基本信息
- 原书名：Principles of Mathematical Analysis,Third Edition
- 原出版社： McGraw-Hill
编辑推荐
这是一部近代的数学名*，一直受到数学界的推崇。作为Rudin的分析学经典*作之一，本书在西方各国乃至我国均有着广泛而深远的影响，被许多高校用做数学分析课的必选教材。本书涵盖了高等微积分学的丰富内容，*精彩的部分集中在基础拓扑结构、函数项序列与级数、多变量函数以及微分形式的积分等章节。第3版经过增删与修订，*加符合学生的阅读习惯与思考方式。本书内容相当精练，结构简单明了，这也是作者*作的一大特色。与其说这是一部教科书，不如说这是一部字典。
内容简介
作译者
Walter Rudin，1953年于杜克大学获得数学博士学位。曾先后执教于麻省理工学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究兴趣集中在调和分析和复变函数。除本书外，他还著有另外两本名著：《Functional Analysis》和《Real and Complex Analysi
目录
Preface
Chapter 1 The Real and Complex Number Systems
Introduction
Ordered Sets
Fields
The Real Field
The Extended Real Number System
The Complex Field
Euclidean Spaces
Appendix
Exercises
Chapter 2 Basic Topology
Finite, Countable, and Uncountable Sets
Metric Spaces
Compact Sets
Perfect Sets
Connected Sets
Exercises
Chapter 3 Numerical Sequences and Series
Convergent Sequences
Chapter 1 The Real and Complex Number Systems
Introduction
Ordered Sets
Fields
The Real Field
The Extended Real Number System
The Complex Field
Euclidean Spaces
Appendix
Exercises
Chapter 2 Basic Topology
Finite, Countable, and Uncountable Sets
Metric Spaces
Compact Sets
Perfect Sets
Connected Sets
Exercises
Chapter 3 Numerical Sequences and Series
Convergent Sequences
前言
This book is intended to serve as a text for the course in analysis that is usuallytaken by advanced undergraduates or by first-year students who study mathematics.
The present edition covers essentially the same topics as the second one,with some additions, a few minor omissions, and considerable rearrangement. I hope that these changes will make the material more accessible amd more attractive to the students who take such a course.
Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed
whenever the time seems ripe.
The material on functions of several variables is almost completely re-written, with many details filled in, and with more examples and more motiva-tion. The proof of the inverse function theorem--the key item in Chapter 9--is simplified by means of the fixed point theorem about contraction mappings.
Differential forms are discussed in much greater detail. Several applications of Stokes' theorem are included.
As regards other changes, the chapter on the Riemann-Stieltjes integral has been trimmed a bit, a short do-it-yourself section on the gamma function has been added to Chapter 8, and there is a large number of new exercises, most of them with fairly detailed hints.
I have also included several references to articles appearing in the American Mathematical Monthly and in Mathematics Magazine, in the hope that students will develop the habit of looking into the journal literature. Most of these references were kindly supplied by R. B. Burckel.
Over the years, many people, students as well as teachers, have sent me corrections, criticisms, and other comments concerning the previous editions of this book. I have appreciated these, and I take this opportunity to express my sincere thanks to all who have written me.
WALTER RUDIN
The present edition covers essentially the same topics as the second one,with some additions, a few minor omissions, and considerable rearrangement. I hope that these changes will make the material more accessible amd more attractive to the students who take such a course.
Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed
whenever the time seems ripe.
The material on functions of several variables is almost completely re-written, with many details filled in, and with more examples and more motiva-tion. The proof of the inverse function theorem--the key item in Chapter 9--is simplified by means of the fixed point theorem about contraction mappings.
Differential forms are discussed in much greater detail. Several applications of Stokes' theorem are included.
As regards other changes, the chapter on the Riemann-Stieltjes integral has been trimmed a bit, a short do-it-yourself section on the gamma function has been added to Chapter 8, and there is a large number of new exercises, most of them with fairly detailed hints.
I have also included several references to articles appearing in the American Mathematical Monthly and in Mathematics Magazine, in the hope that students will develop the habit of looking into the journal literature. Most of these references were kindly supplied by R. B. Burckel.
Over the years, many people, students as well as teachers, have sent me corrections, criticisms, and other comments concerning the previous editions of this book. I have appreciated these, and I take this opportunity to express my sincere thanks to all who have written me.
WALTER RUDIN
