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.

2009 年，“基础学科拔尖学生培养试验计划”正式启动，旨在培养中国自己的学术 大师. 探索建立拔尖创新人才培养的有效机制，促进拔尖创新人才脱颖而出，是建设创 新型国家、实现中华民族伟大复兴的历史要求，也是当前对教育改革的迫切要求. 2020 年，部分高校开展基础学科招生改革试点（强基计划）. 强基计划开宗明义：聚焦国家 重大战略需求，在确保公平公正的前提下，探索建立多维度考核学生的评价模式，逐步 形成基础学科拔尖创新人才选拔培养的有效机制，重点解决基础学科领军人才短缺和长 远发展的瓶颈问题. 本书就是对拔尖创新人才培养的重要探索，在工业和信息化部“十 四五”规划教材的建设要求下，坚持以学生为中心，通过深度与广度的探索，强化学生 的数学基础. 近年来，一些高水平研究型大学在拔尖创新人才培养方面进行了有益的尝试和多样化的 探索，并取得了初步成果. 未来技术学院作为哈尔滨工业大学对本科拔尖创新学生实施“精 英教育”的试验田、示范区，秉承“学术型、国际化、高素质、重创新”的培养理念，以探 索路子、选拔尖子、培育苗子为使命宗旨，竭力为拔尖创新学生的科学精神塑造、创新意识 培养、实践能力提高、综合素质提升等搭建平台、提供舞台，培养造就未来的学术带头人、 工程领军人、政企领导人等杰出人才. 本书正是针对哈尔滨工业大学未来技术学院及其他高校拔尖创新学生编写的微积分教 材，增加了传统微积分教材中较少介绍的实数理论、一致收敛、一致连续、高阶微分、广义 重积分、重积分的变量代换、场论初步、外微分形式、含参变量积分、傅里叶积分与傅里叶 变换等方面的内容，力求做到让学生知其然也知其所以然. 本书分上、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. 书中带星号（*）的内容为选学内容. 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. 本书得到了哈尔滨工业大学数学学院教学研究中心各位教师的大力支持，也吸收了哈尔 滨工业大学威海分校数学系、东北林业大学数学系、哈尔滨理工大学数学系等高校教师提出 的宝贵意见和建议，在此一并表示感谢. 由于作者水平有限，加之编写时间仓促，书中错误和不足之处在所难免，恳请读者不吝 赐教

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 ······················································································································.53 8.1.1 二重积分的概念 ···································································································.53 8.1.2 黎曼积分的概念 ···································································································.54 8.1.3 黎曼积分的分类 ···································································································.54 8.1.4 黎曼积分的性质 ···································································································.55 习题 8.1 ·························································································································.57 8.2 二重积分的计算 ··········································································································.58 8.2.1 直角坐标系下二重积分的计算 ················································································.58 8.2.2 极坐标系下二重积分的计算 ···················································································.62 8.2.3 二重积分的对换对称性 ··························································································.66 8.2.4 二重积分的变量代换 ·····························································································.67 习题 8.2 ·························································································································.70 8.3 三重积分的计算 ··········································································································.72 8.3.1 直角坐标系下三重积分的计算 ················································································.72 8.3.2 柱坐标系下三重积分的计算 ···················································································.75 8.3.3 球坐标系下三重积分的计算 ···················································································.77 8.3.4 三重积分一般的变量代换 ······················································································.79 习题 8.3 ·························································································································.81 8.4 第一型曲线积分 ··········································································································.83 习题 8.4 ·························································································································.86 8.5 第一型曲面积分的计算 ·······························································································.87 习题 8.5 ·························································································································.93 8.6 黎曼积分的应用实例 ··································································································.93 8.6.1 物体质心 ·············································································································.94 8.6.2 转动惯量 ·············································································································.96 习题 8.6 ·························································································································.98 8.7 含参变量的积分 ··········································································································.99 习题 8.7 ·······················································································································.101 综合题 ································································································································.101 第 9 章 多元向量值函数积分学 ····························································································.104 9.1 向量场 ························································································································.104 9.1.1 向量场与向量线 ·································································································.104 9.1.2 向径的导数 ········································································································.105 9.2 第二型曲线积分 ········································································································.106 9.2.1 第二型曲线积分的定义 ························································································.106 9.2.2 第二型曲线积分的计算 ························································································.108 9.2.3 两种类型曲线积分的关系 ····················································································.111 习题 9.2 ·······················································································································.112 9.3 格林公式及平面流速场的环量与旋度 ·····································································.113 9.3.1 格林公式 ···········································································································.113 9.3.2 平面流速场的环量与旋度 ····················································································.117 习题 9.3 ·······················································································································.117 9.4 平面曲线积分与路径无关的条件及保守场 ······························································.118 9.4.1 平面曲线积分与路径无关的条件 ··········································································.118 9.4.2 保守场、原函数、全微分方程 ··············································································.123 习题 9.4 ·······················································································································.126 9.5 第二型曲面积分 ········································································································.127 9.5.1 第二型曲面积分的定义 ························································································.127 9.5.2 第二型曲面积分的计算 ························································································.130 习题 9.5 ·······················································································································.135 9.6 高斯公式、通量与散度 ·····························································································.136 9.6.1 高斯公式 ···········································································································.136 9.6.2 向量场的通量与散度 ···························································································.139 9.6.3 第二型曲面积分的三个等价条件 ··········································································.140 习题 9.6 ·······················································································································.142 9.7 斯托克斯公式、环量与旋度 ·····················································································.144 9.7.1 斯托克斯公式 ·····································································································.144 9.7.2 向量场的环量与旋度 ···························································································.146 习题 9.7 ·······················································································································.150 9.8 外微分形式 ················································································································.151 9.8.1 外积和外微分形式 ······························································································.151 9.8.2 外微分运算 ········································································································.151 9.8.3 梯度、旋度与散度的数学意义 ··············································································.152 9.8.4 多变量微积分的基本定理（一般的斯托克斯公式） ················································.154 综合题 ································································································································.154 第 10 章 无穷级数 ·················································································································.156 10.1 数项级数的敛散性 ··································································································.156 10.1.1 敛散性的概念 ···································································································.156 10.1.2 数项级数的几个基本性质 ···················································································.158 10.1.3 柯西收敛原理 ···································································································.160 习题 10.1 ·····················································································································.161 10.2 正项级数的敛散性及其判别法 ···············································································.162 习题 10.2 ·····················································································································.172 10.3 任意项级数的敛散性及其判别法 ···········································································.174 习题 10.3 ·····················································································································.181 10.4 函数项级数 ··············································································································.182 10.4.1 函数项级数的收敛域与和函数 ············································································.182 10.4.2 函数项级数的一致收敛性 ···················································································.183 习题 10.4 ·····················································································································.189 10.5 幂级数 ······················································································································.189 10.5.1 幂级数的收敛半径和收敛域 ···············································································.190 10.5.2 幂级数的运算 ···································································································.194 习题 10.5 ·····················································································································.196 10.6 函数的幂级数展开 ··································································································.197 10.6.1 泰勒级数及直接展开法 ······················································································.197 10.6.2 间接展开法 ······································································································.201 10.6.3 幂级数和函数 ···································································································.204 习题 10.6 ·····················································································································.206 10.7 幂级数的应用举例 ··································································································.208 10.7.1 函数值的近似计算 ····························································································.208 10.7.2 在积分计算中的应用 ·························································································.209 10.7.3 方程的幂级数解法 ····························································································.210 习题 10.7 ·····················································································································.212 10.8 傅里叶级数 ··············································································································.213 10.8.1 三角函数系的正交性 ·························································································.213 10.8.2 傅里叶级数 ······································································································.214 10.8.3 正弦级数和余弦级数 ·························································································.216 10.8.4 以 2l 为周期的函数的傅里叶级数 ········································································.217 10.8.5 有限区间上的函数的傅里叶展开 ·········································································.219 习题 10.8 ·····················································································································.220 综合题 ································································································································.222