Mathematical Analysis II (Universitext) [Hardcover]

V. A. Zorich (Author), R. Cooke (Translator)

Book Description

Publication Date: January 22, 2004 | ISBN-10: 3540406336 | ISBN-13: 978-3540406334 | Edition: 1

The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions.

Editorial Reviews

Review

From the reviews: "... The treatment is indeed rigorous and comprehensive with introductory chapters containing an initial section on logical symbolism (used thoughout the text), through sections on sets and functions with an entire chapter on the real numbers. [...] The formalism and rigour of the presentation will appeal to mathematicians and to those non-specialists who seek a rigorous basis for the mathematics that they use in their daily work. For such, these books are a valuable and welcome addition to existing English-language texts." D.Herbert, University of London, Contemporary Physics 2004, Vol. 45, Issue 6 "The book under consideration is aimed primarily at university students and teachers specializing in mathematics and natural sciences, and at all those who wish to see both the mathematical theory with carefully formulated theorems and rigorous proofs on the one hand, and examples of its effective use in the solution of practical problems on the other hand. The last fact differs this book positively from many traditional expositions and is of great importance especially in connection with the applied character of the future activity of the majority of students. [...]. This two-volume work presents a well thought-out and thoroughly written first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, elliptic functions and distributions. Clarity of exposition, instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books belong also to the distinguished key features of the book. [...] The first volume presents a complete course on one-variable calculus along with the multivariable differential calculus elucidated in an up-to-day, clear manner, with a pleasant geometric flavor. [...] The basic material of the Part 2 consists on the one hand of multiple integrals and line and surface integrals, leading to the generalized Stokes formula and some examples of its application, and on the other hand the machinery of series and integrals depending on a parameter, including Fourier series, the Fourier transform, and the presentation of asymptotic expansions. The presentation of the material is also here very geometric. The second volume is especially unusual for textbooks of modern analysis and such a way of structuring the course can be considered as innovative. [...] Both parts are supplemented by prefaces, problems from the midterm examinations, examination topics,references and subject as well as name Indexes. The book is written excellently, with rigorous proofs, and geometrical explanations. The main text is supplemented with a large collection of examples, and nearly every section ends with a set of problems and exercises that significantly complement the main text (unfortunately there are not solutions to the problems and exercises for the self-control). Each volume ends with a list of topics, questions or problems for midterm examinations and with a list of examination topics. The subject index, name index and index of basic notation round up the book and made it very convenient for use. The book can serve as a foundation for a four semester course for students or can be useful as support for all who are studying or teaching mathematical analysis. The reader will be able to follow the presentation with a minimum previous knowledge. The researcher can find interesting references, in particulary giving access to classical as well as to modern results." I. P. Gavrilyuk, Zeitschrift für Analysis und ihre Anwendungen Volume 23, Issue 4, 2004, p. 861-863 "This is a very nice textbook on mathematical analysis, which will be useful to both the students and the lecturers. [...] About style of explanation one can say that the definitions are motivated and precisely formulated. The proofs of theorems are in appropriate generality, presented in detail and without logical gaps. This is illustrated in many examples (many of them arise in applications) and each section ends with a list of problems and exercises, which extend and supplement the basic text. [...]" European Mathematical Society Newsletter, Sept. 2004, p. 47 "This is the translation of the fourth edition of a well known course on mathematical analysis, taught for several years by the author at the Moscow State University (MSU) and at other universities. Together with V.I.Arnold and S.P.Novkov, the author is one of the organizers of advanced experimental courses at MSU, this experience being reflected in the book too. Written in the good tradition of Russian mathematical textbooks, the present one combines intuition and accessibility with modern mathematical rigor. ... There are a lot of exercises and problems, of varying difficulty, spread through the book, needed for a better understanding of the subject, as well as historical notes about the great names who contributed along the centuries to the building of the edifice of mathematical analysis. This comprehensive course on mathematical analysis provides the readers, first of all students specializing in mathematics, with rigorous proofs of the fundamental theorems, but also with its applications in mathematics itself and outside it. It is correlated with subsequent disciplines relying on its methods and results, as differential equations, differential geometry, functions of a complex variable and functional analysis." T.Trif, Studia Universitatis Babes. Bolyai Mathematica, Vol. XLIX, Issue 3, 2004 "These two big volumes of the well-known advanced course of Calculus written by Professor Vladimier A. Zorich on the base of his lectures to students of Moscow State University. There are four editions of the textbook in Russian: the first of them was printed in 1980 and thus this book has withstood severe test of time; to my mind, the book is one of the best (possibly best) modern textbooks in Analysis. The words of A.N. Kolmogorov "… An entirely logical rigor of discussion … is combined with simplicity and completeness as well as with the development of the habit to work with real problems from natural sciences" are complete and clear characterization of this book. … The author writes: "This book has been aimed primarily at mathematicians desiring to obtain thorough proofs of the fundamental theorems, but who are at the same time interested in the life of these theorems outside of mathematics itself". However, I think that this book will be useful to all beginning mathematicians (students and postgraduate students in mathematics, natural sciences, engineering and technology) who want seriously to study analysis and also all specialists (first and foremost, lecturers and teachers) in analysis and interdisciplinary sciences. Undoubtedly, any mathematical library must have this textbook." Peter Zabreiko, Minsk, Zentralblatt MATH 1071 - 3, 2005  "Let's get one thing straight from the very beginning. I like this two-volume set. It will make an excellent reference for students and provides a vast reservoir of interesting exercises and exam questions for analysis teachers. Get your library order a copy as soon as possible. [...] What special features, beside enormous breadth, distinguish these volumes from other introductory analysis texts? [...] 1. The Foundations Are Carefully Laid. [...] 2. It Is Comprehensive and Encyclopedic. [...] 3. Material Is Carefully Motivated by Practical Considerations. [...] 4. Important Ideas Are Introduced More Than Once. [...] 5. The Pace Accelerates as the Text Progresses. [...] 6. This Two-Volume Set Contains Plenty of Good Examples. [...] 7. It Also Contains Plenty of Exercises. [...] 8. Unusual Touches. [...] [...] William R. Wade, University of Tennessee, SIAM Book Reviews, Vol. 46, No. 4 "This is the translation of the fourth edition of a well known course on mathematical analysis, taught for several years by the author … . Written in the good tradition of Russian mathematical textbooks, the present one combines intuition and accessibility with modern mathematical rigor. The book is divided into two volumes. … There are a lot of exercises and problems, of varying difficulty, spread through the book, needed for a better understanding of the subject, as well as historical notes … ." T.Trif, Studia Universitatis Babes-Bolyai Mathematica, Vol. XLIX (3), 2004 "This is a translation of the fourth edition of a two volume textbook … . The textbook is ‘aimed primarily at university students and teachers specializing in mathematics and natural sciences and at all those who wish to see both the rigorous mathematical theory and examples of its effective use in the solution of real problems of natural science.’ … The formalism and rigour of the presentation will appeal to mathematicians … . these books are a valuable and welcome addition to existing English-language texts." Dr. D. Herbert, Contemporary Physics, Vol. 45 (6), 2004 "This is a very nice textbook on mathematical analysis, which will be useful to both the students and the lecturers. … About style of explanation one can say that the definitions are motivated and precisely formulated. The proofs of theorems are in appropriate generality, presented in detail and without logical gaps. This is illustrated in many examples … and each section ends with a list of problems and exercises, which extend and supplement the basic text." EMS - European Mathematical Society Newsletter, September, 2004 "The book under consideration is aimed primarily at university students and teachers specializing in mathematics and natural sciences … . This two-volume work presents a well thought-out and thoroughly written first course in analysis … . Clarity of exposition, instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books belong also to the distinguished key features of the book. The book is written excellently … . The reader will be able to follow the presentation with a minimum previous knowledge." P.Gavrilyuk, ZAA - Zeitschrift für Analysis und ihre Anwendu...

Product Details

• Hardcover: 703 pages
• Publisher: Springer; 1 edition (January 22, 2004)
• Language: English
• ISBN-10: 3540406336
• ISBN-13: 978-3540406334
• Product Dimensions: 9.4 x 6.5 x 1.6 inches
• Shipping Weight: 2.5 pounds (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #668,710 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

24 of 26 people found the following review helpful:

5.0 out of 5 stars Glowing review, but a correction...., October 7, 2004

By 

Roger L. Cooke "Roger Cooke" (Burlington, Vermont USA) - See all my reviews
(REAL NAME)   

This review is from: Mathematical Analysis II (Universitext) (Hardcover)

Since I am listed by Amazon (but not by Springer) as one of the authors, you should quite properly be skeptical of my 5-star review. But I really mean it: I think the book is outstanding.

Now the correction. I am IN NO SENSE a co-author of this book, merely its translator. The translation was very enjoyable work, and I enjoyed the interaction with the author that it made possible. That, however, does not make me a co-author. (But if you'd like to see some books that I HAVE authored, please search.)

Roger Cooke

15 of 15 people found the following review helpful:

5.0 out of 5 stars Analysis made palatable, even for physicists., April 22, 2004

By 

A. Castro "traca_virtual" (California , USA) - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: Mathematical Analysis I (Universitext) (v. 1) (Hardcover)

The book besides covering a broad material on classical analysis(with a modern touch), exposes the basic core of analysis expected from a mathematics or physics student without making use of pedantic and unnecessary formalism. The author emphasizes the connection of important ideas via concrete and substancial examples more than insisting in pathological and or trivial examples. It has plenty of examples coming from physics and other sciences(following the tradition of the russian school : of teaching mathematics emphasizing links with other areas). We can't forget to mention the many geometrical insights provided. Moreover the book is "filled" w/ good exercises that really colaborates for a solid mathematical education and has also a detailed appendix where an instructor can find some very interesting and challenging problems for a seminar discussion or final exams. Undoubtly an worthwhile reading!

14 of 14 people found the following review helpful:

5.0 out of 5 stars Outstanding, March 19, 2006

By 

Fabio Tonti (Austria) - See all my reviews

This review is from: Mathematical Analysis I (Universitext) (v. 1) (Hardcover)

These two books written by V.A. Zorich represent a great course in analysis, both for people who just started dealing with the subject and for more experienced students. The treatment is thorough and spreads from an entire chapter about real numbers to very advanced problems. It also points out many applications in natural sciences.

A good and rather necessary addition would be the solutions to the problems given in these books. Thus students would have a way to check their work. Nevertheless it's worth more than five stars.