分析方法

出版时间：2010年04月  出版社：世界图书出版公司  作者：斯特里沙兹(Robert S.Strichartz)  页数：739  
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前言

Do not ask permission to understand.Do not wait for the word of authority.Seize reason in your own hand.With your own teeth savor the fruit.Mathematics is more than a collection of theorems, definitions,problems and techniques; it is a way of thought. The same can be said about an individual branch of mathematics, such as analysis. Analysis has its roots in the work of Archimedes and other ancient Greek ge-ometers, who developed techniques to find areas, volumes, centers of gravity, arc lengths, and tangents to curves. In the seventeenth century these techniques were further developed, culminating in the invention of the calculus of Newton and Leibniz. During the eighteenth centu-ry the calculus was fashioned into a tool of bold computational power and applied to diverse problems of practical and theoretical interest.At the same time the foundation of analysis——the logical justification for the success of the methods——was left in limbo. This had practical consequences: for example, Euler——the leading mathematician of the eighteenth century——developed all the techniques needed for the study of Fourier series, but he never carried out the project. On the contrary,he argued in print against the possibility of representing functions as Fourier series, when this proposal was put forth by Daniel Bernoulli,and his argument was based on fundamental misconceptions concerning the nature of functions and infinite series.In the nineteenth century, the problem of the foundation of anal-ysis was faced squarely and resolved. The theory that was developed forms most of the content of this book. We will describe it in its logical order, starting from the most basic concepts such as sets and numbers and building up to the more involved concepts of limits, continuity,derivative, and integral. The actual historical order of discovery was almost the reverse; much like peeling a cabbage, mathematicians be-gan with the outermost layers and worked their way inward. Cauchy and Bolzano began the process in the 1820s by developing the theo-ry of functions without defining the real numbers. The first rigorous definition of the real number system came in the work of Dedekind,Weierstrass, and Heine in the 1860s. Set theory came later in the work of Cantor, Peano, and Frege.The consequences of the nineteenth century foundational work were enormous and are still being felt today. Perhaps the least important consequence was the establishment of a logically valid explanation of the calculus. More important, with the clearing away of the concep-tual murk, new problems emerged with clarity and were developed into important theories. We will give some illustrations of these new nineteenth century discoveries in our discussions of differential equa-tions, Fourier series, higher dimensional calculus, and manifolds. Most important of all, however, the nineteenth century foundational work paved the way for the work of the twentieth century. Analysis today is a subject of vast scope and beauty, ranging from the abstract to the concrete, characterized both by the bold computational power of the eighteenth century and the logical subtlety of the nineteenth century.Most of these developments are beyond the scope of this book or at best merely hinted at. Still, it is my hope that the reader, after hav-ing entered so deeply along the way of analysis, will be encouraged to continue the study.My goal in writing this book is to communicate the mathematical ideas of the subject to the reader. I have tried to be generous with ex-planations. Perhaps there will be places where I belabor the obvious,nevertheless, I think there is enough truly challenging material here to inspire even the strongest students. On the other hand, there will inevitably be places where each reader will find difficulties in follow-ing the arguments. When this happens, I suggest that you write your questions in the margins. Later, when you go over the material, you may find that you can answer the question. If not, be sure to ask your instructor or another student; often, it is a minor misunderstanding that causes confusion and can easily be cleared up. Sometimes, the in-herent difficulty of the material will demand considerable effort on your part to attain understanding. I hope you will not become frustrated in the process; it is something which all students of mathematics must confront. I believe that what you learn through a process of struggle is more likely to stick with you than what you learn without effort.Understanding mathematics is a complex process. It involves not only following the details of an argument and verifying its correctness,but seeing the overall strategy of the argument, the role played by every hypothesis, and understanding how different theorems and definitions fit together to create the whole. It is a long-term process; in a sense,you cannot appreciate the significance of the first theorem until you have learned the last theorem. So please be sure to review old mate-rial; you may find the chapter summaries useful for this purpose. The mathematical ideas presented in this book are of fundamental impor-tance, and you are sure to encounter them again in further studies in both pure and applied mathematics. Learn them well and they will serve you well in the future. It may not be an easy task, but it is a worthy one.

内容概要

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，本书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。

书籍目录

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

章节摘录

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用户评论 (总计7条)

•   经典教材，写得非常好，书如其名，包括了很多国内教材没讲的内容：实数系的建立，任意有限多个变元的微积分等，不过更适合高年级本科生或研究生。
•   能用上就是好书
•   希望此生能通读一遍吧
•   这本书是很好的，就是编辑装订得太差了，边边角角都没打磨好，而且纸张也太差了。
•   非常好的分析入门书籍...
•   这是我看过的“最有道理”的分析书籍，它并不是要告诉你定义是什么，而是告诉你我们需要解决什么样一个问题，这个问题如何解决的，这个解决方法的妙处在哪里，以致于最终形成一个定义。计算机系学生非常适合这样“有道理、有直觉”的讲法，而不是像rudin那样直接给定义，给重要结果的“急功近利”讲法。非常有阅读快感，强烈推荐！
•   剛剛瀏覽了一下amazon.cn上的評論，有讀者將作者比作數學裡的雨果。看過雨果的書的人應該知道，雨果經常在小說中插入冗長的議論（這些議論連歌德所借喻的干柴都算不上，更別說是把柴點燃了）。分析方法一書中常常出現很多類似的東西，因而作者被譏諷為“雨果”。