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。本书则向你阐述了有关其分析方法。主要内容包括实数体系结构;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;隐函数等。
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书名 | 分析方法(修订版) |
分类 | 科学技术-自然科学-数学 |
作者 | (美)斯特里沙兹 |
出版社 | 世界图书出版公司 |
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简介 | 编辑推荐 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。本书则向你阐述了有关其分析方法。主要内容包括实数体系结构;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;隐函数等。 目录 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 Taylor's 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 |
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