本书系统地阐述了微积分学的基本理论。在叙述上，作者尽量作到既严谨而又通俗易懂，并指出概念之间的内在联系和直观背景。本书是《微积分和数学分析引论》的第2卷第2册，为多变量情形，它适合于理工科大学师生、数学工作者和工程技术人员。

Chapter 1 Functions of Several Variables and Their Derivatives

 1.1 Points and Points Sets in the Plane and in Space

a.Sequences of points.Convergence

b.Sets of points in the plane

c.The boundary of a set.Closed and open sets

d.Closure as set of limit points

e.Points and sets of points in space

 1.2 Functions of Several Independent Variables

a.Functions and their domains

b.The simplest types of functions

c.Geometrical representation of functions

 1.3 Continuity

a.Definition

b.The concept of limit of a function of several variables

c.The order to which a function vanishes

 1.4 The Partial Derivatives of a Function

a.Definition.Geometrical representation

b.Examples

c.Continuity and the existence of partial derivatives

d.Change of the order of differentiation

 1.5 The Differential of a Function and Its Geometrical Meaning

a.The concept of differentiability

b.Directional derivatives

c.Geometric interpretation of differentiability,The tangent plane

d.The total differential of a function

e.Application to the calculus of errors

 1.6 Functions of Functions (Compound Functions) and the Introduction of New Independent Variables

a.Compound functions.The chain rule

b.Examples

c.Change of independent variables

 1.7 The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables

a.Preliminary remarks about approximation by polynomials

b.The mean value theorem

c.Taylor's theorem for several independent variables

 1.8 Integrals of a Function Depending on a Parameter

a.Examples and definitions

b.Continuity and differentiability of an integral with respect to the parameter

c.Interchange of integrations.Smoothing of functions

 1.9 Differentials and Line Integrals

a.Linear differential forms

b.Line integrals of linear differential forms

c.Dependence of line integrals on endpoints

 1.10 The Fundamental Theorem on Integrability of Linear Differential Forms

a.Integration of total differentials

b.Necessary conditions for line integrals to depend only on the end points

c.Insufficiency of the integrability conditions

d.Simply connected sets

e.The fundamental theorem

 APPENDIX

 ……

Chapter 2 Vectors, Matrices, Linear Transformations

Chapter 3 Developments and Applications of the Differential Calculus

Chapter 4 Multiple Integrals

Chapter 5 Relations Between Surface and Volume Integrals

Chapter 6 Differential Equations

Chapter 7 Calculus of Variations

Chapter 8 Functions of a Complex Variable

List of Biographical Dates

Index