分析方法

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.

內(nèi)容概要

數(shù)學(xué)主要講述思想的方法，深入理解數(shù)學(xué)比掌握一大堆的定理、定義、問題和技術(shù)顯得更為重要。理論和定義共同作用，本書在介紹實(shí)分析的時(shí)候結(jié)合詳盡、廣泛的闡釋，使得讀者完全理解分析基礎(chǔ)和方法。目次：基礎(chǔ)；實(shí)數(shù)體系結(jié)構(gòu)；實(shí)線拓?fù)?；連續(xù)函數(shù)；微分學(xué)；積分學(xué)；序列和函數(shù)級(jí)數(shù)；超函數(shù)；歐拉空間和矩陣空間；歐拉空間上的微分計(jì)算；常微分方程；傅里葉級(jí)數(shù)；隱函數(shù)、曲線和曲面；勒貝格積分；多重積分?！　∽x者對(duì)象：數(shù)學(xué)專業(yè)的研究生以及相關(guān)的科研人員。

書籍目錄

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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