Overview
 The thoroughness of the coverage, from elementary to very advanced
 The clarity of the exposition
 The originality and variety of the exercises and examples
 Especially good for physics students
 Includes supplementary material: sn.pub/extras
Part of the book series: Universitext (UTX)
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Reviews
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 nonspecialists 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 Englishlanguage 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 twovolume work presents a well thoughtout 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 onevariable calculus along with the multivariable differential calculus elucidated in an uptoday, clear manner, with a pleasant geometric flavor. [...]
The basic material of the Part 2 consists on the one hand ofmultiple 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 selfcontrol). 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 fÃ¼r Analysis und ihre Anwendungen Volume 23, Issue 4, 2004, p. 861863
"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 preciselyformulated. 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 wellknown 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 "â€¦ An entirely logical rigor of discussion â€¦ 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. â€¦
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 twovolume 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 TwoVolume 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 â€¦ . 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. â€¦ 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 â€¦ ."
T.Trif, Studia Universitatis BabesBolyai Mathematica, Vol. XLIX (3), 2004
"This is a translation of the fourth edition of a two volume textbook â€¦ . The textbook is â€˜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.â€™ â€¦ The formalism and rigour of the presentation will appeal to mathematicians â€¦ . these books are a valuable and welcome addition to existing Englishlanguage 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. â€¦ 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 â€¦ 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 underconsideration is aimed primarily at university students and teachers specializing in mathematics and natural sciences â€¦ . This twovolume work presents a well thoughtout and thoroughly written first course in analysis â€¦ . 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 â€¦ . The reader will be able to follow the presentation with a minimum previous knowledge."
P.Gavrilyuk, ZAA  Zeitschrift fÃ¼r Analysis und ihre Anwendungen, Vol. 23 (4), 2004
"Letâ€™s get one thing straight from the very beginning. I like this twovolume 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 to order a copy as soon as possible. â€¦ It is Comprehensive and Encyclopedic. â€¦ One place this work really shines is in its examples. â€¦ The text is further enhanced by the historic notes that are sprinkled throughout."
William R. Wade, SIAM Review, Vol. 46 (4), 2004
"The presentation is always rigorous and thorough â€“ a journey through analysis at its best. â€¦ Zorich succeeds in lively presenting a wealth of reallife examples within nearly each section in order to illuminate the abstract results and to show typical applications in which these results are used. These applications are also carefully worked out and presented so that it is a pleasure to follow the author. â€¦ I can only recommend the volumes to everyone interested in an introductory analysis course â€¦ ."
Thomas Sonar, Monatshefte fÃ¼r Mathematik, Issue 4, 2004
Diese profunde EinfÃ¼hrung [Math.Analysis I und II] in die Analysis sollte in keiner mathematischen Bibliothek fehlen, selbst bei budgetÃ¤ren Restriktionen, trotz der ÃœberfÃ¼lle anEinfÃ¼hrungsbÃ¼chern. Eine genaue, bewuÃŸte LektÃ¼re dieses profunden Werks kÃ¶nnte mÃ¶gliche kÃ¼nftige Autoren mittelmÃ¤ÃŸiger AnalysisbÃ¼cher vielleicht abschrecken.
[...]Meisterhaft wird hier intuitives Verstehen gefÃ¶rdert, vermittelt durch anschauliche geometrische Denkweisen, heuristische Ideen und induktive Vorgangsweisen, ohne ExaktheitsansprÃ¼che hintanzustellen oder konkrete Details oder Anwendungen auch nur ansatzweise zu vernachlÃ¤ssigen. Der Aufbau ist in vieler Hinsicht ungewÃ¶hnlich, erÃ¶ffnet frÃ¼he Einblicke und Weitblicke und regt zum Denken an [...], ist auch der historischen Entwicklung angemessen und bietet eine wichtige Alternative zu den vielen "eleganten" ZugÃ¤ngen, bei denen die Vermittlung wichtiger nÃ¶tiger Entwicklungsschritte fÃ¼r ein aktives VerstÃ¤ndnis zu kurz kommt.
Der umfassende, Nachbardisziplinen laufend berÃ¼hrende Zugang trÃ¤gt reiche FrÃ¼chte, ebenso die facettenreiche FÃ¼lle an ErklÃ¤rungen der Wurzeln und Essenz der grundlegenden Konzepte und Resultate, die Beschreibungen von ZusammenhÃ¤ngen und Ausblicke auf weitere Entwicklungen mit vielen in EinfÃ¼hrungsbÃ¼chern leider eher unÃ¼blichen Anwendungen und QuerbezÃ¼gen [...]. Man erwirbt mit diesem Werk zusÃ¤tzlich ein vollstÃ¤ndiges, umfangreiches und wertvolles "ProblemBuch". Bei aller reichhaltiger FÃ¼lle stellt sich die Mathematik hier aber immer als eine Einheit dar, in ihrer auf den heutigen Stellenwert Bezug nehmenden historischen und philosophischen Entwicklung, geprÃ¤gt durch, an passender Stelle kompetent gewÃ¼rdigte, bedeutende groÃŸe schÃ¶pferische PersÃ¶nlichkeiten. [...] Dieses vorzÃ¼gliche Werk atmet den Geist einer bewunderungswÃ¼rdigen, vielschichtigen Forscher und LehrerpersÃ¶nlichkeit."
H.Rindler, Monatshefte fÃ¼r Mathematik 146, Issue 4, 2005
"Die vorliegenden zwei BÃ¤nde sind die englische Ãœbersetzung eines russischen Werkes, das bereits Anfang der achtziger Jahre erschienen ist und inzwischen bereits zumvierten Mal aufgelegt wurde. Die BÃ¼cher beinhalten auf Ã¼ber 1200 Seiten die klassische Analysis in einer zeitgemÃ¤ÃŸen Darstellung sowie Querverbindungen zu Algebra, Differenzailgleichungen, Differenzialgeometrie, komplexe Analysis und Funktionalanlaysis. Addressaten sind Studenten (und Lehrende), die neben einer strengen mathematischen Theorie auch konkrete Anwendungen suchen...
Dieses ausgezeichnete Werk kann StudienanfÃ¤ngern und fortgeschrittenen Studierenden uneingeschrÃ¤nkt empfohlen werden, aber auch Lehrende werden viele Anregungen darin finden."
M.Kronfellner (Wien), IMN  Internationale Mathematische Nachrichten 59, Issue 198, 2005, S. 3637
Authors and Affiliations
Bibliographic Information
Book Title: Mathematical Analysis II
Authors: Vladimir A. Zorich
Series Title: Universitext
Publisher: Springer Berlin, Heidelberg

eBook Packages: Springer Book Archive
Copyright Information: SpringerVerlag Berlin Heidelberg 2004
Softcover ISBN: 9783540874539Published: 21 November 2008
Series ISSN: 01725939
Series EISSN: 21916675
Edition Number: 1
Number of Pages: XV, 688
Additional Information: Original Russian edition published by MCCME, Moscow, Russia, 2002
Topics: Analysis, Theoretical, Mathematical and Computational Physics