本書按照一元函數微分學、微分方程、空間解析幾何與向量代數、多元函數微分學和無窮級數的順序編排，每節包括四個部分：重要知識點、例題辨析、真題演練和真題演練解析。
第1 章 函數與極限····························································································1
1.1 映射與函數···························································································1
1.2 數列的極限···························································································4
1.3 函數的極限························································································.10
1.4 無窮小與無窮大··················································································.14
1.5 極限運算法則·····················································································.17
1.6 極限存在準則和兩個重要極限·································································.23
1.7 無窮小的比較·····················································································.26
1.8 函數的連續性與間斷點·········································································.32
1.9 連續函數的運算及初等函數的連續性························································.36
1.10 閉區間上連續函數的性質·····································································.38
第2 章 導數與微分·························································································.41
2.1 導數概念···························································································.41
2.2 函數的求導法則··················································································.48
2.3 高階導數···························································································.52
2.4 隱函數及由參數方程所確定函數的導數和相關變化率···································.55
2.5 函數的微分························································································.62
第3 章 微分中值定理與導數的應用····································································.67
3.1 微分中值定理·····················································································.67
3.2 洛必達法則························································································.73
3.3 泰勒公式···························································································.79
3.4 函數的單調性與曲線的凹凸性·································································.86
3.5 函數的極值與最大值、最小值·································································.91
3.6 函數圖形的描繪··················································································.96
3.7 曲率·································································································.99
第4 章 不定積分····························································································102
4.1 不定積分的概念與性質·········································································102
4.2 換元積分法························································································105
4.3 分部積分法························································································108
4.4 有理函數的積分··················································································113
第5 章 定積分·······························································································117
5.1 定積分的概念與性質············································································117
5.2 微積分基本公式··················································································122
5.3 定積分的換元法和分部積分法·································································127
5.4 反常積分···························································································134
第6 章 定積分的應用······················································································138
6.1 定積分在幾何學上的應用······································································138
6.2 定積分在物理學上的應用······································································145
第7 章 微分方程····························································································151
7.1 微分方程的基本概念············································································151
7.2 可分離變量的微分方程·········································································153
7.3 齊次方程···························································································157
7.4 一階線性微分方程···············································································159
7.5 可降階的高階微分方程·········································································164
7.6 高階線性微分方程···············································································167
7.7 常系數齊次線性微分方程······································································170
7.8 常系數非齊次線性微分方程····································································173
第8 章 向量代數與空間解析幾何·······································································177
8.1 向量及其線性運算···············································································177
8.2 數量積、向量積和混合積······································································180
8.3 平面及其方程·····················································································182
8.4 空間直線及其方程···············································································185
8.5 曲面及其方程·····················································································190
8.6 空間曲線及其方程···············································································192
第9 章 多元函數微分法及其應用·······································································195
9.1 多元函數的基本概念············································································195
9.2 偏導數······························································································199
9.3 全微分······························································································204
9.4 多元復合函數的求導法則······································································210
9.5 隱函數的求導公式···············································································216
9.6 多元函數微分學的幾何應用····································································225
9.7 方向導數與梯度··················································································229
9.8 多元函數的極值及求法·········································································233
第10 章 重積分······························································································240
10.1 二重積分的概念與性質········································································240
10.2 二重積分的計算法··············································································244
10.3 三重積分··························································································252
10.4 重積分的應用····················································································257
第11 章 曲線積分與曲面積分············································································262
11.1 對弧長的曲線積分··············································································262
11.2 對坐標的曲線積分··············································································265
11.3 格林公式及其應用··············································································270
11.4 對面積的曲面積分··············································································277
11.5 對坐標的曲面積分··············································································281
11.6 高斯公式、通量與散度········································································286
11.7 斯托克斯公式、環流量與旋度·······························································292
第12 章 無窮級數···························································································296
12.1 常數項級數的概念和性質·····································································296
12.2 常數項級數的審斂法···········································································300
12.3 冪級數·····························································································308
12.4 函數展開成冪級數··············································································317
12.5 函數的冪級數展開式的應用··································································321
12.6 傅里葉級數·······················································································323
12.7 一般周期函數的傅里葉級數··································································327
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