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在 " }, { "type": "equation_inline", "content": "P_0(1,1, - 1, - 1)" }, { "type": "text", "content": " 的某邻域中能确定隐函数组 " }, { "type": "equation_inline", "content": "u = u(x,y),v = v(x,y)" }, { "type": "text", "content": " ，并求 " }, { "type": "equation_inline", "content": "\\frac{\\partial u}{\\partial x},\\frac{\\partial v}{\\partial y}" } ] }, "bbox": [ 87, 847, 712, 877 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "111" } ] }, "bbox": [ 485, 954, 510, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.3 隐函数定理" } ] }, "bbox": [ 781, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 设 " }, { "type": "equation_inline", "content": "u = u(x, y), v = v(x, y)" }, { "type": "text", "content": " 由方程组" } ] }, "bbox": [ 122, 85, 411, 99 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} u = f (u x, v + y) \\\\ v = g (u - x, v ^ {2} y) \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/52ceb7b53056a16fd9226c3e0df740b52cffc7a733f2bee329d249190f287080.jpg" } }, "bbox": [ 416, 104, 561, 149 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "给出的隐函数，求 " }, { "type": "equation_inline", "content": "u_{x}, v_{x}" } ] }, "bbox": [ 89, 153, 275, 168 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 已知如下方程组，求 " }, { "type": "equation_inline", "content": "z_{x}" }, { "type": "text", "content": " :" } ] }, "bbox": [ 122, 172, 329, 186 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} x = u + v \\\\ y = u ^ {2} + v ^ {2} \\\\ z = u ^ {3} + v ^ {3} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/e9abbc2d3f065bc5430b8be714ff383ccb3aaa030771442cc585ffe57e919f70.jpg" } }, "bbox": [ 438, 185, 542, 252 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 令 " }, { "type": "equation_inline", "content": "u = f(x - ut, y - ut, z - ut), g(x, y, z) = 0" }, { "type": "text", "content": "，求 " }, { "type": "equation_inline", "content": "u_x, u_y" } ] }, "bbox": [ 119, 262, 561, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1) " }, { "type": "equation_inline", "content": "F=x^{2}+y^{2}+u^{2}-v^{2}-2,G=x+y+u+v" }, { "type": "text", "content": " ，则显然F,G对每个分量都有连续偏导数， " }, { "type": "equation_inline", "content": "F(P_{0})=G(P_{0})=0" }, { "type": "text", "content": " ，方便起见，直接求出所有偏导：" } ] }, "bbox": [ 90, 279, 905, 312 ] }, { "type": "equation_interline", "content": { "math_content": "\\left[ \\begin{array}{c c c c} F _ {x} & F _ {y} & F _ {u} & F _ {v} \\\\ G _ {x} & G _ {y} & G _ {u} & G _ {v} \\end{array} \\right] = \\left[ \\begin{array}{c c c c} 2 x & 2 y & 2 u & - 2 v \\\\ 1 & 1 & 1 & 1 \\end{array} \\right] \\Rightarrow \\frac {\\partial (F , G)}{\\partial (u , v)} = \\left| \\begin{array}{c c} 2 u & - 2 v \\\\ 1 & 1 \\end{array} \\right| = 2 v + 2 u", "math_type": "latex", "image_source": { "path": "images/57e271d2f87b2881b54558e9de5c5016598378506f3ea72dfdafe880dec14ac8.jpg" } }, "bbox": [ 176, 315, 823, 355 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "J(P_0) \\neq 0" }, { "type": "text", "content": "，隐函数组存在，根据隐函数组求导：" } ] }, "bbox": [ 90, 359, 482, 373 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {\\partial u}{\\partial x} = - \\frac {1}{J} \\frac {\\partial (F , G)}{\\partial (x , v)} = - \\frac {1}{2 u + 2 v} \\cdot (2 x + 2 v) = - \\frac {x + v}{u + v}, \\frac {\\partial v}{\\partial y} = - \\frac {y - u}{u + v}", "math_type": "latex", "image_source": { "path": "images/73a30104a0229a7aac5299a04a6b76c3392530cee78633261fec5d2647239304.jpg" } }, "bbox": [ 248, 376, 747, 407 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 直接两侧求导，得到" } ] }, "bbox": [ 122, 411, 304, 425 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} u _ {x} = f _ {1} \\cdot (u _ {x} x + u) + f _ {2} v _ {x} \\Rightarrow (1 - x f _ {1}) u _ {x} - f _ {2} v _ {x} = u f _ {1} \\\\ v _ {x} = g _ {1} \\cdot (u _ {x} - 1) + g _ {2} 2 v v _ {x} y \\Rightarrow g _ {1} u _ {x} + (2 v y g _ {2} - 1) v _ {x} = g _ {1} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/bac7fb0c4fd2ff1c5707782d8c6290684ef84a0a23c468977979d9847cb9cd93.jpg" } }, "bbox": [ 268, 430, 710, 474 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解线性方程组得到" } ] }, "bbox": [ 90, 478, 230, 492 ] }, { "type": "equation_interline", "content": { "math_content": "u _ {x} = \\frac {(2 v y g _ {2} - 1) u f _ {1} + f _ {2} g _ {1}}{(1 - x f _ {1}) (2 v y g _ {2} - 1) + f _ {2} g _ {1}}, v _ {x} = \\frac {(1 - x f _ {1}) g _ {1} - u f _ {1} g _ {1}}{(1 - x f _ {1}) (2 v y g _ {2} - 1) + f _ {2} g _ {1}}", "math_type": "latex", "image_source": { "path": "images/012366d9b189e4bacb31cebbd9f2d35fbe0a1542917c80e252fb4ea82db2935f.jpg" } }, "bbox": [ 243, 494, 749, 525 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 令 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 为自变量， " }, { "type": "equation_inline", "content": "u, v, z" }, { "type": "text", "content": " 为因变量，则" } ] }, "bbox": [ 122, 527, 433, 542 ] }, { "type": "equation_interline", "content": { "math_content": "z _ {x} = - \\frac {\\frac {\\partial (F , G , H)}{\\partial (x , u , v)}}{\\frac {\\partial (F , G , H)}{\\partial (z , u , v)}}", "math_type": "latex", "image_source": { "path": "images/ca4cff2919a656212cd914eef1571f37ff8c6f77ef431bd947a36faeb4a42422.jpg" } }, "bbox": [ 436, 545, 557, 588 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.4 多元无条件极值问题" } ], "level": 2 }, "bbox": [ 95, 609, 373, 630 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.7 (极值与极值点)" } ], "level": 2 }, "bbox": [ 122, 651, 312, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "开集 " }, { "type": "equation_inline", "content": "D \\subset \\mathbb{R}^n" }, { "type": "text", "content": "，函数 " }, { "type": "equation_inline", "content": "f: D \\to \\mathbb{R}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x_0 \\in D" }, { "type": "text", "content": "，若存在球 " }, { "type": "equation_inline", "content": "B_r(x_0) \\subset D" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\forall x \\in B_r(x_0)" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "f(x) \\geq f(x_0)" }, { "type": "text", "content": "，则称 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 的一个极小值点。" } ] }, "bbox": [ 112, 671, 880, 706 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.8 (驻点、稳定点)" } ], "level": 2 }, "bbox": [ 122, 728, 312, 745 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "P_0(x_0,y_0)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f_{x}(x_{0},y_{0}) = f_{y}(x_{0},y_{0}) = 0" }, { "type": "text", "content": " ，则称其为 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 的驻点/稳定点。" } ] }, "bbox": [ 114, 749, 702, 766 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.4 多元无条件极值问题" } ] }, "bbox": [ 714, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "112" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.9 (Hesse 矩阵)" } ], "level": 2 }, "bbox": [ 122, 86, 290, 101 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f:\\mathbb{R}^n\\to \\mathbb{R}" }, { "type": "text", "content": " ，则其Hesse矩阵为：" } ] }, "bbox": [ 115, 107, 359, 121 ] }, { "type": "equation_interline", "content": { "math_content": "H = \\left[ \\begin{array}{c c c c} \\frac {\\partial^ {2} f}{\\partial x _ {1} ^ {2}} & \\frac {\\partial^ {2} f}{\\partial x _ {1} \\partial x _ {2}} & \\dots & \\frac {\\partial^ {2} f}{\\partial x _ {1} \\partial x _ {n}} \\\\ \\frac {\\partial^ {2} f}{\\partial x _ {2} \\partial x _ {1}} & \\frac {\\partial^ {2} f}{\\partial x _ {2} ^ {2}} & \\dots & \\frac {\\partial^ {2} f}{\\partial x _ {2} \\partial x _ {n}} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\frac {\\partial^ {2} f}{\\partial x _ {n} \\partial x _ {1}} & \\frac {\\partial^ {2} f}{\\partial x _ {n} \\partial x _ {2}} & \\dots & \\frac {\\partial^ {2} f}{\\partial x _ {n} ^ {2}} \\end{array} \\right]", "math_type": "latex", "image_source": { "path": "images/50f7ee24a7fd87c8a90ee7105e1aef06fa53b1c493d85e24b1ea2133784996bf.jpg" } }, "bbox": [ 334, 123, 658, 237 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.10 (正定、负定、不定)" } ], "level": 2 }, "bbox": [ 122, 262, 354, 277 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "矩阵 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 是对称矩阵，考虑 " }, { "type": "equation_inline", "content": "Q(x) = \\mathbf{x}^T A\\mathbf{x}" }, { "type": "text", "content": "，分为以下三种情况：" } ] }, "bbox": [ 115, 282, 573, 297 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 正(负)定: 对 " }, { "type": "equation_inline", "content": "\\forall \\mathbf{x}, Q(\\mathbf{x}) \\geq (\\leq)0" } ] }, "bbox": [ 139, 302, 391, 316 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 不定： " }, { "type": "equation_inline", "content": "\\exists \\mathbf{p}, \\mathbf{q} \\in \\mathbb{R}^n" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "Q(\\mathbf{p}) \\leq 0 \\leq Q(\\mathbf{p})" } ] }, "bbox": [ 139, 319, 436, 335 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.11 (正定的判别法)" } ], "level": 2 }, "bbox": [ 122, 360, 319, 374 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 为对称矩阵，则 " }, { "type": "equation_inline", "content": "A" }, { "type": "text", "content": " 严格正定当且仅当其各阶顺序主子式均大于0" } ] }, "bbox": [ 114, 380, 606, 395 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.4.1 无条件极值" } ], "level": 2 }, "bbox": [ 95, 432, 263, 450 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.12 (极值点必要条件)" } ], "level": 2 }, "bbox": [ 122, 464, 336, 480 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "P_0(x_0,y_0)" }, { "type": "text", "content": " 存在偏导数，且在 " }, { "type": "equation_inline", "content": "P_0" }, { "type": "text", "content": " 取到极值，则 " }, { "type": "equation_inline", "content": "f_{x}(x_{0},y_{0}) = f_{y}(x_{0},y_{0}) = 0" } ] }, "bbox": [ 114, 485, 751, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 用反证法即可。" } ] }, "bbox": [ 90, 521, 245, 536 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.13 (极值点充分条件)" } ], "level": 2 }, "bbox": [ 122, 546, 336, 561 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f:\\mathbb{R}^2\\to \\mathbb{R},(x_0,y_0)" }, { "type": "text", "content": " 为其驻点，且 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0,y_0)" }, { "type": "text", "content": " 某一邻域中有二阶连续导数，" } ] }, "bbox": [ 114, 565, 682, 582 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若Hesse矩阵 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 是严格正(负)定矩阵，则 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 的严格极小(大)值点" } ] }, "bbox": [ 139, 586, 685, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若Hesse矩阵 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 可逆但为不定方阵，则 " }, { "type": "equation_inline", "content": "\\mathbf{x}_0" }, { "type": "text", "content": " 不是 " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 的极值点" } ] }, "bbox": [ 139, 604, 598, 618 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 若 Hesse 矩阵 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 不可逆，则不能判别" } ] }, "bbox": [ 139, 623, 436, 637 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 Taylor 公式:" } ] }, "bbox": [ 90, 657, 263, 671 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = f (x _ {0}, y _ {0}) + \\frac {1}{2} \\left( \\begin{array}{c c} \\Delta x & \\Delta y \\end{array} \\right) \\left( \\begin{array}{c c} f _ {x x} (P _ {0}) & f _ {x y} (P _ {0}) \\\\ f _ {x y} (P _ {0}) & f _ {y y} (P _ {0}) \\end{array} \\right) \\binom{\\Delta x}{\\Delta y} + o ((\\Delta x) ^ {2} + (\\Delta y) ^ {2})", "math_type": "latex", "image_source": { "path": "images/f422db058761231d12228c211ca8bbe878f1e501976bdc06ef4148f635ed952a.jpg" } }, "bbox": [ 163, 675, 831, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此如果黑塞矩阵为正定，则 " }, { "type": "equation_inline", "content": "f(x,y) > f(x_0,y_0)" }, { "type": "text", "content": "，若负定则 " }, { "type": "equation_inline", "content": "f(x,y) < f(x_0,y_0)" }, { "type": "text", "content": "，若不定则不取极值，若黑塞矩阵行列式为0，则不能判别" } ] }, "bbox": [ 90, 718, 905, 751 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 13.2 (二元极值判别法)" } ], "level": 2 }, "bbox": [ 122, 760, 329, 776 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "f: \\mathbb{R}^2 \\to \\mathbb{R}" }, { "type": "text", "content": " 的驻点， " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(x_0, y_0)" }, { "type": "text", "content": " 某个邻域中有连续二阶偏导，记" } ] }, "bbox": [ 115, 781, 680, 796 ] }, { "type": "equation_interline", "content": { "math_content": "a = \\frac {\\partial^ {2} f}{\\partial x ^ {2}} (x _ {0}, y _ {0}), b = \\frac {\\partial^ {2} f}{\\partial x \\partial y} (x _ {0}, y _ {0}), c = \\frac {\\partial^ {2} f}{\\partial y ^ {2}} (x _ {0}, y _ {0})", "math_type": "latex", "image_source": { "path": "images/e14790305d8e956477defcc9da8f0fe9d2468b6c2efd5dedb2c6ef750aaba703.jpg" } }, "bbox": [ 310, 799, 685, 831 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "ac - b^{2} > 0" }, { "type": "text", "content": " 且 " }, { "type": "equation_inline", "content": "a > 0" }, { "type": "text", "content": ": 极小值" } ] }, "bbox": [ 141, 834, 376, 848 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "ac - b^{2} > 0" }, { "type": "text", "content": " 且 " }, { "type": "equation_inline", "content": "a < 0" }, { "type": "text", "content": ": 极大值" } ] }, "bbox": [ 141, 852, 376, 866 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "ac - b^{2} < 0" }, { "type": "text", "content": " ：无极值" } ] }, "bbox": [ 141, 871, 312, 885 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "ac - b^{2} = 0" }, { "type": "text", "content": " ：不确定" } ] }, "bbox": [ 141, 890, 310, 904 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.4 多元无条件极值问题" } ] }, "bbox": [ 715, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "113" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 极小值即顺序主子式大于0；极大值需要负定，而负定等价于所有元素取负号后为正定，因此注意 " }, { "type": "equation_inline", "content": "ac - b^{2} > 0" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "ac - b^{2} < 0" }, { "type": "text", "content": " 等价于不负定也不正定，而 " }, { "type": "equation_inline", "content": "ac - b^{2} = 0" }, { "type": "text", "content": " 等价于不可逆。" } ] }, "bbox": [ 87, 84, 905, 118 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.15无穷极大、无极小证明 " }, { "type": "equation_inline", "content": "f(x,y) = (1 + e^{y})\\cos x - ye^{y}" }, { "type": "text", "content": " 有无穷个极大值点，但无极小值点。证明(1)先计算驻点： " }, { "type": "equation_inline", "content": "f_{x} = -(1 + e^{y})\\sin x = 0,f_{y} = e^{y}\\cos x - e^{y} - ye^{y} = 0" }, { "type": "text", "content": " ，需要 " }, { "type": "equation_inline", "content": "\\sin x = 0,y = \\cos x - 1" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "x = n\\pi ,y = (-1)^{n} - 1" } ] }, "bbox": [ 60, 140, 905, 192 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 判断极值类型: " }, { "type": "equation_inline", "content": "f_{xx} = -(1 + e^{y})\\cos x, f_{xy} = -e^{y}\\sin x, f_{yy} = e^{y}(\\cos x - 2 - y)" }, { "type": "text", "content": ", 考虑 " }, { "type": "equation_inline", "content": "n = 2m" }, { "type": "text", "content": " 时, " }, { "type": "equation_inline", "content": "P_{m}(2m\\pi,0)" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "f_{xx}(P_{m}) = -2, f_{xy} = 0, f_{yy} = -1, f_{xx}f_{yy} - f_{xy}^{2} = 2 > 0" }, { "type": "text", "content": ", 因此为极大值点。考虑 " }, { "type": "equation_inline", "content": "n = 2m + 1, P_{m}((2m + 1)\\pi, -2), f_{xx}(P_{m}) = 1 + e^{-2} > 0, f_{xy} = 0, f_{yy} = -e^{-2}, f_{xx}f_{yy} - f_{xy}^{2} = -e^{-2}(1 + e^{-2}) < 0" }, { "type": "text", "content": " 不取极值。" } ] }, "bbox": [ 87, 195, 905, 249 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当无法直接使用 Hesse 矩阵进行判断时，可以先求出驻点，再使用配方等方法判断驻点是否最小，下面的题目便有用到" } ] }, "bbox": [ 90, 269, 905, 303 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.16计算极值点与极值(1)求 " }, { "type": "equation_inline", "content": "f(x,y) = \\frac{1}{2} x^2 +xy + \\frac{1}{2} y^2 -2x - 2y + 5" }, { "type": "text", "content": " 的全部极值点与极值" } ] }, "bbox": [ 58, 317, 811, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 求 " }, { "type": "equation_inline", "content": "f(x, y, z) = x^2 + y^2 + z^2 - xz - yz - 3x + y + 4z + 7" }, { "type": "text", "content": " 的极值点与极值解 (1) 先算驻点: " }, { "type": "equation_inline", "content": "f_x = f_y = x + y - 2" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "x + y = 2" }, { "type": "text", "content": " 的点为驻点。再判断极值性: " }, { "type": "equation_inline", "content": "f_{xx} = 1, f_{xy} = 1, f_{yy} = 1" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "f_{xx}f_{yy} - f_{xy}^2 = 0" }, { "type": "text", "content": ", 无法判断用 Hesse 矩阵判断。而 " }, { "type": "equation_inline", "content": "f(x, y) = \\frac{1}{2}(x + y)^2 - 2(x + y) + 5 = \\frac{1}{2}[(x + y) - 2]^2 + 3" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "x + y = 2" }, { "type": "text", "content": " 时不仅是极值点, 而且是最值点。" } ] }, "bbox": [ 85, 342, 905, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 先算驻点: " }, { "type": "equation_inline", "content": "f_x = 2x - z - 3, f_y = 2y - z + 1, f_z = 2z - x - y + 4" }, { "type": "text", "content": ", 解线性方程组得到 " }, { "type": "equation_inline", "content": "x = 0, y = -2, z = -3" }, { "type": "text", "content": "。先尝试用 Hesse 矩阵: " }, { "type": "equation_inline", "content": "f_{xx} = 2, f_{xy} = 0, f_{xz} = -1, f_{yy} = 2, f_{yz} = -1, f_{zz} = 2" }, { "type": "text", "content": ", 即" } ] }, "bbox": [ 90, 419, 905, 453 ] }, { "type": "equation_interline", "content": { "math_content": "H = \\left[ \\begin{array}{r r r} 2 & 0 & - 1 \\\\ 0 & 2 & - 1 \\\\ - 1 & - 1 & 2 \\end{array} \\right]", "math_type": "latex", "image_source": { "path": "images/4c40732c6a858528d00f2b2f5a470265c9c78e01c4bdecece89948fbbcf666dc.jpg" } }, "bbox": [ 406, 456, 584, 516 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "顺序主子式都大于0，因此为极小值点。" } ] }, "bbox": [ 90, 519, 384, 533 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.4.2 最值问题：极值点 + 边界" } ], "level": 2 }, "bbox": [ 94, 556, 384, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "要求某区域内函数的最值，只需要考虑区域内部的极值点以及边界即可。" } ] }, "bbox": [ 122, 583, 655, 598 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.4.3 多元极值理论证明" } ], "level": 2 }, "bbox": [ 95, 621, 324, 637 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.17 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 D 上连续且有连续偏导，若 " }, { "type": "equation_inline", "content": "f_{x} + f_{y} = f, \\quad f\\bigg|_{\\partial D} = 0" }, { "type": "text", "content": " ，证明 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 D 上恒等于 0\n\n证明 反证：若 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 不恒为 0，则 " }, { "type": "equation_inline", "content": "\\exists(x_{0},y_{0}) \\in D^{o}" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "f(x_{0},y_{0}) \\neq 0" }, { "type": "text", "content": " ，不妨设 " }, { "type": "equation_inline", "content": "f(x_{0},y_{0}) > 0" }, { "type": "text", "content": " ，设 f 在 " }, { "type": "equation_inline", "content": "(x_{0},y_{0})" }, { "type": "text", "content": " 处取正最大值，由于在区域内部，最大值一定是极值点，即 " }, { "type": "equation_inline", "content": "f_{x}(x_{0},y_{0}) = f_{x}(x_{0},y_{0}) = 0" }, { "type": "text", "content": " ，则与 " }, { "type": "equation_inline", "content": "f(x_{0},y_{0}) > 0" }, { "type": "text", "content": " 矛盾。" } ] }, "bbox": [ 58, 651, 905, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习13.18构造辅助函数设 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在单位圆盘 " }, { "type": "equation_inline", "content": "D:\\{x^{2} + y^{2}\\leq 1\\}" }, { "type": "text", "content": " 内具有连续一阶偏导，且对 " }, { "type": "equation_inline", "content": "\\forall (x,y)\\in D" }, { "type": "text", "content": " ，有 " }, { "type": "equation_inline", "content": "|f(x,y)|\\leq 1" }, { "type": "text", "content": " ，证明： " }, { "type": "equation_inline", "content": "\\exists (x_0,y_0)\\in D^o" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "f_{x}^{2}(x_{0},y_{0}) + f_{y}^{2}(x_{0},y_{0}) < 16" }, { "type": "text", "content": " 证明构造 " }, { "type": "equation_inline", "content": "g(x,y) = f(x,y) + 2(x^{2} + y^{2})" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "g(0,0) = f(0,0)\\leq 1" }, { "type": "text", "content": " ，而在边界上 " }, { "type": "equation_inline", "content": "g(x,y) = f(x,y) + 2\\geq 1" }, { "type": "text", "content": " ，因此要么 " }, { "type": "equation_inline", "content": "g(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上恒等于1，要么在 " }, { "type": "equation_inline", "content": "D^o" }, { "type": "text", "content": " 某点取极小值，总 " }, { "type": "equation_inline", "content": "\\exists (x_0,y_0)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "g_{x}(x_{0},y_{0}) = g_{y}(x_{0},y_{0}) = 0" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "f_{x}(x_{0},y_{0}) = -4x_{0},f_{y}(x_{0},y_{0}) = -4y_{0}" }, { "type": "text", "content": " ，进而 " }, { "type": "equation_inline", "content": "f_{x}^{2}(x_{0},y_{0}) + f_{y}^{2}(x_{0},y_{0})\\leq 16(x_{0}^{2} + y_{0}^{2})\\leq 16" } ] }, "bbox": [ 58, 733, 905, 825 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.5 条件极值" } ] }, "bbox": [ 798, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "114" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.5 条件极值" } ], "level": 2 }, "bbox": [ 95, 87, 253, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "13.5.1 Lagrange 乘数法" } ], "level": 2 }, "bbox": [ 95, 127, 310, 143 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下面考虑目标函数为 " }, { "type": "equation_inline", "content": "y = f(x_{1}, x_{2}, \\cdots, x_{n})" }, { "type": "text", "content": " ，条件组 " }, { "type": "equation_inline", "content": "\\varphi_{k}(x_{1}, x_{2}, \\cdots, x_{n}) = 0, k = 1, 2, \\cdots, m (m < n)" } ] }, "bbox": [ 124, 155, 850, 171 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 13.11 (Lagrange 函数)" } ], "level": 2 }, "bbox": [ 122, 181, 327, 197 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "针对上述目标函数与条件组，定义 Lagrange 函数" } ] }, "bbox": [ 115, 203, 478, 217 ] }, { "type": "equation_interline", "content": { "math_content": "L (x _ {1}, x _ {2}, \\dots , x _ {n}, \\lambda_ {1}, \\lambda_ {2}, \\dots , \\lambda_ {m}) = f (x _ {1}, x _ {2}, \\dots , x _ {n}) + \\sum_ {k = 1} ^ {m} \\lambda_ {k} \\varphi_ {k} (x _ {1}, x _ {2}, \\dots , x _ {n})", "math_type": "latex", "image_source": { "path": "images/03c9aa1c53e7756170d0d0fb02bf3e87c12727540946ee35bffb9abed2208c67.jpg" } }, "bbox": [ 205, 219, 791, 258 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 13.14 (Lagrange 乘数法)" } ], "level": 2 }, "bbox": [ 122, 282, 342, 298 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f, \\varphi_{i}" }, { "type": "text", "content": " 在 D 上有一阶连续偏导， " }, { "type": "equation_inline", "content": "P_{0}(x_{1}^{(0)}, \\cdots, x_{n}^{(0)})" }, { "type": "text", "content": " 是 f 极值点，Jacobi 矩阵 " }, { "type": "equation_inline", "content": "\\left(\\frac{\\partial \\varphi_{k}}{\\partial x_{i}}(P_{0})\\right)_{m \\times n}" }, { "type": "text", "content": " 秩为 m，则存在 " }, { "type": "equation_inline", "content": "\\lambda_{1}^{(0)}, \\lambda_{2}^{(1)}, \\cdots, \\lambda_{m}^{(0)}" }, { "type": "text", "content": " ，使得 " }, { "type": "equation_inline", "content": "(x_{1}^{(0)}, x_{2}^{(0)}, \\cdots, x_{n}^{(0)}, \\lambda_{1}^{(0)}, \\cdots, \\lambda_{m}^{(0)})" }, { "type": "text", "content": " 是 Lagrange 函数的稳定点：" } ] }, "bbox": [ 112, 303, 882, 355 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} L _ {x _ {1}} = L _ {x _ {2}} = \\dots = L _ {x _ {n}} = 0 \\\\ L _ {\\lambda_ {1}} = L _ {\\lambda_ {2}} = \\dots = L _ {\\lambda_ {m}} = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/a5d078545deb0dca5a6995fed878f435d2ee06d3d47e1a38bb37da8d22de4dcf.jpg" } }, "bbox": [ 378, 357, 601, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.19Lagrange 乘数法的直接使用 (1) 讨论 " }, { "type": "equation_inline", "content": "f(x,y,z)=xyz" }, { "type": "text", "content": " 在条件 " }, { "type": "equation_inline", "content": "x^{2}+y^{2}+z^{2}=1" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "x+y+z=0" }, { "type": "text", "content": " 条件下的最值。" } ] }, "bbox": [ 58, 441, 905, 474 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.20Lagrange 乘数法证明不等式 (1) 用 Lagrange 乘数法证明 " }, { "type": "equation_inline", "content": "\\sqrt[n]{x_1 x_2 \\cdots x_n} \\leq \\frac{x_1 + \\cdots + x_n}{n}" } ] }, "bbox": [ 58, 474, 801, 499 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)x,y>0，证明 " }, { "type": "equation_inline", "content": "\\frac{x^{n}+y^{n}}{2}\\geq\\left(\\frac{x+y}{2}\\right)^{n}" } ] }, "bbox": [ 122, 497, 418, 527 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 设 " }, { "type": "equation_inline", "content": "x_{1} + \\cdots + x_{n} = a" }, { "type": "text", "content": " ，目标即求 " }, { "type": "equation_inline", "content": "f = x_{1} \\cdots x_{n}" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x_{1} + \\cdots + x_{n} = a" }, { "type": "text", "content": " 条件下的最值。则构造 " }, { "type": "equation_inline", "content": "L = x_{1}x_{2} \\cdots x_{n} + \\lambda(x_{1} + \\cdots + x_{n} - a)" }, { "type": "text", "content": " ，解方程组：" } ] }, "bbox": [ 87, 527, 905, 562 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} L _ {x _ {1}} = \\frac {x _ {1} \\cdots x _ {n}}{x _ {1}} + \\lambda = 0 \\\\ \\vdots \\\\ L _ {x _ {n}} = \\frac {x _ {1} \\cdots x _ {n}}{x _ {n}} + \\lambda = 0 \\\\ L _ {\\lambda} = x _ {1} + x _ {2} + \\dots + x _ {n} - a = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/e7a912eb12fe49044aa09e1095df11f7f816f48cfd60374346de01965072017c.jpg" } }, "bbox": [ 361, 565, 618, 662 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解得 " }, { "type": "equation_inline", "content": "x_{1}=x_{2}=\\cdots=x_{n}=\\frac{a}{n}" }, { "type": "text", "content": " 。f 在区域边界上取到最小值，在极值点取到最大值，因此" } ] }, "bbox": [ 90, 668, 742, 693 ] }, { "type": "equation_interline", "content": { "math_content": "x _ {1} \\dots x _ {n} \\leq \\left(\\frac {a}{n}\\right) ^ {n} = \\left(\\frac {x _ {1} + \\cdots + x _ {n}}{n}\\right) ^ {n}", "math_type": "latex", "image_source": { "path": "images/d9d9641d0b1faf0e9da8ace94ee6c6ada039c6b9243ed648d662d37830451410.jpg" } }, "bbox": [ 352, 695, 643, 726 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 13.21Lagrange 乘数法的应用 (1) 抛物面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = z" }, { "type": "text", "content": " 被平面 " }, { "type": "equation_inline", "content": "x + y + z = 1" }, { "type": "text", "content": " 截成一个椭圆，求椭圆到原点的最近和最远距离" } ] }, "bbox": [ 58, 744, 905, 777 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)目标函数 " }, { "type": "equation_inline", "content": "f(x,y,z) = x^{2} + y^{2} + z^{2}" }, { "type": "text", "content": "，条件为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} - z = 0" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "x + y + z - 1 = 0" }, { "type": "text", "content": "。根据Lagrange乘数法，令" } ] }, "bbox": [ 89, 781, 905, 797 ] }, { "type": "equation_interline", "content": { "math_content": "L (x, y, z, \\lambda , \\mu) = x ^ {2} + y ^ {2} + z ^ {2} + \\lambda (x ^ {2} + y ^ {2} - z) + \\mu (x + y + z - 1)", "math_type": "latex", "image_source": { "path": "images/c6761e4a51930ae061476bbf07fe9e2a50ceb2c2deb35972a6f4836a74aeb80f.jpg" } }, "bbox": [ 253, 804, 741, 821 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.5 条件极值" } ] }, "bbox": [ 798, 54, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "115" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 求一阶偏导数，且令它们都为零，则有" } ] }, "bbox": [ 90, 85, 416, 99 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} L _ {x} = 2 x + 2 x \\lambda + \\mu = 0 \\\\ L _ {y} = 2 y + 2 y \\lambda + \\mu = 0 \\\\ L _ {z} = 2 z - \\lambda + \\mu = 0 \\\\ L _ {\\lambda} = x ^ {2} + y ^ {2} - z = 0 \\\\ L _ {\\mu} = x + y + z - 1 = 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/c98ab97b73e6d7d127d9d1fbfec24e60974cbe9e44e14d4da1d2bcd8d52ea85c.jpg" } }, "bbox": [ 396, 105, 584, 212 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "首先前两个方程有对称性，因此相减得到 " }, { "type": "equation_inline", "content": "2(x - y) + 2(x - y)\\lambda = (x - y)(1 + \\lambda) = 0" }, { "type": "text", "content": " ，要求 " }, { "type": "equation_inline", "content": "x = y" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "\\lambda = -1" }, { "type": "text", "content": " 。若 " }, { "type": "equation_inline", "content": "\\lambda = -1" }, { "type": "text", "content": " ，代入第一个式子得到 " }, { "type": "equation_inline", "content": "\\mu = 0" }, { "type": "text", "content": " ，代入第三个式子得到 " }, { "type": "equation_inline", "content": "z = -\\frac{1}{2}" }, { "type": "text", "content": " ，而第四个式子要求 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = z" }, { "type": "text", "content": " ，因此无解。若 " }, { "type": "equation_inline", "content": "x = y" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "z = 2x^{2}, z = -2x + 1" }, { "type": "text", "content": " ，解出 " }, { "type": "equation_inline", "content": "x = y = \\frac{-1 \\pm \\sqrt{3}}{2}, z = 2 \\mp \\sqrt{3}" }, { "type": "text", "content": " ，而" } ] }, "bbox": [ 90, 218, 907, 286 ] }, { "type": "equation_interline", "content": { "math_content": "f \\left(\\frac {- 1 \\pm \\sqrt {3}}{2}, \\frac {- 1 \\pm \\sqrt {3}}{2}, 2 \\mp \\sqrt {3}\\right) = 9 \\mp 5 \\sqrt {3}", "math_type": "latex", "image_source": { "path": "images/136d5bd713070d62296429aa2ad7e2b79c965f9b04d956c852693e4c96bfc9cf.jpg" } }, "bbox": [ 332, 291, 665, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "最大值为 " }, { "type": "equation_inline", "content": "9 + 5\\sqrt{3}" }, { "type": "text", "content": "，最小值为 " }, { "type": "equation_inline", "content": "9 - 5\\sqrt{3}" } ] }, "bbox": [ 90, 332, 376, 347 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "13.5 条件极值" } ] }, "bbox": [ 800, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "116" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 14 章 含参积分" } ], "level": 2 }, "bbox": [ 388, 87, 626, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.1 含参正常积分" } ], "level": 2 }, "bbox": [ 95, 145, 302, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.1.1 含参正常积分基本概念与定理" } ], "level": 2 }, "bbox": [ 94, 184, 425, 202 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 14.1 (含参正常积分)" } ], "level": 2 }, "bbox": [ 119, 216, 312, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x,t)" }, { "type": "text", "content": " 是定义于 " }, { "type": "equation_inline", "content": "G = \\{\\alpha (t)\\leq x\\leq \\beta (t),a\\leq t\\leq b\\}" }, { "type": "text", "content": " 区域的二元函数，当对 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 积分时，该积分为 " }, { "type": "equation_inline", "content": "t" }, { "type": "text", "content": " 的函数：" } ] }, "bbox": [ 114, 236, 878, 253 ] }, { "type": "equation_interline", "content": { "math_content": "F (t) = \\int_ {\\alpha (t)} ^ {\\beta (t)} f (x, t) \\mathrm{d} x, t \\in [ a, b ]", "math_type": "latex", "image_source": { "path": "images/59d21f3576806f82b4bdb7a7c0c93d12a10f5ce5fb378f23404bfaa389f35358.jpg" } }, "bbox": [ 379, 255, 616, 291 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.1 (规则区域含参正常积分性质)" } ], "level": 2 }, "bbox": [ 122, 315, 413, 331 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x,t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]\\times [c,d]" }, { "type": "text", "content": " 连续，则" } ] }, "bbox": [ 114, 335, 371, 351 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 连续性：含参积分连续，即对 " }, { "type": "equation_inline", "content": "\\forall t_0 \\in [c, d]" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\lim_{t \\to t_0} \\int_a^b f(x, t) \\mathrm{d}x = \\int_a^b f(x, t_0) \\mathrm{d}x" } ] }, "bbox": [ 137, 349, 732, 382 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 积分交换： " }, { "type": "equation_inline", "content": "\\int_{c}^{d}\\mathrm{d}t\\int_{a}^{b}f(x,t)\\mathrm{d}x = \\int_{a}^{b}\\mathrm{d}x\\int_{c}^{d}f(x,t)\\mathrm{d}x" } ] }, "bbox": [ 139, 380, 547, 413 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 可导：若 " }, { "type": "equation_inline", "content": "f_{t}(x,t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]\\times [c,d]" }, { "type": "text", "content": " 也连续，则 " }, { "type": "equation_inline", "content": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\int_{a}^{b}f(x,t)\\mathrm{d}x\\right) = \\int_{a}^{b}f_{t}(x,t)\\mathrm{d}x" } ] }, "bbox": [ 137, 412, 746, 450 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.2 (不规则区域的连续与可微性)" } ], "level": 2 }, "bbox": [ 122, 474, 413, 489 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "f(x,t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[\\alpha (t),\\beta (t)]\\times [c,d]" }, { "type": "text", "content": " 连续， " }, { "type": "equation_inline", "content": "\\alpha (t),\\beta (t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[c,d]" }, { "type": "text", "content": " 上连续，则" } ] }, "bbox": [ 115, 494, 610, 511 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 连续性： " }, { "type": "equation_inline", "content": "\\varphi (t) = \\int_{\\alpha (t)}^{\\beta (t)}f(x,t)\\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[c,d]" }, { "type": "text", "content": " 上连续，即" } ] }, "bbox": [ 137, 508, 537, 544 ] }, { "type": "equation_interline", "content": { "math_content": "\\forall t _ {0} \\in [ c, d ], \\lim _ {t \\rightarrow t _ {0}} \\varphi (t) = \\int_ {\\alpha (t _ {0})} ^ {\\beta (t _ {0})} f (x, t _ {0}) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/b13f2651b6c7eb1b385efb044f347991cfba7dacfbe6b2007d148b33ae47e506.jpg" } }, "bbox": [ 366, 546, 670, 581 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 可微性：若 " }, { "type": "equation_inline", "content": "f_{t}(x,t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]\\times [c,d]" }, { "type": "text", "content": " 上连续， " }, { "type": "equation_inline", "content": "\\alpha (t),\\beta (t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[c,d]" }, { "type": "text", "content": " 上可导，则(将被积分变量用上下限代替)" } ] }, "bbox": [ 139, 583, 882, 618 ] }, { "type": "equation_interline", "content": { "math_content": "\\varphi^ {\\prime} (t) = f (\\beta (t), t) \\beta^ {\\prime} (t) - f (\\alpha (t), t) \\alpha^ {\\prime} (t) + \\int_ {\\alpha (t)} ^ {\\beta (t)} f _ {t} (x, t) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/f354c4a607821b8b1abdb5a0703ca6be81120b7a0455263b8b5d4b3c45a7b4da.jpg" } }, "bbox": [ 307, 615, 729, 651 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 14.1 含参积分求导 (1) 重点: 已知 " }, { "type": "equation_inline", "content": "f(x) \\in C[a, b]" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "F(y) = \\int_{a}^{b} f(x) |y - x| \\mathrm{d}x, y \\in [a, b]" }, { "type": "text", "content": ", 计算 " }, { "type": "equation_inline", "content": "F''(y)" } ] }, "bbox": [ 58, 682, 853, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)首先得改写为不规则区域积分： " }, { "type": "equation_inline", "content": "F(y)=\\int_{a}^{y}f(x)(y-x)\\mathrm{d}x+\\int_{y}^{b}f(x)(x-y)\\mathrm{d}x" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 87, 713, 764, 747 ] }, { "type": "equation_interline", "content": { "math_content": "F ^ {\\prime} (y) = f (y) (y - y) \\cdot 1 + \\int_ {a} ^ {y} f (x) \\mathrm{d} x - f (y) (y - y) - \\int_ {y} ^ {b} f (x) \\mathrm{d} x = \\int_ {a} ^ {y} f (x) \\mathrm{d} x - \\int_ {y} ^ {b} f (x) \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/b3195c66c11383623a40380c9a7f75658f0445a7ea62e4d4409b8dcf4fb1ced9.jpg" } }, "bbox": [ 159, 749, 835, 785 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "进而得到 " }, { "type": "equation_inline", "content": "F''(y) = f(y) + f(y) = 2f(y)" } ] }, "bbox": [ 90, 787, 379, 803 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.2含参积分求导的应用(1)计算极限 " }, { "type": "equation_inline", "content": "\\lim_{x\\to 0}\\frac{\\int_0^{x^2}\\sqrt{1 + t^2}\\mathrm{d}t}{e^x\\sin x - \\sin x}" }, { "type": "text", "content": " 解(1)方法一：使用含参积分求导，分母等价于 " }, { "type": "equation_inline", "content": "(1 + x)x - x = x^2" }, { "type": "text", "content": " ，使用洛必达法则得到(注意分子是含参积分求导):" } ] }, "bbox": [ 58, 818, 905, 884 ] }, { "type": "equation_interline", "content": { "math_content": "\\lim _ {x \\to 0} \\frac {2 x \\sqrt {1 + x ^ {4}}}{2 x} = \\lim _ {x \\to 0} \\sqrt {1 + x ^ {4}} = 1", "math_type": "latex", "image_source": { "path": "images/c0cc2a3928b5d5ef02ff715630db5455f84b56bf3b2cd2ddbafb800ce214d276.jpg" } }, "bbox": [ 364, 883, 631, 913 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "方法二：Taylor展开 " }, { "type": "equation_inline", "content": "+" }, { "type": "text", "content": " 直接积分：分母等价于 " }, { "type": "equation_inline", "content": "x^{2}" }, { "type": "text", "content": "，分子用 " }, { "type": "equation_inline", "content": "\\sqrt{1 + t^2} = 1 + \\frac{t^2}{2} + o(t^2)" }, { "type": "text", "content": " 得到：" } ] }, "bbox": [ 124, 82, 783, 111 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\lim _ {x \\to 0} \\frac {\\int_ {0} ^ {x ^ {2}} 1 + \\frac {t ^ {2}}{2} + o (t ^ {2}) \\mathrm{d} t}{x ^ {2}} = \\lim _ {x \\to 0} \\frac {x ^ {2} + \\frac {x ^ {6}}{6} + o (x ^ {6})}{x ^ {2}} = 1", "math_type": "latex", "image_source": { "path": "images/97a664204b35c6293d74607d394582f0ac03821c4854a1166ccb0e58aad25540.jpg" } }, "bbox": [ 287, 114, 707, 149 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.1.2 常用含参正常积分的计算" } ], "level": 2 }, "bbox": [ 94, 167, 384, 185 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "引理14.1" } ], "level": 2 }, "bbox": [ 122, 200, 196, 214 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "|a| < 1" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\int \\frac{\\mathrm{d}x}{1 + a\\cos x} = \\frac{2}{\\sqrt{1 - a^2}}\\arctan \\left(\\sqrt{\\frac{1 - a}{1 + a}}\\tan \\frac{x}{2}\\right) + C," }, { "type": "text", "content": " 特别地， " }, { "type": "equation_inline", "content": "\\int_0^\\pi \\frac{\\mathrm{d}x}{1 + a\\cos x} = \\frac{\\pi}{\\sqrt{1 - x^2}}" } ] }, "bbox": [ 110, 217, 882, 256 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 使用万能公式，在前面三角函数不定积分中涉及过。" } ] }, "bbox": [ 87, 275, 515, 291 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.3积分：涉及 " }, { "type": "equation_inline", "content": "\\frac{x^b - x^a}{\\ln x}" }, { "type": "text", "content": " 型(1)计算 " }, { "type": "equation_inline", "content": "\\int_0^1\\frac{x^b - x^a}{\\ln x}\\mathrm{d}x,0 < a < b" } ] }, "bbox": [ 58, 306, 586, 338 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "\\int_0^1 \\sin \\left(\\ln \\frac{1}{x}\\right) \\frac{x^b - x^a}{\\ln x} \\, \\mathrm{d}x" } ] }, "bbox": [ 122, 337, 379, 369 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 计算 " }, { "type": "equation_inline", "content": "F(a) = \\int_{0}^{\\frac{\\pi}{2}} \\ln \\frac{1 + a \\cos x}{1 - a \\cos x} \\cdot \\frac{1}{\\cos x} \\, \\mathrm{d}x" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "a \\in (-1, 1)" } ] }, "bbox": [ 122, 370, 579, 399 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 由于 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} x^{y} \\mathrm{d}y = \\frac{x^{y}}{\\ln x}\\bigg|_{a}^{b} = \\frac{x^{b} - x^{a}}{\\ln x}" }, { "type": "text", "content": "，因此根据交换积分顺序得到" } ] }, "bbox": [ 87, 400, 605, 432 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {1} \\frac {x ^ {b} - x ^ {a}}{\\ln x} \\mathrm{d} t = \\int_ {0} ^ {1} \\mathrm{d} x \\int_ {a} ^ {b} x ^ {y} \\mathrm{d} y = \\int_ {a} ^ {b} \\mathrm{d} y \\int_ {0} ^ {1} x ^ {y} \\mathrm{d} x = \\ln \\frac {1 + b}{1 + a}", "math_type": "latex", "image_source": { "path": "images/7b60bdb23d9d8363292476b71f7e8999ebdd48cb75e132a889124767cddcf7be.jpg" } }, "bbox": [ 270, 436, 724, 470 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} x^{y} \\mathrm{d}y = \\frac{x^{b} - x^{a}}{\\ln x}" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "x = 0" }, { "type": "text", "content": " 为瑕点，考虑 " }, { "type": "equation_inline", "content": "\\lim_{x \\to 0} \\sin \\left(\\ln \\frac{1}{x}\\right) x^{y} = 0" }, { "type": "text", "content": "，因此可视为连续。从而" } ] }, "bbox": [ 122, 472, 863, 505 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {1} \\sin \\left(\\ln {\\frac {1}{x}}\\right) \\frac {x ^ {b} - x ^ {a}}{\\ln x} \\mathrm{d} x = \\int_ {a} ^ {b} \\mathrm{d} y \\int_ {0} ^ {1} \\sin \\left(\\ln {\\frac {1}{x}}\\right) x ^ {y} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/2ff2c9f213a57aaa1c6c33c702dda54120b6830d277e96462a09148de406437d.jpg" } }, "bbox": [ 294, 508, 702, 541 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "令 " }, { "type": "equation_inline", "content": "u = \\ln \\frac{1}{x}" }, { "type": "text", "content": " 即可转化为 " }, { "type": "equation_inline", "content": "\\int_{a}^{b}\\mathrm{d}y\\int_{0}^{+\\infty}\\sin ue^{-(y + 1)u}\\mathrm{d}u" }, { "type": "text", "content": " ，结果为 " }, { "type": "equation_inline", "content": "\\arctan (b + 1) - \\arctan (a + 1)" } ] }, "bbox": [ 90, 544, 761, 577 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 注意到" } ] }, "bbox": [ 124, 575, 203, 589 ] }, { "type": "equation_interline", "content": { "math_content": "\\ln \\frac {1 + a \\cos x}{1 - a \\cos x} \\cdot \\frac {1}{\\cos x} \\mathrm{d} x = \\frac {\\ln (1 + a \\cos x) - \\ln (1 - a \\cos x)}{\\cos x} = \\int_ {- a} ^ {a} \\frac {\\mathrm{d} y}{1 + y \\cos x}", "math_type": "latex", "image_source": { "path": "images/ff6c01a15710c0a416167a65a7791485c7c6e6137bc76932e15b4aece6727979.jpg" } }, "bbox": [ 218, 590, 774, 621 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "\\frac{y}{1 + y\\cos x}" }, { "type": "text", "content": " 连续，因此" } ] }, "bbox": [ 90, 627, 285, 653 ] }, { "type": "equation_interline", "content": { "math_content": "F (a) = \\int_ {- a} ^ {a} \\mathrm{d} y \\int_ {0} ^ {\\frac {\\pi}{2}} \\frac {1}{1 + y \\cos x} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/719ad294d0d2066d69ea642814fa7fa44ffa81640c4f933ae84ba7e6948048a8.jpg" } }, "bbox": [ 374, 656, 621, 689 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "最终积分结果为 " }, { "type": "equation_inline", "content": "\\pi \\arcsin a" } ] }, "bbox": [ 90, 693, 287, 707 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "14.2 含参反常积分与一致收敛" } ] }, "bbox": [ 680, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "118" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.2 含参反常积分与一致收敛" } ], "level": 2 }, "bbox": [ 95, 87, 423, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.2.1 基本概念" } ], "level": 2 }, "bbox": [ 95, 127, 243, 143 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 14.2 (含参反常积分)" } ], "level": 2 }, "bbox": [ 122, 159, 312, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x,t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[c, +\\infty) \\times I" }, { "type": "text", "content": " 上有定义，对 " }, { "type": "equation_inline", "content": "\\forall t \\in I" }, { "type": "text", "content": " ，反常积分 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} f(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则记 " }, { "type": "equation_inline", "content": "\\Phi(t) = \\int_{c}^{+\\infty} f(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 称其为含参反常积分。" } ] }, "bbox": [ 112, 177, 894, 228 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 14.3 (含参反常积分一致收敛)" } ], "level": 2 }, "bbox": [ 121, 248, 379, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若对 " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists N > c, \\forall M > N, \\forall t \\in I" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|\\int_{M}^{+\\infty} f(x,t) \\mathrm{d}x| < \\epsilon" }, { "type": "text", "content": "，则称 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} f(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 一致收敛于 " }, { "type": "equation_inline", "content": "\\Phi(t)" } ] }, "bbox": [ 114, 267, 821, 305 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.3 (一致收敛基本充要条件)" } ], "level": 2 }, "bbox": [ 121, 326, 379, 341 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "含参反常积分一致收敛有以下两个基本充要条件：" } ] }, "bbox": [ 114, 346, 480, 360 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- Cauchy 收敛准则： " }, { "type": "equation_inline", "content": "\\forall \\epsilon, \\exists M > c, \\forall A_1, A_2 > M, \\forall t \\in I" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|\\int_{A_1}^{A_2} f(x, t) \\mathrm{d}x| < \\epsilon" } ] }, "bbox": [ 137, 360, 722, 392 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 上确界判别法： " }, { "type": "equation_inline", "content": "\\lim_{A\\to +\\infty}\\sup_{t\\in I}|\\int_{A}^{+\\infty}f(x,t)\\mathrm{d}x| = 0" }, { "type": "text", "content": "。" } ] }, "bbox": [ 139, 391, 519, 424 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "笔记 证明不一致收敛一般用上确界判别法，具体见下面的例子" } ], "level": 2 }, "bbox": [ 87, 443, 552, 458 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 14.4 重要例子：确界判别 证明 " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\sin xy}{y}dy" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[\\delta,+\\infty)" }, { "type": "text", "content": " 上一致收敛，但在 " }, { "type": "equation_inline", "content": "(0,+\\infty)" }, { "type": "text", "content": " 上不一致收敛\n\n证明 (1) 一致收敛：做变量替换 u=xy 得到 " }, { "type": "equation_inline", "content": "\\int_{A}^{+\\infty}\\frac{\\sin xy}{y}dy=\\int_{Ax}^{+\\infty}\\frac{\\sin u}{u}du" }, { "type": "text", "content": " ，根据反常积分 Dirichlet 判别法可知 " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\sin u}{u}du" }, { "type": "text", "content": " 收敛，因此 " }, { "type": "equation_inline", "content": "\\forall\\epsilon,\\exists M,\\forall A'>M" }, { "type": "text", "content": " 时，有" } ] }, "bbox": [ 58, 473, 905, 564 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {A ^ {\\prime}} ^ {+ \\infty} \\frac {\\sin u}{u} \\mathrm{d} u \\right| < \\epsilon", "math_type": "latex", "image_source": { "path": "images/08107167afcfcb723e80ae111b570defae899828692b3f444ff2916244b7f4df.jpg" } }, "bbox": [ 425, 567, 571, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对题中的 " }, { "type": "equation_inline", "content": "\\delta" }, { "type": "text", "content": " ，取 " }, { "type": "equation_inline", "content": "A > \\frac{M}{\\delta}" }, { "type": "text", "content": " ，当 " }, { "type": "equation_inline", "content": "x \\geq \\delta" }, { "type": "text", "content": " 时有" } ] }, "bbox": [ 90, 600, 386, 630 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {A} ^ {+ \\infty} \\frac {\\sin x y}{y} \\mathrm{d} y \\right| < \\epsilon", "math_type": "latex", "image_source": { "path": "images/b090b7f1a4325d5df7d16f37ef6ab1fcaa5eeb5cefa032cca5455dd2ffd4e33e.jpg" } }, "bbox": [ 420, 627, 576, 661 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此根据确界判别法可知一致收敛。" } ] }, "bbox": [ 90, 668, 356, 682 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 不一致收敛：因为" } ] }, "bbox": [ 122, 686, 287, 701 ] }, { "type": "equation_interline", "content": { "math_content": "\\sup _ {x \\in (0, + \\infty)} \\left| \\int_ {A} ^ {+ \\infty} \\frac {\\sin x y}{y} \\mathrm{d} y \\right| = \\sup _ {x \\in (0, + \\infty)} \\left| \\int_ {A x} ^ {+ \\infty} \\frac {\\sin u}{u} \\mathrm{d} u \\right| \\geq \\left| \\int_ {0} ^ {+ \\infty} \\frac {\\sin u}{u} \\mathrm{d} u \\right| = \\frac {\\pi}{2}", "math_type": "latex", "image_source": { "path": "images/b9dd62ec894a0269e080bb4b4abba474788a37ab4212084b605077961804656d.jpg" } }, "bbox": [ 218, 703, 776, 738 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当 " }, { "type": "equation_inline", "content": "A \\to +\\infty" }, { "type": "text", "content": " 时，极限不趋于0，因此根据上确界准则可知不一致收敛。" } ] }, "bbox": [ 90, 741, 613, 756 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 本题在 " }, { "type": "equation_inline", "content": "[\\delta, +\\infty)" }, { "type": "text", "content": " 上也可以用 Dirichlet 判别法，在 " }, { "type": "equation_inline", "content": "(0, +\\infty)" }, { "type": "text", "content": " 无法用的原因在于 " }, { "type": "equation_inline", "content": "\\int_0^N \\sin xy \\, \\mathrm{d}y = \\left. \\frac{\\cos xy}{x} \\right|_0^N" }, { "type": "text", "content": " 对 " }, { "type": "equation_inline", "content": "x \\in [0, +\\infty)" }, { "type": "text", "content": " 不一致有界。" } ] }, "bbox": [ 65, 753, 905, 803 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.5不一致收敛例子(1)讨论 " }, { "type": "equation_inline", "content": "\\varphi (x) = \\int_0^{+\\infty}xe^{-xy}\\mathrm{d}y" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, + \\infty)" }, { "type": "text", "content": " 上的一致收敛性解 " }, { "type": "equation_inline", "content": "(1)\\varphi (x) = \\int_0^{+\\infty}e^{-xy}d(xy) = e^{-xy}|_{+\\infty}^0 = 1" }, { "type": "text", "content": " ，但是其不一致收敛，原因是 " }, { "type": "equation_inline", "content": "\\sup_{x\\in (0, + \\infty)}|\\int_M^{+\\infty}xe^{-xy}\\mathrm{d}y| = \\sup_{x\\in (0, + \\infty)}e^{-Mx} = 1" }, { "type": "text", "content": " 不趋于0。(如果 " }, { "type": "equation_inline", "content": "x\\in (a, + \\infty),a > 0" }, { "type": "text", "content": " 则一致收敛)" } ] }, "bbox": [ 58, 814, 951, 894 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "14.2 含参反常积分与一致收敛" } ] }, "bbox": [ 680, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "119" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.4 (M 判别法)" } ], "level": 2 }, "bbox": [ 122, 86, 282, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "g(x)" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "|f(x,t)| \\leq g(x), x \\in [c, +\\infty), t \\in I" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} g(x) \\mathrm{d}x" }, { "type": "text", "content": " 收敛，则 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} f(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛。" } ] }, "bbox": [ 112, 104, 890, 151 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.5 (比较判别法)" } ], "level": 2 }, "bbox": [ 122, 173, 295, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x,t)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[c, +\\infty) \\times I" }, { "type": "text", "content": " 上连续， " }, { "type": "equation_inline", "content": "|f(x,t)| \\leq F(x,t)" }, { "type": "text", "content": "，若 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} F(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛，则 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} f(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上绝对一致收敛" } ] }, "bbox": [ 110, 191, 892, 240 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据 Cauchy 收敛准则可知 " }, { "type": "equation_inline", "content": "\\forall\\epsilon,\\exists M,\\forall A_{1},A_{2}>M,\\forall t,|\\int_{A_{1}}^{A_{2}}F(x,t)\\mathrm{d}x|<\\epsilon" }, { "type": "text", "content": " ，因此推出 " }, { "type": "equation_inline", "content": "\\int_{A_{1}}^{A_{2}}|f(x,t)|\\mathrm{d}x<\\epsilon" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 87, 249, 900, 285 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.6M判别法(1)证明 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\cos xy}{1 + x^2}\\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(- \\infty , + \\infty)" }, { "type": "text", "content": " 上一致收敛" } ] }, "bbox": [ 58, 288, 610, 319 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 证明 " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty} e^{-x^2 y} \\, \\mathrm{d}y" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x \\in [a, b]" }, { "type": "text", "content": " 一致收敛" } ] }, "bbox": [ 121, 318, 453, 350 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 对任意 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\left|\\frac{\\cos xy}{1 + x^2}\\right| \\leq \\frac{1}{1 + x^2}" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty} \\frac{\\mathrm{d}x}{1 + x^2}" }, { "type": "text", "content": " 收敛，因此由 " }, { "type": "equation_inline", "content": "M" }, { "type": "text", "content": " 判别法可知一致收敛。" } ] }, "bbox": [ 87, 348, 805, 380 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于 " }, { "type": "equation_inline", "content": "e^{-x^2y} \\leq e^{-a^2y}" }, { "type": "text", "content": ", 且根据比较原则 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty} e^{-a^2y} \\mathrm{d}y" }, { "type": "text", "content": " 收敛" } ] }, "bbox": [ 121, 382, 573, 412 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 即使含参反常积分是常数，也不代表其一致收敛。" } ] }, "bbox": [ 87, 414, 492, 430 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.2.2 Dirichlet 与 Abel 判别法" } ], "level": 2 }, "bbox": [ 95, 451, 373, 469 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.6 (Dirichlet 与 Abel 判别法)" } ], "level": 2 }, "bbox": [ 122, 485, 388, 499 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty}f(x,t)g(x,t)\\mathrm{d}x" }, { "type": "text", "content": " ，满足下面的条件则一致收敛- Dirichlet 判别法: (1) " }, { "type": "equation_inline", "content": "\\forall N > c, \\int_{c}^{N} f(x, t) \\mathrm{d}x" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "t" }, { "type": "text", "content": " 一致有界 (2) 固定 " }, { "type": "equation_inline", "content": "t \\in I" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "g(x, t)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 单调, 且 " }, { "type": "equation_inline", "content": "x \\to \\infty" }, { "type": "text", "content": " 时关于 " }, { "type": "equation_inline", "content": "t" }, { "type": "text", "content": " 一致收敛 (即与 " }, { "type": "equation_inline", "content": "t" }, { "type": "text", "content": " 的选取无关) 到 0" } ] }, "bbox": [ 114, 505, 529, 537 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 139, 536, 880, 580 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- Abel 判别法: (1) " }, { "type": "equation_inline", "content": "\\int_{c}^{+\\infty} f(x,t) \\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛 (2) 固定 " }, { "type": "equation_inline", "content": "t \\in I" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "g(x,t)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 单调, " }, { "type": "equation_inline", "content": "g(x,t)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "t \\in I" }, { "type": "text", "content": " 一致有界" } ] }, "bbox": [ 139, 583, 889, 626 ] }, { "type": "image", "content": { "image_source": { "path": "images/3f126d83b6ea9697d9f0e348d015e579832e02fcc703d48ac76fdd9deaed2701.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 637, 78, 656 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 注意这里单调都是关于 x 的，一致都是关于 t 的。" } ] }, "bbox": [ 87, 642, 490, 657 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.7Dirichlet判别法(1)证明 " }, { "type": "equation_inline", "content": "\\int_1^{+\\infty}\\frac{\\sin x}{1 + xe^y}\\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, + \\infty)" }, { "type": "text", "content": " 一致收敛" } ] }, "bbox": [ 58, 670, 623, 703 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 证明 " }, { "type": "equation_inline", "content": "\\int_{1}^{+\\infty} \\frac{y \\sin xy}{1 + y^2} \\, \\mathrm{d}y" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "(0, +\\infty)" }, { "type": "text", "content": " 上内闭一致收敛" } ] }, "bbox": [ 122, 703, 517, 733 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "\\left|\\int_{1}^{N}\\sin x\\mathrm{d}x\\right|" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 一致有界。 " }, { "type": "equation_inline", "content": "\\frac{1}{1 + xe^y}" }, { "type": "text", "content": "，一旦固定 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": "，肯定关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 单调。由于 " }, { "type": "equation_inline", "content": "0 < \\frac{1}{1 + xe^y} \\leq \\frac{1}{1 + x} \\to 0" }, { "type": "text", "content": "，因此一致收敛到0。由Dirichlet判别法可知。" } ] }, "bbox": [ 89, 733, 905, 783 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 对于 " }, { "type": "equation_inline", "content": "[a, b] \\subset (0, +\\infty)" }, { "type": "text", "content": "，对 " }, { "type": "equation_inline", "content": "\\forall x \\in [a, b]" }, { "type": "text", "content": "，有" } ] }, "bbox": [ 122, 787, 448, 802 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {1} ^ {N} \\sin x y \\mathrm{d} y \\right| = \\left| - \\frac {\\cos x y}{x} \\right| _ {1} ^ {N} \\leq \\frac {2}{a}", "math_type": "latex", "image_source": { "path": "images/aa3b14f4de5e7c0b3d7238b0332ef69d6fa10af568f7fbbc3af6065d4695d590.jpg" } }, "bbox": [ 371, 804, 623, 843 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "\\left|\\frac{y}{1 + y^2}\\right| \\leq \\frac{1}{2}" }, { "type": "text", "content": " 一致有界，且" } ] }, "bbox": [ 90, 846, 317, 878 ] }, { "type": "equation_interline", "content": { "math_content": "\\left(\\frac {y}{1 + y ^ {2}}\\right) ^ {\\prime} = \\frac {1 - y ^ {2}}{(1 + y ^ {2}) ^ {2}} \\leq 0", "math_type": "latex", "image_source": { "path": "images/006a9b5a3aca536216ec145149335c15f72973a7b47d3dfbd9fa67f9d88c9fde.jpg" } }, "bbox": [ 394, 875, 603, 910 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "14.2 含参反常积分与一致收敛" } ] }, "bbox": [ 680, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "120" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "\\frac{y}{1 + y^2}" }, { "type": "text", "content": " 单调减， " }, { "type": "equation_inline", "content": "y \\to +\\infty" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\frac{y}{1 + y^2} \\to 0" }, { "type": "text", "content": "，根据Dirichlet判别法可知" } ] }, "bbox": [ 90, 80, 642, 110 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.8Abel判别法(1)证明 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-xy}\\frac{\\sin x}{x}\\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[0, + \\infty)" }, { "type": "text", "content": " 上一致收敛" } ] }, "bbox": [ 60, 114, 625, 146 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\varphi(y) = \\int_{0}^{+\\infty}\\frac{\\sin x^2}{1 + xy}\\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "y\\in [0, + \\infty)" }, { "type": "text", "content": " 一致收敛" } ] }, "bbox": [ 122, 143, 500, 175 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "\\int_{0}^{+\\infty}\\frac{\\sin x}{x}dx" }, { "type": "text", "content": " 关于 y 一致收敛。 " }, { "type": "equation_inline", "content": "e^{-xy}" }, { "type": "text", "content": " 关于 x 单调，且 " }, { "type": "equation_inline", "content": "0 0" }, { "type": "text", "content": " ，则含参积分 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-px}\\frac{\\sin bx - \\sin ax}{x}\\mathrm{d}x = \\arctan \\frac{b}{p} -\\arctan \\frac{a}{p}" } ] }, "bbox": [ 114, 760, 655, 793 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 首先 " }, { "type": "equation_inline", "content": "\\frac{\\sin bx - \\sin ax}{x} = \\int_{a}^{b}\\cos xy\\mathrm{d}y" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 90, 804, 443, 838 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {+ \\infty} \\mathrm{d} x \\int_ {a} ^ {b} e ^ {- p x} \\cos x y \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/dca7ed3a7a7750a9e6a58f93a291dc9a69ebee9a587d9c706bca6e27e645907d.jpg" } }, "bbox": [ 384, 840, 613, 873 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 " }, { "type": "equation_inline", "content": "|e^{-px}\\cos xy|\\leq e^{-px}" }, { "type": "text", "content": " ，显然 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-px}\\mathrm{d}x" }, { "type": "text", "content": " 绝对收敛，因此 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}e^{-px}\\cos xy\\mathrm{d}x,y\\in [a,b]" }, { "type": "text", "content": " 一致收敛，交换积" } ] }, "bbox": [ 90, 875, 905, 910 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "121" } ] }, "bbox": [ 485, 953, 510, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "14.3 含参反常积分的性质" } ] }, "bbox": [ 714, 55, 904, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "分顺序得到" } ] }, "bbox": [ 90, 85, 179, 99 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {a} ^ {b} \\mathrm{d} y \\int_ {0} ^ {+ \\infty} e ^ {- p x} \\cos x y \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/27fada42d2cb856c06cdea64ca3cf5ec7eafb385465ec231b751b7d24fe6631a.jpg" } }, "bbox": [ 384, 101, 613, 135 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据表格法得到:" } ] }, "bbox": [ 90, 137, 220, 151 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} e ^ {- p x} \\cos x y \\mathrm{d} x = - \\frac {1}{p} e ^ {- p x} \\cos x y \\bigg | _ {0} ^ {+ \\infty} + \\frac {y}{p ^ {2}} e ^ {- p x} \\sin x y \\bigg | _ {0} ^ {+ \\infty} - \\int_ {0} ^ {+ \\infty} \\frac {y ^ {2}}{p ^ {2}} e ^ {- p x} \\cos x y \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/6a8236a77d15ad3ad18b9c47f19ac2f7d35a2908913e121092747c77a2e60748.jpg" } }, "bbox": [ 176, 154, 818, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此得到 " }, { "type": "equation_inline", "content": "\\frac{y^2 + p^2}{p^2}\\int_0^{+\\infty}\\cos xy\\mathrm{d}y = \\frac{1}{p}" } ] }, "bbox": [ 90, 191, 368, 223 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {a} ^ {b} \\frac {p}{p ^ {2} + y ^ {2}} \\mathrm{d} y = \\arctan {\\frac {b}{p}} - \\arctan {\\frac {a}{p}}", "math_type": "latex", "image_source": { "path": "images/72d49d773aa8a7b3a67667091cd289b19a7b90fe2d3bcfd94768b59f5a75f2d3.jpg" } }, "bbox": [ 344, 225, 650, 259 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.11 (Dirichlet 积分)" } ], "level": 2 }, "bbox": [ 122, 274, 321, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "积分 " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin x}{x}\\mathrm{d}x = \\frac{\\pi}{2}" }, { "type": "text", "content": " ，进一步地， " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{\\sin ax}{x}\\mathrm{d}x = \\frac{\\pi}{2}\\mathrm{sgn}(a)" } ] }, "bbox": [ 114, 294, 584, 326 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.12 (正态积分公式)" } ], "level": 2 }, "bbox": [ 122, 350, 319, 365 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {+ \\infty} e ^ {- x ^ {2}} \\mathrm{d} x = \\frac {\\sqrt {\\pi}}{2}", "math_type": "latex", "image_source": { "path": "images/2acc45fdeb17306e83bb66490103943d1a35cefd4cc1dbb834db93b0a8626a38.jpg" } }, "bbox": [ 115, 370, 265, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习14.9相关练习(1) " }, { "type": "equation_inline", "content": "\\int_0^{+\\infty}\\frac{e^{-a^2x^2} - e^{-b^2x^2}}{x^e}\\mathrm{d}x" } ] }, "bbox": [ 58, 432, 443, 468 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "14.3.3 含参反常积分与函数项级数的关系" } ], "level": 2 }, "bbox": [ 95, 486, 463, 502 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 14.13 (一致收敛级数表示法 (ZJU 考研考过))" } ], "level": 2 }, "bbox": [ 121, 519, 489, 533 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\int_{c}^{+\\infty}f(x,y)\\mathrm{d}y" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛的充要条件为对任一趋于无穷的递增数列 " }, { "type": "equation_inline", "content": "A_{n}" }, { "type": "text", "content": " (其中 " }, { "type": "equation_inline", "content": "A_{1} = c" }, { "type": "text", "content": "), " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}\\int_{A_n}^{A_{n+1}}f(x,y)\\mathrm{d}y" }, { "type": "text", "content": " 作为函数项级数 " }, { "type": "equation_inline", "content": "\\sum_{n=1}^{\\infty}u_n(x)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 上一致收敛" } ] }, "bbox": [ 110, 536, 894, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 左推右：由于一致收敛，因此 " }, { "type": "equation_inline", "content": "\\forall\\epsilon,\\exists M>c" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "A^{\\prime\\prime}>A^{\\prime}>M" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "\\forall x" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "\\left|\\int_{A^{\\prime}}^{A^{\\prime\\prime}}f(x,y)\\mathrm{d}y\\right|<\\epsilon" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "A_{n}" }, { "type": "text", "content": " 趋于无穷， " }, { "type": "equation_inline", "content": "\\exists N,\\forall m>n>N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "A_{m}>A_{n}>M" }, { "type": "text", "content": " ，对 " }, { "type": "equation_inline", "content": "\\forall x\\in I" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|u_{n}(x)+\\cdots+u_{m}(x)|=|\\int_{A_{n}}^{A_{m+1}}f(x,y)\\mathrm{d}y|<\\epsilon" }, { "type": "text", "content": " ，即右侧级数一致收敛" } ] }, "bbox": [ 87, 623, 905, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 右推左：反设含参积分不一致收敛， " }, { "type": "equation_inline", "content": "\\exists \\epsilon_0, \\forall M > c, \\exists A'' > A' > M, x' \\in I" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 122, 701, 715, 715 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {A ^ {\\prime}} ^ {A ^ {\\prime \\prime}} f (x ^ {\\prime}, y) \\mathrm{d} y \\right| \\geq \\epsilon_ {0}", "math_type": "latex", "image_source": { "path": "images/5d8d4aebf0dd23f1144cd8e17a688c9f07041450522eb81eb92b5bf7ce002ff4.jpg" } }, "bbox": [ 416, 719, 581, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "先取 " }, { "type": "equation_inline", "content": "M_{1} = \\max \\{1,c\\}" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\exists A_2,A_1" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "A_{2} > A_{1} > M_{1}" }, { "type": "text", "content": " ，以及 " }, { "type": "equation_inline", "content": "x_{1}\\in I" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 89, 760, 631, 776 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {A _ {1}} ^ {A _ {2}} f (x _ {1}, y) \\mathrm{d} y \\right| \\geq \\epsilon_ {0}", "math_type": "latex", "image_source": { "path": "images/8baa73125ad92c02d7a94e9618f4c1bca9bcf37ea2f7992c71ded2a397ef77ed.jpg" } }, "bbox": [ 416, 780, 581, 816 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理取 " }, { "type": "equation_inline", "content": "M_{n} = \\max \\{n,A_{2n - 2}\\}" }, { "type": "text", "content": " ，有 " }, { "type": "equation_inline", "content": "A_{2n} > A_{2n - 1} > M_n,x_n\\in I" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 90, 820, 584, 835 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\int_ {A _ {2 n - 1}} ^ {A _ {2 n}} f (x _ {n}, y) \\mathrm{d} y \\right| \\geq \\epsilon_ {0}", "math_type": "latex", "image_source": { "path": "images/9761a12b815081a7574be63f5311c5efff001f71880007ad69c7b69b49578a71.jpg" } }, "bbox": [ 408, 839, 589, 877 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "此时 " }, { "type": "equation_inline", "content": "\\exists \\epsilon_0, \\forall N, \\exists n > N, \\exists x_n \\in I" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "|u_{2n}(x_n)| = |\\int_{A_{2n}}^{A_{2n + 1}} f(x_n, y) \\mathrm{d}y| \\geq \\epsilon_0" }, { "type": "text", "content": "，这与函数项级数一致收敛条件矛盾" } ] }, "bbox": [ 90, 881, 905, 916 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "14.3 含参反常积分的性质" } ] }, "bbox": [ 714, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "122" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(因为一致收敛要求所有 x 都趋于 0)。" } ] }, "bbox": [ 90, 85, 364, 99 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "14.3 含参反常积分的性质" } ] }, "bbox": [ 715, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "123" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 15 章 多重积分" } ], "level": 2 }, "bbox": [ 388, 87, 626, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.1 二重积分" } ], "level": 2 }, "bbox": [ 95, 145, 253, 164 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.1.1 二重积分的概念与计算" } ], "level": 2 }, "bbox": [ 94, 184, 364, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 15.1 (二重积分)" } ], "level": 2 }, "bbox": [ 121, 216, 278, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "D \\subset R^{2}" }, { "type": "text", "content": " 是有界集，对 D 做分割 " }, { "type": "equation_inline", "content": "T = \\{\\sigma_{1}, \\cdots, \\sigma_{n}\\}" }, { "type": "text", "content": " ，则二重积分定义为" } ] }, "bbox": [ 114, 236, 636, 253 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {D} f (x, y) \\mathrm{d} x \\mathrm{d} y = \\lim _ {| | T | | \\rightarrow 0} \\sum_ {i = 1} ^ {n} f (\\xi_ {i}, \\eta_ {i}) s (\\sigma_ {i})", "math_type": "latex", "image_source": { "path": "images/5c257928cb2b1346f9e80976d5c1c7517f6f79bf886c1901f60a7831890914f1.jpg" } }, "bbox": [ 341, 256, 653, 293 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.1 (二重积分与累次积分)" } ], "level": 2 }, "bbox": [ 122, 318, 363, 334 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "D = [a,b]\\times [c,d]" }, { "type": "text", "content": " 或者 " }, { "type": "equation_inline", "content": "D = [a,b]\\times [f_1(x),f_2(x)]" }, { "type": "text", "content": " ，方形边界二重积分可随意转换为累次积分，函数边界先积分函数边界" } ] }, "bbox": [ 114, 337, 882, 370 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {D} f (x, y) \\mathrm{d} x \\mathrm{d} y = \\int_ {a} ^ {b} \\mathrm{d} x \\int_ {c} ^ {d} f (x, y) \\mathrm{d} y, \\quad \\iint_ {D} f (x, y) \\mathrm{d} x \\mathrm{d} y = \\int_ {a} ^ {b} \\mathrm{d} x \\int_ {f _ {1} (x)} ^ {f _ {2} (x)} f (x, y) \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/01e95310432256bdfaa2a2d7226181d3cd3bd23541efd18934050ee3a82693f4.jpg" } }, "bbox": [ 189, 374, 806, 409 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 二重积分的直接计算要特别关注其区域的对称性。而累次积分的顺序一般看函数，那边容易积就先积 (也得靠经验觉得哪个好积)。" } ] }, "bbox": [ 75, 428, 902, 462 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习15.1对称性计算(1)计算 " }, { "type": "equation_inline", "content": "\\iint_{D}x^{2}y\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "D:|x| + |y|\\leq 1" } ] }, "bbox": [ 58, 476, 584, 508 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)由于 D 关于 x 轴对称，且 " }, { "type": "equation_inline", "content": "f(x,y)=x^{2}y" }, { "type": "text", "content": " 关于 y 为奇函数，因此 " }, { "type": "equation_inline", "content": "\\iint_{D}x^{2}y\\,dx\\,dy=0" } ] }, "bbox": [ 87, 506, 734, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 15.2 转换为累次积分 (1) 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} e^{-y^{2}} dx dy" }, { "type": "text", "content": " ，其中 D 是 y = x 与 " }, { "type": "equation_inline", "content": "y = x^{\\frac{1}{3}}" }, { "type": "text", "content": " 围成的区域" } ] }, "bbox": [ 58, 542, 784, 574 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} \\frac{\\sin x}{x} \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "y = x" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "y = x^2" }, { "type": "text", "content": " 围成的区域" } ] }, "bbox": [ 121, 571, 623, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 圆周边界：计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} \\frac{\\mathrm{d}x \\, \\mathrm{d}y}{\\sqrt{2a - x}}" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x = 0, y = 0" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "(x - a)^2 + (y - a)^2 = a^2" }, { "type": "text", "content": " 所围成的区域" } ] }, "bbox": [ 121, 599, 890, 629 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)注意要先积 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " ，因为 " }, { "type": "equation_inline", "content": "e^{-y^2}" }, { "type": "text", "content": " 先积 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 比较方便，答案为 " }, { "type": "equation_inline", "content": "\\frac{1}{e}" } ] }, "bbox": [ 89, 627, 534, 656 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 注意要先积 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": "，因为显然 " }, { "type": "equation_inline", "content": "\\frac{\\sin x}{x}" }, { "type": "text", "content": " 不好积，答案为 " }, { "type": "equation_inline", "content": "1 - \\sin 1" } ] }, "bbox": [ 121, 653, 561, 675 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 这里 " }, { "type": "equation_inline", "content": "\\int \\frac{1}{\\sqrt{2a - x}} \\mathrm{d}x" }, { "type": "text", "content": " 看成 " }, { "type": "equation_inline", "content": "\\int (2a - x)^{-\\frac{1}{2}} \\mathrm{d}x" }, { "type": "text", "content": " 就是个幂函数，不要被吓到。先积 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " ，得到" } ] }, "bbox": [ 122, 677, 771, 707 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {a} \\mathrm{d} x \\int_ {0} ^ {a - \\sqrt {2 a x - x ^ {2}}} \\frac {1}{\\sqrt {2 a - x}} \\mathrm{d} y = \\int_ {0} ^ {a} \\frac {a - \\sqrt {2 a x - x ^ {2}}}{\\sqrt {2 a - x}} \\mathrm{d} x = (2 \\sqrt {2} - \\frac {8}{3}) a ^ {\\frac {3}{2}}", "math_type": "latex", "image_source": { "path": "images/8ab3164ead49a1d8286c4aa55ca3035f3f53e9b0f7e904f057425b9e2f479a03.jpg" } }, "bbox": [ 213, 712, 778, 746 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 15.3 几个进阶题 (1) 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} |xy - \\frac{1}{4}| \\, dx \\, dy" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "D : [0, 1] \\times [0, 1]" }, { "type": "text", "content": " (补区域技巧)" } ] }, "bbox": [ 58, 756, 744, 787 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} \\sqrt{|y - x^2|} \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "D: -1 \\leq x \\leq 1, 0 \\leq y \\leq 2" }, { "type": "text", "content": " (对称性)" } ] }, "bbox": [ 122, 787, 668, 816 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)要画出 " }, { "type": "equation_inline", "content": "xy = \\frac{1}{4}" }, { "type": "text", "content": " 线，分成上下两个部分，记上面的为 " }, { "type": "equation_inline", "content": "D_{1}" }, { "type": "text", "content": "，下面的是 " }, { "type": "equation_inline", "content": "D_{2}" }, { "type": "text", "content": "，但是 " }, { "type": "equation_inline", "content": "D_{2}" }, { "type": "text", "content": " 过于不规则，可以为 " }, { "type": "equation_inline", "content": "D_{2}" }, { "type": "text", "content": " 补上 " }, { "type": "equation_inline", "content": "D_{1}" }, { "type": "text", "content": " 的部分，这样算起来就方便不少。即 " }, { "type": "equation_inline", "content": "I = \\iint_{D_1}(xy - \\frac{1}{4})\\mathrm{d}x\\mathrm{d}y - \\iint_{D_2}(xy - \\frac{1}{4})\\mathrm{d}x\\mathrm{d}y = 2\\iint_{D_1}(xy - \\frac{1}{4})\\mathrm{d}x\\mathrm{d}y - \\iint_{D}(xy - \\frac{1}{4})\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": "，最终答案为 " }, { "type": "equation_inline", "content": "\\frac{3}{32} +\\frac{1}{8}\\ln 2" } ] }, "bbox": [ 87, 816, 905, 898 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 画出 " }, { "type": "equation_inline", "content": "y = x^2" }, { "type": "text", "content": "，根据关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 为偶，只需要考虑 " }, { "type": "equation_inline", "content": "x > 0" }, { "type": "text", "content": "，关于 " }, { "type": "equation_inline", "content": "y = x^2" }, { "type": "text", "content": " 分为两个区域，上面记 " }, { "type": "equation_inline", "content": "D_1" }, { "type": "text", "content": "，下面记 " }, { "type": "equation_inline", "content": "D_2" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "D_1, D_2" }, { "type": "text", "content": " 都算规则，直接计算即可。要注意 " }, { "type": "equation_inline", "content": "(x^2)^{\\frac{2}{3}} = |x|^3!" } ] }, "bbox": [ 89, 84, 905, 118 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.1.2 二重积分可积性" } ], "level": 2 }, "bbox": [ 94, 141, 304, 156 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习15.4两道经典题 " }, { "type": "equation_inline", "content": "(1)D = [0,1]\\times [0,1]" }, { "type": "text", "content": " ，对 " }, { "type": "equation_inline", "content": "x\\in \\mathbb{R}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "q_{x}" }, { "type": "text", "content": " 表示 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 化成既约分数后的分母，证明下面 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上二重积分存在但两个累次积分不存在" } ] }, "bbox": [ 58, 168, 905, 200 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = \\left\\{ \\begin{array}{l l} \\frac {1}{q _ {x}} + \\frac {1}{q _ {y}}, & (x, y) \\text {为有理点} \\\\ 0, & (x, y) \\text {非有理点} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/06c466715a0ffa66e74e8e968e98a97bf3a352995d9a532b6e3db19b332bbde0.jpg" } }, "bbox": [ 354, 203, 638, 254 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "D=[0,1]\\times[0,1]" }, { "type": "text", "content": " ，证明下面 " }, { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 在D上二重积分不存在但两个累次积分存在" } ] }, "bbox": [ 122, 259, 737, 275 ] }, { "type": "equation_interline", "content": { "math_content": "f (x, y) = \\left\\{ \\begin{array}{l l} 1, & (x, y) \\text {为有理点且} q _ {x} = q _ {y} \\\\ 0, & \\text {其他情形} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/b08ef8c8f435c55e55c10eabf08bd0612034711683dda6fde26d3bb5134970a2.jpg" } }, "bbox": [ 346, 279, 645, 322 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) " }, { "type": "equation_inline", "content": "\\forall \\epsilon" }, { "type": "text", "content": "，仅有有限个点满足 " }, { "type": "equation_inline", "content": "f(x, y) > \\epsilon" }, { "type": "text", "content": "，因此存在分割 " }, { "type": "equation_inline", "content": "T" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "S(T) - s(T) < \\epsilon" }, { "type": "text", "content": "，二重积分存在且等于0。下证先 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 不存在： " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 取无理时， " }, { "type": "equation_inline", "content": "f(x, y) \\equiv 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\int_{0}^{1} f(x, y) \\mathrm{d}x = 0" }, { "type": "text", "content": " 积分存在。若 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 取有理，则在 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 无理处 " }, { "type": "equation_inline", "content": "f(x, y) = 0" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 有理处 " }, { "type": "equation_inline", "content": "f(x, y) = \\frac{1}{q_x} + \\frac{1}{q_y}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "f(x, y)" }, { "type": "text", "content": " 在任意区间上振幅总大于 " }, { "type": "equation_inline", "content": "\\frac{1}{q_y}" }, { "type": "text", "content": "，因此积分不存在。同理可证先 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 不存在" } ] }, "bbox": [ 87, 326, 907, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 中任意部分均有振幅为1，因此二重积分不存在。若固定 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " ，若 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 无理，则 " }, { "type": "equation_inline", "content": "f(x,y) = 0" }, { "type": "text", "content": " ，显然关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 可积，若 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 无理，则 " }, { "type": "equation_inline", "content": "f(x,y) > \\epsilon" }, { "type": "text", "content": " 对于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 而言只有有限处，因此也关于 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 可积，所以先 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 可积。另一侧同理。" } ] }, "bbox": [ 87, 419, 905, 453 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.1.3 二重积分换元法" } ], "level": 2 }, "bbox": [ 95, 474, 304, 491 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.2 (二重积分换元法)" } ], "level": 2 }, "bbox": [ 121, 506, 329, 521 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "原二重积分坐标轴 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": "，定义于 " }, { "type": "equation_inline", "content": "R" }, { "type": "text", "content": " 上，令 " }, { "type": "equation_inline", "content": "u = u(x, y), v = v(x, y)" }, { "type": "text", "content": " 反解出 " }, { "type": "equation_inline", "content": "x = x(u, v), y = y(u, v)" }, { "type": "text", "content": "，变换后的区域为 " }, { "type": "equation_inline", "content": "(u, v) \\in \\Omega" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 112, 526, 880, 560 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {R} f (x, y) \\mathrm{d} x \\mathrm{d} y = \\iint_ {\\Omega} f (x (u, v), y (u, v)) \\left| \\frac {\\partial (x , y)}{\\partial (u , v)} \\right| \\mathrm{d} u \\mathrm{d} v", "math_type": "latex", "image_source": { "path": "images/862b00cc92c524e6b08cabc5015fc1b6397f138ef3f9cddfec0dfcc629c3f04a.jpg" } }, "bbox": [ 295, 563, 702, 596 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "定义 " }, { "type": "equation_inline", "content": "J(u,v)=\\frac{\\partial(x,y)}{\\partial(u,v)}" } ] }, "bbox": [ 115, 598, 282, 629 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记一、注意Jacobi行列式外面的是绝对值！二、可以先计算 " }, { "type": "equation_inline", "content": "I = \\frac{\\partial(u,v)}{\\partial(x,y)}" }, { "type": "text", "content": "，计算 " }, { "type": "equation_inline", "content": "|J|" }, { "type": "text", "content": " 时只需要取倒数加绝对值，即 " }, { "type": "equation_inline", "content": "|J| = |\\frac{1}{I}|" }, { "type": "text", "content": " 。三、计算变换后的范围可以尝试将原本的各个边界转化到新变量，即代入原边界，计算新边界各个变量需要满足的关系。" } ] }, "bbox": [ 82, 643, 907, 709 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习15.5一般换元计算题(1)计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D}e^{\\frac{x - y}{x + y}}dxdy" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 是由 " }, { "type": "equation_inline", "content": "x = 0,y = 0" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "x + y = a(a > 0)" }, { "type": "text", "content": " 围成的区域" } ] }, "bbox": [ 58, 725, 905, 755 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} \\frac{(x + y) \\ln(1 + \\frac{y}{x})}{\\sqrt{1 - x - y}} \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 是由 " }, { "type": "equation_inline", "content": "x = 0, y = 0, x + y = 1" }, { "type": "text", "content": " 围成的区域" } ] }, "bbox": [ 121, 753, 779, 785 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D} \\frac{3x}{y^2 + xy^3} \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 是由 " }, { "type": "equation_inline", "content": "xy = 1, xy = 3, y^2 = x, y^2 = 3x" }, { "type": "text", "content": " 围成的区域" } ] }, "bbox": [ 122, 785, 781, 813 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 (1) 设 " }, { "type": "equation_inline", "content": "x + y = u, x - y = v" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "x = \\frac{u + v}{2}, y = \\frac{u - v}{2}" }, { "type": "text", "content": "，考虑边界转化， " }, { "type": "equation_inline", "content": "x = 0, y = 0" }, { "type": "text", "content": " 等价于 " }, { "type": "equation_inline", "content": "u = -v, u = v" }, { "type": "text", "content": "，而 " }, { "type": "equation_inline", "content": "y = -x + a" }, { "type": "text", "content": " 等价于：" } ] }, "bbox": [ 89, 813, 904, 851 ] }, { "type": "equation_interline", "content": { "math_content": "u = a, v = 2 x + a", "math_type": "latex", "image_source": { "path": "images/3501df17c7426a259abc92068dd08569615e81165ccda1b377aad35a9def5e66.jpg" } }, "bbox": [ 431, 856, 566, 868 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "此时 u = a，而 v 任意。" } ] }, "bbox": [ 90, 881, 267, 896 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "125" } ] }, "bbox": [ 485, 954, 512, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "15.1 二重积分" } ] }, "bbox": [ 798, 55, 904, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此区域变为 " }, { "type": "equation_inline", "content": "u + v = 0, u - v = 0, u = a" }, { "type": "text", "content": " 围成的区域，即 " }, { "type": "equation_inline", "content": "0 \\leq u \\leq a, -u \\leq v \\leq u, J = \\left| \\begin{array}{cc} \\frac{1}{2} & \\frac{1}{2} \\\\ \\frac{1}{2} & -\\frac{1}{2} \\end{array} \\right| = -\\frac{1}{2}" }, { "type": "text", "content": "，得到最终结果为 " }, { "type": "equation_inline", "content": "\\frac{a^2}{4}(e - e^{-1})" } ] }, "bbox": [ 90, 82, 905, 161 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 设 " }, { "type": "equation_inline", "content": "x + y = u, x = v" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "x = v, y = u - v" }, { "type": "text", "content": " 即可，区域变为 " }, { "type": "equation_inline", "content": "v = 0, u = v, u = 1" }, { "type": "text", "content": " 围成的区域，答案为 " }, { "type": "equation_inline", "content": "\\frac{16}{15}" } ] }, "bbox": [ 121, 160, 890, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 做 " }, { "type": "equation_inline", "content": "xy = u, \\frac{y^2}{x} = v" } ] }, "bbox": [ 122, 185, 285, 215 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.1.4 极坐标换元" } ], "level": 2 }, "bbox": [ 95, 233, 263, 250 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.3 (二重积分极坐标换元)" } ], "level": 2 }, "bbox": [ 122, 265, 363, 280 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于二重积分 " }, { "type": "equation_inline", "content": "\\iint_{S} f(x, y) \\mathrm{d}x \\mathrm{~d}y" }, { "type": "text", "content": "，做极坐标换元 " }, { "type": "equation_inline", "content": "x = r \\cos \\theta, y = r \\sin \\theta" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "J(r, \\theta) = r" }, { "type": "text", "content": "；极坐标换元 " }, { "type": "equation_inline", "content": "x = ar \\cos \\theta, y = br \\sin \\theta" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "J(r, \\theta) = abr" }, { "type": "text", "content": "。积分时先 " }, { "type": "equation_inline", "content": "\\mathrm{d}r" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "\\mathrm{d}\\theta" }, { "type": "text", "content": "，具体如下图所示" } ] }, "bbox": [ 112, 287, 892, 330 ] }, { "type": "image", "content": { "image_source": { "path": "images/5b7b42b6a99544f36af749cad867071bd860f5a296b8bf3c0ba808703c397c2b.jpg" }, "content": "", "image_caption": [ { "type": "text", "content": "图 15.1: 二重积分极坐标换元" } ], "image_footnote": [] }, "bbox": [ 174, 356, 821, 543 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习15.6规则极坐标与广义极坐标(1)计算 " }, { "type": "equation_inline", "content": "I = \\iint_{D}\\frac{\\mathrm{d}x\\mathrm{d}y}{\\sqrt{1 - x^2 - y^2}}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2}\\leq 1" } ] }, "bbox": [ 58, 595, 726, 626 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)广义极坐标：计算 " }, { "type": "equation_inline", "content": "\\frac{x^2}{a^2} +\\frac{y^2}{b^2} +\\frac{z^2}{c^2}\\leq 1" }, { "type": "text", "content": " 的体积" } ] }, "bbox": [ 119, 624, 480, 656 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)是8倍第一卦限积分，该区域顶面为 " }, { "type": "equation_inline", "content": "z = c\\sqrt{1 - \\frac{x^2}{a^2} - \\frac{y^2}{b^2}}" }, { "type": "text", "content": " ，底面为" } ] }, "bbox": [ 121, 665, 638, 697 ] }, { "type": "equation_interline", "content": { "math_content": "D = \\left\\{(x, y): 0 \\leq y \\leq b \\sqrt {1 - \\frac {x ^ {2}}{a ^ {2}}}, 0 \\leq x \\leq a \\right\\}", "math_type": "latex", "image_source": { "path": "images/6cd5f6981b6636f796497a599d91f736ac5183617ce0d0a6ba3d265bfbec013f.jpg" } }, "bbox": [ 329, 700, 665, 737 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此 " }, { "type": "equation_inline", "content": "V = 8\\iint_{D}c\\sqrt{1 - \\frac{x^2}{a^2} - \\frac{y^2}{b^2}}\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " ，利用广义极坐标变换得到 " }, { "type": "equation_inline", "content": "z = c\\sqrt{1 - r^2}" }, { "type": "text", "content": " ，故" } ] }, "bbox": [ 90, 741, 700, 775 ] }, { "type": "equation_interline", "content": { "math_content": "V = 8 \\int_ {0} ^ {\\frac {\\pi}{2}} \\mathrm{d} \\theta \\int_ {0} ^ {1} c \\sqrt {1 - r ^ {2}} a b r \\mathrm{d} r = \\frac {4 \\pi}{3} a b c", "math_type": "latex", "image_source": { "path": "images/70fa8efbcaf1356178954db37956ef731b851fd5d0a05b1f855bc32765daf19d.jpg" } }, "bbox": [ 342, 778, 655, 810 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习15.7几道非规则边界(1) " }, { "type": "equation_inline", "content": "f" }, { "type": "text", "content": " 为连续函数，将 " }, { "type": "equation_inline", "content": "I = \\iint_{D}f\\left(\\frac{y}{x}\\right)\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " 转化为定积分，其中 " }, { "type": "equation_inline", "content": "D:\\frac{1}{4}\\leq x^2 +y^2\\leq x" }, { "type": "text", "content": " (2)将 " }, { "type": "equation_inline", "content": "I = \\iint_{D}f(x,y)\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " 化为极坐标形式，其中 " }, { "type": "equation_inline", "content": "D:[0,1]\\times [0,1]" } ] }, "bbox": [ 58, 820, 897, 877 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 考虑 " }, { "type": "equation_inline", "content": "x = r\\cos \\theta, y = r\\sin \\theta" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\frac{1}{4} \\leq r^2 \\leq r\\cos \\theta" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "\\frac{1}{2} \\leq r \\leq \\cos \\theta" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "-\\frac{\\pi}{3} \\leq \\theta \\leq \\frac{\\pi}{3}" }, { "type": "text", "content": "。即 " }, { "type": "equation_inline", "content": "I =" } ] }, "bbox": [ 87, 878, 905, 907 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "15.1 二重积分" } ] }, "bbox": [ 800, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "126" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "equation_interline", "content": { "math_content": "\\int_ {- \\frac {\\pi}{3}} ^ {\\frac {\\pi}{3}} \\mathrm{d} \\theta \\int_ {\\frac {1}{2}} ^ {\\cos \\theta} f (\\tan \\theta) r \\mathrm{d} r = \\frac {1}{2} \\int_ {- \\frac {\\pi}{3}} ^ {\\frac {\\pi}{3}} f (\\tan \\theta) (\\cos^ {2} \\theta - \\frac {1}{4}) \\mathrm{d} \\theta", "math_type": "latex", "image_source": { "path": "images/18d4a53f09c2a616b382234053d0b7a13c27dfa7271707000fbce79f43b2a653.jpg" } }, "bbox": [ 89, 80, 519, 116 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)一般都是先 " }, { "type": "equation_inline", "content": "r" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "\\theta" }, { "type": "text", "content": " ，将 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 分为左上，右下两个三角形，则" } ] }, "bbox": [ 122, 117, 574, 131 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {\\frac {\\pi}{4}} \\mathrm{d} \\theta \\int_ {0} ^ {\\frac {1}{\\cos \\theta}} f (r \\cos \\theta , r \\sin \\theta) r \\mathrm{d} r + \\int_ {\\frac {\\pi}{4}} ^ {\\frac {\\pi}{2}} \\mathrm{d} \\theta \\int_ {0} ^ {\\frac {1}{\\sin \\theta}} f (r \\cos \\theta , r \\sin \\theta) r \\mathrm{d} r", "math_type": "latex", "image_source": { "path": "images/2f87c38590919a9ecf199edbbb7e5b76c2bdf6f090d69a7d72dc29fe2471f98f.jpg" } }, "bbox": [ 225, 133, 769, 170 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "先 " }, { "type": "equation_inline", "content": "\\theta" }, { "type": "text", "content": " 后 " }, { "type": "equation_inline", "content": "r" }, { "type": "text", "content": " 会比较麻烦" } ] }, "bbox": [ 90, 174, 247, 187 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "三类特殊的圆" } ], "level": 2 }, "bbox": [ 90, 210, 196, 225 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.4 (三类特殊圆极坐标)" } ], "level": 2 }, "bbox": [ 122, 235, 344, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "a > 0" }, { "type": "text", "content": " ，极坐标 " }, { "type": "equation_inline", "content": "x = r\\cos \\theta ,y = r\\sin \\theta" }, { "type": "text", "content": " ，以下特殊圆的极坐标公式：" } ] }, "bbox": [ 114, 255, 618, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "(x - a)^2 + y^2 = a^2" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "x^2 + y^2 = 2ax" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "r = 2a\\cos \\theta, \\theta \\in [-\\frac{\\pi}{2}, \\frac{\\pi}{2}]" } ] }, "bbox": [ 139, 271, 653, 292 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "x^{2} + (y - a)^{2} = a^{2}" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = 2ay" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "r = 2a\\sin \\theta, \\theta \\in [0, \\pi]" } ] }, "bbox": [ 139, 291, 631, 307 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "(x - a)^2 + (y - a)^2 = 2a^2" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "x^2 + y^2 = 2a(x + y)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "r = 2a(\\cos \\theta + \\sin \\theta), \\theta \\in [-\\frac{\\pi}{4}, \\frac{3\\pi}{4}]" } ] }, "bbox": [ 139, 307, 818, 332 ] }, { "type": "image", "content": { "image_source": { "path": "images/b54fa69c6f03c09c287c78d1c9f720c960f3f0469a6172b2465da531dea1784b.jpg" }, "content": "", "image_caption": [ { "type": "text", "content": "图 15.2: 三个特殊的圆" } ], "image_footnote": [] }, "bbox": [ 255, 359, 742, 507 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.2 三重积分" } ], "level": 2 }, "bbox": [ 95, 571, 253, 590 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.2.1 投影法" } ], "level": 2 }, "bbox": [ 95, 609, 223, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "将三重积分转换为累次积分进行计算，常用的方法有投影法与截面法" } ] }, "bbox": [ 121, 638, 633, 653 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.5 (投影法)" } ], "level": 2 }, "bbox": [ 122, 663, 263, 678 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "积分区域 " }, { "type": "equation_inline", "content": "V \\subset R^{3}" }, { "type": "text", "content": " ，对应 x, y 坐标平面投影区域 D，且 " }, { "type": "equation_inline", "content": "V = \\{(x, y, z) : (x, y) \\in D, z_{1}(x, y) \\leq z \\leq z_{2}(x, y)\\}" }, { "type": "text", "content": " ，则：" } ] }, "bbox": [ 114, 683, 882, 715 ] }, { "type": "equation_interline", "content": { "math_content": "\\iiint_ {V} f (x, y, z) \\mathrm{d} V = \\iint_ {D} \\mathrm{d} x \\mathrm{d} y \\int_ {z _ {1} (x, y)} ^ {z _ {2} (x, y)} f (x, y, z) \\mathrm{d} z", "math_type": "latex", "image_source": { "path": "images/0eb84657eb4aac68d33bcc099896bb06ee8820a8a0462450800fcf48893d21ac.jpg" } }, "bbox": [ 314, 715, 682, 750 ] }, { "type": "image", "content": { "image_source": { "path": "images/ad5fd2105a5935951c414ac9ded62f37636325f2db703c2dd4021028281fbc93.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 58, 763, 78, 782 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 一般而言投影法比截面法更常用一点" } ] }, "bbox": [ 87, 769, 401, 784 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 15.8 投影法 (1) 计算 " }, { "type": "equation_inline", "content": "I = \\iiint_{V} (x + y + z) \\, \\mathrm{d}V" }, { "type": "text", "content": " ，其中 V 是 " }, { "type": "equation_inline", "content": "x + y + z = 1" }, { "type": "text", "content": " 与坐标面围成的区域" } ] }, "bbox": [ 58, 800, 801, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "\\iiint_{V} (x^{2} + y^{2} + z) \\mathrm{d}V" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\left\\{ \\begin{array}{l} z = y \\\\ x = 0 \\end{array} \\right." }, { "type": "text", "content": " 绕着 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴旋转一圈而成的曲面与 " }, { "type": "equation_inline", "content": "z = 1" }, { "type": "text", "content": " 围成的区域" } ] }, "bbox": [ 121, 829, 880, 873 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 用投影法 " }, { "type": "equation_inline", "content": "I = \\iint_{D} \\mathrm{d}x \\, \\mathrm{d}y \\int_{0}^{1 - x - y} x + y + z \\, \\mathrm{d}z = \\frac{1}{8}" } ] }, "bbox": [ 87, 873, 497, 907 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "127" } ] }, "bbox": [ 485, 953, 512, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "15.2 三重积分" } ] }, "bbox": [ 798, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 旋转面为 " }, { "type": "equation_inline", "content": "z = \\sqrt{x^2 + y^2}" }, { "type": "text", "content": " (画个图就看得出来)，投影为 " }, { "type": "equation_inline", "content": "x^2 + y^2 \\leq 1" }, { "type": "text", "content": "，因此积分为" } ] }, "bbox": [ 122, 84, 727, 99 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} I = \\iint_ {D} \\mathrm{d} x \\mathrm{d} y \\int_ {\\sqrt {x ^ {2} + y ^ {2}}} ^ {1} (x ^ {2} + y ^ {2} + z) \\mathrm{d} z \\\\ = \\iint_ {D} (x ^ {2} + y ^ {2}) (1 - \\sqrt {x ^ {2} + y ^ {2}}) \\mathrm{d} x \\mathrm{d} y + \\frac {1}{2} \\iint_ {D} [ 1 - (x ^ {2} + y ^ {2}) ] \\mathrm{d} x \\mathrm{d} y \\\\ = \\int_ {0} ^ {2 \\pi} \\mathrm{d} \\theta \\int_ {0} ^ {1} r ^ {3} (1 - r) \\mathrm{d} r + \\frac {1}{2} \\int_ {0} ^ {2 \\pi} \\mathrm{d} \\theta \\int_ {0} ^ {1} (1 - r ^ {2}) r \\mathrm{d} r \\\\ = \\frac {\\pi}{1 0} + \\frac {\\pi}{4} = \\frac {7 \\pi}{2 0} \\end{array}", "math_type": "latex", "image_source": { "path": "images/b35a4079ff078f7b6f21d06db76602923eead664b57979b00de0cc3c220be600.jpg" } }, "bbox": [ 253, 104, 741, 234 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.2.2 截面法" } ], "level": 2 }, "bbox": [ 95, 252, 223, 268 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.6 (截面法)" } ], "level": 2 }, "bbox": [ 122, 284, 262, 299 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "积分区域 " }, { "type": "equation_inline", "content": "V \\subset \\mathbb{R}^3" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 被垂直于 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴平面截取的平面为 " }, { "type": "equation_inline", "content": "D_z" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 114, 304, 576, 319 ] }, { "type": "equation_interline", "content": { "math_content": "\\iiint_ {V} f (x, y, z) \\mathrm{d} V = \\int_ {a} ^ {b} \\mathrm{d} z \\iint_ {D _ {z}} f (x, y, z) \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/679e46fe62127f26e88fb99921590a17571494cba360a97b70ec0dbbb6762309.jpg" } }, "bbox": [ 329, 323, 668, 357 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.2.3 三重积分换元法" } ], "level": 2 }, "bbox": [ 95, 394, 302, 411 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.7 (三重积分换元)" } ], "level": 2 }, "bbox": [ 122, 426, 312, 442 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若坐标替换 " }, { "type": "equation_inline", "content": "x = x(u,v,w),y = y(u,v,w),z = z(u,v,w)" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 115, 447, 552, 462 ] }, { "type": "equation_interline", "content": { "math_content": "\\iiint_ {V} f (x, y, z) \\mathrm{d} x \\mathrm{d} y \\mathrm{d} z = \\iiint_ {V ^ {\\prime}} f (x, y, z) \\left| \\frac {\\partial (x , y , z)}{\\partial (u , v , w)} \\right| \\mathrm{d} u \\mathrm{d} v \\mathrm{d} w", "math_type": "latex", "image_source": { "path": "images/5e5454d0ad60673177346986ab1f8b2b295ee1a8bcb6f95b1006d0171f19f7be.jpg" } }, "bbox": [ 278, 464, 717, 497 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\frac {\\partial (x , y , z)}{\\partial (u , v , w)} \\right| = \\left| \\begin{array}{l l l} \\frac {\\partial x}{\\partial u} & \\frac {\\partial x}{\\partial v} & \\frac {\\partial x}{\\partial w} \\\\ \\frac {\\partial y}{\\partial u} & \\frac {\\partial y}{\\partial v} & \\frac {\\partial y}{\\partial w} \\\\ \\frac {\\partial z}{\\partial u} & \\frac {\\partial z}{\\partial v} & \\frac {\\partial z}{\\partial w} \\end{array} \\right|", "math_type": "latex", "image_source": { "path": "images/5bb189ac78cc8a6cc14e380ec86261fb0f7244bde6e1c4d62b26f7d6d40c0a0a.jpg" } }, "bbox": [ 376, 499, 611, 577 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 一方面还是注意 Jacobi 行列式外面的是绝对值。另一方面可以先算 " }, { "type": "equation_inline", "content": "\\frac{\\partial(u, v, w)}{\\partial(x, y, z)}" }, { "type": "text", "content": "，再取倒数和绝对值计算 " }, { "type": "equation_inline", "content": "\\left|\\frac{\\partial(x, y, z)}{\\partial(u, v, w)}\\right|" } ] }, "bbox": [ 89, 590, 905, 650 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "15.2.4 柱面坐标变换与球坐标变换" } ], "level": 2 }, "bbox": [ 95, 668, 403, 684 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.8 (柱面变换)" } ], "level": 2 }, "bbox": [ 122, 700, 278, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "做 " }, { "type": "equation_inline", "content": "x = r\\cos \\theta ,y = r\\sin \\theta ,z = z" }, { "type": "text", "content": " 变换时， " }, { "type": "equation_inline", "content": "J(u,v,w) = r" }, { "type": "text", "content": " 。往往确定区域：" } ] }, "bbox": [ 114, 721, 640, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "V^{\\prime} = \\{a\\leq r\\leq b,c\\leq \\theta \\leq d,\\alpha (r,\\theta)\\leq z\\leq \\beta (r,\\theta)\\}" }, { "type": "text", "content": " 因此用投影法 " }, { "type": "equation_inline", "content": "\\iint_{D}\\mathrm{d}r\\mathrm{d}\\theta \\int_{\\alpha (r,\\theta)}^{\\beta (r,\\theta)}z\\mathrm{d}z" }, { "type": "text", "content": " ，即一般积分顺序按照 " }, { "type": "equation_inline", "content": "\\mathrm{dz},\\mathrm{dr},\\mathrm{d}\\theta" } ] }, "bbox": [ 114, 743, 677, 797 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 15.9 柱坐标变换 (1) 计算 " }, { "type": "equation_inline", "content": "I = \\iiint_{V} (x^{2} + y^{2}) \\, \\mathrm{d}V" }, { "type": "text", "content": " ，其中 V 是由曲面 " }, { "type": "equation_inline", "content": "2(x^{2} + y^{2}) = z" }, { "type": "text", "content": " 与 z = 4 为界面的区域。\n解 (1) 为抛物面与平面围成的区域，令 " }, { "type": "equation_inline", "content": "x = r \\cos \\theta, y = r \\sin \\theta, z = z" }, { "type": "text", "content": " 得到区域为" } ] }, "bbox": [ 58, 832, 899, 872 ] }, { "type": "equation_interline", "content": { "math_content": "V ^ {\\prime} = \\left\\{(r, \\theta , z): 0 \\leq r \\leq 2, 0 \\leq \\theta \\leq 2 \\pi , 2 r ^ {2} \\leq z \\leq 4 \\right\\}", "math_type": "latex", "image_source": { "path": "images/175b9f0a49bd2976920d84db7b3fa971203068e477e94f9075139463e5b8a093.jpg" } }, "bbox": [ 305, 879, 687, 897 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "15.2 三重积分" } ] }, "bbox": [ 800, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "128" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对应积分 " }, { "type": "equation_inline", "content": "\\iiint_{V'} r^2 \\cdot r \\, \\mathrm{d}V = \\int_0^{2\\pi} \\, \\mathrm{d}\\theta \\int_0^{\\sqrt{2}} \\, \\mathrm{d}r \\int_{2r^2}^4 r^3 \\, \\mathrm{d}z = \\frac{8\\pi}{3}" } ] }, "bbox": [ 90, 84, 514, 118 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.9 (球坐标变换)" } ], "level": 2 }, "bbox": [ 122, 135, 295, 149 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "做 " }, { "type": "equation_inline", "content": "x = r\\sin \\varphi \\cos \\theta ,y = r\\sin \\varphi \\sin \\theta ,z = r\\cos \\varphi" }, { "type": "text", "content": " 时 " }, { "type": "equation_inline", "content": "J = r^2\\sin \\varphi" }, { "type": "text", "content": " 。积分时先按照 " }, { "type": "equation_inline", "content": "\\mathrm{d}r,\\mathrm{d}\\varphi ,\\mathrm{d}\\theta" }, { "type": "text", "content": " 顺序积分" } ] }, "bbox": [ 114, 155, 831, 172 ] }, { "type": "image", "content": { "image_source": { "path": "images/93315663037de58e7e2eda9f87b39ab445a1eaf012a8cff06679212e4a2f18a2.jpg" }, "content": "", "image_caption": [ { "type": "text", "content": "图 15.3: 三重积分球坐标换元" } ], "image_footnote": [] }, "bbox": [ 173, 197, 821, 359 ] }, { "type": "image", "content": { "image_source": { "path": "images/fc68779cbf73ff0e464878fe775b9e52163a5b057fb1e986b9ed4d5fe07d16e1.jpg" }, "content": "", "image_caption": [], "image_footnote": [] }, "bbox": [ 57, 392, 80, 412 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 使用球坐标变换的原则：(1) 被积函数为 " }, { "type": "equation_inline", "content": "f(x^{2} + y^{2} + z^{2}), f(x^{2} + y^{2})(2)" }, { "type": "text", "content": " 积分区域为球、锥或者其一部分" } ] }, "bbox": [ 58, 395, 884, 414 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 15.10 (广义球坐标变换)" } ], "level": 2 }, "bbox": [ 122, 439, 337, 455 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于以 " }, { "type": "equation_inline", "content": "(x_0, y_0, z_0)" }, { "type": "text", "content": "，以 " }, { "type": "equation_inline", "content": "a, b, c" }, { "type": "text", "content": " 为半轴长的椭球面所围成的区域 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": "，做如下广义球坐标变换：" } ] }, "bbox": [ 114, 460, 774, 475 ] }, { "type": "equation_interline", "content": { "math_content": "T: \\left\\{ \\begin{array}{l} x = x _ {0} + a r \\sin \\varphi \\cos \\theta \\\\ y = y _ {0} + b r \\sin \\varphi \\sin \\theta \\\\ z = z _ {0} + c r \\cos \\varphi \\end{array} , \\quad r \\geq 0, \\varphi \\in [ 0, \\pi ], \\theta \\in [ 0, 2 \\pi ] \\right.", "math_type": "latex", "image_source": { "path": "images/4ccfc58dd69182bca7856193f8a81011a230d46961c61bf9ee0d46e9cebbfc19.jpg" } }, "bbox": [ 273, 481, 721, 546 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到" } ] }, "bbox": [ 114, 551, 154, 564 ] }, { "type": "equation_interline", "content": { "math_content": "| J | = \\frac {\\partial (x , y , z)}{\\partial (r , \\varphi , \\theta)} = a b c r ^ {2} \\sin \\varphi", "math_type": "latex", "image_source": { "path": "images/f642393a799efd07d96475bd1bd57c8337d0f5fe463a6367c0a44d2b005fd6ca.jpg" } }, "bbox": [ 389, 561, 608, 593 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习15.10广义球坐标变换(1)计算 " }, { "type": "equation_inline", "content": "I = \\iiint_{V}z\\mathrm{d}x\\mathrm{d}y\\mathrm{d}z" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\frac{x^2}{a^2} +\\frac{y^2}{b^2} +\\frac{z^2}{c^2}\\leq 1" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "z\\geq 0" }, { "type": "text", "content": " 围成的区域解(1)做球坐标变换，积分变为：" } ] }, "bbox": [ 58, 621, 870, 668 ] }, { "type": "equation_interline", "content": { "math_content": "\\iiint_ {V ^ {\\prime}} c r \\cos \\varphi \\cdot a b c r ^ {2} \\sin \\varphi \\mathrm{d} r \\mathrm{d} \\varphi \\mathrm{d} \\theta = a b c ^ {2} \\int_ {0} ^ {\\frac {\\pi}{2}} \\mathrm{d} \\varphi \\int_ {0} ^ {2 \\pi} \\mathrm{d} \\theta \\int_ {0} ^ {1} r ^ {3} \\sin \\varphi \\cos \\varphi \\mathrm{d} r = \\frac {\\pi}{4} a b c ^ {2}", "math_type": "latex", "image_source": { "path": "images/f477a32fdadec46efd44d485da7fca6bcb32884700752c230805af806db8f1ab.jpg" } }, "bbox": [ 189, 671, 806, 703 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "15.2 三重积分" } ] }, "bbox": [ 798, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "129" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 16 章 曲线积分" } ], "level": 2 }, "bbox": [ 388, 87, 626, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.1 曲线的参数化" } ], "level": 2 }, "bbox": [ 95, 145, 302, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.2 第一型曲线积分" } ], "level": 2 }, "bbox": [ 95, 185, 324, 205 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.2.1 第一型曲线积分的基本概念" } ], "level": 2 }, "bbox": [ 94, 224, 403, 241 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 16.1 (第一型曲线积分)" } ], "level": 2 }, "bbox": [ 122, 256, 329, 272 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "R^{3}" }, { "type": "text", "content": " 中的可求长曲线, " }, { "type": "equation_inline", "content": "f:\\Gamma\\to R" }, { "type": "text", "content": ", 将 " }, { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 划分为分段 " }, { "type": "equation_inline", "content": "s_{1},\\cdots,s_{n}" }, { "type": "text", "content": ", 每段的弧长为 " }, { "type": "equation_inline", "content": "\\Delta s_{i},\\xi_{i}=(x_{i},y_{i},z_{i})\\in s_{i}" }, { "type": "text", "content": ", 定义" } ] }, "bbox": [ 112, 275, 882, 310 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {\\Gamma} f (x, y, z) \\mathrm{d} s = \\lim _ {\\Delta s _ {i} \\rightarrow 0} \\sum_ {i = 1} ^ {n} f (x _ {i}, y _ {i}, z _ {i}) \\Delta s _ {i}", "math_type": "latex", "image_source": { "path": "images/f0dbe9ae5a59d04960b8ed2a974453c8493a7bfeeb656434f7bb14dc5c4b69b5.jpg" } }, "bbox": [ 341, 307, 653, 344 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.1 (第一型曲线积分的计算)" } ], "level": 2 }, "bbox": [ 122, 370, 379, 385 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\mathbb{R}^3" }, { "type": "text", "content": " 中的光滑曲线，且曲线可表示为 " }, { "type": "equation_inline", "content": "\\mathbf{r} = x(t)\\mathbf{i} + y(t)\\mathbf{j} + z(t)\\mathbf{k}, t \\in [a, b]" }, { "type": "text", "content": "，则曲线积分为" } ] }, "bbox": [ 112, 390, 778, 406 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {\\Gamma} f (x, y, z) \\mathrm{d} s = \\int_ {a} ^ {b} f (x (t), y (t), z (t)) \\sqrt {x ^ {\\prime 2} (t) + y ^ {\\prime 2} (t) + z ^ {\\prime 2} (t)} \\mathrm{d} t", "math_type": "latex", "image_source": { "path": "images/fad74ad05a2889cc320726f844f5e9678c437b84cce4f1f0d80a01ec356b0e5c.jpg" } }, "bbox": [ 260, 410, 734, 444 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 设 " }, { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 上的分点对应 " }, { "type": "equation_inline", "content": "r_{0}, r_{1}, \\cdots, r_{n}" }, { "type": "text", "content": " ，对应 " }, { "type": "equation_inline", "content": "a = t_{0} < t_{1} < \\cdots < t_{n-1} < t_{n} = b" }, { "type": "text", "content": " 。首先根据弧长公式得到 " }, { "type": "equation_inline", "content": "\\Delta s_{i} = \\int_{t_{i-1}}^{t_{i}} \\sqrt{x'^{2}(t) + y'^{2}(t) + z'^{2}(t)} \\mathrm{d}t" }, { "type": "text", "content": " ，根据积分中值定理得到 " }, { "type": "equation_inline", "content": "\\Delta s_{i} = \\sqrt{x'^{2}(\\zeta_{i}) + y'^{2}(\\zeta_{i}) + z'^{2}(\\zeta_{i})} \\Delta t_{i}, \\zeta_{i} \\in [t_{i-1}, t_{i}]" }, { "type": "text", "content": " ，代入第一型曲线积分的定义得到 " }, { "type": "equation_inline", "content": "\\int_{\\Gamma} f(x, y, z) \\, ds = \\lim_{||T|| \\to 0} \\sum_{i=1}^{n} f(x_{i}, y_{i}, z_{i}) \\sqrt{x'^{2}(\\zeta_{i}) + y'^{2}(\\zeta_{i}) + z'^{2}(\\zeta_{i})} \\Delta t_{i}" }, { "type": "text", "content": " ，当 " }, { "type": "equation_inline", "content": "\\Delta t_{i} \\to 0" }, { "type": "text", "content": " 时即 Riemann 积分，因此得到结论。" } ] }, "bbox": [ 89, 461, 905, 558 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 16.1 (二维第一型曲线积分的计算)" } ], "level": 2 }, "bbox": [ 122, 568, 413, 583 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 " }, { "type": "equation_inline", "content": "\\Gamma \\subset \\mathbb{R}^2" }, { "type": "text", "content": " ，若平面曲线 " }, { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 可表示为 " }, { "type": "equation_inline", "content": "y = \\varphi (x)" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "\\varphi" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "[a,b]" }, { "type": "text", "content": " 连续，则" } ] }, "bbox": [ 114, 589, 623, 605 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {\\Gamma} f (x, y) \\mathrm{d} s = \\int_ {a} ^ {b} f (x, \\varphi (x)) \\sqrt {1 + \\varphi^ {\\prime 2} (x)} \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/fc2ad573347b12e687305ad05687032434548b11a30bce574851a4fc1f9a9551.jpg" } }, "bbox": [ 334, 609, 660, 642 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 第一型曲线积分没有方向一说，例如 " }, { "type": "equation_inline", "content": "A, B" }, { "type": "text", "content": " 曲线上参数 " }, { "type": "equation_inline", "content": "t \\in [a, b]" }, { "type": "text", "content": "，那么积分时 " }, { "type": "equation_inline", "content": "AB" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "BA" }, { "type": "text", "content": " 的积分都是 " }, { "type": "equation_inline", "content": "\\int_{a}^{b} f(x, y) \\mathrm{d}t" }, { "type": "text", "content": "，不能写为 " }, { "type": "equation_inline", "content": "\\int_{b}^{a} f(x, y) \\mathrm{d}t" } ] }, "bbox": [ 87, 659, 904, 709 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.1极坐标格式 " }, { "type": "equation_inline", "content": "(1)L:x^{2} + y^{2} = 1" }, { "type": "text", "content": " ，求 " }, { "type": "equation_inline", "content": "\\int_{L}(x^{2} + y^{2})\\mathrm{d}s,\\int_{L}(x^{2} + 2y^{2})\\mathrm{d}s" } ] }, "bbox": [ 58, 713, 648, 745 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\int_{L} \\sqrt{2y^2 + z^2} \\, \\mathrm{d}s" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "x^2 + y^2 + z^2 = a^2" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "x = y" }, { "type": "text", "content": " 相交的圆周" } ] }, "bbox": [ 121, 743, 640, 772 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "\\int_{L} (x^{2} + y^{2} + z^{2}) \\, \\mathrm{d}s" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 是螺旋线 " }, { "type": "equation_inline", "content": "x = a \\cos t, y = a \\sin t, z = bt, t \\in [0, 2\\pi]" }, { "type": "text", "content": " 的一段" } ] }, "bbox": [ 122, 772, 776, 800 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 积分较难算： " }, { "type": "equation_inline", "content": "\\int_{L} xy \\, \\mathrm{d}s" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1" }, { "type": "text", "content": " 在第一象限的部分解 (1) " }, { "type": "equation_inline", "content": "\\int_{L} (x^{2} + y^{2}) \\, \\mathrm{d}s = \\int_{L} 1 \\, \\mathrm{d}s = 2\\pi" }, { "type": "text", "content": "。记 " }, { "type": "equation_inline", "content": "x = \\cos \\theta, y = \\sin \\theta" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "(x'(\\theta))^{2} + (y'(\\theta))^{2} = 1" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\int_{L} (x^{2} + 2y^{2}) \\, \\mathrm{d}s = \\int_{0}^{2\\pi} (1 + \\sin^{2}\\theta) \\, \\mathrm{d}\\theta = 2\\pi + 4 \\cdot \\frac{1}{2} \\cdot \\frac{\\pi}{2} = 3\\pi" } ] }, "bbox": [ 122, 797, 633, 829 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 87, 828, 905, 889 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 显然 " }, { "type": "equation_inline", "content": "2y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\int_{L} \\sqrt{2y^2 + z^2} \\, \\mathrm{d}s = a \\cdot 2a\\pi = 2\\pi a^2" } ] }, "bbox": [ 122, 888, 591, 916 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "x^{2}+y^{2}+z^{2}=a^{2}+b^{2}t^{2},\\quad(x^{\\prime})^{2}+(y^{\\prime})^{2}+(z^{\\prime})^{2}=a^{2}+b^{2}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "I=\\int_{0}^{2\\pi}(a^{2}+b^{2}t^{2})\\sqrt{a^{2}+b^{2}}dt=" }, { "type": "equation_inline", "content": "\\sqrt{a^{2}+b^{2}}(2\\pi a^{2}+\\frac{8}{3}\\pi^{3}b^{2})" } ] }, "bbox": [ 90, 84, 905, 137 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 令 " }, { "type": "equation_inline", "content": "x = a \\cos t, y = b \\sin t" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\int_{L} xy \\, \\mathrm{d}s = ab \\int_{0}^{\\frac{\\pi}{2}} \\sin \\theta \\cos \\theta \\sqrt{a^2 \\sin^2 \\theta + b^2 \\cos^2 \\theta} \\, \\mathrm{d}\\theta" }, { "type": "text", "content": "，可化简为：" } ] }, "bbox": [ 122, 137, 826, 171 ] }, { "type": "equation_interline", "content": { "math_content": "{\\frac {a b}{2 (a ^ {2} - b ^ {2})}} \\int_ {0} ^ {\\frac {\\pi}{2}} \\sqrt {(a ^ {2} - b ^ {2}) \\sin^ {2} \\theta + b ^ {2}} \\mathrm{d} [ (a ^ {2} - b ^ {2}) \\sin^ {2} \\theta ] = {\\frac {a b (a ^ {2} + a b + b ^ {2})}{3 (a + b)}}", "math_type": "latex", "image_source": { "path": "images/755a8c3943774ed871ce3579c1503def93bc59bd8e46a616d54d7d3001927cff.jpg" } }, "bbox": [ 226, 174, 769, 206 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.2折线形式(1)计算 " }, { "type": "equation_inline", "content": "\\int_{L}y\\mathrm{d}s" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "y^{2} = 4x" }, { "type": "text", "content": " 从(0,0)到(1,2)的一段" } ] }, "bbox": [ 58, 217, 678, 247 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "\\int_{L} (x + y) \\, \\mathrm{d}s" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 是以 " }, { "type": "equation_inline", "content": "O(0,0), A(1,0), B(0,1)" }, { "type": "text", "content": " 为顶点的三角形" } ] }, "bbox": [ 119, 247, 675, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 " }, { "type": "equation_inline", "content": "(1)x = \\frac{1}{4} y^2, y \\in [0,2]" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\int_{L} y \\, \\mathrm{d}s = \\int_{0}^{2} y\\sqrt{1 + \\frac{1}{4} y^2} \\, \\mathrm{d}y = \\frac{4}{3}(2\\sqrt{2} - 1)" } ] }, "bbox": [ 87, 275, 628, 307 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 分三段进行考虑。 " }, { "type": "equation_inline", "content": "\\int_{OA} (x + y) \\mathrm{d}s = \\int_{0}^{1} (x) \\mathrm{d}x = \\frac{1}{2}" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "\\int_{OB} (x + y) \\mathrm{d}s = \\frac{1}{2}" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "AB" }, { "type": "text", "content": " 上 " }, { "type": "equation_inline", "content": "y = 1 - x, x \\in [0,1]" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "\\int_{AB} (x + y) \\mathrm{d}s = \\int_{0}^{1} 1\\sqrt{1 + 1} \\mathrm{d}x = \\sqrt{2}" }, { "type": "text", "content": ", 结果为 " }, { "type": "equation_inline", "content": "1 + \\sqrt{2}" } ] }, "bbox": [ 87, 307, 905, 369 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.2.2 使用对称性" } ], "level": 2 }, "bbox": [ 94, 387, 263, 404 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.3 镜面对称性 (1) 计算 " }, { "type": "equation_inline", "content": "\\int_{L} xy ds" }, { "type": "text", "content": " ，其中 L 为上半椭圆 " }, { "type": "equation_inline", "content": "\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (y \\geq 0)" } ] }, "bbox": [ 58, 413, 680, 445 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)重要：计算 " }, { "type": "equation_inline", "content": "\\int_{L}|y|\\mathrm{d}s" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为双组线 " }, { "type": "equation_inline", "content": "(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})" } ] }, "bbox": [ 122, 447, 620, 470 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 计算 " }, { "type": "equation_inline", "content": "\\oint_{L} |x|^{\\frac{1}{3}} \\mathrm{d}s" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为星形线 " }, { "type": "equation_inline", "content": "x^{\\frac{2}{3}} + y^{\\frac{2}{3}} = a^{\\frac{2}{3}}" } ] }, "bbox": [ 122, 474, 502, 502 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)由于 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 轴对称，而 " }, { "type": "equation_inline", "content": "xy" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 轴对称，因此 " }, { "type": "equation_inline", "content": "\\int_{L} xy \\, \\mathrm{d}s = 0" } ] }, "bbox": [ 87, 501, 601, 530 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 显然 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 轴均对称，因此将 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 分为四段，设 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 是第一象限的部分，因此 " }, { "type": "equation_inline", "content": "\\int_{L} |y| \\mathrm{~d}s = 4 \\int_{L_{1}} y \\mathrm{~d}s" }, { "type": "text", "content": " 考虑 " }, { "type": "equation_inline", "content": "x = r \\cos \\theta, y = r \\sin \\theta" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 极坐标方程为 " }, { "type": "equation_inline", "content": "r^{2} = a^{2} \\cos 2\\theta" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 方程为 " }, { "type": "equation_inline", "content": "r = a \\sqrt{\\cos 2\\theta}" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\theta \\in [0, \\frac{\\pi}{4}]" }, { "type": "text", "content": "，即：" } ] }, "bbox": [ 89, 530, 905, 579 ] }, { "type": "equation_interline", "content": { "math_content": "4 \\int_ {L _ {1}} y \\mathrm{d} s = 4 \\int_ {0} ^ {\\frac {\\pi}{4}} a \\sqrt {\\cos 2 \\theta} \\sin \\theta \\sqrt {a ^ {2} \\cos 2 \\theta + a ^ {2} \\frac {\\sin^ {2} 2 \\theta}{\\cos 2 \\theta}} \\mathrm{d} \\theta = 4 a ^ {2} \\int_ {0} ^ {\\frac {\\pi}{4}} \\sin \\theta \\mathrm{d} \\theta = (4 - 2 \\sqrt {2}) a ^ {2}", "math_type": "latex", "image_source": { "path": "images/9ca6a4d9dc744d304580117c51da1956d39e5f0dd0cbf14541ba21fd5c634c73.jpg" } }, "bbox": [ 164, 586, 830, 623 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 显然关于 " }, { "type": "equation_inline", "content": "x, y" }, { "type": "text", "content": " 轴对称，因此 " }, { "type": "equation_inline", "content": "\\int_{L} |x|^{\\frac{1}{3}} \\mathrm{d}s = 4 \\int_{L_1} x^{\\frac{1}{3}} \\mathrm{d}s" }, { "type": "text", "content": "。设 " }, { "type": "equation_inline", "content": "x = a \\cos^3 \\theta, y = a \\sin^3 \\theta, \\theta \\in [0, \\frac{\\pi}{2}]" }, { "type": "text", "content": "，此时 " }, { "type": "equation_inline", "content": "x'^2 y'^2 = 9a^2 \\sin^2 \\cos^2 \\theta" }, { "type": "text", "content": "，结果为：" } ] }, "bbox": [ 87, 627, 902, 671 ] }, { "type": "equation_interline", "content": { "math_content": "I = 4 \\int_ {L _ {1}} x ^ {\\frac {1}{3}} \\mathrm{d} s = 1 2 a ^ {\\frac {4}{3}} \\int_ {0} ^ {\\frac {\\pi}{2}} \\cos^ {2} \\theta \\sin \\theta \\mathrm{d} \\theta = 4 a ^ {\\frac {4}{3}}", "math_type": "latex", "image_source": { "path": "images/114e8da6e0ec80842187310cb69a22ab4c74297c3d093e85f1a08adceef75150.jpg" } }, "bbox": [ 322, 675, 673, 708 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.4 轮换对称性 (1) L 是 " }, { "type": "equation_inline", "content": "x + y + z = 0, x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 的交线，计算 " }, { "type": "equation_inline", "content": "\\int_{L} x ds, \\int_{L} x^{2} ds, \\int_{L} y dz ds" }, { "type": "text", "content": " 的值" } ] }, "bbox": [ 58, 718, 863, 747 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)L是 " }, { "type": "equation_inline", "content": "x + y + z = 0,x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 的交线，计算 " }, { "type": "equation_inline", "content": "\\int_{L}x^{2} + yz\\mathrm{d}s" } ] }, "bbox": [ 121, 747, 618, 775 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)考虑轮换对称性，查看交线，发现 " }, { "type": "equation_inline", "content": "x, y, z" }, { "type": "text", "content": " 轴都等价，因此变量可以随意轮换" } ] }, "bbox": [ 87, 775, 694, 790 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l} \\int_ {L} x \\mathrm{d} s = \\frac {1}{3} \\int_ {L} x + y + z \\mathrm{d} s = 0 \\\\ \\int_ {L} x ^ {2} \\mathrm{d} s = \\frac {1}{3} \\int_ {L} x ^ {2} + y ^ {2} + z ^ {2} \\mathrm{d} s = \\frac {1}{3} \\cdot 2 \\pi = \\frac {2 \\pi}{3} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/61d67d8667bfa3929a0f29083f5702ffa0a27184718190f01ea9830b19892d98.jpg" } }, "bbox": [ 312, 793, 667, 848 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "第三个：由于 " }, { "type": "equation_inline", "content": "(x + y + z) = 0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "(x + y + z)^2 = 0" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "(x + y + z)^2 -(x^2 +y^2 +z^2) = 2(xy + yz + xz) = -1," } ] }, "bbox": [ 90, 852, 905, 869 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "131" } ] }, "bbox": [ 485, 954, 510, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.2 第一型曲线积分" } ] }, "bbox": [ 747, 55, 904, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即 " }, { "type": "equation_inline", "content": "xy + yz + xz = -\\frac{1}{2}" }, { "type": "text", "content": "，因此：" } ] }, "bbox": [ 90, 80, 317, 110 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {L} y z \\mathrm{d} s = \\frac {1}{3} \\int_ {L} x y + y z + x z \\mathrm{d} s = \\frac {1}{3} \\cdot - \\frac {1}{6} \\cdot 2 \\pi = - \\frac {\\pi}{3}", "math_type": "latex", "image_source": { "path": "images/34862c8a66ffeec93ab8e80c4ddcff3a93bab82caeb6781dc1b42decf1c5fb01.jpg" } }, "bbox": [ 304, 111, 692, 143 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 和第一题本质一样" } ] }, "bbox": [ 122, 143, 287, 159 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.2.3 曲线积分中值问题" } ], "level": 2 }, "bbox": [ 95, 181, 324, 198 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.2 (曲线积分中值定理)" } ], "level": 2 }, "bbox": [ 121, 214, 346, 229 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "f(x,y)" }, { "type": "text", "content": " 是光滑曲线 " }, { "type": "equation_inline", "content": "L:x = x(t),y = y(t),t\\in [a,b]" }, { "type": "text", "content": " 上的连续函数，设 " }, { "type": "equation_inline", "content": "\\Delta L" }, { "type": "text", "content": " 是曲线 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 的弧长，则 " }, { "type": "equation_inline", "content": "\\exists (x_0,y_0)\\in L," }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 112, 233, 880, 267 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {L} f (x, y) \\mathrm{d} s = f (x _ {0}, y _ {0}) \\Delta L", "math_type": "latex", "image_source": { "path": "images/675ab06692e6a4d1038341c21b84d1ddd22e65f5d0896c247e8ae4be06f37889.jpg" } }, "bbox": [ 394, 263, 601, 296 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 根据计算公式得到 " }, { "type": "equation_inline", "content": "\\int_{L} f(x, y) \\mathrm{d}s = \\int_{a}^{b} f(x(t), y(t)) \\sqrt{(x'(t))^2 + (y'(t))^2} \\mathrm{d}t" }, { "type": "text", "content": "，根据积分第一中值定理得到等于 " }, { "type": "equation_inline", "content": "f(x(t_0), y(t_0)) \\int_{a}^{b} \\sqrt{(x'(t))^2 + (y'(t))^2} \\mathrm{d}t = f(x_0, y_0) \\Delta L" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "(x_0, y_0) = (x(t_0), y(t_0))" } ] }, "bbox": [ 89, 307, 905, 373 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.5 积分第一中值定理的曲线推广 " }, { "type": "equation_inline", "content": "f(x,y,z)" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "g(x,y,z)" }, { "type": "text", "content": " 在可求长连续曲线 L: " }, { "type": "equation_inline", "content": "x = x(t)" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "y = y(t)" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "z = z(t)" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "t \\in [a,b]" }, { "type": "text", "content": " (曲线不一定光滑！)，上连续，且 " }, { "type": "equation_inline", "content": "g(x,y,z)" }, { "type": "text", "content": " 非负，证明： " }, { "type": "equation_inline", "content": "\\exists(\\xi,\\eta,\\zeta) \\in L" }, { "type": "text", "content": " 使得" } ] }, "bbox": [ 58, 382, 905, 418 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {L} f (x, y, z) g (x, y, z) \\mathrm{d} s = f (\\xi , \\eta , \\zeta) \\int_ {L} g (x, y, z) \\mathrm{d} s", "math_type": "latex", "image_source": { "path": "images/c7d07d2bc8ef251dbefd9a2e65ef70ee73cbd9ca40cae612b767e4de94b2b482.jpg" } }, "bbox": [ 314, 420, 682, 451 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 由于不一定光滑，因此不能用含参形式计算。由于 " }, { "type": "equation_inline", "content": "f(x,y,z)" }, { "type": "text", "content": " 连续，设在 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 上 " }, { "type": "equation_inline", "content": "m \\leq f(x(t), y(t), z(t)) \\leq M" }, { "type": "text", "content": "，由于 " }, { "type": "equation_inline", "content": "g" }, { "type": "text", "content": " 非负，因此 " }, { "type": "equation_inline", "content": "mg(x,y,z) \\leq f(x,y,z)g(x,y,z) \\leq Mg(x,y,z)" }, { "type": "text", "content": "，两侧积分得到：" } ] }, "bbox": [ 89, 453, 905, 488 ] }, { "type": "equation_interline", "content": { "math_content": "m \\int_ {L} g (x, y, z) \\mathrm{d} s \\leq \\int_ {L} f (x, y, z) g (x, y, z) \\mathrm{d} s \\leq M \\int_ {L} g (x, y, z) \\mathrm{d} s", "math_type": "latex", "image_source": { "path": "images/6e258d2771dc4452d2f9b03100fc410c146656aa3007d07dba205a5ef4646689.jpg" } }, "bbox": [ 265, 491, 729, 521 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "\\int_{L} g(x, y, z) \\mathrm{d}s = 0" }, { "type": "text", "content": "，则全部为零， " }, { "type": "equation_inline", "content": "(\\xi, \\eta, \\zeta)" }, { "type": "text", "content": " 任选。若不为 0，则同除以 " }, { "type": "equation_inline", "content": "\\int_{L} g(x, y, z) \\mathrm{d}s" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "m \\leq \\frac{\\int_{L} f(x, y, z) g(x, y, z) \\mathrm{d}s}{\\int_{L} f(x, y, z) \\mathrm{d}s} \\leq M" }, { "type": "text", "content": "，根据连续性介值性定理可证。" } ] }, "bbox": [ 89, 524, 952, 571 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.3 第二型曲线积分" } ], "level": 2 }, "bbox": [ 94, 596, 324, 617 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.3.1 基本概念与参数计算" } ], "level": 2 }, "bbox": [ 95, 636, 344, 653 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 16.2 (第二型曲线积分)" } ], "level": 2 }, "bbox": [ 122, 669, 329, 683 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "\\mathbb{R}^3" }, { "type": "text", "content": " 中的有向曲线，其端点为 " }, { "type": "equation_inline", "content": "A, B \\in \\mathbb{R}^3" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "F(x, y, z) = P(x, y, z)\\mathbf{i} + Q(x, y, z)\\mathbf{j} + R(x, y, z)\\mathbf{k}" }, { "type": "text", "content": "，将 " }, { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 划分为 " }, { "type": "equation_inline", "content": "A = \\mathbf{r}_0 \\to \\mathbf{r}_1 \\to \\mathbf{r}_{n-1} \\to \\mathbf{r}_n = B" }, { "type": "text", "content": "，第二型曲线积分定义为：" } ] }, "bbox": [ 112, 688, 880, 722 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {\\Gamma} P \\mathrm{d} x + Q \\mathrm{d} y + R \\mathrm{d} z = \\lim _ {| | T | | \\to 0} P (\\xi_ {i}, \\eta_ {i}, \\zeta_ {i}) \\Delta x _ {i} + Q (\\xi_ {i}, \\eta_ {i}, \\zeta_ {i}) \\Delta y _ {i} + R (\\xi_ {i}, \\eta_ {i}, \\zeta_ {i}) \\Delta z _ {i}", "math_type": "latex", "image_source": { "path": "images/bbb2c93433822cec4bcafcabdb38effd141f9da4cfc91d1201d0cc9abc341b0b.jpg" } }, "bbox": [ 196, 727, 798, 757 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.3 (第二型曲线积分的计算)" } ], "level": 2 }, "bbox": [ 122, 781, 379, 797 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若曲线 " }, { "type": "equation_inline", "content": "\\Gamma" }, { "type": "text", "content": " 由 " }, { "type": "equation_inline", "content": "\\mathbf{r}(t) = x(t)\\mathbf{i} + y(t)\\mathbf{j} + z(t)\\mathbf{k}, t \\in [a, b]" }, { "type": "text", "content": " 给出，则" } ] }, "bbox": [ 114, 801, 544, 816 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {\\Gamma} P \\mathrm{d} x + Q \\mathrm{d} y + R \\mathrm{d} z = \\int_ {a} ^ {b} [ P x ^ {\\prime} (t) + Q y ^ {\\prime} (t) + R z ^ {\\prime} (t) ] \\mathrm{d} t", "math_type": "latex", "image_source": { "path": "images/e1348229068d51b28d1fd4c7cb3b1c05a87ab4732d49e7fafb98b8722cafa7a8.jpg" } }, "bbox": [ 292, 822, 702, 854 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.6折线积分(1)计算 " }, { "type": "equation_inline", "content": "\\int_{L}x\\mathrm{d}y - y\\mathrm{d}x" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为三角形 " }, { "type": "equation_inline", "content": "O(0,0),A(1,0),B(1,2)" }, { "type": "text", "content": " 为顶点的三角形逆时针曲线" } ] }, "bbox": [ 58, 883, 905, 916 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.3 第二型曲线积分" } ] }, "bbox": [ 747, 54, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "132" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "\\int_{L} x dx + y dy + z dz" }, { "type": "text", "content": " ，其中 L 是 " }, { "type": "equation_inline", "content": "(1, 1, 1)" }, { "type": "text", "content": " 到 " }, { "type": "equation_inline", "content": "(2, 3, 4)" }, { "type": "text", "content": " 的直线段\n\n解 (1) 分三段计算： " }, { "type": "equation_inline", "content": "\\int_{OA} x \\, dx - y \\, dx = 0" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "\\int_{AB} x \\, dy - y \\, dx" }, { "type": "text", "content": " 以 y 为参数，即 " }, { "type": "equation_inline", "content": "\\int_{0}^{2} 1 \\, dy - y \\cdot 0 = 2" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "\\int_{BO} x \\, dy - y \\, dx" }, { "type": "text", "content": " ，以 x 为参数，y = 2x，即 " }, { "type": "equation_inline", "content": "\\int_{1}^{0} x \\cdot 2 - 2x \\, dx = 0" }, { "type": "text", "content": " ，最终结果为 2。" } ] }, "bbox": [ 89, 84, 905, 173 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 以 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 为参数, " }, { "type": "equation_inline", "content": "y = 2x - 1, z = 3x - 2" }, { "type": "text", "content": ", 因此结果为 " }, { "type": "equation_inline", "content": "\\int_{1}^{2} x + 2(2x - 1) + 3(3x - 2) \\mathrm{d}x = 13" } ] }, "bbox": [ 121, 173, 796, 205 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.7 椭圆、圆周积分 (1) 计算 " }, { "type": "equation_inline", "content": "\\int_{L}(x^{2}+2xy)\\mathrm{d}y" }, { "type": "text", "content": " ，其中 L 为 " }, { "type": "equation_inline", "content": "\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(y\\geq0)" }, { "type": "text", "content": " 方向逆时针" } ] }, "bbox": [ 58, 208, 793, 240 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L} \\frac{-x \\, \\mathrm{d}x + y \\, \\mathrm{d}y}{x^2 + y^2}" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为圆周 " }, { "type": "equation_inline", "content": "x^2 + y^2 = 1" }, { "type": "text", "content": "，方向逆时针解 (1) 设 " }, { "type": "equation_inline", "content": "x = a\\cos \\theta, y = b\\sin \\theta, \\theta \\in [0, \\pi]" }, { "type": "text", "content": "。则 " }, { "type": "equation_inline", "content": "\\int_{L} (x^2 + 2xy)\\mathrm{d}y = \\int_{0}^{\\pi} a^2\\cos^3\\theta + 2ab^2\\sin \\theta\\cos^2\\theta\\mathrm{d}\\theta = \\frac{4}{3}ab^2" }, { "type": "text", "content": " (2) 设 " }, { "type": "equation_inline", "content": "x = \\cos \\theta, y = \\sin \\theta, \\theta \\in [0, 2\\pi]" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "I = \\int_{0}^{2\\pi} (\\cos \\theta\\sin \\theta + \\sin \\theta\\cos \\theta)\\mathrm{d}\\theta = 0" } ] }, "bbox": [ 121, 237, 647, 269 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 90, 265, 860, 329 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.8 交线积分: 化为单变量 (1) 计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L} (y - z) \\, \\mathrm{d}x + (z - x) \\, \\mathrm{d}y + (x - y) \\, \\mathrm{d}z" }, { "type": "text", "content": " ，其中 L 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = 1, x + z = 1" }, { "type": "text", "content": " ，从 z 轴正无穷看为顺时针" } ] }, "bbox": [ 58, 337, 905, 376 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 重要：计算 " }, { "type": "equation_inline", "content": "\\oint_{L} xyz \\, \\mathrm{d}z" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "y = z" }, { "type": "text", "content": " 相交的圆，方向为 1,2,7,8 卦限。" } ] }, "bbox": [ 119, 378, 828, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 设 " }, { "type": "equation_inline", "content": "x = \\cos \\theta, y = \\sin \\theta, z = 1 - \\cos \\theta" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\theta \\in [2\\pi, 0]" }, { "type": "text", "content": " (不是 " }, { "type": "equation_inline", "content": "[0, 2\\pi]!" }, { "type": "text", "content": "。)。则" } ] }, "bbox": [ 89, 403, 652, 419 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {2 \\pi} ^ {0} [ (\\sin \\theta + \\cos \\theta - 1) (- \\sin \\theta) + (1 - 2 \\cos \\theta) \\cos \\theta + (\\cos \\theta - \\sin \\theta) \\sin \\theta ] \\mathrm{d} \\theta = 4 \\pi", "math_type": "latex", "image_source": { "path": "images/d18c4da6eb31115a502050637602cd60534592994cc9c5bb714de4cb93761188.jpg" } }, "bbox": [ 188, 423, 806, 455 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 由于 " }, { "type": "equation_inline", "content": "x^{2} + 2y^{2} = 1" }, { "type": "text", "content": "，设 " }, { "type": "equation_inline", "content": "x = \\cos \\theta, y = z = \\frac{1}{\\sqrt{2}} \\sin \\theta" }, { "type": "text", "content": "，由于从第一卦限出发， " }, { "type": "equation_inline", "content": "x, y, z" }, { "type": "text", "content": " 均为正，而 " }, { "type": "equation_inline", "content": "\\theta" }, { "type": "text", "content": " 为小正数时，位于第一卦限，因此 " }, { "type": "equation_inline", "content": "\\theta \\in [0, 2\\pi]" }, { "type": "text", "content": "。因此：" } ] }, "bbox": [ 87, 458, 904, 500 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {0} ^ {2 \\pi} \\frac {1}{2} \\cos \\theta \\sin^ {2} \\theta \\frac {1}{\\sqrt {2}} \\cos \\theta \\mathrm{d} \\theta = \\frac {\\sqrt {2} \\pi}{1 6}", "math_type": "latex", "image_source": { "path": "images/0cff7b461bf9583b6d6c0bb3af5ace845e360ed67dccb686f10e82ca7e119ba9.jpg" } }, "bbox": [ 344, 502, 650, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 第二题虽然是球面的一部分，但是曲线积分都要化成一个变量的，因此不能用球坐标变换！" } ] }, "bbox": [ 58, 542, 794, 558 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.9二次曲面交线计算 " }, { "type": "equation_inline", "content": "\\int_{L}y\\mathrm{d}x + z\\mathrm{d}y + x\\mathrm{d}z" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为椭球与柱面交线：" } ] }, "bbox": [ 58, 574, 668, 605 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {x ^ {2}}{a ^ {2}} + \\frac {y ^ {2}}{b ^ {2}} + \\frac {z ^ {2}}{c ^ {2}} = 1, \\frac {x}{a} + \\frac {z}{c} = 1, x, y, z \\geq 0", "math_type": "latex", "image_source": { "path": "images/2428ee3243b5da92e87a09d53dfa4e572885bdf0d83d88e7bc01a33b4b037821.jpg" } }, "bbox": [ 344, 607, 652, 638 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "方向从 " }, { "type": "equation_inline", "content": "(a,0,0)" }, { "type": "text", "content": " 到 " }, { "type": "equation_inline", "content": "(0,0,c)" } ] }, "bbox": [ 89, 638, 280, 653 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 直接用极坐标极度不方便，先令 " }, { "type": "equation_inline", "content": "x = a\\overline{x}, y = b\\overline{y}, z = c\\overline{z}" }, { "type": "text", "content": "，得到 " }, { "type": "equation_inline", "content": "\\overline{x}^2 + \\overline{y}^2 + \\overline{z}^2 = 1, \\overline{x} + \\overline{z} = 1" }, { "type": "text", "content": "。将前者配凑为 " }, { "type": "equation_inline", "content": "(\\overline{x} + \\overline{z})^2 + (\\overline{x} - \\overline{z})^2 + 2\\overline{y}^2 = 2" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "(\\overline{x} - \\overline{z})^2 + 2\\overline{y}^2 = 1" }, { "type": "text", "content": "，因此设 " }, { "type": "equation_inline", "content": "\\overline{x} - \\overline{z} = \\cos \\theta, \\sqrt{2}\\overline{y} = \\sin \\theta" }, { "type": "text", "content": "，得出 " }, { "type": "equation_inline", "content": "\\overline{x} = \\frac{1}{2}(1 + \\cos \\theta), \\overline{y} = \\frac{1}{\\sqrt{2}}\\sin \\theta, \\overline{z} = \\frac{1}{2}(1 - \\cos \\theta)" }, { "type": "text", "content": "，根据方向以及端点可判断 " }, { "type": "equation_inline", "content": "\\theta \\in [0, \\pi]" }, { "type": "text", "content": "（具体含义难以判断）。因此代入计算可知结果为 " }, { "type": "equation_inline", "content": "\\frac{ac}{2} - \\frac{\\pi b}{4\\sqrt{2}}(a + c)" } ] }, "bbox": [ 87, 656, 905, 746 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.3.2 使用对称性" } ], "level": 2 }, "bbox": [ 95, 766, 263, 783 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "和第一型曲线积分一样，下面考虑镜像对称以及轮换对称性" } ] }, "bbox": [ 121, 794, 566, 809 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 镜像对称：如 " }, { "type": "equation_inline", "content": "\\int_{S} f(x, y, z) \\mathrm{d}x" }, { "type": "text", "content": "，要同时考虑 " }, { "type": "equation_inline", "content": "f(x, y, z)" }, { "type": "text", "content": " 和 " }, { "type": "equation_inline", "content": "\\mathrm{d}x" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 上的符号" } ] }, "bbox": [ 115, 810, 670, 834 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 轮换对称：要保证积分区域和积分表达式均轮换对称。轮换有两种使用方法：(1) 保持积分区域不变，积分表达式轮换 (2) 保持积分表达式不变，积分区域轮换 (具体见轮换对称练习)" } ] }, "bbox": [ 114, 835, 905, 869 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.3 第二型曲线积分" } ] }, "bbox": [ 747, 55, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "133" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.4 (dx 的符号)" } ], "level": 2 }, "bbox": [ 122, 87, 285, 102 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若沿着给定方向， " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 由大变小，则 " }, { "type": "equation_inline", "content": "\\mathrm{dx}" }, { "type": "text", "content": " 为负，同理若变大则 " }, { "type": "equation_inline", "content": "\\mathrm{dx}" }, { "type": "text", "content": " 为正，以此判断对称性。如果不容易直接判断，则画出投影曲线，再在平面曲线上判断方向。" } ] }, "bbox": [ 114, 105, 884, 141 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.10镜像对称性(1)证明 " }, { "type": "equation_inline", "content": "I = \\oint_{L}\\frac{-x\\mathrm{d}x + y\\mathrm{d}y}{x^{2} + y^{2}} = 0" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为圆周 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = 1" }, { "type": "text", "content": " ，方向逆时针。" } ] }, "bbox": [ 58, 170, 821, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "I = \\int_{L} (-2xe^{-x^2} \\sin y) \\, \\mathrm{d}x + (e^{-x^2} \\cos y + x^4) \\, \\mathrm{d}y" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 (1,0) 到 (-1,0) 的半圆 " }, { "type": "equation_inline", "content": "y = \\sqrt{1 - x^2}" }, { "type": "text", "content": " 解 (1) " }, { "type": "equation_inline", "content": "I = -\\int_{L} x \\, \\mathrm{d}x + \\int_{L} y \\, \\mathrm{d}y" }, { "type": "text", "content": "，前者分为 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 轴上下两部分，上下 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 符号大小均相等，由于上方 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 从大变小，下方从小变大，因此 " }, { "type": "equation_inline", "content": "\\mathrm{d}x" }, { "type": "text", "content": " 符号相反，得到 " }, { "type": "equation_inline", "content": "-\\int_{L} x \\, \\mathrm{d}x = 0" }, { "type": "text", "content": "。后者同理根据 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 轴上下两部分得到结果为 0。" } ] }, "bbox": [ 87, 202, 904, 282 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 关于 " }, { "type": "equation_inline", "content": "y" }, { "type": "text", "content": " 轴对称。第一项 " }, { "type": "equation_inline", "content": "\\mathrm{d}x" }, { "type": "text", "content": " 同号， " }, { "type": "equation_inline", "content": "-2xe^{-x^2}\\sin y" }, { "type": "text", "content": " 异号，因此第一个积分为0。第二个 " }, { "type": "equation_inline", "content": "\\mathrm{d}y" }, { "type": "text", "content": " 异号， " }, { "type": "equation_inline", "content": "e^{-x^2}\\cos y + x^4" }, { "type": "text", "content": " 同号等值，因此第二个积分也为0。综上整个积分为0。" } ] }, "bbox": [ 89, 285, 905, 319 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.11轮换对称性(1)计算 " }, { "type": "equation_inline", "content": "\\int_{L}x\\mathrm{d}x + y\\mathrm{d}y + z\\mathrm{d}z" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "x + y + z = 0" }, { "type": "text", "content": " 相交圆周， " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 从第一卦限来看是逆时针的。(2)计算 " }, { "type": "equation_inline", "content": "\\int_{L}(y^{2} - z^{2})\\mathrm{d}x + (z^{2} - x^{2})\\mathrm{d}y + (x^{2} - y^{2})\\mathrm{d}z" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 在第一卦限的边界曲线，方向为 " }, { "type": "equation_inline", "content": "xy" }, { "type": "text", "content": " 平面， " }, { "type": "equation_inline", "content": "yz" }, { "type": "text", "content": " 平面， " }, { "type": "equation_inline", "content": "zx" }, { "type": "text", "content": " 平面解(1)由于 " }, { "type": "equation_inline", "content": "x,y,z" }, { "type": "text", "content": " 可以随便互换， " }, { "type": "equation_inline", "content": "I" }, { "type": "text", "content": " 可以写出 " }, { "type": "equation_inline", "content": "3\\int_{L}x\\mathrm{d}x" }, { "type": "text", "content": " 或 " }, { "type": "equation_inline", "content": "y,z" }, { "type": "text", "content": " 对应积分，要选择一个好算的。考虑 " }, { "type": "equation_inline", "content": "z = -(x + y)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + (x + y)^{2} = 1" }, { "type": "text", "content": " ，得到 " }, { "type": "equation_inline", "content": "(x + \\frac{1}{2} y)^{2} + \\frac{3}{4} y^{2} = \\frac{1}{2}" }, { "type": "text", "content": " 。因此令 " }, { "type": "equation_inline", "content": "x + \\frac{1}{2} y = \\frac{1}{\\sqrt{2}}\\cos \\theta, \\frac{\\sqrt{3}}{2} y = \\frac{1}{\\sqrt{2}}\\sin \\theta" }, { "type": "text", "content": " ，因此得到参数方程为： " }, { "type": "equation_inline", "content": "x = \\frac{1}{\\sqrt{2}}\\cos \\theta - \\frac{1}{\\sqrt{6}}\\sin \\theta, y = \\frac{2}{\\sqrt{6}}\\sin \\theta, z = -\\frac{1}{\\sqrt{2}}\\cos \\theta - \\frac{1}{\\sqrt{6}}\\sin \\theta, \\theta \\in [0,2\\pi]" } ] }, "bbox": [ 58, 338, 907, 518 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "可以算 " }, { "type": "equation_inline", "content": "\\int_{L} y \\, \\mathrm{d}y" }, { "type": "text", "content": " 比较简单，因此最终结果为 0。" } ] }, "bbox": [ 90, 523, 425, 554 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 设三个部分曲线分别为 " }, { "type": "equation_inline", "content": "L_{1}, L_{2}, L_{3}" }, { "type": "text", "content": ", 由于 " }, { "type": "equation_inline", "content": "x, y, z" }, { "type": "text", "content": " 可以随便互换, 若 " }, { "type": "equation_inline", "content": "(x, y, z) \\to (z, x, y)" }, { "type": "text", "content": ", 则积分表达式不变, 而 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 部分在该变换下变为 " }, { "type": "equation_inline", "content": "L_{2}" }, { "type": "text", "content": ", 因此 " }, { "type": "equation_inline", "content": "L_{1}, L_{2}, L_{3}" }, { "type": "text", "content": " 积分值相等。因此 " }, { "type": "equation_inline", "content": "I = 3 \\int_{L_{1}} (y^{2} - z^{2}) \\mathrm{d}x + (z^{2} - x^{2}) \\mathrm{d}y + (x^{2} - y^{2}) \\mathrm{d}z =" }, { "type": "text", "content": ", 对于 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 而言, 虽然积分表达式轮换对称, 但是积分区域不轮换对称, 因此只能硬算了。取 " }, { "type": "equation_inline", "content": "x = \\cos \\theta, y = \\sin \\theta, z = 0, \\theta \\in [0, \\frac{\\pi}{2}]" }, { "type": "text", "content": ", 得到结果为 " }, { "type": "equation_inline", "content": "-4" }, { "type": "text", "content": "。" } ] }, "bbox": [ 87, 552, 907, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.12 双曲面交线 + 对称性 重点：计算 " }, { "type": "equation_inline", "content": "I=\\int_{L}y^{2}dx+z^{2}dy+x^{2}dz" }, { "type": "text", "content": " ，其中 L 是维维安尼曲线 " }, { "type": "equation_inline", "content": "x^{2}+y^{2}+z^{2}=a^{2}, x^{2}+y^{2}=ax (z\\geq0, a>0)" }, { "type": "text", "content": " ，从 x 轴 " }, { "type": "equation_inline", "content": "+\\infty" }, { "type": "text", "content": " 看去，L 方向为逆时针\n\n解 注意 " }, { "type": "equation_inline", "content": "x^{2}+y^{2}=ax" }, { "type": "text", "content": " 即 " }, { "type": "equation_inline", "content": "(x-\\frac{a}{2})^{2}+y^{2}=\\frac{a^{2}}{4}" }, { "type": "text", "content": " 是圆柱面，且过原点。考虑关于 xz 平面对称，积分表达式显然不变，dx 相反，dy 相同（需要把投影的圆画出来看看），dz 相反，因此只需要计算 " }, { "type": "equation_inline", "content": "I=\\int_{L}z^{2}dy=2\\int_{L_{1}}z^{2}dy" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 是 L 在 " }, { "type": "equation_inline", "content": "y\\geq0" }, { "type": "text", "content": " 的部分，设 " }, { "type": "equation_inline", "content": "x=a\\sin\\varphi\\cos\\theta, y=a\\sin\\varphi\\sin\\theta, z=a\\cos\\varphi, \\theta\\in[0, \\frac{\\pi}{2}] (\\theta" }, { "type": "text", "content": " 的范围具体见三重积分球坐标变换），代入柱面公式得到 " }, { "type": "equation_inline", "content": "\\sin\\varphi=\\cos\\theta" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "\\theta=\\frac{\\pi}{2}-\\varphi" }, { "type": "text", "content": " ，综上 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 的参数方程为：" } ] }, "bbox": [ 58, 645, 905, 788 ] }, { "type": "equation_interline", "content": { "math_content": "x = a \\sin^ {2} \\varphi , y = a \\sin \\varphi \\cos \\varphi , z = a \\cos \\varphi , \\varphi \\in [ \\frac {\\pi}{2}, 0 ]", "math_type": "latex", "image_source": { "path": "images/de2254356b0d95f6f0062f215c6cfa1795fbb7a5621bb6be9a6b599f5b22290f.jpg" } }, "bbox": [ 307, 793, 687, 816 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到最终结果为 " }, { "type": "equation_inline", "content": "-\\frac{\\pi}{4} a^3" } ] }, "bbox": [ 92, 819, 260, 844 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.3 第二型曲线积分" } ] }, "bbox": [ 747, 55, 902, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "134" } ] }, "bbox": [ 485, 954, 512, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.3.3 两类曲线积分的关系" } ], "level": 2 }, "bbox": [ 95, 85, 344, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.5 (光滑曲线切向量)" } ], "level": 2 }, "bbox": [ 122, 118, 329, 133 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "考虑光滑曲线 " }, { "type": "equation_inline", "content": "\\mathbf{r}(s) = (x(s), y(s)), s \\in [0, l]" }, { "type": "text", "content": "，满足 " }, { "type": "equation_inline", "content": "(x'(s))^2 + (y'(x))^2 = 1" }, { "type": "text", "content": "。则其单位切向量为 " }, { "type": "equation_inline", "content": "\\mathbf{t} = (x'(s), y'(s)) = (\\cos(\\mathbf{t}, x), \\cos(\\mathbf{t}, y))" }, { "type": "text", "content": "，即" } ] }, "bbox": [ 114, 137, 880, 174 ] }, { "type": "equation_interline", "content": { "math_content": "\\mathrm{d} x = \\cos (\\mathbf {t}, x) \\mathrm{d} s, \\quad \\mathrm{d} y = \\cos (\\mathbf {t}, y) \\mathrm{d} s", "math_type": "latex", "image_source": { "path": "images/aa1cb4f275b87c0b8ec9554b0046679cc537e3c6158ac69f470b30721594a1bc.jpg" } }, "bbox": [ 364, 181, 631, 197 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.6 (两类曲线积分的关系)" } ], "level": 2 }, "bbox": [ 121, 221, 363, 236 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "根据光滑曲线切向量的结论，可知：" } ] }, "bbox": [ 114, 241, 379, 255 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {L} P \\mathrm{d} x + Q \\mathrm{d} y = \\int_ {L} \\left[ P (x, y) \\cos (\\mathbf {t}, x) + Q (x, y) \\cos (\\mathbf {t}, y) \\right] \\mathrm{d} s", "math_type": "latex", "image_source": { "path": "images/6da9469ab6467487fbcf0461cc0da4020872b5ce940cc0da01731ada592193ef.jpg" } }, "bbox": [ 278, 256, 717, 288 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.4 Green 公式" } ], "level": 2 }, "bbox": [ 95, 332, 275, 351 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.4.1 Green 公式及其基本应用" } ], "level": 2 }, "bbox": [ 95, 370, 379, 387 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.7 (Green 公式)" } ], "level": 2 }, "bbox": [ 122, 403, 295, 418 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\Omega \\subset \\mathbb{R}^2" }, { "type": "text", "content": " 是有限条分段光滑曲线连接成的闭曲线， " }, { "type": "equation_inline", "content": "P(x,y), Q(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "\\Omega" }, { "type": "text", "content": " 上连续，且有连续偏导 " }, { "type": "equation_inline", "content": "\\frac{\\partial Q}{\\partial x}, \\frac{\\partial P}{\\partial y}" }, { "type": "text", "content": "，沿着 " }, { "type": "equation_inline", "content": "\\partial \\Omega" }, { "type": "text", "content": " 的方向， " }, { "type": "equation_inline", "content": "\\Omega" }, { "type": "text", "content": " 一直在左侧，则" } ] }, "bbox": [ 112, 423, 880, 464 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {\\partial \\Omega} P \\mathrm{d} x + Q \\mathrm{d} y = \\iint_ {\\Omega} (\\frac {\\partial Q}{\\partial x} - \\frac {\\partial P}{\\partial y}) \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/1586a0f265f28dbc548b05a4704292a9ee4551a5b1864cd41532bc8a58936915.jpg" } }, "bbox": [ 351, 467, 645, 499 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 使用 Green 公式的基本条件是曲线光滑和偏导存在，一般在 P, Q 本身复杂，但 " }, { "type": "equation_inline", "content": "\\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y}" }, { "type": "text", "content": " 简单时使用。记忆时可记忆为：" } ] }, "bbox": [ 87, 511, 905, 552 ] }, { "type": "equation_interline", "content": { "math_content": "\\left| \\begin{array}{c c} \\frac {\\partial}{\\partial x} & \\frac {\\partial}{\\partial y} \\\\ P & Q \\end{array} \\right|", "math_type": "latex", "image_source": { "path": "images/a4a8978d56471c648e277d692e441473f54d368ca27daa32c81725cb87d99628.jpg" } }, "bbox": [ 448, 549, 546, 596 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.13Green 公式基本应用 (1)a, b 为正数，L 为从 (2a, 0) 到 (0, 0) 的半圆 " }, { "type": "equation_inline", "content": "y = \\sqrt{2ax - x^{2}}" }, { "type": "text", "content": " ，计算" } ] }, "bbox": [ 58, 614, 825, 630 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {L} [ e ^ {x} \\sin y - b (x + y) ] \\mathrm{d} x + (e ^ {x} \\cos y - a x) \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/a1b3960b2a21670dd34c10f5647079650cb2818bea8f066d102b4b4c90b37571.jpg" } }, "bbox": [ 315, 634, 678, 663 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)L 为 " }, { "type": "equation_inline", "content": "\\sin x" }, { "type": "text", "content": " 从 " }, { "type": "equation_inline", "content": "(0,0)" }, { "type": "text", "content": " 到 " }, { "type": "equation_inline", "content": "(\\pi,0)" }, { "type": "text", "content": " 的一段，计算：" } ] }, "bbox": [ 121, 665, 457, 681 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {L} \\sqrt {x ^ {2} + y ^ {2}} \\mathrm{d} x + y \\left[ x y + \\ln \\left(x + \\sqrt {x ^ {2} + y ^ {2}}\\right) \\right] \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/edaaa6187e8b6e7549909cef31db1360b885b1328c344782f6484f62731eca86.jpg" } }, "bbox": [ 300, 684, 695, 714 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为曲线 " }, { "type": "equation_inline", "content": "y = \\sin x" }, { "type": "text", "content": " 从 " }, { "type": "equation_inline", "content": "(0,0)" }, { "type": "text", "content": " 到 " }, { "type": "equation_inline", "content": "(\\pi ,0)" }, { "type": "text", "content": " 的一段，计算" } ] }, "bbox": [ 122, 717, 515, 733 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {L} e ^ {x} \\left[ (1 - \\cos y) \\mathrm{d} x - (y - \\sin y) \\mathrm{d} y \\right]", "math_type": "latex", "image_source": { "path": "images/9e4f8833d46fc9c9d05f69e8fd88b5e8b2bf40d5d90dc8ed1827201d1eb5e212.jpg" } }, "bbox": [ 349, 736, 645, 766 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)显然积分本身很麻烦，考虑 " }, { "type": "equation_inline", "content": "P = e^{x}\\sin y - b(x + y),Q = e^{x}\\cos y - ax" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "Q_{x} - P_{y} = b - a" }, { "type": "text", "content": " 很方便，因此考虑用Green公式把圆周补上，记 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "(0,0)\\rightarrow (2a,0)" }, { "type": "text", "content": " 直线。则 " }, { "type": "equation_inline", "content": "I = \\int_{L + L_1} - \\int_{L_1} = \\iint_Db - a\\mathrm{d}x\\mathrm{d}y - \\int_{L_1} - bx\\mathrm{d}x =" }, { "type": "equation_inline", "content": "\\frac{1}{2}\\pi a^2 (b - a) + 2a^2 b" } ] }, "bbox": [ 89, 766, 905, 838 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 考虑 " }, { "type": "equation_inline", "content": "Q_x - P_y = y^2 + \\frac{y + \\frac{xy}{\\sqrt{x^2 + y^2}}}{x + \\sqrt{x^2 + y^2}} - \\frac{y}{\\sqrt{x^2 + y^2}} = y^2" }, { "type": "text", "content": "，记 " }, { "type": "equation_inline", "content": "L_1" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "(\\pi, 0)" }, { "type": "text", "content": " 到 " }, { "type": "equation_inline", "content": "(0, 0)" }, { "type": "text", "content": " 的直线段，则 " }, { "type": "equation_inline", "content": "\\int_L P \\, \\mathrm{d}x + Q \\, \\mathrm{d}y =" } ] }, "bbox": [ 121, 837, 905, 877 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.4 Green 公式" } ] }, "bbox": [ 786, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "135" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "#" } ] }, "bbox": [ 884, 612, 900, 624 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\int_{\\pi}^{0}x\\mathrm{d}x = -\\frac{1}{2}\\pi^2" }, { "type": "text", "content": " ，因此：" } ] }, "bbox": [ 90, 83, 275, 114 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\int_ {L + L _ {1}} - \\int_ {L _ {1}} = - \\iint_ {D} y ^ {2} \\mathrm{d} x \\mathrm{d} y + \\frac {1}{2} \\pi^ {2} = - \\int_ {0} ^ {\\pi} \\mathrm{d} x \\int_ {0} ^ {\\sin x} y ^ {2} \\mathrm{d} y + \\frac {1}{2} \\pi^ {2} = \\frac {1}{2} \\pi^ {2} - \\frac {4}{9}", "math_type": "latex", "image_source": { "path": "images/6dc9f36297e9c733015005bf5a2f89efd5d79c66b4fc676ba4a102f09275c04c.jpg" } }, "bbox": [ 194, 118, 801, 151 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 首先 " }, { "type": "equation_inline", "content": "Q_x - P_y = -ye^x" }, { "type": "text", "content": "，记 " }, { "type": "equation_inline", "content": "L_1" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "(\\pi, 0) \\to (0, 0)" }, { "type": "text", "content": " 的直线，最终结果为 " }, { "type": "equation_inline", "content": "\\frac{1}{5} (e^\\pi - 1)" }, { "type": "text", "content": "。中间 " }, { "type": "equation_inline", "content": "\\frac{1}{2} \\int_0^\\pi e^x \\sin^2 x \\, \\mathrm{d}x = \\frac{1}{4} \\int_0^\\pi e^x (1 - \\cos 2x) \\, \\mathrm{d}x = \\frac{1}{4} e^x \\bigg|_0^\\pi - \\frac{1}{4} \\frac{e^x}{4 + 1} (\\cos 2x + 2\\sin 2x) \\bigg|_0^\\pi" } ] }, "bbox": [ 89, 155, 905, 216 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.14 一道经典题 (1) 计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L} \\frac{x \\, dy - y \\, dx}{x^{2} + y^{2}}" }, { "type": "text", "content": " ，其中 L 为任一不包含原点区域的边界 (包含原点情况在后面讨论)。\n解(1) 结果为 0" } ] }, "bbox": [ 58, 221, 905, 282 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.8 (区域面积转换为曲线积分)" } ], "level": 2 }, "bbox": [ 121, 310, 396, 325 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "平面区域 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 的面积为：" } ] }, "bbox": [ 115, 329, 284, 343 ] }, { "type": "equation_interline", "content": { "math_content": "S _ {D} = \\iint_ {D} \\mathrm{d} S = \\frac {1}{2} \\oint_ {L} x \\mathrm{d} y - y \\mathrm{d} x", "math_type": "latex", "image_source": { "path": "images/092f026a117e015f3884585e45dd733b0fd0af849782a69536738d8bf3b0fb57.jpg" } }, "bbox": [ 379, 340, 616, 372 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.15 曲面面积应用 计算抛物线 " }, { "type": "equation_inline", "content": "(x+y)^{2}=ax(a>0)" }, { "type": "text", "content": " 与 x 轴围成的面积解结果： " }, { "type": "equation_inline", "content": "\\frac{1}{6}a^{2}" }, { "type": "text", "content": " ，见华东师大课本" } ] }, "bbox": [ 58, 409, 663, 449 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.4.2 内部有瑕点" } ], "level": 2 }, "bbox": [ 95, 469, 262, 486 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.9 (瑕点 Green 公式)" } ], "level": 2 }, "bbox": [ 122, 502, 332, 516 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "P(x,y), Q(x,y)" }, { "type": "text", "content": " 除瑕点外满足 " }, { "type": "equation_inline", "content": "Q_x - P_y = 0" }, { "type": "text", "content": "，边界 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 围成的区域 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 包含瑕点 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": "，则可作包含 " }, { "type": "equation_inline", "content": "x_0" }, { "type": "text", "content": " 的小区域 " }, { "type": "equation_inline", "content": "D_1" }, { "type": "text", "content": "，其边界 " }, { "type": "equation_inline", "content": "L_1" }, { "type": "text", "content": "，方向逆时针，根据 " }, { "type": "equation_inline", "content": "\\int_L - \\int_{L_1} = \\iint_{D - D_1} = 0" }, { "type": "text", "content": " 得出：" } ] }, "bbox": [ 114, 521, 880, 568 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {L} P \\mathrm{d} x + Q \\mathrm{d} y = \\int_ {L _ {1}} P \\mathrm{d} x + Q \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/3667d6399272a821c79272f55c14bd7114244af0d98873698db7a40749c55241.jpg" } }, "bbox": [ 374, 570, 621, 601 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "保证 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 曲线表达式的特殊性即可快速计算出结果。" } ] }, "bbox": [ 114, 604, 490, 619 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.16 最经典题目 重点：L 为简单封闭光滑曲线，且不经过原点，a, b > 0，计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L} \\frac{x \\, dy - y \\, dx}{ax^{2} + by^{2}}" }, { "type": "text", "content": " 。\n\n解 显然 P, Q 在不包含原点的区域上有连续偏导数， " }, { "type": "equation_inline", "content": "P_{y} = \\frac{-(ax^{2} + by^{2}) + y \\cdot 2by}{(ax^{2} + by^{2})^{2}}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "Q_{x} = \\frac{by^{2} - ax^{2}}{(ax^{2} + by^{2})^{2}}" }, { "type": "text", "content": " ，若 L 围成的区域不包含原点，则根据 " }, { "type": "equation_inline", "content": "Q_{x} - P_{y} = 0" }, { "type": "text", "content": " 得到积分为 0。若 D 包含原点，考虑挖去原点的小区域，其边界 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "ax^{2} + by^{2} = r^{2}" }, { "type": "text", "content": " ，方向逆时针。则 " }, { "type": "equation_inline", "content": "\\int_{L} P \\, dx + Q \\, dy = \\int_{L_{1}} P \\, dx + Q \\, dy = \\frac{1}{r^{2}} \\int_{L_{1}} x \\, dy - y \\, dx" }, { "type": "text", "content": " ，再次使用 Green 公式得到结果为 " }, { "type": "equation_inline", "content": "\\frac{1}{r^{2}} \\iint_{D_{1}} 2 \\, dx \\, dy = \\frac{2\\pi}{\\sqrt{ab}}" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 58, 633, 905, 765 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 16.2 (常用函数 1)" } ], "level": 2 }, "bbox": [ 122, 775, 292, 790 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "P_0(x_0,y_0)" }, { "type": "text", "content": " 则：" } ] }, "bbox": [ 117, 795, 240, 810 ] }, { "type": "equation_interline", "content": { "math_content": "\\oint_ {L} \\frac {(x - x _ {0}) \\mathrm{d} y - (y - y _ {0}) \\mathrm{d} x}{a (x - x _ {0}) ^ {2} + b (y - y _ {0}) ^ {2}} = \\left\\{ \\begin{array}{l l} \\frac {2 \\pi}{\\sqrt {a b}}, & L \\text {围成的区域包含} P _ {0} \\\\ 0, & L \\text {围成的区域不包含} P _ {0} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/74d6111d9b13dd5d0e5cbd1baae6f5b62ea1ce89174a78e2906129c0bca2f55f.jpg" } }, "bbox": [ 263, 812, 727, 865 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.4 Green 公式" } ] }, "bbox": [ 786, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "136" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习16.17推广题1计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L}\\frac{(x - y)\\mathrm{d}x + (x + 4y)\\mathrm{d}y}{x^2 + 4y^2}" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = 1" }, { "type": "text", "content": "，方向逆时针解 " }, { "type": "equation_inline", "content": "P_{y} = Q_{x} = \\frac{-x^{2} - 8xy + 4y^{2}}{(x^{2} + 4y^{2})^{2}}" }, { "type": "text", "content": "，记 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + 4y^{2} = r^{2}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "I = \\int_{L_1} P\\mathrm{d}x + Q\\mathrm{d}y = \\frac{1}{r^2}\\int_{L_1}(x - y)\\mathrm{d}x + (x + 4y)\\mathrm{d}y," }, { "type": "text", "content": " 再用Green公式得到 " }, { "type": "equation_inline", "content": "I = \\frac{1}{r^2}\\iint_{D_1}2\\mathrm{d}x\\mathrm{d}y = \\frac{2}{r^2}\\cdot \\frac{\\pi r^2}{2} = \\pi" } ] }, "bbox": [ 60, 82, 905, 175 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 16.3 (常用函数 2)" } ], "level": 2 }, "bbox": [ 124, 185, 290, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "c_{1},c_{2}>0" }, { "type": "text", "content": " 时， " }, { "type": "equation_inline", "content": "a_{2}=-b_{1},\\frac{a_{1}}{c_{1}}=\\frac{b_{2}}{c_{2}}" }, { "type": "text", "content": " 有：" } ] }, "bbox": [ 114, 206, 389, 235 ] }, { "type": "equation_interline", "content": { "math_content": "\\oint_ {L} \\frac {\\left(a _ {1} x + a _ {2} y\\right) \\mathrm{d} x + \\left(b _ {1} x + b _ {2} y\\right) \\mathrm{d} y}{c _ {1} x ^ {2} + c _ {2} y ^ {2}} = \\left\\{ \\begin{array}{l l} \\frac {2 b _ {1} \\pi}{\\sqrt {c _ {1} c _ {2}}}, & L \\text {围成区域包含原点} \\\\ 0, & L \\text {围成区域不包含原点} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/48d09cebdf0f61746611f2c8adff53cdc7fd60aa63c06e25643411098eb140c2.jpg" } }, "bbox": [ 233, 236, 757, 288 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "引理 16.1 (推广椭圆面积)" } ], "level": 2 }, "bbox": [ 124, 332, 312, 349 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设 " }, { "type": "equation_inline", "content": "a, c > 0, ac - b^2 > 0" }, { "type": "text", "content": "，考虑 " }, { "type": "equation_inline", "content": "ax^2 + 2bxy + cy^2 = 1" }, { "type": "text", "content": "，则该曲线为椭圆，围成的面积为 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{\\sqrt{ac - b^2}}" } ] }, "bbox": [ 114, 351, 818, 384 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明写为 " }, { "type": "equation_inline", "content": "a(x + \\frac{b}{a} y)^2 +(c - \\frac{b^2}{a})y^2 = 1" }, { "type": "text", "content": " ，等价于 " }, { "type": "equation_inline", "content": "\\frac{x^2}{1 / a} +\\frac{y^2}{a / (ac - b^2)} = 1" }, { "type": "text", "content": " ，根据椭圆面积公式 " }, { "type": "equation_inline", "content": "(\\pi \\cdot ab)" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "\\frac{\\pi}{\\sqrt{ac - b^2}}" }, { "type": "text", "content": " 练习16.18推广题2重点：计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L}\\frac{x\\mathrm{d}y - y\\mathrm{d}x}{x^{2} + xy + y^{2}}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 是圆周 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = 1" }, { "type": "text", "content": " ，逆时针方向。" } ] }, "bbox": [ 62, 391, 902, 451 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解分母即 " }, { "type": "equation_inline", "content": "(x + \\frac{1}{2} y)^2 +\\frac{3}{4} y^2 = 0" }, { "type": "text", "content": " ，若分母为0，则 " }, { "type": "equation_inline", "content": "x = y = 0" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "P_{y} = Q_{x} = \\frac{y^{2} - x^{2}}{(x^{2} + xy + y^{2})^{2}}" }, { "type": "text", "content": " ，取 " }, { "type": "equation_inline", "content": "L_{1}" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + xy + y^{2} = r^{2}" }, { "type": "text", "content": " 该图形为椭圆，面积为 " }, { "type": "equation_inline", "content": "\\frac{2\\pi r^2}{\\sqrt{3}}" }, { "type": "text", "content": " ，使用Green公式得到结果为 " }, { "type": "equation_inline", "content": "\\frac{4\\pi}{\\sqrt{3}}" } ] }, "bbox": [ 84, 450, 905, 514 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "推论 16.4 (常用函数 3)" } ], "level": 2 }, "bbox": [ 122, 523, 290, 538 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "a, b, c" }, { "type": "text", "content": " 满足 " }, { "type": "equation_inline", "content": "a, c > 0, ac - b^2 > 0" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 115, 543, 396, 558 ] }, { "type": "equation_interline", "content": { "math_content": "\\oint_ {L} \\frac {x \\mathrm{d} y - y \\mathrm{d} x}{a x ^ {2} + 2 b x y + c y ^ {2}} = \\frac {2 \\pi}{\\sqrt {a c - b ^ {2}}}", "math_type": "latex", "image_source": { "path": "images/59bff5c3ef3954af2e00c8b5acb37cdfeff0d11d0f984ecda3964d1488ed3b8d.jpg" } }, "bbox": [ 374, 561, 621, 592 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 16.19 其他形式 L 为不过原点的简单光滑封闭曲线，计算：" } ] }, "bbox": [ 58, 630, 557, 645 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\oint_ {L} {\\frac {e ^ {y}}{x ^ {2} + y ^ {2}}} \\left[ (x \\sin x + y \\cos x) \\mathrm{d} x + (y \\sin x - x \\cos x) \\mathrm{d} y \\right]", "math_type": "latex", "image_source": { "path": "images/2f294d70133213a9ff5364ce29a494a82d1987b35b9e24a4729ba87d770d9c28.jpg" } }, "bbox": [ 278, 646, 717, 677 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "16.4.3 曲线积分与路径无关性" } ], "level": 2 }, "bbox": [ 95, 695, 363, 713 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 16.10 (路径无关性)" } ], "level": 2 }, "bbox": [ 122, 728, 302, 743 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "D 为单连通区域, 若 " }, { "type": "equation_inline", "content": "P(x,y)" }, { "type": "text", "content": " , " }, { "type": "equation_inline", "content": "Q(x,y)" }, { "type": "text", "content": " 在 D 连续且有一阶连续偏导, L 为 D 内任意按段光滑曲线则 " }, { "type": "equation_inline", "content": "\\int_{L} P \\, dx + Q \\, dy" }, { "type": "text", "content": " 与路径无关当且仅当:" } ] }, "bbox": [ 114, 750, 880, 791 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 内处处有 " }, { "type": "equation_inline", "content": "P_{y} = Q_{x}" } ] }, "bbox": [ 139, 795, 314, 810 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 内任意按段光滑闭合曲线 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": ", 有 " }, { "type": "equation_inline", "content": "\\oint_{L} P \\mathrm{~d} x + Q \\mathrm{~d} y = 0" } ] }, "bbox": [ 139, 809, 556, 837 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 内存在原函数 " }, { "type": "equation_inline", "content": "u(x,y)" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "\\mathrm{du} = P\\mathrm{dx} + Q\\mathrm{dy}" } ] }, "bbox": [ 139, 837, 502, 852 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "16.4 Green 公式" } ] }, "bbox": [ 786, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "137" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 17 章 曲面积分" } ], "level": 2 }, "bbox": [ 389, 87, 625, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.1 曲面的面积" } ], "level": 2 }, "bbox": [ 95, 145, 278, 165 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.1 (曲面面积公式)" } ], "level": 2 }, "bbox": [ 122, 186, 312, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 是可求面积平面有界区域， " }, { "type": "equation_inline", "content": "z = f(x,y)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上有连续一阶偏导，则曲面 " }, { "type": "equation_inline", "content": "z = f(x,y),(x,y)\\in D" }, { "type": "text", "content": " 的面积为：" } ] }, "bbox": [ 114, 205, 882, 237 ] }, { "type": "equation_interline", "content": { "math_content": "\\Delta S = \\iint_ {D} \\sqrt {1 + f _ {x} ^ {2} + f _ {y} ^ {2}} \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/20b26417699715aca641c2d9b4459decf67448372e527247cf71df5afa02750b.jpg" } }, "bbox": [ 386, 237, 610, 267 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.2 (参数方程曲面面积)" } ], "level": 2 }, "bbox": [ 122, 291, 344, 306 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 为可求面积平面有界区域， " }, { "type": "equation_inline", "content": "x = x(u,v),y = y(u,v),z = z(u,v)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 上有连续一阶偏导，则曲面 " }, { "type": "equation_inline", "content": "\\mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v))" }, { "type": "text", "content": " 的面积公式为：" } ] }, "bbox": [ 112, 311, 880, 346 ] }, { "type": "equation_interline", "content": { "math_content": "\\Delta S = \\iint_ {D} \\sqrt {E G - F ^ {2}} \\mathrm{d} u \\mathrm{d} v, \\left\\{ \\begin{array}{l} E = x _ {u} ^ {2} + y _ {u} ^ {2} + z _ {u} ^ {2} \\\\ F = x _ {u} x _ {v} + y _ {u} y _ {v} + z _ {u} z _ {v} \\\\ G = x _ {v} ^ {2} + y _ {v} ^ {2} + z _ {v} ^ {2} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/0d61ecf2d8bfa0797899af3a1b0138965212a2e3a1053c7773115491ab7b5d4a.jpg" } }, "bbox": [ 285, 350, 690, 416 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.1曲面面积计算(1)证明 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = R^{2}" }, { "type": "text", "content": " 的表面积为 " }, { "type": "equation_inline", "content": "4\\pi R^2" }, { "type": "text", "content": " 解(1)考虑上半平面 " }, { "type": "equation_inline", "content": "z = \\sqrt{R^2 - x^2 - y^2}" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "D" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} \\leq R^{2}" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "z_x = -\\frac{x}{z}, z_y = -\\frac{y}{z}" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "\\sqrt{1 + z_x^2 + z_y^2} = \\frac{R}{z}" }, { "type": "text", "content": "。得到球面面积为" } ] }, "bbox": [ 60, 455, 905, 507 ] }, { "type": "equation_interline", "content": { "math_content": "S = 2 \\iint_ {D} \\frac {R}{z} \\mathrm{d} x \\mathrm{d} y = 2 \\int_ {0} ^ {R} \\mathrm{d} r \\int_ {0} ^ {2 \\pi} \\frac {R}{\\sqrt {R ^ {2} - r ^ {2}}} r \\mathrm{d} \\theta = 4 \\pi R \\sqrt {R ^ {2} - r ^ {2}} \\bigg | _ {R} ^ {0} = 4 \\pi R ^ {2}", "math_type": "latex", "image_source": { "path": "images/7421a4191f621d3a4b4a0e44d32b19ea7b28017f75a408f71424fcf0daee1afa.jpg" } }, "bbox": [ 216, 508, 778, 545 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 对于 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": "，有 " }, { "type": "equation_inline", "content": "\\sqrt{1 + z_x^2 + z_y^2} = \\frac{a}{z}" }, { "type": "text", "content": "，这个结论在计算球面曲面积分时经常用到！" } ] }, "bbox": [ 55, 549, 798, 574 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.2 第一型曲面积分" } ], "level": 2 }, "bbox": [ 95, 595, 324, 615 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.2.1 基本概念与计算公式" } ], "level": 2 }, "bbox": [ 95, 634, 342, 651 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 17.1 (第一型曲面积分)" } ], "level": 2 }, "bbox": [ 122, 665, 329, 681 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为光滑曲面片，对 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 做分割 " }, { "type": "equation_inline", "content": "S = \\{\\Delta S_1, \\dots, \\Delta S_n\\}" }, { "type": "text", "content": "，记 " }, { "type": "equation_inline", "content": "\\| \\Delta \\| = \\max \\{s(\\Delta S_i)\\}" }, { "type": "text", "content": "，则定义" } ] }, "bbox": [ 114, 686, 757, 702 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} f (x, y, z) \\mathrm{d} S = \\lim _ {| | \\Delta | | \\rightarrow 0} \\sum_ {i = 1} ^ {n} f (x _ {i}, y _ {i}, z _ {i}) \\Delta S _ {i}", "math_type": "latex", "image_source": { "path": "images/319401719bf2d2737dfcd1858db86c380e75809b541731da573bffbc442dbe55.jpg" } }, "bbox": [ 334, 707, 662, 741 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.3 (投影法)" } ], "level": 2 }, "bbox": [ 122, 766, 260, 783 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 可表示为 " }, { "type": "equation_inline", "content": "z = z(x,y),(x,y)\\in D" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 115, 787, 421, 803 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} f (x, y, z) \\mathrm{d} S = \\iint_ {D} f (x, y, z (x, y)) \\sqrt {1 + (\\frac {\\partial z}{\\partial x}) ^ {2} + (\\frac {\\partial z}{\\partial y}) ^ {2}} \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/87e51fdeb047decbcb396a901c8138c8d947fd5062f957a185f999bd9ddc961b.jpg" } }, "bbox": [ 265, 809, 731, 844 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.4 (球坐标变换结论)" } ], "level": 2 }, "bbox": [ 122, 86, 327, 101 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " ，令 " }, { "type": "equation_inline", "content": "z = \\sqrt{a^2 - x^2 - y^2}" }, { "type": "text", "content": " ，则得到" } ] }, "bbox": [ 114, 107, 564, 124 ] }, { "type": "equation_interline", "content": { "math_content": "\\sqrt {1 + z _ {x} ^ {2} + z _ {y} ^ {2}} = \\frac {a}{z}", "math_type": "latex", "image_source": { "path": "images/40ec7fe527d34a6b459c94723507fa278788be35d84deebde36e7b8c574452f2.jpg" } }, "bbox": [ 428, 128, 566, 154 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明 直接计算可验证" } ] }, "bbox": [ 90, 172, 255, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 球面可以使用投影法或极坐标，本质上两者都可以用，而且相差不大。" } ] }, "bbox": [ 87, 191, 643, 205 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.2投影法(1)计算 " }, { "type": "equation_inline", "content": "\\iint_{S} \\frac{\\mathrm{d} S}{z}" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 是球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " 被平面 " }, { "type": "equation_inline", "content": "z = h" }, { "type": "text", "content": " 所截的顶部。" } ] }, "bbox": [ 58, 219, 786, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 计算 " }, { "type": "equation_inline", "content": "A = \\int_{S} (x + y + z) \\, \\mathrm{d}S" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 表示 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2} (z \\geq 0)" }, { "type": "text", "content": "，即球上表面" } ] }, "bbox": [ 122, 252, 737, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)做投影法 " }, { "type": "equation_inline", "content": "z = \\sqrt{a^2 - x^2 - y^2}" }, { "type": "text", "content": "，积分变为 " }, { "type": "equation_inline", "content": "\\iint_{D} \\frac{a}{z^2} \\, \\mathrm{d}x \\, \\mathrm{d}y = \\iint_{D} \\frac{a}{a^2 - x^2 - y^2} \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": "，做极坐标变换得到" } ] }, "bbox": [ 87, 277, 858, 307 ] }, { "type": "equation_interline", "content": { "math_content": "\\int_ {0} ^ {2 \\pi} \\mathrm{d} \\theta \\int_ {0} ^ {\\sqrt {a ^ {2} - h ^ {2}}} \\frac {a}{a ^ {2} - r ^ {2}} r \\mathrm{d} r = 2 \\pi a \\ln \\frac {a}{h}", "math_type": "latex", "image_source": { "path": "images/6df5f81217ce3777dc24e69af279bc5773d43f957b4bca28b6adb057c4352bb9.jpg" } }, "bbox": [ 349, 311, 647, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据对称性只剩一项，即 " }, { "type": "equation_inline", "content": "A = \\iint_{S} z \\, \\mathrm{d}S" }, { "type": "text", "content": "，因此根据 " }, { "type": "equation_inline", "content": "z = \\sqrt{a^2 - x^2 - y^2}" }, { "type": "text", "content": " 代入计算得到：" } ] }, "bbox": [ 122, 348, 771, 378 ] }, { "type": "equation_interline", "content": { "math_content": "A = \\iint_ {x ^ {2} + y ^ {2} \\leq a ^ {2}} z \\cdot \\frac {a}{z} \\mathrm{d} x \\mathrm{d} y = \\iint_ {x ^ {2} + y ^ {2} \\leq a ^ {2}} a \\mathrm{d} x \\mathrm{d} y = a ^ {3} \\pi", "math_type": "latex", "image_source": { "path": "images/4d947b6be2c7a77171dacc3b92f76f95c386a4745731841bd8a07a35717d1ca0.jpg" } }, "bbox": [ 300, 381, 692, 413 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.5 (参数方程法)" } ], "level": 2 }, "bbox": [ 122, 432, 295, 448 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 由参数方程 " }, { "type": "equation_inline", "content": "x(u,v),y(u,v),z(u,v)" }, { "type": "text", "content": " 给出， " }, { "type": "equation_inline", "content": "(u,v)\\in D" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 115, 453, 564, 469 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} f (x, y, z) \\mathrm{d} S = \\iint_ {D} f [ x (u, v), y (u, v), z (u, v) ] \\sqrt {E G - F ^ {2}} \\mathrm{d} u \\mathrm{d} v, \\text {其中:} \\left\\{ \\begin{array}{l l} E = x _ {u} ^ {2} + y _ {u} ^ {2} + z _ {u} ^ {2} \\\\ F = x _ {u} x _ {v} + y _ {u} y _ {v} + z _ {u} z _ {v} \\\\ G = x _ {v} ^ {2} + y _ {v} ^ {2} + z _ {v} ^ {2} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/29177573ed172ca37c77f4784eb807612c9d8506f3ee219b56131e66d8c8fb04.jpg" } }, "bbox": [ 124, 473, 850, 539 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.6 (球坐标极坐标变换)" } ], "level": 2 }, "bbox": [ 122, 565, 344, 580 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " ，令 " }, { "type": "equation_inline", "content": "x = a\\sin \\varphi \\cos \\theta ,y = a\\sin \\varphi \\sin \\theta ,z = a\\cos \\varphi" }, { "type": "text", "content": " ，则" } ] }, "bbox": [ 114, 586, 712, 602 ] }, { "type": "equation_interline", "content": { "math_content": "\\left\\{ \\begin{array}{l l} E = a ^ {2} \\sin^ {2} \\varphi \\\\ F = 0 & \\Rightarrow \\sqrt {E G - F ^ {2}} = a ^ {2} \\sin \\varphi \\\\ G = a ^ {2} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/a375beedfc26a4f13c706c52a3f233e2bded15752512c40bcdfb0f616807534c.jpg" } }, "bbox": [ 339, 606, 660, 671 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.3参数方程法(1)重点：计算 " }, { "type": "equation_inline", "content": "I = \\iint_{S}\\frac{\\mathrm{d}S}{\\sqrt{x^2 + y^2 + (z + a)^2}}" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}(z\\geq 0)" }, { "type": "text", "content": " 解(1)采用球坐标变换，积分变为 " }, { "type": "equation_inline", "content": "I = \\iint_{S'}\\frac{a^2\\sin\\varphi}{\\sqrt{a^2\\sin^2\\varphi + a^2(\\cos\\varphi + 1)^2}}\\mathrm{d}\\theta \\mathrm{d}\\varphi" }, { "type": "text", "content": " ，区域为 " }, { "type": "equation_inline", "content": "\\varphi \\in [0,\\frac{\\pi}{2}],\\theta \\in [0,2\\pi ]" }, { "type": "text", "content": " ，进一步化简得到： " }, { "type": "equation_inline", "content": "I = \\iint_{S'}\\frac{-a\\sin\\varphi}{\\sqrt{2}\\sqrt{1 + \\cos\\varphi}}\\mathrm{d}\\theta \\mathrm{d}\\varphi = -\\sqrt{2}\\pi a\\int_0^{\\frac{\\pi}{2}}\\frac{\\mathrm{d}(\\cos\\varphi)}{\\sqrt{1 + \\cos\\varphi}} = 4\\pi a - 2\\sqrt{2}\\pi a" } ] }, "bbox": [ 58, 705, 902, 822 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "17.2.2 使用对称性" } ] }, "bbox": [ 95, 841, 263, 857 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.4积分曲面对称(1)计算 " }, { "type": "equation_inline", "content": "\\iint_{S} (x + \\sin yz) \\mathrm{d}S" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" } ] }, "bbox": [ 58, 866, 719, 898 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "139" } ] }, "bbox": [ 485, 953, 512, 965 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.2 第一型曲面积分" } ] }, "bbox": [ 747, 54, 904, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\iint_{S} (x + y + z)^{3} \\mathrm{d} S" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "|x| + |y| + |z| = 1" }, { "type": "text", "content": "解(1)根据 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "yz" }, { "type": "text", "content": " 平面对称得到 " }, { "type": "equation_inline", "content": "\\iint_{S}x\\mathrm{d}s = 0" }, { "type": "text", "content": " ，根据 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "xz" }, { "type": "text", "content": " 平面对称得到 " }, { "type": "equation_inline", "content": "\\iint_{S}\\sin yz\\mathrm{d}s = 0" }, { "type": "text", "content": " (2)根据关于原点中心对称性得到为0。" } ] }, "bbox": [ 122, 82, 500, 112 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 89, 110, 794, 149 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 根据关于原点中心对称性得到为 0。" } ] }, "bbox": [ 122, 139, 408, 153 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.5轮换对称性(1)计算 " }, { "type": "equation_inline", "content": "I = \\iint_{S}x^{2}\\mathrm{d}S" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "J = \\iint xy\\mathrm{d}S" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " 解(1)由于 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 有轮换对称性，得到 " }, { "type": "equation_inline", "content": "I = \\frac{1}{3}\\iint_{S}x^{2} + y^{2} + z^{2}\\mathrm{d}S = \\frac{4}{3}\\pi a^4" }, { "type": "text", "content": " 。 " }, { "type": "equation_inline", "content": "J" }, { "type": "text", "content": " 没必要考虑轮换对称性，由于 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "yz" }, { "type": "text", "content": " 平面对称，因此 " }, { "type": "equation_inline", "content": "J = 0" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 58, 167, 905, 241 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.3 第二型曲面积分" } ], "level": 2 }, "bbox": [ 95, 266, 324, 286 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.3.1 基本概念、投影法、参数方程法" } ], "level": 2 }, "bbox": [ 94, 305, 443, 322 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 17.2 (第二型曲面积分)" } ], "level": 2 }, "bbox": [ 122, 337, 329, 353 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于向量场 " }, { "type": "equation_inline", "content": "F(x,y,z) = (P(x,y,z),Q(x,y,z),R(x,y,z))" }, { "type": "text", "content": " ，其通过曲面 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 的通量定义为第二型曲面积分(向外为正):" } ] }, "bbox": [ 112, 357, 880, 391 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} P (x, y, z) \\mathrm{d} y \\mathrm{d} z + Q (x, y, z) \\mathrm{d} z \\mathrm{d} x + R (x, y, z) \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/76a4c8245836becff0636bf76825bdef2781cca8dcaa83ef5b07dc4db8eca828.jpg" } }, "bbox": [ 304, 388, 692, 420 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.7 (投影法)" } ], "level": 2 }, "bbox": [ 122, 442, 262, 457 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若曲面 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 的方程为 " }, { "type": "equation_inline", "content": "z = z(x,y)" }, { "type": "text", "content": "，在 " }, { "type": "equation_inline", "content": "xy" }, { "type": "text", "content": " 平面投影区域为 " }, { "type": "equation_inline", "content": "D_{xy}" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 115, 462, 584, 478 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} R (x, y, z) \\mathrm{d} x \\mathrm{d} y = \\pm \\iint_ {D _ {x y}} R (x, y, z (x, y)) \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/6fc77eeb703aa33f5fa0cf4d4191799bcb51328f35b3404edf745a72479114ed.jpg" } }, "bbox": [ 315, 482, 680, 512 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "正负取决于 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 的法向量与 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴正向的夹角，若锐角则取正，钝角取负，直角积分为0。" } ] }, "bbox": [ 115, 516, 747, 531 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 若曲面 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "z = a" }, { "type": "text", "content": " 的一部分，则 " }, { "type": "equation_inline", "content": "\\iint_{S} P \\, \\mathrm{d}y \\, \\mathrm{d}z + Q \\, \\mathrm{d}z \\, \\mathrm{d}x + R \\, \\mathrm{d}x \\, \\mathrm{d}y = \\iint_{S} R(x, y, z) \\, \\mathrm{d}x \\, \\mathrm{d}y = \\pm \\iint_{D} R(x, y, a) \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": "。若计算 " }, { "type": "equation_inline", "content": "\\mathrm{d}y \\, \\mathrm{d}z, \\mathrm{d}z \\, \\mathrm{d}x" }, { "type": "text", "content": "，则需要考虑在 " }, { "type": "equation_inline", "content": "yz, zx" }, { "type": "text", "content": " 平面的投影" } ] }, "bbox": [ 87, 542, 905, 587 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.6投影法练习(1)计算 " }, { "type": "equation_inline", "content": "\\iint_{S}xyz\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 是球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x\\geq 0,y\\geq 0" }, { "type": "text", "content": " 部分，且取球面外侧(2)考虑 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x = y = z = 0,x = y = z = a" }, { "type": "text", "content": " 六个平面所围成的立方体表面，外侧为正，计算：" } ] }, "bbox": [ 58, 604, 905, 645 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} (x y - y z) \\mathrm{d} y \\mathrm{d} z + x ^ {2} \\mathrm{d} z \\mathrm{d} x + (y ^ {2} + x z) \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/34866699d0df918b10dd21d8952b6368dc03e5ed88fab751941c5cacb3f88c98.jpg" } }, "bbox": [ 329, 648, 665, 677 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)积分等于 " }, { "type": "equation_inline", "content": "2\\iint_{D_{xy}}xy\\sqrt{1 - x^2 - y^2}\\mathrm{d}x\\mathrm{d}y = 2\\int_0^{\\frac{\\pi}{2}}\\mathrm{d}\\theta \\int_0^1 r^3\\cos \\theta \\sin \\theta \\sqrt{1 - r^2}\\mathrm{d}r = \\frac{2}{15}" } ] }, "bbox": [ 87, 678, 742, 713 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 可以用 Gauss 公式，也可以一个一个考虑。例如考虑 " }, { "type": "equation_inline", "content": "z = a" }, { "type": "text", "content": " 平面 " }, { "type": "equation_inline", "content": "S_{1}" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "\\iint_{S_1}(y^2 + xz)\\mathrm{d}x\\mathrm{d}y = \\iint_D(y^2 + ax)\\mathrm{d}x\\mathrm{d}y = \\frac{5}{6} a^4" }, { "type": "text", "content": "，其余同理。" } ] }, "bbox": [ 89, 714, 905, 769 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.8 (参数方程法)" } ], "level": 2 }, "bbox": [ 122, 790, 295, 804 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 可表示为参数方程 " }, { "type": "equation_inline", "content": "x = x(u,v),y = y(u,v),z = z(u,v),(u,v) \\in D" }, { "type": "text", "content": "，记 " }, { "type": "equation_inline", "content": "A = \\left|\\frac{\\partial(y,z)}{\\partial(u,v)}\\right|,B = \\left|\\frac{\\partial(z,x)}{\\partial(u,v)}\\right|,C = \\left|\\frac{\\partial(x,y)}{\\partial(u,v)}\\right|" }, { "type": "text", "content": "，则" } ] }, "bbox": [ 114, 810, 882, 871 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} P \\mathrm{d} y \\mathrm{d} z + Q \\mathrm{d} z \\mathrm{d} x + R \\mathrm{d} x \\mathrm{d} y = \\pm \\iint_ {D} (P A + Q B + R C) \\mathrm{d} u \\mathrm{d} v", "math_type": "latex", "image_source": { "path": "images/b80fb5956c8eb40f63bc341d4e00770c7139546ee829d8e59953aef255678df7.jpg" } }, "bbox": [ 268, 873, 727, 904 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.3 第二型曲面积分" } ] }, "bbox": [ 747, 55, 902, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "140" } ] }, "bbox": [ 485, 953, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若向量 " }, { "type": "equation_inline", "content": "(A, B, C)" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 法向量方向一致则取正，否则取负。" } ] }, "bbox": [ 114, 95, 541, 111 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.7参数方程法(1) " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "\\frac{x^2}{a^2} +\\frac{y^2}{b^2} +\\frac{z^2}{c^2} = 1" }, { "type": "text", "content": " 上半部并取外侧，计算 " }, { "type": "equation_inline", "content": "\\iint_{S}x^{3}\\mathrm{d}y\\mathrm{d}z" }, { "type": "text", "content": " 解" } ] }, "bbox": [ 58, 139, 705, 186 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.3.2 利用对称性" } ], "level": 2 }, "bbox": [ 94, 208, 263, 225 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "以 " }, { "type": "equation_inline", "content": "\\iint_{S} R(x, y, z) \\mathrm{d}x \\mathrm{~d}y" }, { "type": "text", "content": " 为例， " }, { "type": "equation_inline", "content": "\\mathrm{d}x \\mathrm{~d}y" }, { "type": "text", "content": " 的正负取决于对称点处 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 外法向量与 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴正向的夹角，结合 " }, { "type": "equation_inline", "content": "R(x, y, z)" }, { "type": "text", "content": " 关于 " }, { "type": "equation_inline", "content": "xy" }, { "type": "text", "content": " 平面或者原点的对称性可分解 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 上的曲面积分， " }, { "type": "equation_inline", "content": "\\mathrm{d}y \\mathrm{~d}z" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "\\mathrm{d}z \\mathrm{~d}x" }, { "type": "text", "content": " 同理。" } ] }, "bbox": [ 90, 237, 905, 280 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.8 方向判断训练 设 " }, { "type": "equation_inline", "content": "S: |x| + |y| + |z| = 1" }, { "type": "text", "content": " ，方向朝外，判断下面哪些积分值为 0:" } ] }, "bbox": [ 58, 300, 734, 316 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) " }, { "type": "equation_inline", "content": "\\iint_{S} z \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": " (2) " }, { "type": "equation_inline", "content": "\\iint_{S} x^{2} \\sin y \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": " (3) " }, { "type": "equation_inline", "content": "\\iint_{S} y^{2} \\sin z \\, \\mathrm{d}x \\, \\mathrm{d}y" }, { "type": "text", "content": " (4) " }, { "type": "equation_inline", "content": "\\iint_{S} \\sin x \\sin^{2} y \\sin^{3} z \\, \\mathrm{d}x \\, \\mathrm{d}y" } ] }, "bbox": [ 176, 319, 818, 350 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)关于 " }, { "type": "equation_inline", "content": "xy" }, { "type": "text", "content": " 平面对称， " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 相反，由于法向量与 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴正向夹角相反， " }, { "type": "equation_inline", "content": "\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " 也为负，因此非零" } ] }, "bbox": [ 89, 351, 774, 367 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 关于 " }, { "type": "equation_inline", "content": "xz" }, { "type": "text", "content": " 平面对称， " }, { "type": "equation_inline", "content": "\\sin y" }, { "type": "text", "content": " 相反， " }, { "type": "equation_inline", "content": "\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " 为正，因此积分为0" } ] }, "bbox": [ 122, 370, 574, 385 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3) 无论怎么考虑都非零" } ] }, "bbox": [ 122, 388, 304, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(4) 考虑 yz， " }, { "type": "equation_inline", "content": "\\sin x" }, { "type": "text", "content": " 相反，dxdy 为正，因此积分为 0。" } ] }, "bbox": [ 122, 407, 507, 423 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.9 基本计算 (1) 计算 " }, { "type": "equation_inline", "content": "I_{1} = \\iint_{S} z \\, dx \\, dy, I_{2} = \\iint_{S} z^{2} \\, dx \\, dy" }, { "type": "text", "content": " ，其中 S 为球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " ，方向取外侧解 (1) " }, { "type": "equation_inline", "content": "I_{2}" }, { "type": "text", "content": " 考虑关于 xy 平面对称， " }, { "type": "equation_inline", "content": "z^{2}" }, { "type": "text", "content": " 相等，dx dy 相反，因此积分为 0。 " }, { "type": "equation_inline", "content": "I_{1}" }, { "type": "text", "content": " 可转换为两倍的上半球面， " }, { "type": "equation_inline", "content": "I_{1} = 2 \\iint_{S_{1}} z \\, dx \\, dy" }, { "type": "text", "content": " ，由于 " }, { "type": "equation_inline", "content": "z = \\sqrt{a^{2} - x^{2} - y^{2}}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "D : x^{2} + y^{2} \\leq a^{2}" }, { "type": "text", "content": " ，因此" } ] }, "bbox": [ 58, 437, 905, 514 ] }, { "type": "equation_interline", "content": { "math_content": "I _ {1} = \\iint_ {D} \\sqrt {a ^ {2} - x ^ {2} - y ^ {2}} \\mathrm{d} x \\mathrm{d} y = 2 \\int_ {0} ^ {a} \\mathrm{d} r \\int_ {0} ^ {2 \\pi} \\sqrt {a ^ {2} - r ^ {2}} r \\mathrm{d} \\theta = \\frac {4 \\pi a ^ {3}}{3}", "math_type": "latex", "image_source": { "path": "images/7f67a81dd42bbb56c992d8c3245d7ef9493197eea51cdadf83d984821c5fd437.jpg" } }, "bbox": [ 252, 514, 742, 548 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.3.3 两类曲面积分的联系" } ], "level": 2 }, "bbox": [ 94, 567, 344, 583 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.9 (两类曲面积分的联系)" } ], "level": 2 }, "bbox": [ 122, 600, 363, 615 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为光滑曲面，正侧单位法向量 " }, { "type": "equation_inline", "content": "\\mathbf{n} = (\\cos \\alpha, \\cos \\beta, \\cos \\gamma)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "\\cos \\alpha \\mathrm{d}S = \\mathrm{d}y \\mathrm{d}z, \\cos \\beta \\mathrm{d}S = \\mathrm{d}z \\mathrm{d}x, \\cos \\gamma \\mathrm{d}S = \\mathrm{d}x \\mathrm{d}y" }, { "type": "text", "content": "，即" } ] }, "bbox": [ 112, 619, 882, 653 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} P \\mathrm{d} y \\mathrm{d} z + Q \\mathrm{d} z \\mathrm{d} x + R \\mathrm{d} x \\mathrm{d} y = \\iint_ {S} (P \\cos \\alpha + Q \\cos \\beta + R \\cos \\gamma) \\mathrm{d} S", "math_type": "latex", "image_source": { "path": "images/c283309ff24e90395dc4849714c6041352fd52f41884c6c047b06d7ac079619e.jpg" } }, "bbox": [ 247, 650, 749, 681 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记对一般曲面 " }, { "type": "equation_inline", "content": "F(x,y,z) = 0" }, { "type": "text", "content": " ，其梯度 " }, { "type": "equation_inline", "content": "(F_{x},F_{y},F_{z})" }, { "type": "text", "content": " 就是曲面法向量，因此单位法向量的计算方式为：" } ] }, "bbox": [ 87, 699, 838, 715 ] }, { "type": "equation_interline", "content": { "math_content": "\\mathbf {n} = \\pm \\frac {1}{\\sqrt {F _ {x} ^ {2} + F _ {y} ^ {2} + F _ {z} ^ {2}}} (F _ {x}, F _ {y}, F _ {z})", "math_type": "latex", "image_source": { "path": "images/0911c550bf05476bd8c645744445eadaade9b35a55fc75e124416755ab628f5b.jpg" } }, "bbox": [ 366, 719, 628, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习17.10重新计算第二型曲面积分(1)重新计算 " }, { "type": "equation_inline", "content": "\\iint_{S}xyz\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " ，其中 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 是球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x\\geq 0,y\\geq 0" }, { "type": "text", "content": " 部分，方向朝外解(1)法向量 " }, { "type": "equation_inline", "content": "\\mathbf{n} = \\frac{1}{\\sqrt{3}} (1,1,1)" }, { "type": "text", "content": " ，因此可以转换为第一型曲面积分：" } ] }, "bbox": [ 58, 763, 907, 833 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} x y z \\cdot x \\cdot \\frac {a}{z} \\mathrm{d} S = 2 \\iint_ {D} r ^ {4} \\cos^ {2} \\theta \\sin \\theta \\mathrm{d} r \\mathrm{d} \\theta = \\frac {2}{1 5}", "math_type": "latex", "image_source": { "path": "images/386cf02cc5263b3fb51c529d5c716dee8a86515d9697dd5bdab1f8b660f97073.jpg" } }, "bbox": [ 315, 835, 680, 866 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这里乘上2的原因是关于 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 正负有两块。" } ] }, "bbox": [ 90, 869, 389, 884 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.3 第二型曲面积分" } ] }, "bbox": [ 747, 54, 904, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "141" } ] }, "bbox": [ 485, 954, 510, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.11 用曲面积分联系计算第二型曲面积分 S 为 " }, { "type": "equation_inline", "content": "x + y + z = 1" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 4x" }, { "type": "text", "content": " 内部的部分，取上侧为正，计算：" } ] }, "bbox": [ 58, 84, 905, 117 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\iint_ {S} y \\mathrm{d} y \\mathrm{d} z + z \\mathrm{d} z \\mathrm{d} x + x \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/e4b0940eebf701526d66685efe44d1185edf9082278ef8d10ea2c0a17ffc9629.jpg" } }, "bbox": [ 374, 114, 621, 146 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 (1) 已知 " }, { "type": "equation_inline", "content": "F(x, y, z) = x + y + z - 1" }, { "type": "text", "content": "，则梯度为 " }, { "type": "equation_inline", "content": "\\nabla F = (1, 1, 1)" }, { "type": "text", "content": "，单位法向量 " }, { "type": "equation_inline", "content": "\\mathbf{n} = \\frac{1}{\\sqrt{3}}(1, 1, 1)" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "I = \\iint_{S} \\frac{1}{\\sqrt{3}}(x + y + z) dS = \\frac{1}{\\sqrt{3}} S" }, { "type": "text", "content": "，计算截面圆面积即可，结果为 " }, { "type": "equation_inline", "content": "\\frac{11}{3\\sqrt{3}}\\pi" } ] }, "bbox": [ 87, 153, 907, 212 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.4 Gauss 公式" } ], "level": 2 }, "bbox": [ 94, 233, 275, 252 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.4.1 Gauss 公式与基本使用" } ], "level": 2 }, "bbox": [ 94, 272, 359, 288 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.10 (Gauss 公式)" } ], "level": 2 }, "bbox": [ 121, 304, 304, 319 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "V \\subset \\mathbb{R}^3" }, { "type": "text", "content": " 是有界闭区域， " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 边界是分片光滑的曲面 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "P(x, y, z), Q(x, y, z), R(x, y, z)" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 内有连续偏导，则" } ] }, "bbox": [ 112, 324, 882, 357 ] }, { "type": "equation_interline", "content": { "math_content": "\\oiint_ {S _ {\\text {外}}} P \\mathrm{d} y \\mathrm{d} z + Q \\mathrm{d} z \\mathrm{d} x + R \\mathrm{d} x \\mathrm{d} y = \\iiint_ {V} (\\frac {\\partial P}{\\partial x} + \\frac {\\partial Q}{\\partial y} + \\frac {\\partial R}{\\partial z}) \\mathrm{d} x \\mathrm{d} y \\mathrm{d} z", "math_type": "latex", "image_source": { "path": "images/9ddbe4fec42fe2f869f565e394796421ceb3824d91f3a7655a52d636b8eab39c.jpg" } }, "bbox": [ 258, 354, 737, 387 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 使用 Gauss 公式需要保证 V 内任意位置，P, Q, R 都有连续偏导。" } ] }, "bbox": [ 85, 407, 603, 423 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.12 补面计算 (1) 东南大学 2022: 计算 " }, { "type": "equation_inline", "content": "I = \\iint_{S} \\frac{(z + 1)\\mathrm{d}y\\mathrm{d}z + (y + 1)\\mathrm{d}z\\mathrm{d}x + (x + 1)\\mathrm{d}x\\mathrm{d}y}{x^{2} + y^{2} + z^{2}}" }, { "type": "text", "content": " ，其中 S 为 " }, { "type": "equation_inline", "content": "z = \\sqrt{9 - x^{2} - y^{2}}" }, { "type": "text", "content": " 曲面，取下侧\n\n解 (1) 补面 " }, { "type": "equation_inline", "content": "S_{1} = \\{z = 0, x^{2} + y^{2} \\leq 9\\}" }, { "type": "text", "content": " ，方向朝上，则 " }, { "type": "equation_inline", "content": "I = \\frac{1}{9}I_{1} - \\frac{1}{9}I_{2} = \\frac{1}{9}\\iint_{S + S_{1}} - \\frac{1}{9}\\iint_{S_{1}}" }, { "type": "text", "content": " ，根据 Gauss 公式计算 " }, { "type": "equation_inline", "content": "I_{1}" }, { "type": "text", "content": " :" } ] }, "bbox": [ 58, 437, 905, 525 ] }, { "type": "equation_interline", "content": { "math_content": "I _ {1} = - \\iiint_ {V} (0 + 1 + 0) \\mathrm{d} x \\mathrm{d} y \\mathrm{d} z = - 1 8 \\pi", "math_type": "latex", "image_source": { "path": "images/0d178195302199f5ee35dc97f2cde912230318aa7cad80c82f118f1178546507.jpg" } }, "bbox": [ 352, 526, 642, 556 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算 " }, { "type": "equation_inline", "content": "I_{2}" }, { "type": "text", "content": " ，由于在平面上，曲面积分直接变为二重积分，且 dz=0，：" } ] }, "bbox": [ 89, 558, 586, 574 ] }, { "type": "equation_interline", "content": { "math_content": "I _ {2} = \\iint_ {x ^ {2} + y ^ {2} \\leq 9} (x + 1) \\mathrm{d} x \\mathrm{d} y = \\int_ {0} ^ {2 \\pi} \\mathrm{d} \\theta \\int_ {0} ^ {3} r ^ {2} \\cos + r \\mathrm{d} r = 9 \\pi", "math_type": "latex", "image_source": { "path": "images/e31850e4bf690eaf6890a8cc021161128a849102a4cc9f8f7c93355a35b437c1.jpg" } }, "bbox": [ 280, 577, 715, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此结果为 " }, { "type": "equation_inline", "content": "I = \\frac{1}{9} (-18\\pi - 9\\pi) = -3\\pi" } ] }, "bbox": [ 90, 613, 374, 639 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.13 菱形区域：变量替换！(1) 重点：S 为封闭曲面 " }, { "type": "equation_inline", "content": "|x - y + z| + |y - z + x| + |z - x + y| = 1" }, { "type": "text", "content": " ，方向朝外，计算 " }, { "type": "equation_inline", "content": "I = \\iint_{S} (x + y - z) \\, \\mathrm{d}y \\, \\mathrm{d}z + [2y + \\sin(z + x)] \\, \\mathrm{d}z \\, \\mathrm{d}x + (3z + e^{x + y}) \\, \\mathrm{d}x \\, \\mathrm{d}y" } ] }, "bbox": [ 58, 637, 905, 680 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2)(东北师大) 求 " }, { "type": "equation_inline", "content": "I = \\iint_{S} (x + \\cos(y - z)) \\, \\mathrm{d}y \\, \\mathrm{d}z + (2y + e^{x + z}) \\, \\mathrm{d}z \\, \\mathrm{d}x + (3z + \\sin(x - y) \\, \\mathrm{d}x \\, \\mathrm{d}y)" }, { "type": "text", "content": "，这里 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为封闭曲面，方向取外侧" } ] }, "bbox": [ 89, 683, 905, 722 ] }, { "type": "equation_interline", "content": { "math_content": "\\vert x - y + z \\vert + \\vert y - z + x \\vert + \\vert z - x + y \\vert = 1", "math_type": "latex", "image_source": { "path": "images/65c4696d54621c545d5be6d5cab17da867684258c1e6ef37ee36a4d0e0f24361.jpg" } }, "bbox": [ 342, 726, 653, 741 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)使用 Gauss 公式即 " }, { "type": "equation_inline", "content": "I=\\iiint_{V}(1+2+3)\\mathrm{d}x\\mathrm{d}y\\mathrm{d}z" }, { "type": "text", "content": " ，考虑坐标变换 " }, { "type": "equation_inline", "content": "u=x-y+z, v=y-z+x, w=z-x+y" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "|u|+|v|+|w|\\leq1" }, { "type": "text", "content": " ，而 " }, { "type": "equation_inline", "content": "\\frac{\\partial(u,v,w)}{\\partial(x,y,z)}=4" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "J=\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}=\\frac{1}{4}" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "I=\\iint_{V'}\\frac{3}{2}\\mathrm{d}u\\mathrm{d}v\\mathrm{d}w=2" }, { "type": "text", "content": " ，这里 " }, { "type": "equation_inline", "content": "V'=\\frac{4}{3}" }, { "type": "text", "content": " 是个锥体。" } ] }, "bbox": [ 89, 750, 905, 820 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 使用 Gauss 公式即 " }, { "type": "equation_inline", "content": "I = \\iiint_{V} 6 \\mathrm{~d} V" }, { "type": "text", "content": "，使用坐标变换" } ] }, "bbox": [ 121, 816, 505, 848 ] }, { "type": "equation_interline", "content": { "math_content": "u = x + z - y, v = y + x - z, w = y + z - x", "math_type": "latex", "image_source": { "path": "images/8e9cea8fd2971d874d2b859a7415267efb1f4e48f7575ade1830d7d760fc4ed2.jpg" } }, "bbox": [ 336, 850, 660, 863 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "得到 " }, { "type": "equation_inline", "content": "\\frac{\\partial(u,v,w)}{\\partial(x,y,z)} = 4" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "\\frac{\\partial(x,y,z)}{\\partial(u,v,w)} = \\frac{1}{4}" }, { "type": "text", "content": " ，积分区域为 " }, { "type": "equation_inline", "content": "|u| + |v| + |w|\\leq 1" }, { "type": "text", "content": " ，分为8个小锥体，每个锥体体积 " }, { "type": "equation_inline", "content": "\\frac{1}{6}" }, { "type": "text", "content": " ，因" } ] }, "bbox": [ 90, 865, 905, 896 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "142" } ] }, "bbox": [ 485, 953, 512, 964 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.4 Gauss 公式" } ] }, "bbox": [ 786, 55, 905, 68 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "此最终结果为:" } ] }, "bbox": [ 90, 85, 205, 99 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\frac {3}{2} \\times 8 \\times \\frac {1}{6} = 2", "math_type": "latex", "image_source": { "path": "images/894671dfdf5f12d1b1c2134191fe00cd31eae19e797a3b46bfb8a114d34e1b92.jpg" } }, "bbox": [ 425, 96, 571, 124 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.4.2 内部有瑕点" } ], "level": 2 }, "bbox": [ 94, 142, 263, 159 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.14 经典题目 S 为不经过原点的光滑封闭曲面，计算" } ] }, "bbox": [ 58, 170, 534, 185 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\iint_ {S} {\\frac {x}{(a x ^ {2} + b y ^ {2} + c z ^ {2}) ^ {\\frac {3}{2}}}} \\mathrm{d} y \\mathrm{d} z + {\\frac {y}{(a x ^ {2} + b y ^ {2} + c z ^ {2}) ^ {\\frac {3}{2}}}} \\mathrm{d} z \\mathrm{d} x + {\\frac {z}{(a x ^ {2} + b y ^ {2} + c z ^ {2}) ^ {\\frac {3}{2}}}} \\mathrm{d} x \\mathrm{d} y", "math_type": "latex", "image_source": { "path": "images/530c888239f46cf6a081862224af3cbbd98a232585fc61b0bd41ff9ce1e3978e.jpg" } }, "bbox": [ 176, 187, 818, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解记 " }, { "type": "equation_inline", "content": "r^2 = ax^2 + by^2 + cz^2" }, { "type": "text", "content": "，则两侧对 " }, { "type": "equation_inline", "content": "x" }, { "type": "text", "content": " 求导可以得到" } ] }, "bbox": [ 87, 221, 494, 237 ] }, { "type": "equation_interline", "content": { "math_content": "r _ {x} = \\frac {a x}{r}", "math_type": "latex", "image_source": { "path": "images/f5503cc2e002b60454888b258d36708e8591239f671e8adb07f1b57513383a05.jpg" } }, "bbox": [ 463, 241, 532, 267 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理可以得到 " }, { "type": "equation_inline", "content": "r_y, r_z" }, { "type": "text", "content": "。同时 " }, { "type": "equation_inline", "content": "P = \\frac{x}{r^3}, Q = \\frac{y}{r^3}, R = \\frac{z}{r^3}" }, { "type": "text", "content": "，因此可以计算得到" } ] }, "bbox": [ 89, 267, 623, 294 ] }, { "type": "equation_interline", "content": { "math_content": "P _ {x} + Q _ {y} + R _ {z} = 0", "math_type": "latex", "image_source": { "path": "images/11dd1de899ccc50ac74e3923fea4b430d3e9830de36b3be0fa82d6abce4b61c4.jpg" } }, "bbox": [ 428, 296, 569, 312 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如果 S 围成的区域不包含原点，则积分为 0。如果包含原点，则挖掉 " }, { "type": "equation_inline", "content": "V_{1}: ax^{2} + by^{2} + cz^{2} \\leq \\epsilon^{2}" }, { "type": "text", "content": " ， " }, { "type": "equation_inline", "content": "S_{1}" }, { "type": "text", "content": " 面方向定义为 " }, { "type": "equation_inline", "content": "V - V_{1}" }, { "type": "text", "content": " 朝向 " }, { "type": "equation_inline", "content": "V_{1}" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\iint_{S + S_{1}} P \\, dy \\, dz + Q \\, dz \\, dx + R \\, dx \\, dy = \\iiint_{V - V_{1}} 0 \\, dx \\, dy \\, dz = 0" }, { "type": "text", "content": " ，因此 " }, { "type": "equation_inline", "content": "I = -\\iint_{S_{1}} P \\, dy \\, dz + Q \\, dz \\, dx + R \\, dx \\, dy" }, { "type": "text", "content": " ，用 Gauss 公式得到" } ] }, "bbox": [ 89, 318, 905, 375 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\iiint_ {V _ {1}} \\frac {3}{\\epsilon^ {3}} \\mathrm{d} x \\mathrm{d} y \\mathrm{d} z = \\frac {3}{\\epsilon^ {3}} \\cdot \\frac {4}{3} \\cdot \\frac {\\pi \\epsilon^ {3}}{\\sqrt {a b c}} = \\frac {4 \\pi}{\\sqrt {a b c}}", "math_type": "latex", "image_source": { "path": "images/586e5d2cd90a1a540f50fee0c62e216829845c8ab3625c0d22fb70aa272baec8.jpg" } }, "bbox": [ 331, 376, 665, 409 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记最常见的形式为： " }, { "type": "equation_inline", "content": "\\iint_{S}\\frac{x\\mathrm{d}y\\mathrm{d}z + y\\mathrm{d}z\\mathrm{d}x + z\\mathrm{d}x\\mathrm{d}y}{(x^2 + y^2 + z^2)^{\\frac{3}{2}}} = \\left\\{ \\begin{array}{ll}4\\pi , & S\\text{围成区域包含原点}\\\\ 0, & S\\text{围成区域不包含原点} \\end{array} \\right." } ] }, "bbox": [ 55, 414, 715, 458 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.15 推广形式 S 为上半椭球面 " }, { "type": "equation_inline", "content": "2021x^{2} + 2022y^{2} + 2023z^{2} = 1 (z \\geq 0)" }, { "type": "text", "content": " ，方向取向上，计算" } ] }, "bbox": [ 58, 466, 794, 483 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\iint_ {S} \\frac {x \\mathrm{d} y \\mathrm{d} z + y \\mathrm{d} z \\mathrm{d} x + z \\mathrm{d} x \\mathrm{d} y}{\\left(x ^ {2} + y ^ {2} + z ^ {2}\\right) ^ {\\frac {3}{2}}}", "math_type": "latex", "image_source": { "path": "images/64dcf08b45bea22e63bfcfe7c5f519b64e1fb7537b9865e934c735b58a6bba98.jpg" } }, "bbox": [ 373, 485, 621, 517 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 记 " }, { "type": "equation_inline", "content": "r = \\sqrt{x^2 + y^2 + z^2}" }, { "type": "text", "content": "，显然除原点外 " }, { "type": "equation_inline", "content": "P_x + Q_y + R_z = 0" }, { "type": "text", "content": "，记 " }, { "type": "equation_inline", "content": "V: 2021x^2 + 2022y^2 + 2023z^2 \\leq 1 (z \\geq 0)" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "V_1: x^2 + y^2 + z^2 \\leq \\epsilon^2 (z \\geq 0)" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "V - V_1" }, { "type": "text", "content": " 的表面分为三个部分 " }, { "type": "equation_inline", "content": "S + S_1 + S_2" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "S_1" }, { "type": "text", "content": " 是小椭球体表面， " }, { "type": "equation_inline", "content": "S_2" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "z = 0" }, { "type": "text", "content": " 内两个直底面， " }, { "type": "equation_inline", "content": "S_1, S_2" }, { "type": "text", "content": " 方向朝下。根据 Gauss 公式得到 " }, { "type": "equation_inline", "content": "\\iint_{S + S_1 + S_2} P \\mathrm{d}y \\mathrm{d}z + Q \\mathrm{d}z \\mathrm{d}x + R \\mathrm{d}x \\mathrm{d}y = 0" }, { "type": "text", "content": "，即" } ] }, "bbox": [ 87, 521, 905, 585 ] }, { "type": "equation_interline", "content": { "math_content": "\\iint_ {S} P \\mathrm{d} y \\mathrm{d} z + Q \\mathrm{d} z \\mathrm{d} x + R \\mathrm{d} x \\mathrm{d} y = - \\iint_ {S _ {1}} - \\iint_ {S _ {2}}", "math_type": "latex", "image_source": { "path": "images/dcc878858e679db5e4daf484ce3671f2f72f78d28b73ea9eefbbb784f6fce08d.jpg" } }, "bbox": [ 324, 587, 668, 620 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\iint_{S_2}" }, { "type": "text", "content": " 由于 " }, { "type": "equation_inline", "content": "z = 0" }, { "type": "text", "content": " 得到积分为0， " }, { "type": "equation_inline", "content": "\\iint_{S_1}" }, { "type": "text", "content": " 得到 " }, { "type": "equation_inline", "content": "r = \\epsilon" }, { "type": "text", "content": " ，即 " }, { "type": "equation_inline", "content": "P = \\frac{x}{\\epsilon^3},Q = \\frac{y}{\\epsilon^3},R = \\frac{z}{\\epsilon^3}" }, { "type": "text", "content": " ，再补上 " }, { "type": "equation_inline", "content": "V_{1}" }, { "type": "text", "content": " 下面的小平面 " }, { "type": "equation_inline", "content": "S_{3}" }, { "type": "text", "content": " ，用Gauss公式得到 " }, { "type": "equation_inline", "content": "I = \\frac{1}{\\epsilon^3}\\iint_{-S_1 + S_3}x\\mathrm{d}y\\mathrm{d}z + y\\mathrm{d}z\\mathrm{d}x + z\\mathrm{d}x\\mathrm{d}y" }, { "type": "text", "content": " ，最终结果为 " }, { "type": "equation_inline", "content": "2\\pi" } ] }, "bbox": [ 87, 621, 905, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 最后一步也可以用两类曲面积分的关系进行计算，当第二型不方便算时，要善于转化为第一型进行计算" } ] }, "bbox": [ 58, 686, 887, 701 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.5 Stokes 公式" } ], "level": 2 }, "bbox": [ 94, 728, 278, 747 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.5.1 Stokes 公式与基本使用" } ], "level": 2 }, "bbox": [ 94, 768, 363, 784 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.11 (Stokes 公式)" } ], "level": 2 }, "bbox": [ 121, 797, 304, 813 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设曲面 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 的边界 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 是按段光滑的连续曲线，若 " }, { "type": "equation_inline", "content": "P, Q, R" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 上连续，且有一阶偏导，则" } ] }, "bbox": [ 112, 819, 761, 834 ] }, { "type": "equation_interline", "content": { "math_content": "\\oint_ {L} P \\mathrm{d} x + Q \\mathrm{d} y + R \\mathrm{d} z = \\iint_ {S} \\left| \\begin{array}{c c c} \\mathrm{d} y \\mathrm{d} z & \\mathrm{d} z \\mathrm{d} x & \\mathrm{d} x \\mathrm{d} y \\\\ \\frac {\\partial}{\\partial x} & \\frac {\\partial}{\\partial y} & \\frac {\\partial}{\\partial z} \\\\ P & Q & R \\end{array} \\right| = \\iint_ {S} \\left| \\begin{array}{c c c} \\cos \\alpha & \\cos \\beta & \\cos \\gamma \\\\ \\frac {\\partial}{\\partial x} & \\frac {\\partial}{\\partial y} & \\frac {\\partial}{\\partial z} \\\\ P & Q & R \\end{array} \\right| \\mathrm{d} S", "math_type": "latex", "image_source": { "path": "images/c25df4e153299cde4d31e264ea7af8802f7908b8cee7e27759df30eb3b63da61.jpg" } }, "bbox": [ 183, 837, 811, 903 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.5 Stokes 公式" } ] }, "bbox": [ 784, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "143" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即将空间第二型曲线积分转换为第二或第一型曲面积分，这里 " }, { "type": "equation_inline", "content": "\\mathbf{n} = (\\cos \\alpha, \\cos \\beta, \\cos \\gamma)" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 正侧的外单位法向量。注意求法向的曲面 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 和最终积分的 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 要是同一个曲面。" } ] }, "bbox": [ 112, 96, 887, 130 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记Stokes公式一般用在三维第一型曲线积分，且积分曲线是两个曲面的交线，往往转换为第一型曲面积分更好算：外法向量能用平面就用平面(为常数)，圆面则为圆心到 " }, { "type": "equation_inline", "content": "(x,y,z)" }, { "type": "text", "content": " 的单位向量。" } ] }, "bbox": [ 87, 148, 905, 183 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "笔记 确定单位法向量的方向：若积分曲线方向是正向(区域在左)，则单位法向朝曲面朝外，否则朝内。" } ] }, "bbox": [ 85, 185, 843, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.16 一次项 Stokes 公式 利用 Stokes 公式计算下面的第一型曲线积分:" } ] }, "bbox": [ 58, 223, 650, 237 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(1) 平面与球面：计算 " }, { "type": "equation_inline", "content": "I = \\oint_{L} y \\, \\mathrm{d}x + z \\, \\mathrm{d}y + x \\, \\mathrm{d}z" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "x + y + z = 0" }, { "type": "text", "content": " 的交线，从 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴 " }, { "type": "equation_inline", "content": "+\\infty" }, { "type": "text", "content": " 看为逆时针。" } ] }, "bbox": [ 87, 240, 905, 278 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) 球面与柱面：计算 " }, { "type": "equation_inline", "content": "\\oint_{L} (x - 1) \\mathrm{d}y - (y + 1) \\mathrm{d}x + z \\mathrm{d}z" }, { "type": "text", "content": "，其中 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 1" }, { "type": "text", "content": " 上半球与柱面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = x" }, { "type": "text", "content": " 的交线，从 " }, { "type": "equation_inline", "content": "z" }, { "type": "text", "content": " 轴正向来看为逆时针方向。" } ] }, "bbox": [ 89, 278, 905, 318 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解(1)记 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "x + y + z = 0" }, { "type": "text", "content": " 位于球面 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = a^{2}" }, { "type": "text", "content": " 内的部分，取上侧 " }, { "type": "equation_inline", "content": "\\mathbf{n} = \\frac{1}{\\sqrt{3}} (1,1,1)" }, { "type": "text", "content": "，则根据Stokes公式：" } ] }, "bbox": [ 87, 317, 907, 344 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\iint_ {S} \\left| \\begin{array}{c c c} \\frac {1}{\\sqrt {3}} & \\frac {1}{\\sqrt {3}} & \\frac {1}{\\sqrt {3}} \\\\ \\frac {\\partial}{\\partial x} & \\frac {\\partial}{\\partial y} & \\frac {\\partial}{\\partial z} \\\\ y & z & x \\end{array} \\right| = \\frac {1}{\\sqrt {3}} \\iint_ {S} (- 1 - 1 - 1) \\mathrm{d} S = - \\sqrt {3} \\pi a ^ {2}", "math_type": "latex", "image_source": { "path": "images/0282f47fe6755e0dc5e59a8f622a6cec37577d46a4356d142b2b981cef217fd4.jpg" } }, "bbox": [ 257, 347, 737, 420 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2) " }, { "type": "equation_inline", "content": "\\mathbf{n} = (x, y, z)" }, { "type": "text", "content": "，因此根据Stokes公式得到" } ] }, "bbox": [ 122, 426, 440, 439 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\iint_ {S} \\left| \\begin{array}{c c c} x & y & z \\\\ \\frac {\\partial}{\\partial x} & \\frac {\\partial}{\\partial y} & \\frac {\\partial}{\\partial z} \\\\ - y - 1 & x - 1 & z \\end{array} \\right| \\mathrm{d} S = \\iint_ {S} 2 z \\mathrm{d} S = \\iint_ {D} 2 \\mathrm{d} S = \\frac {\\pi}{2}", "math_type": "latex", "image_source": { "path": "images/9d81532fda049e0d744e69d2874f716b9119761130b52afb64487ba8c8c42c5c.jpg" } }, "bbox": [ 262, 445, 732, 508 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "练习 17.17 二次项 Stokes 公式 经典：L 是 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} + z^{2} = 4x" }, { "type": "text", "content": " 与 " }, { "type": "equation_inline", "content": "x^{2} + y^{2} = 2x" }, { "type": "text", "content": " 的交线在 " }, { "type": "equation_inline", "content": "z \\geq 0" }, { "type": "text", "content": " 的部分，方向为原点进入第一卦限，计算：" } ] }, "bbox": [ 58, 521, 905, 555 ] }, { "type": "equation_interline", "content": { "math_content": "I = \\oint_ {L} (y ^ {2} + z ^ {2}) \\mathrm{d} x + (x ^ {2} + z ^ {2}) \\mathrm{d} y + (x ^ {2} + y ^ {2}) \\mathrm{d} z", "math_type": "latex", "image_source": { "path": "images/483c6e74fd405830bf0659f438013f208a01f40e363c88e3472d787d1619b3ce.jpg" } }, "bbox": [ 319, 552, 673, 583 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "解 曲面 S 是 " }, { "type": "equation_inline", "content": "(x-2)^{2}+y^{2}+z^{2}=4" }, { "type": "text", "content": " 的一部分，注意原点进第一卦限时区域在右侧，因此方向朝内。单位法向量为 " }, { "type": "equation_inline", "content": "\\mathbf{n}=-\\frac{1}{2}(x-2,y,z)" }, { "type": "text", "content": " ，因此根据 Stokes 公式：" } ] }, "bbox": [ 87, 590, 902, 630 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} I = \\oint_ {L} (y ^ {2} + z ^ {2}) \\mathrm{d} x + (x ^ {2} + z ^ {2}) \\mathrm{d} y + (x ^ {2} + y ^ {2}) \\mathrm{d} z \\\\ = - \\iint_ {S} [ (x - 2) (y - z) + y (z - x) + z (x - y) ] \\mathrm{d} S = 2 \\iint_ {S} (y - z) \\mathrm{d} S \\end{array}", "math_type": "latex", "image_source": { "path": "images/259af58534d492c7ea7021297b342469c8509fe7b207b9e38d0c9fdf274eb1da.jpg" } }, "bbox": [ 245, 634, 751, 696 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于 S 关于 xz 平面对称，因此 " }, { "type": "equation_inline", "content": "\\iint_{S} y \\, dS = 0" }, { "type": "text", "content": " ，此时" } ] }, "bbox": [ 90, 699, 468, 728 ] }, { "type": "equation_interline", "content": { "math_content": "I = - 2 \\iint_ {S} z \\mathrm{d} S", "math_type": "latex", "image_source": { "path": "images/af76adcf990f8dd2ccb5feb4390527e69d87a50a436c2b075a7423f606aee4b3.jpg" } }, "bbox": [ 438, 730, 557, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 方程为 " }, { "type": "equation_inline", "content": "z = \\sqrt{4 - (x - 2)^2 - y^2}" }, { "type": "text", "content": "，投影为 " }, { "type": "equation_inline", "content": "D = \\{(x,y):(x - 1)^2 +y^2\\leq 1\\}" }, { "type": "text", "content": "，且 " }, { "type": "equation_inline", "content": "\\sqrt{1 + z_x^2 + z_y^2} = \\frac{2}{z}" }, { "type": "text", "content": "，得到" } ] }, "bbox": [ 90, 765, 882, 790 ] }, { "type": "equation_interline", "content": { "math_content": "I = - 2 \\iint_ {S} z \\mathrm{d} S = - 2 \\iint_ {D} z \\cdot \\frac {2}{z} \\mathrm{d} x \\mathrm{d} y = - 4 \\pi", "math_type": "latex", "image_source": { "path": "images/897a628482cb758f9f788e1bd23dcbf75fbd37c9f34dadd5a961c3f30460c128.jpg" } }, "bbox": [ 337, 791, 658, 824 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.5 Stokes 公式" } ] }, "bbox": [ 784, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "144" } ] }, "bbox": [ 485, 954, 512, 965 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "17.5.2 空间第二型曲线积分与路径无关性" } ], "level": 2 }, "bbox": [ 95, 85, 463, 102 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 17.12 (空间第二型曲线积分与路径无关性)" } ], "level": 2 }, "bbox": [ 122, 118, 472, 134 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "V \\subseteq \\mathbb{R}^3" }, { "type": "text", "content": " 是单连通区域， " }, { "type": "equation_inline", "content": "P, Q, R" }, { "type": "text", "content": " 在 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 上连续，且有一阶偏导数， " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": " 为 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 内任意光滑曲线，则 " }, { "type": "equation_inline", "content": "\\int_{L} P \\mathrm{d}x + Q \\mathrm{d}y + R \\mathrm{d}z" }, { "type": "text", "content": " 与路径无关与下面条件等价：" } ] }, "bbox": [ 114, 140, 880, 181 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 任意 " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 内光滑封闭曲线 " }, { "type": "equation_inline", "content": "L" }, { "type": "text", "content": ", 满足 " }, { "type": "equation_inline", "content": "\\oint_{L} P \\mathrm{~d} x + Q \\mathrm{~d} y + R \\mathrm{~d} z = 0" } ] }, "bbox": [ 139, 181, 594, 206 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- " }, { "type": "equation_inline", "content": "V" }, { "type": "text", "content": " 内处处成立: " }, { "type": "equation_inline", "content": "P_{y} = Q_{x}, Q_{z} = R_{y}, R_{x} = P_{z}" } ] }, "bbox": [ 139, 206, 484, 222 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "17.5 Stokes 公式" } ] }, "bbox": [ 784, 55, 905, 68 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "145" } ] }, "bbox": [ 485, 954, 512, 964 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "第 18 章 实数完备性定理" } ], "level": 1 }, "bbox": [ 346, 87, 668, 111 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定义 18.1 (聚点)" } ], "level": 2 }, "bbox": [ 122, 145, 245, 160 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若存在各项互异的收敛数列 " }, { "type": "equation_inline", "content": "\\{x_{n}\\} \\subseteq S" }, { "type": "text", "content": " ，使得 " }, { "type": "equation_inline", "content": "\\lim_{n\\to \\infty}x_n = \\xi" }, { "type": "text", "content": " ，则 " }, { "type": "equation_inline", "content": "\\xi" }, { "type": "text", "content": " 称为 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 的一个聚点。" } ] }, "bbox": [ 115, 164, 731, 189 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定理 18.1 (实数七大定理)" } ], "level": 2 }, "bbox": [ 122, 211, 312, 225 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "七大定理如下：" } ], "level": 2 }, "bbox": [ 117, 231, 228, 244 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 确界定理： " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 是非空实数集，若 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 有上界，则 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 必有上确界" } ] }, "bbox": [ 139, 249, 601, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 单调有界定理" } ] }, "bbox": [ 141, 268, 263, 282 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- Cauchy 收敛准则" } ] }, "bbox": [ 139, 287, 284, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 闭区间套定理" } ] }, "bbox": [ 141, 305, 263, 318 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 有限开覆盖定理： " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 是实数子集， " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 为开区间族，若 " }, { "type": "equation_inline", "content": "\\forall x \\in S" }, { "type": "text", "content": " 都含于 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 至少一个开区间中，则称 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 开覆盖。若 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "[a, b]" }, { "type": "text", "content": " 的开覆盖，则可从 " }, { "type": "equation_inline", "content": "H" }, { "type": "text", "content": " 中选出有限个区间覆盖 " }, { "type": "equation_inline", "content": "[a, b]" } ] }, "bbox": [ 139, 323, 880, 356 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "- 聚点定理：实轴上任一有界无限点集 " }, { "type": "equation_inline", "content": "S" }, { "type": "text", "content": " 至少有一个聚点" } ] }, "bbox": [ 139, 359, 569, 374 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "致密性定理：有界数列必有收敛子列" } ] }, "bbox": [ 139, 378, 431, 393 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下面采取确界原理-单调有界定理-闭区间套定理-有限覆盖定理-聚点定理/致密性定理-Cauchy收敛准则的形式循环证明其等价性：" } ] }, "bbox": [ 90, 431, 905, 464 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "1. 确界证明单调有界定理：设 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 单增有上界，根据确界原理有上确界，取 " }, { "type": "equation_inline", "content": "a = \\sup \\{a_{n}\\}" }, { "type": "text", "content": "，只需证明 " }, { "type": "equation_inline", "content": "a" }, { "type": "text", "content": " 是 " }, { "type": "equation_inline", "content": "a_{n}" }, { "type": "text", "content": " 极限。根据上确界定义有 " }, { "type": "equation_inline", "content": "\\exists N, a - \\epsilon < a_{N}" }, { "type": "text", "content": "，再由单增和极限定义显然。" } ] }, "bbox": [ 110, 468, 904, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2. 单调有界证明闭区间套： " }, { "type": "equation_inline", "content": "a_{n}, b_{n}" }, { "type": "text", "content": " 都单调有极限，所以夹住的点存在。唯一性用 " }, { "type": "equation_inline", "content": "|\\xi - \\xi'| \\leq \\lim_{n \\to \\infty} (b_{n} - a_{n}) = 0" }, { "type": "text", "content": " 证明" } ] }, "bbox": [ 110, 504, 905, 537 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3. 闭区间套证明有限覆盖：反设不成立，则 " }, { "type": "equation_inline", "content": "[a, \\frac{a + b}{2}], [\\frac{a + b}{2}, b]" }, { "type": "text", "content": " 中至少有一个没有有限覆盖，不停分下去，根据闭区间套有 " }, { "type": "equation_inline", "content": "\\xi \\in [a_n, b_n]" }, { "type": "text", "content": "。当 " }, { "type": "equation_inline", "content": "n" }, { "type": "text", "content": " 足够大时，一定存在开覆盖集中的区间包含它，因此与假设矛盾。" } ] }, "bbox": [ 110, 539, 905, 575 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "4. 有限覆盖证明致密性定理：设 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 有界，若无穷项相等，则显然。若相等的只有有限项，可知 " }, { "type": "equation_inline", "content": "\\exists x_0 \\in [a, b]" }, { "type": "text", "content": " 使得 " }, { "type": "equation_inline", "content": "(x_0 - \\delta, x_0 + \\delta)" }, { "type": "text", "content": " 中有无限项，否则根据有限开覆盖，总共 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 只有有限项，矛盾。因此在 " }, { "type": "equation_inline", "content": "(x_0 - \\frac{1}{k}, x_0 + \\frac{1}{k})" }, { "type": "text", "content": " 中取 " }, { "type": "equation_inline", "content": "x_{k}" }, { "type": "text", "content": " 即有 " }, { "type": "equation_inline", "content": "\\lim_{k \\to \\infty} x_{k} = x_{0}" } ] }, "bbox": [ 110, 577, 905, 634 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "5. 聚点定理证明 Cauchy 收敛准则：设 " }, { "type": "equation_inline", "content": "x_{n}" }, { "type": "text", "content": " 收敛， " }, { "type": "equation_inline", "content": "\\lim_{n \\to \\infty} x_{n} = a" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "\\exists N, \\forall n, m > N" }, { "type": "text", "content": " 有 " }, { "type": "equation_inline", "content": "|x_{n} - a| < \\epsilon, |x_{m} - a| < \\epsilon" }, { "type": "text", "content": "，因此 " }, { "type": "equation_inline", "content": "|a_{n} - a_{m}| \\leq |a_{n} - a| + |a - a_{m}| = 2\\epsilon" } ] }, "bbox": [ 110, 636, 905, 671 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "6. Cauchy 收敛准则证明确界原理: b 为上界, " }, { "type": "equation_inline", "content": "a \\in S" }, { "type": "text", "content": " , 不断二分, 构造 " }, { "type": "equation_inline", "content": "a_{n}, b_{n}" }, { "type": "text", "content": " , 最后 " }, { "type": "equation_inline", "content": "|a_{n+p} - a_{n}| \\leq |b_{n} - a_{n}| < \\epsilon" }, { "type": "text", "content": " , 因此有极限 a, 满足 " }, { "type": "equation_inline", "content": "\\exists N, a_{N} + \\epsilon > a" }, { "type": "text", "content": " 。" } ] }, "bbox": [ 110, 674, 905, 707 ] } ] ]