本书主要针对拔尖创新人才培养而编写，分上、下两册. 上册内容包括极限与连续、导数与微分、微分 中值定理及导数应用、不定积分、定积分及其应用、微分方程；下册内容包括多元函数微分学、多元数量值函数积分学、多元向量值函数积分学、无穷级数. 本书可作为高等学校理工科专业微积分课程的教材，也适合准备考研的学生参考.
尤超教授，理学博士，现任哈尔滨工业大学数学学院数学教学研究中心副主任。主要研究方向为算子代数，主讲工科数学分析、数学竞赛辅导等课程。主持、参与国家自然科学基金项目2项，主持、参与省部级教改项目2项。曾获2022年宝钢优秀教师奖，2021年黑龙江省青年教学名师奖。
第 7 章 多元函数微分学 ············································································································.1
7.1 多元函数的基本概念及其极限 ·····················································································.1
7.1.1 n 维欧式空间 ········································································································.1
7.1.2 多元函数 ···············································································································.2
7.1.3 多元函数的重极限与连续 ························································································.3
7.1.4 多元函数的累次极限 ·······························································································.7
习题 7.1 ···························································································································.8
7.2 偏导数与高阶偏导数 ····································································································.9
7.2.1 偏导数的定义 ·········································································································.9
7.2.2 高阶偏导数 ··········································································································.11
习题 7.2 ·························································································································.14
7.3 全微分 ··························································································································.14
习题 7.3 ·························································································································.18
7.4 复合函数求导法 ··········································································································.19
习题 7.4 ·························································································································.25
7.5 隐函数求导法 ··············································································································.26
习题 7.5 ·························································································································.30
7.6 偏导数的几何应用 ······································································································.31
7.6.1 空间曲线的切线与法平面 ······················································································.31
7.6.2 曲面的切平面与法线 ·····························································································.33
7.6.3 二元函数全微分的几何意义 ···················································································.35
习题 7.6 ·························································································································.35
7.7 多元函数的极值 ··········································································································.36
7.7.1 多元函数的无条件极值 ··························································································.36
7.7.2 条件极值与拉格朗日乘数法 ···················································································.42
习题 7.7 ·························································································································.46
7.8 方向导数与梯度 ··········································································································.46
7.8.1 场的基本概念 ·······································································································.46
7.8.2 方向导数 ·············································································································.46
7.8.3 梯度 ····················································································································.49
习题 7.8 ·························································································································.51
综合题 ··································································································································.52
第 8 章 多元数量值函数积分学 ······························································································.53
8.1 黎曼积分 ·················································
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