卓里奇数学分析教程 第2卷（第2版）

　　

内容简介

《卓里奇数学分析教程》是作者在莫斯科大学力学数学系从60年代开始教授数学分析课程不断积累的基础上写成的，自1981年第１版出版以来，已畅销全球40年，并在一直修订增补。在此教程中作者加强了分析学、代数学和几何学等现代数学课程之间的联系，重点关注一般数学中非常有本质意义的概念和方法，采用适当接近现代数学文献的语言进行叙述，在保持数学一般理论叙述严谨性的同时，也尽量体现数学在自然科学中的各种应用。

《卓里奇数学分析教程》共两卷，第２卷内容包括：连续映射的一般理论、赋范空间中的微分学、重积分、Rn中的曲面和微分形式、曲线积分与曲面积分、向量分析与场论、微分形式在流形上的积分、级数和含参变量的函数族的一致收敛性和基本运算、含参变量的积分、傅里叶级数与傅里叶变换、渐近展开式。

《卓里奇数学分析教程》观点较高，内容丰富新颖，所选习题极具特色，是教材理论部分的有益补充。这套教程书可作为综合性大学和师范大学数学、物理、力学及相关专业的教师和学生的教材或主要参考书，也可供工科大学应用数学专业的教师和学生参考使用。

作者简介

弗拉基米尔·卓里奇（Vladimir A. Zorich）是莫斯科国立大学教授，主要从事分析、保角几何、拟共形映照方面的研究工作。他解决了空间拟共形映照下的球面同胚问题，并因该研究成果获得了“青年数学家国家奖”。作为莫斯科国立大学数学力学系高级实验课程的组织者之一，他在一些大学中开设并教授现代分析学课程，并发表了大量的数学研究成果。

编辑推荐

本书是世图“俄罗斯数学经典”书系中的一种，被沃尔夫奖得主、俄罗斯科学院院士阿诺尔德（V. I. Arnold）誉为现有数学分析现代教材的best。与其他数学分析教科书相比，它更多地运用了现代数学（包括代数学、几何学和拓扑学）的思想和方法，而且也更贴近自然科学（特别是物理学和力学）的应用。本书被清华大学数理基础科学班精品课程选用为授课教材。

Science has not stood still in the years since the first English edition of this book was published. For example， Fermat's last theorem has been proved， the Poincare conjecture is now a theorem， and the Higgs boson has been discovered. Other events in science， while not directly related to the contents of a textbook in classical mathematical analysis， have indirectly led the author to learn something new， to think over something familiar， or to extend his knowledge and understanding. All of this additional knowledge and understanding end up being useful even when one speaks about something apparently completely unrelated.
　　In addition to the original Russian edition， the book has been published in English， German， and Chinese. Various attentive multilingual readers have detected many errors in the text. Luckily， these are local errors， mostly misprints. They have assuredly all been corrected in this new edition.
　　But the main difference between the second and first English editions is the addition of a series of appendices to each volume. There are six of them in the first and five of them in the second. So as not to disturb the original text， they are placed at the end of each volume. The subjects of the appendices are diverse. They are meant to be useful to students （in mathematics and physics） as well as to teachers， who may be motivated by different goals. Some of the appendices are surveys， both prospective and retrospective. The final survey contains the most important conceptual achievements of the whole course， which establish connections between analysis and other parts of mathematics as a whole.

图书目录

Prefaces

9. Continuous Mappings (General Theory)

10. Differential Calculus from a General Viewpoint

11. Multiple Integrals

12. Surfaces and Differential Forms in Rn

13. Line and Surface Integrals

14. Elements of Vector Analysis and Field Theory

15. Integration of Differential Forms on Manifolds

16. Uniform Convergence and Basic Operations of Analysis

17. Integrals Depending on a Parameter

18. Fourier Series and the Fourier Transform

19. Asymptotic Expansions

Topics and Questions for Midterm Examinations

Examination Topics

Examination Problems (Series and Integrals Depending on a Parameter)

Intermediate Problems (Integral Calculus of Several Variables)

Appendix A. Series as a Tool (Introductory Lecture)

Appendix B. Change of Variables in Multiple Integrals

Appendix C. Multidimensional Geometry and Functions of a Very Large Number of Variables

Appendix D. Operators of Field Theory in Curvilinear Coordinates

Appendix E. Modern Formula of Newton-Leibniz

References

Index of Basic Notation

Subject Index

Name Index