1610912322-b551b095a53deaf3d3fbd1ed05ae9b84 (Зорич 2-е издание (на английском))

UniversitextVladimir A. ZorichMathematicalAnalysis ISecond EditionUniversitextUniversitextSeries EditorsSheldon AxlerSan Francisco State University, San Francisco, CA, USAVincenzo CapassoUniversità degli Studi di Milano, Milano, ItalyCarles CasacubertaUniversitat de Barcelona, Barcelona, SpainAngus MacIntyreQueen Mary University of London, London, UKKenneth RibetUniversity of California, Berkeley, CA, USAClaude SabbahCNRS, École Polytechnique, Palaiseau, FranceEndre SüliUniversity of Oxford, Oxford, UKWojbor A. WoyczyńskiCase Western Reserve University, Cleveland, OH, USAUniversitext is a series of textbooks that presents material from a wide variety ofmathematical disciplines at master’s level and beyond. The books, often well classtested by their author, may have an informal, personal even experimental approachto their subject matter.

Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts.Thus as research topics trickle down into graduate-level teaching, first textbookswritten for new, cutting-edge courses may make their way into Universitext.For further volumes:www.springer.com/series/223Vladimir A. ZorichMathematical Analysis ISecond EditionVladimir A.

ZorichDepartment of MathematicsMoscow State UniversityMoscow, RussiaTranslators:Roger Cooke (first English edition translated from the 4th Russian edition)Burlington, Vermont, USAandOctavio Paniagua T. (Appendices A–F and new problems of the 6th Russian edition)Berlin, GermanyOriginal Russian edition: Matematicheskij Analiz (Part I, 6th corrected edition, Moscow,2012) MCCME (Moscow Center for Continuous Mathematical Education Publ.)ISSN 0172-5939UniversitextISBN 978-3-662-48790-7DOI 10.1007/978-3-662-48792-1ISSN 2191-6675 (electronic)ISBN 978-3-662-48792-1 (eBook)Library of Congress Control Number: 2016930048Mathematics Subject Classification (2010): 26-01, 26Axx, 26Bxx, 42-01Springer Heidelberg New York Dordrecht London© Springer-Verlag Berlin Heidelberg 2004, 2015This work is subject to copyright.

All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc.

in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.Printed on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)PrefacesPreface to the Second English EditionScience has not stood still in the years since the first English edition of this bookwas published.

For example, Fermat’s last theorem has been proved, the Poincaréconjecture is now a theorem, and the Higgs boson has been discovered. Other eventsin 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 thinkover something familiar, or to extend his knowledge and understanding.

All of thisadditional knowledge and understanding end up being useful even when one speaksabout something apparently completely unrelated.1In addition to the original Russian edition, the book has been published in English, German, and Chinese. Various attentive multilingual readers have detectedmany errors in the text. Luckily, these are local errors, mostly misprints. They haveassuredly 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 andfive of them in the second.

So as not to disturb the original text, they are placed at theend of each volume. The subjects of the appendices are diverse. They are meant to beuseful to students (in mathematics and physics) as well as to teachers, who may bemotivated by different goals. Some of the appendices are surveys, both prospectiveand retrospective. The final survey contains the most important conceptual achievements of the whole course, which establish connections between analysis and otherparts of mathematics as a whole.1 There is a story about Erdős, who, like Hadamard, lived a very long mathematical and humanlife.

When he was quite old, a journalist who was interviewing him asked him about his age. Erdősreplied, after deliberating a bit, “I remember that when I was very young, scientists established thatthe Earth was two billion years old. Now scientists assert that the Earth is four and a half billionyears old. So, I am approximately two and a half billion years old.”vviPrefacesI was happy to learn that this book has proven to be useful, to some extent, notonly to mathematicians, but also to physicists, and even to engineers from technicalschools that promote a deeper study of mathematics.It is a real pleasure to see a new generation that thinks bigger, understands moredeeply, and is able to do more than the generation on whose shoulders it grew.Moscow, Russia2015V.

ZorichPreface to the First English EditionAn entire generation of mathematicians has grown up during the time between theappearance of the first edition of this textbook and the publication of the fourthedition, a translation of which is before you. The book is familiar to many people,who either attended the lectures on which it is based or studied out of it, and whonow teach others in universities all over the world.

I am glad that it has becomeaccessible to English-speaking readers.This textbook consists of two parts. It is aimed primarily at university studentsand teachers specializing in mathematics and natural sciences, and at all those whowish to see both the rigorous mathematical theory and examples of its effective usein the solution of real problems of natural science.Note that Archimedes, Newton, Leibniz, Euler, Gauss, Poincaré, who are heldin particularly high esteem by us, mathematicians, were more than mere mathematicians. They were scientists, natural philosophers.

In mathematics resolvingof important specific questions and development of an abstract general theory areprocesses as inseparable as inhaling and exhaling. Upsetting this balance leads toproblems that sometimes become significant both in mathematical education and inscience in general.The textbook exposes classical analysis as it is today, as an integral part of theunified Mathematics, in its interrelations with other modern mathematical coursessuch as algebra, differential geometry, differential equations, complex and functional analysis.Rigor of discussion is combined with the development of the habit of workingwith real problems from natural sciences. The course exhibits the power of concepts and methods of modern mathematics in exploring specific problems.

Variousexamples and numerous carefully chosen problems, including applied ones, forma considerable part of the textbook. Most of the fundamental mathematical notionsand results are introduced and discussed along with information, concerning theirhistory, modern state and creators. In accordance with the orientation toward naturalsciences, special attention is paid to informal exploration of the essence and roots ofthe basic concepts and theorems of calculus, and to the demonstration of numerous,sometimes fundamental, applications of the theory.For instance, the reader will encounter here the Galilean and Lorentz transforms,the formula for rocket motion and the work of nuclear reactor, Euler’s theoremPrefacesviion homogeneous functions and the dimensional analysis of physical quantities, theLegendre transform and Hamiltonian equations of classical mechanics, elements ofhydrodynamics and the Carnot’s theorem from thermodynamics, Maxwell’s equations, the Dirac delta-function, distributions and the fundamental solutions, convolution and mathematical models of linear devices, Fourier series and the formulafor discrete coding of a continuous signal, the Fourier transform and the Heisenberguncertainty principle, differential forms, de Rham cohomology and potential fields,the theory of extrema and the optimization of a specific technological process, numerical methods and processing the data of a biological experiment, the asymptoticsof the important special functions, and many other subjects.Within each major topic the exposition is, as a rule, inductive, sometimes proceeding from the statement of a problem and suggestive heuristic considerationsconcerning its solution, toward fundamental concepts and formalisms.

Detailed atfirst, the exposition becomes more and more compressed as the course progresses.Beginning ab ovo the book leads to the most up-to-date state of the subject.Note also that, at the end of each of the volumes, one can find the list of the maintheoretical topics together with the corresponding simple, but nonstandard problems(taken from the midterm exams), which are intended to enable the reader both determine his or her degree of mastery of the material and to apply it creatively inconcrete situations.More complete information on the book and some recommendations for its use inteaching can be found below in the prefaces to the first and second Russian editions.Moscow, Russia2003V. ZorichviiiPrefacesPreface to the Sixth Russian EditionOn my own behalf and on behalf of future readers, I thank all those, living in different countries, who had the possibility to inform the publisher or me personallyabout errors (typos, errors, omissions), found in Russian, English, German and Chinese editions of this textbook.As it turned out, the book has been also very useful to physicists; I am veryhappy about that.

In any case, I really seek to accompany the formal theory withmeaningful examples of its application both in mathematics and outside of it.The sixth edition contains a series of appendices that may be useful to studentsand lecturers. Firstly, some of the material is actually real lectures (for example,the transcription of two introductory survey lectures for students of first and thirdsemesters), and, secondly, this is some mathematical information (sometimes of current interest, such as the relation between multidimensional geometry and the theoryof probability), lying close to the main subject of the textbook.Moscow, Russia2011V.