数学分析原理 英文版·第3版PDF电子书下载

Chapter 1 The Real and Complex Number Systems 1

Introduction 1

Ordered Sets 3

Fields 5

The Real Field 8

The Extended Real Number System 11

The Complex Field 12

Euclidean Spaces 16

Appendix 17

Exercises 21

Chapter 2 Basic Topology 24

Finite,Countable,and Uncountable Sets 24

Metric Spaces 30

Compact Sets 36

Perfect Sets 41

Connected Sets 42

Exercises 43

Chapter 3 Numerical Sequences and Series 47

Convergent Sequences 47

Subsequences 51

Cauchy Sequences 52

Upper and Lower Limits 55

Some Special Sequences 57

Series 58

Series of Nonnegative Terms 61

The Number e 63

The Root and Ratio Tests 65

Power Series 69

Summation by Parts 70

Absolute Convergence 71

Addition and Multiplication of Series 72

Rearrangements 75

Exercises 78

Chapter 4 Continuity 83

Limits of Functions 83

Continuous Functions 85

Continuity and Compactness 89

Continuity and Connectedness 93

Discontinuities 94

Monotonic Functions 95

Infinite Limits and Limits at Infinity 97

Exercises 98

Chapter 5 Differentiation 103

The Derivative of a Real Function 103

Mean Value Theorems 107

The Continuity of Derivatives 108

L'Hospital's Rule 109

Derivatives of Higher Order 110

Taylor's Theorem 110

Differentiation of Vector-valued Functions 111

Exercises 114

Chapter 6 The Riemann-Stieltjes Integral 120

Definition and Existence of the Integral 120

Properties of the Integral 128

Integration and Differentiation 133

Integration of Vector-valued Functions 135

Rectifiable Curves 136

Exercises 138

Chapter 7 Sequences and Series of Functions 143

Discussion of Main Problem 143

Uniform Convergence 147

Uniform Convergence and Continuity 149

Uniform Convergence and Integration 151

Uniform Convergence and Differentiation 152

Equicontinuous Families of Functions 154

The Stone-Weierstrass Theorem 159

Exercises 165

Chapter 8 Some Special Functions 172

Power Series 172

The Exponential and Logarithmic Functions 178

The Trigonometric Functions 182

The Algebraic Completeness of the Complex Field 184

Fourier Series 185

The Gamma Function 192

Exercises 196

Chapter 9 Functions of Several Variables 204

Linear Transformations 204

Differentiation 211

The Contraction Principle 220

The Inverse Function Theorem 221

The Implicit Function Theorem 223

The Rank Theorem 228

Determinants 231

Derivatives of Higher Order 235

Differentiation of Integrals 236

Exercises 239

Chapter 10 Integration of Differential Forms 245

Integration 245

Primitive Mappings 248

Partitions of Unity 251

Change of Variables 252

Differential Forms 253

Simplexes and Chains 266

Stokes'Theorem 273

Closed Forms and Exact Forms 275

Vector Analysis 280

Exercises 288

Chapter 11 The Lebesgue Theory 300

Set Functions 300

Construction of the Lebesgue Measure 302

Measure Spaces 310

Measurable Functions 310

Simple Functions 313

Integration 314

Comparison with the Riemann Integral 322

Integration of Complex Functions 325

Functions of Class ？2 325

Exercises 332

Bibliography 335

List of Special Symbols 337

Index 339