本书按照一元函数微分学、微分方程、空间解析几何与向量代数、多元函数微分学和无穷级数的顺序编排，每节包括四个部分：重要知识点、例题辨析、真题演练和真题演练解析。

第1 章 函数与极限····························································································1
1.1 映射与函数···························································································1
1.2 数列的极限···························································································4
1.3 函数的极限························································································.10
1.4 无穷小与无穷大··················································································.14
1.5 极限运算法则·····················································································.17
1.6 极限存在准则和两个重要极限·································································.23
1.7 无穷小的比较·····················································································.26
1.8 函数的连续性与间断点·········································································.32
1.9 连续函数的运算及初等函数的连续性························································.36
1.10 闭区间上连续函数的性质·····································································.38
第2 章 导数与微分·························································································.41
2.1 导数概念···························································································.41
2.2 函数的求导法则··················································································.48
2.3 高阶导数···························································································.52
2.4 隐函数及由参数方程所确定函数的导数和相关变化率···································.55
2.5 函数的微分························································································.62
第3 章 微分中值定理与导数的应用····································································.67
3.1 微分中值定理·····················································································.67
3.2 洛必达法则························································································.73
3.3 泰勒公式···························································································.79
3.4 函数的单调性与曲线的凹凸性·································································.86
3.5 函数的极值与最大值、最小值·································································.91
3.6 函数图形的描绘··················································································.96
3.7 曲率·································································································.99
第4 章 不定积分····························································································102
4.1 不定积分的概念与性质·········································································102
4.2 换元积分法························································································105
4.3 分部积分法························································································108
4.4 有理函数的积分··················································································113
第5 章 定积分·······························································································117
5.1 定积分的概念与性质············································································117
5.2 微积分基本公式··················································································122
5.3 定积分的换元法和分部积分法·································································127
5.4 反常积分···························································································134
第6 章 定积分的应用······················································································138
6.1 定积分在几何学上的应用······································································138
6.2 定积分在物理学上的应用······································································145
第7 章 微分方程····························································································151
7.1 微分方程的基本概念············································································151
7.2 可分离变量的微分方程·········································································153
7.3 齐次方程···························································································157
7.4 一阶线性微分方程···············································································159
7.5 可降阶的高阶微分方程·········································································164
7.6 高阶线性微分方程···············································································167
7.7 常系数齐次线性微分方程······································································170
7.8 常系数非齐次线性微分方程····································································173
第8 章 向量代数与空间解析几何·······································································177
8.1 向量及其线性运算···············································································177
8.2 数量积、向量积和混合积······································································180
8.3 平面及其方程·····················································································182
8.4 空间直线及其方程···············································································185
8.5 曲面及其方程·····················································································190
8.6 空间曲线及其方程···············································································192
第9 章 多元函数微分法及其应用·······································································195
9.1 多元函数的基本概念············································································195
9.2 偏导数······························································································199
9.3 全微分······························································································204
9.4 多元复合函数的求导法则······································································210
9.5 隐函数的求导公式···············································································216
9.6 多元函数微分学的几何应用····································································225
9.7 方向导数与梯度··················································································229
9.8 多元函数的极值及求法·········································································233
第10 章 重积分······························································································240
10.1 二重积分的概念与性质········································································240
10.2 二重积分的计算法··············································································244
10.3 三重积分··························································································252
10.4 重积分的应用····················································································257
第11 章 曲线积分与曲面积分············································································262
11.1 对弧长的曲线积分··············································································262
11.2 对坐标的曲线积分··············································································265
11.3 格林公式及其应用··············································································270
11.4 对面积的曲面积分··············································································277
11.5 对坐标的曲面积分··············································································281
11.6 高斯公式、通量与散度········································································286
11.7 斯托克斯公式、环流量与旋度·······························································292
第12 章 无穷级数···························································································296
12.1 常数项级数的概念和性质·····································································296
12.2 常数项级数的审敛法···········································································300
12.3 幂级数·····························································································308
12.4 函数展开成幂级数··············································································317
12.5 函数的幂级数展开式的应用··································································321
12.6 傅里叶级数·······················································································323
12.7 一般周期函数的傅里叶级数··································································327