卓里奇數學分析教程‧第2卷(第2版)（簡體書）

內容簡介

《卓裡奇數學分析教程》是作者在莫斯科大學力學數學系從60年代開始教授數學分析課程不斷積累的基礎上寫成的，自1981年第１版出版以來，已暢銷全球40年，並在一直修訂增補。在此教程中作者加強了分析學、代數學和幾何學等現代數學課程之間的聯繫，重點關注一般數學中非常有本質意義的概念和方法，採用適當接近現代數學文獻的語言進行敘述，在保持數學一般理論敘述嚴謹性的同時，也盡量體現數學在自然科學中的各種應用。

《卓裡奇數學分析教程》共兩卷，第２卷內容包括：連續映射的一般理論、賦範空間中的微分學、重積分、Rn中的曲面和微分形式、曲線積分與曲面積分、向量分析與場論、微分形式在流形上的積分、級數和含參變量的函數族的一致收斂性和基本運算、含參變量的積分、傅裡葉級數與傅裡葉變換、漸近展開式。

《卓裡奇數學分析教程》觀點較高，內容豐富新穎，所選習題極具特色，是教材理論部分的有益補充。這套教程書可作為綜合性大學和師範大學數學、物理、力學及相關專業的教師和學生的教材或主要參考書，也可供工科大學應用數學專業的教師和學生參考使用。

作者簡介

弗拉基米爾·卓裡奇（Vladimir A. Zorich）是莫斯科國立大學教授，主要從事分析、保角幾何、擬共形映照方面的研究工作。他解決了空間擬共形映照下的球面同胚問題，並因該研究成果獲得了“青年數學家國家獎”。作為莫斯科國立大學數學力學系高級實驗課程的組織者之一，他在一些大學中開設並教授現代分析學課程，並發表了大量的數學研究成果。

本書是世圖“俄羅斯數學經典”書系中的一種，被沃爾夫獎得主、俄羅斯科學院院士阿諾爾德（V. I. Arnold）譽為現有數學分析現代教材的best。與其他數學分析教科書相比，它更多地運用了現代數學（包括代數學、幾何學和拓撲學）的思想和方法，而且也更貼近自然科學（特別是物理學和力學）的應用。本書被清華大學數理基礎科學班精品課程選用為授課教材。

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.

圖書目錄

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