微积分原理（下）

定　　价：69 元

• 作者：崔建莲
• 出版时间：2023/10/1
• ISBN：9787121464249
• 出 版 社：电子工业出版社

• 中图法分类：O172 
• 页码：356
• 版次：01
• 开本：16开

内容简介

微积分是理工科高等学校非数学类专业最基础、重要的一门核心课程。许多后继数学课程及物理和各种工程学课程都是在微积分课程的基础上展开的，因此学好这门课程对每一位理工科学生来说都非常重要。本套教材在传授微积分知识的同时，注重培养学生的数学思维、语言逻辑和创新能力，弘扬数学文化，培养科学精神。本套教材分上、下两册。上册内容包括实数集与初等函数、数列极限、函数极限与连续、导数与微分、微分学基本定理及应用、不定积分、定积分、广义积分和常微分方程。下册内容包括多元函数的极限与连续、多元函数微分学及其应用、重积分、曲线积分、曲面积分、数项级数、函数项级数、傅里叶级数和含参积分。

作者介绍

崔建莲，清华大学数学系副教授。2002年7月获得中科院数学研究所博士学位，2004年4月北京大学博士后出站，香港大学访问学者，韩国首尔大学访问学者，美国威廉玛丽学院访问学者。2004年4月入职清华大学数学系，现为数学系副教授，主要研究方向为算子代数、算子理论及在量子信息中的应用。发表学术论文60多篇，SCI收录50多篇。

目 录

目录
第10 章 多元函数的极限与连续··········1
10.1 n ? 中的点集拓扑和点列··········.1
10.1.1 n ? 中的点集拓扑···················1
10.1.2 n ? 中的点列·························6
10.1.3 n ? 的完备性·························7
*10.1.4 n ? 中的等价范数···················8
习题10.1 ··································.10
10.2 多元函数与多元向量值函数····.11
10.2.1 多元函数的概念··················.11
10.2.2 二元函数的图像··················.12
10.2.3 多元向量值函数··················.16
习题10.2 ··································.17
10.3 多元函数的极限···················.18
10.3.1 多元函数的重极限···············.18
10.3.2 多元函数的累次极限············.19
10.3.3 向量值函数的极限···············.21
习题10.3 ··································.23
10.4 多元函数和向量值函数的
连续性·······························.24
10.4.1 多元函数连续的概念············.24
10.4.2 多元函数对各个变量的分别
连续·······························.26
10.4.3 多元连续函数的性质············.27
习题10.4 ··································.28
第11 章 多元函数微分学················.30
11.1 多元函数的偏导数与全微分····.30
11.1.1 多元函数的偏导数···············.30
11.1.2 多元函数的全微分···············.32
11.1.3 函数可微的条件··················.34
11.1.4 全微分在函数近似计算中的
应用······························.37
习题11.1 ··································.38
11.2 高阶偏导数与复合函数的
微分··································.39
11.2.1 高阶偏导数·······················.39
11.2.2 复合函数的微分··················.41
11.2.3 一阶全微分的形式不变性·······.43
习题11.2 ··································.44
11.3 方向导数与梯度···················.46
11.3.1 方向导数·························.46
11.3.2 梯度······························.48
习题11.3 ··································.50
11.4 向量值函数的微分················.51
11.4.1 向量值函数的微分···············.51
11.4.2 复合映射的微分··················.54
习题11.4 ··································.55
11.5 隐函数微分法与逆映射微分法··.56
11.5.1 隐函数的微分····················.56
11.5.2 逆映射的微分····················.64
习题11.5 ··································.64
第12 章 多元函数微分学应用··········.67
12.1 多元函数微分学的几何应用····.67
12.1.1 空间曲线·························.67
12.1.2 空间曲面的切平面与法线·······.69
12.1.3 空间曲线的切线与法平面·······.72
习题12.1 ··································.76
12.2 高阶全微分与泰勒公式··········.77
12.2.1 高阶全微分·······················.77
12.2.2 泰勒公式·························.79
习题12.2 ··································.82
12.3 多元函数的极值···················.82
12.3.1 无条件极值·······················.83
12.3.2 条件极值·························.87
习题12.3 ··································.95
第13 章 重积分····························.98
13.1 二重积分的概念及性质··········.98
13.1.1 二重积分的概念··················.98
13.1.2 可积的条件·······················100
13.1.3 二重积分的性质··················101
习题13.1 ··································103
13.2 二重积分的计算···················104
13.2.1 直角坐标系·······················104
13.2.2 二重积分的坐标变换············108
习题13.2 ·································.114
13.3 三重积分···························.116
13.3.1 直角坐标系······················.117
13.3.2 一般坐标变换···················.119
13.3.3 柱坐标变换·······················120
13.3.4 球坐标变换·······················122
习题13.3 ··································124
13.4 重积分在几何和物理中的
应用··································125
13.4.1 空间曲面的面积··················126
13.4.2 重积分在物理中的应用··········128
习题13.4 ··································131
*13.5 n 重积分····························132
13.5.1 若当测度的定义··················132
13.5.2 若当可测的等价条件············134
13.5.3 若当测度的运算性质············135
13.5.4 n 重积分··························138
13.5.5 n 维球坐标变换··················139
第14 章 曲线积分·························143
14.1 第一型曲线积分——关于弧长
的曲线积分·························143
14.1.1 第一型曲线积分的概念··········143
14.1.2 第一型曲线积分的性质·········.145
14.1.3 第一型曲线积分的计算·········.146
14.1.4 柱面侧面积的计算··············.148
习题14.1 ·································.149
14.2 第二型曲线积分——关于坐标
的曲线积分························.150
14.2.1 第二型曲线积分的概念·········.150
14.2.2 两类曲线积分之间的关系······.151
14.2.3 第二型曲线积分的计算·········.151
习题14.2 ·································.155
14.3 格林公式···························.157
14.3.1 格林公式························.157
14.3.2 曲线积分与积分路径无关的
条件·····························.160
14.3.3 求微分式的原函数··············.161
14.3.4 全微分方程······················.164
习题14.3 ·································.166
第15 章 曲面积分························.170
15.1 第一型曲面积分——关于面积
的曲面积分························.170
15.1.1 第一型曲面积分的概念·········.170
15.1.2 第一型曲面积分的计算·········.171
习题15.1 ·································.174
15.2 第二型曲面积分——关于坐标
的曲面积分························.175
15.2.1 第二型曲面积分的概念·········.175
15.2.2 第二型曲面积分的计算·········.178
习题15.2 ·································.181
15.3 高斯公式和斯托克斯公式······.182
15.3.1 高斯公式························.182
15.3.2 斯托克斯公式···················.185
15.3.3 空间曲线积分与积分路径无关
的条件···························.189
习题15.3 ·································.190
15.4 场论初步···························.192
15.4.1 梯度场···························.192
15.4.2 散度场···························.193
15.4.3 旋度场···························.195
15.4.4 三种运算的联合运用············196
15.4.5 平面向量场·······················196
*15.4.6 曲线坐标系·······················198
15.4.7 正交曲线坐标系下的梯度、旋度、
散度和拉普拉斯算子············200
习题15.4 ··································204
第16 章 数项级数·························206
16.1 级数的敛散性······················207
16.1.1 级数收敛与发散的概念··········207
16.1.2 收敛级数的性质··················208
习题16.1 ··································210
16.2 正项级数···························.211
习题16.2 ··································220
16.3 任意项级数·························221
16.3.1 莱布尼茨（Leibniz）判别法····221
16.3.2 绝对收敛级数的性质············222
16.3.3 条件收敛级数的两个判别法·····226
*16.3.4 无穷乘积·························229
习题16.3 ··································229
第17 章 函数项级数······················232
17.1 函数列·······························232
17.1.1 函数列的一致收敛···············232
17.1.2 函数列极限函数的分析性·······237
习题17.1 ··································238
17.2 函数项级数·························239
17.2.1 函数项级数的收敛域············239
17.2.2 函数项级数的一致收敛性·······240
17.2.3 和函数的分析性··················243
*17.2.4 两个例子·························247
习题17.2 ··································251
17.3 幂级数·······························252
17.3.1 幂级数的收敛域与收敛半径·····252
17.3.2 幂级数和函数的分析性··········255
习题17.3 ··································261
17.4 函数的幂级数展开················262
17.4.1 泰勒级数、麦克劳林级数·······263
17.4.2 函数可展开为泰勒级数的条件····264
17.4.3 基本初等函数的麦克劳林级数··.265
17.4.4 利用幂级数求数的近似值······.268
习题17.4 ·································.270
第18 章 傅里叶级数·····················.271
18.1 函数的傅里叶级数···············.272
18.1.1 以2π 为周期函数的傅里叶级数··.272
18.1.2 以2l 为周期函数的傅里叶级数··.278
习题18.1 ·································.280
18.2 傅里叶级数的逐点收敛性······.281
18.2.1 傅里叶级数的性质··············.281
18.2.2 傅里叶级数的逐点收敛·········.284
习题18.2 ·································.291
18.3 傅里叶级数的平方平均收敛···.292
18.3.1 正交投影及Bessel 不等式······.292
18.3.2 三角多项式······················.295
18.3.3 Fejér 核与一致逼近·············.296
18.3.4 均方收敛························.299
习题18.3 ·································.306
18.4 傅里叶积分简介··················.308
18.4.1 傅里叶级数的复数形式·········.308
18.4.2 傅里叶积分：启发式介绍······.309
18.4.3 傅里叶积分：严格理论·········.312
习题18.4 ·································.318
18.5 函数逼近定理·····················.319
18.5.1 魏尔斯特拉斯第一逼近定理····.319
18.5.2 魏尔斯特拉斯第二逼近定理····.325
习题18.5 ·································.327
第19 章 含参积分························.328
19.1 含参定积分························.328
习题19.1 ·································.332
19.2 含参广义积分·····················.333
19.2.1 含参广义积分的一致收敛性····.333
19.2.2 含参广义积分的分析性·········.336
19.2.3 欧拉积分：伽马函数与贝塔
函数·····························.342
习题19.2 ·································.346
参考文献······································.348