微积分和数学分析引论．Ⅱ．2

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 Taylors Theorem for Functions of Several Variables

a.Preliminary remarks about approximation by polynomials

b.The mean value theorem

c.Taylors 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

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