分析方法 修訂版(英文版)（簡體書）

數學主要講述思想的方法，深入理解數學比掌握一大堆的定理、定義、問題和技術顯得更為重要。理論和定義共同作用，本書在介紹實分析的時候結合詳盡、廣泛的闡釋，使得讀者完全理解分析基礎和方法。目次：基礎；實數體系結構；實線拓撲；連續函數；微分學；積分學；序列和函數級數；超函數；歐拉空間和矩陣空間；歐拉空間上的微分計算；常微分方程；傅里葉級數；隱函數、曲線和曲面；勒貝格積分；多重積分。
讀者對象：數學專業的研究生以及相關的科研人員。

Preface
1 Preliminaries
　1.1 The Logic of Quantifiers
　1.2 Infinite Sets
　1.3 Proofs
　1.4 The Rational Number System
　1.5 The Axiom of Choice*
2 Construction of the Real Number System
　2.1 Cauchy Sequences
　2.2 The Reals as an Ordered Field
　2.3 Limits and Completeness
　2.4 Other Versions and Visions
　2.5 Summary
3 Topology of the Real Line
　3.1 The Theory of Limits
　3.2 Open Sets and Closed Sets
　3.3 Compact Sets
　3.4 Summary
4 Continuous Functions
　4.1 Concepts of Continuity
5 Differential Calculus
　5.1 Concepts of the Derivative
　5.2 Properties of the Derivative
　5.3 The Calculus of Derivatives
　5.4 Higher Derivatives and Taylors Theorem
　5.5 Summary
6 Integral Calculus
　6.1 Integrals of Continuous Functions
　6.2 The Riemann Integral
　6.3 Improper Integrals*
　6.4 Summary
7 Sequences and Series of Functions
　7.1 Complex Numbers
　7.2 Numerical Series and Sequences
　7.3 Uniform Convergence
　7.4 Power Series
　7.5 Approximation by Polynomials
　7.6 Equicontinuity
　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