本书分上、下两册. 上册聚焦一元函数微积分学，涵盖函数、极限与连续、导数与微分、微分的应用、不定积分、定积分及其应用、微分方程等内容；下册围绕多元函数微积分学展开，内容包括多元函数微分学、黎曼积分、向量微积分、无穷级数等. 本书每章均配有不同难度的大量习题，供读者巩固所学、强化训练.本书适合高等学校理工科专业学生（尤其是英语基础扎实的学生及海外留学生）学习微积分课程使用，也可作为工科硕士研究生入学考试的复习资料以及工程相关从业者的参考书.

张宇，哈尔滨工业大学数学学院副教授，主要从事低维拓扑与几何学领域的研究工作。他于美国纽约州立大学布法罗分校获得博士学位，之后回到哈工大数学学院任教。在科研方面，张宇曾主持国家自然科学基金青年项目“Haken流形的判定及Virtually Haken猜想”，并在Journal of Knot Theory and Its Ramifications、Topology and its Applications等学术期刊发表多篇关于三维流形不变量与纽结理论的研究论文。他多次主持国际拓扑学研讨会并作学术报告，活跃于学术交流前沿。

Chapter 1 Single-Variable Functions ·································································1
1.1 Single-Variable Functions ··········································································1
1.2 Some Important Concepts of Functions ··························································7
1.3 Elementary Functions ············································································.12
1.4 Polar Coordinates ·················································································.20
Exercises 1································································································.22
Chapter 2 Limits and Continuity·····································································.25
2.1 The Limit of a Sequence·········································································.25
2.2 The Limit of a Function··········································································.30
2.3 Properties of Limits and Criteria for Existence of Limits···································.35
2.4 Limit Laws, Two Important Limits and Infinitesimals ······································.41
2.5 Continuity··························································································.57
Exercises 2································································································.65
Chapter 3 Derivatives and Differentials ···························································.71
3.1 The Concepts of Derivatives ····································································.71
3.2 Differentiation Rules ·············································································.77
3.3 Higher Derivatives················································································.92
3.4 Differentials ·······················································································.98
Exercises 3································································································103
Chapter 4 Applications of Differentiation··························································110
4.1 The Mean Value Theorems ······································································110
4.2 Indeterminate Forms and l’Hospital’s Rule ···················································118
4.3 Taylor’s Formula··················································································124
4.4 Maximum and Minimum Values ·······························································131
4.5 Curve Sketching ··················································································137
4.6 Arc Length Element and Curvature ····························································144
Exercises 4································································································148
Chapter 5 Indefinite Integrals ········································································157
5.1 Indefinite Integrals················································································157
5.2 The Substitution Rule ············································································160
5.3 Integration by Parts···············································································167
5.4 Integration of Several Types of Functions·····················································171
Exercises 5································································································177
Chapter 6 Definite Integrals and Their Applications ···········································181
6.1 Concepts and Properties ·········································································181
6.2 The Fundamental Theorem of Calculus ·······················································188
6.3 Evaluating Definite Integrals····································································192
6.4 Improper Integrals ················································································197
6.5 Applications of Integration ······································································205
Exercises 6································································································218
Chapter 7 Differential Equations ····································································228
7.1 Basic Concepts of Differential Equations ·····················································228
7.2 First-Order Differential Equations ·····························································231
7.3 Linear Differential Equations and the Structure of Their General Solutions ·············245
7.4 Linear Differential Equations with Constant Coefficients ··································250
Exercises 7································································································264