分析方法-轻识

数学主要讲述思想的方法，深入理解数学比掌握一大堆的定理、定义、问题和技术显得更为重要。理论和定义共同作用，《分析方法(修订版)(英文版)》在介绍实分析的时候结合详尽、广泛的阐释，使得读者完全理解分析基础和方法。目次：基础；实数体系结构；实线拓扑；连续函数；微分学；积分学；序列和函数级数；超函数；欧拉空间和矩阵空间；欧拉空间上的微分计算；常微分方程；傅里叶级数；隐函数、曲线和曲面；勒贝格积分；多重积分。读者对象：数学专业的研究生以及相关的科研人员。

Preface

1 Preliminaries

1.1 The Logic of Quantifiers

1.1.1 Rules of Quantifiers

1.1.2 Examples

1.1.3 Exercises

1.2 Infinite Sets

1.2.1 Countable Sets

1...

Preface

1 Preliminaries

1.1 The Logic of Quantifiers

1.1.1 Rules of Quantifiers

1.1.2 Examples

1.1.3 Exercises

1.2 Infinite Sets

1.2.1 Countable Sets

1.2.2 Uncountable Sets

1.2.3 Exercises

1.3 Proofs

1.3.1 How to Discover Proofs

1.3.2 How to Understand Proofs

1.4 The Rational Number System

1.5 The Axiom of Choice

2 Construction of the Real Number System

2.1 Cauchy Sequences

2.1.1 Motivation

2.1.2 The Definition

2.1.3 Exercises

2.2 The Reals as an Ordered Field

2.2.1 Defining Arithmetic

2.2.2 The Field Axioms

2.2.3 Order

2.2.4 Exercises

2.3 Limits and Completeness

2.3.1 Proof of Completeness

2.3.2 Square Roots

2.3.3 Exercises

2.4 Other Versions and Visions

2.4.1 Infinite Decimal Expansion

2.4.2 Dedekind Cuts

2.4.3 Non-Standard Analysis

2.4.4 Constructive Analysis

2.4.5 Exercises

2.5 Summary

3 Topology of the Real Line

3.1 The Theory of Limits

3.1.1 Limits, Sups, and Infs

3.1.2 Limit Points

3.1.3 Exercises

3.2 Open Sets and Closed Sets

3.2.1 Open Sets

3.2.2 Closed Sets

3.2.3 Exercises

3.3 Compact Sets

3.3.1 Exercises

3.4 Summary

4 Continuous Functions

4.1 Concepts of Continuity

4.1.1 Definitions

4.1.2 Limits of Functions and Limits of Sequences

4.1.3 Inverse Images of Open Sets

4.1.4 Related Definitions

4.1.5 Exercises

4.2 Properties of Continuous Functions

4.2.1 Basic Properties

4.2.2 Continuous Functions on Compact Domains

4.2.3 Monotone Functions

4.2.4 Exercises

4.3 Summary

5 Differential Calculus

5.1 Concepts of the Derivative

5.1.1 Equivalent Definitions

5.1.2 Continuity and Continuous Differentiability

5.1.3 Exercises

5.2 Properties of the Derivative

5.2.1 Local Properties

5.2.2 Intermediate Value and Mean Value Theorems

5.2.3 Global Properties

5.2.4 Exercises

5.3 The Calculus of Derivatives

5.3.1 Product and Quotient Rules

5.3.2 The Chain Rule

5.3.3 Inverse Function Theorem

5.3,4 Exercises

5.4 Higher Derivatives and Taylor's Theorem

5.4.1 Interpretations of the Second Derivative

5.4.2 Taylor's Theorem

5.4.3 L'HSpital's Rule

5.4.4 Lagrange Remainder Formula

5.4.5 Orders of Zeros

5.4.6 Exercises

5.5 Summary

6 Integral Calculus

6.1 Integrals of Continuous Functions

6.1.1 Existence of the Integral

6.1.2 Fundamental Theorems of Calculus

6.1.3 Useful Integration Formulas

6.1.4 Numerical Integration

6.1.5 Exercises

6.2 The Riemann Integral

6.2.1 Definition of the Integral

6.2.2 Elementary Properties of the Integral

6.2.3 Functions with a Countable Number of Discon-tinuities

6.2.4 Exercises

6.3 Improper Integrals

6.3.1 Definitions and Examples

6.3.2 Exercises

6.4 Summary

7 Sequences and Series of Functions

7.1 Complex Numbers

7.1.1 Basic Properties of C

7.1.2 Complex-Valued Functions

7.1.3 Exercises

7.2 Numerical Series and Sequences

7.2.1 Convergence and Absolute Convergence

7.2.2 Rearrangements

7.2.3 Summation by Parts

7.2.4 Exercises

7.3 Uniform Convergence

7.3.1 Uniform Limits and Continuity

7.3.2 Integration and Differentiation of Limits

7.3.3 Unrestricted Convergence

7.3.4 Exercises

7.4 Power Series

7.4.1 The Radius of Convergence

7.4.2 Analytic Continuation

7.4.3 Analytic Functions on Complex Domains

7.4.4 Closure Properties of Analytic Functions

7.4.5 Exercises

7.5 Approximation by Polynomials

7.5.1 Lagrange Interpolation

7.5.2 Convolutions and Approximate Identities

7.5.3 The Weierstrass Approximation Theorem

7.5.4 Approximating Derivatives

7.5.5 Exercises

7.6 Eouicontinuity

7.6.1 The Definition of Equicontinuity

7.6.2 The Arzela-Ascoli Theorem

7.6.3 Exercises

7.7 Summary

8 Transcendental Functions

8.1 The Exponential and Logarithm

8.2 Trigonometric Functions

8.3 Summary

9 Euclidean Space and Metric Spaces

9.1 Structures on Euclidean Space

9.2 Topology of Metric Spaces

9.3 Continuous Functions on Metric Spaces

9.4 Summary

10 Differential Calculus in Euclidean Space

10.1 The Differential

10.2 Higher Derivatives

10.3 Summary

11 Ordinary Differential Equations

11.1 Existence and Uniqueness

11.2 Other Methods of Solution

11.3 Vector Fields and Flows

11.4 Summary

12 Fourier Series

12.1 Origins of Fourier Series

12.2 Convergence of Fourier Series

12.3 Summary

13 Implicit Functions, Curves, and Surfaces

13.1 The Implicit Function Theorem

13.2 Curves and Surfaces

13.3 Maxima and Minima on Surfaces

13.4 Arc Length

13.5 Summary

14 The Lebesgue Integral

14.1 The Concept of Measure

14.2 Proof of Existence of Measures

14.3 The Integral

14.4 The Lebesgue Spaces L1 and L2

14.5 Summary

15 Multiple Integrals

15.1 Interchange of Integrals

15.2 Change of Variable in Multiple Integrals

15.3 Summary

Index