分析方法 在線電子書 圖書標籤: 數學 分析 Analysis Math 實分析7 實分析 Mathematics
發表於2024-12-25
分析方法 在線電子書 pdf 下載 txt下載 epub 下載 mobi 下載 2024
這本書可以說讓我手不釋捲,難得的大傢之作。作者知道優秀的數學書籍在於充分介紹intuition
評分這本書可以說讓我手不釋捲,難得的大傢之作。作者知道優秀的數學書籍在於充分介紹intuition
評分這本書可以說讓我手不釋捲,難得的大傢之作。作者知道優秀的數學書籍在於充分介紹intuition
評分這本書可以說讓我手不釋捲,難得的大傢之作。作者知道優秀的數學書籍在於充分介紹intuition
評分這本書可以說讓我手不釋捲,難得的大傢之作。作者知道優秀的數學書籍在於充分介紹intuition
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,《分析方法(修訂版)(英文版)》在介紹實分析的時候結閤詳盡、廣泛的闡釋,使得讀者完全理解分析基礎和方法。目次:基礎;實數體係結構;實綫拓撲;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;傅裏葉級數;隱函數、麯綫和麯麵;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。
目錄
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
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分析方法 在線電子書 pdf 下載 txt下載 epub 下載 mobi 下載 2024