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This softcover edition of a very popular two-volume work presents a thorough 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. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to…mehr

Produktbeschreibung
This softcover edition of a very popular two-volume work presents a thorough 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. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books. The first volume constitutes 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.
  • Produktdetails
  • uniext
  • Verlag: Springer, Berlin
  • Originaltitel: Matematicheskij Analiz
  • Artikelnr. des Verlages: 12525393
  • Erscheinungstermin: 21. November 2008
  • Englisch
  • Abmessung: 239mm x 164mm x 37mm
  • Gewicht: 884g
  • ISBN-13: 9783540874515
  • ISBN-10: 3540874518
  • Artikelnr.: 24733068
Inhaltsangabe
CONTENTS OF VOLUME I Prefaces
Preface to the English edition
Prefaces to the fourth and third editions
Preface to the second edition
From the preface to the first edition 1. Some General Mathematical Concepts and Notation 1.1 Logical symbolism
1.1.1 Connectives and brackets
1.1.2 Remarks on proofs
1.1.3 Some special notation
1.1.4 Concluding remarks
1.1.5 Exercises 1.2 Sets and elementary operations on them
1.2.1 The concept of a set
1.2.2 The inclusion relation
1.2.3 Elementary operations on sets
1.2.4 Exercises 1.3 Functions
1.3.1 The concept of a function (mapping)
1.3.2 Elementary classification of mappings
1.3.3 Composition of functions. Inverse mappings
1.3.4 Functions as relations. The graph of a function
1.3.5 Exercises 1.4 Supplementary material
1.4.1 The cardinality of a set (cardinal numbers)
1.4.2 Axioms for set theory
1.4.3 Set-theoretic language for propositions
1.4.4 Exercises
2. The Real Numbers 2.1 Axioms and properties of real numbers
2.1.1 Definition of the setof real numbers
2.1.2 Some general algebraic properties of real numbers
a. Consequences of the addition axioms
b. Consequences of the multiplication axioms
c. Consequences of the axiom connecting addition and
multiplication
d. Consequences of the order axioms
e. Consequences of the axioms connecting order with addition and
multiplication
2.1.3 The completeness axiom. Least upper bound 2.2 Classes of real numbers and computations
2.2.1 The natural numbers. Mathematical induction
a. Definition of the set of natural numbers
b. The principle of mathematical induction
2.2.2 Rational and irrational numbers
a. The integers
b. The rational numbers
c. The irrational numbers
2.2.3 The principle of Archimedes
Corollaries
2.2.4 Geometric interpretation. Computational aspects
a. The real line
b. Defining a number by successive approximations
c. The positional computation system 2.2.5 Problems and exercises 2.3 Basic lemmas on completeness 
Rezensionen
Diese profunde Einführung [Math.Analysis I und II] in die Analysis sollte in keiner mathematischen Bibliothek fehlen, selbst bei budgetären Restriktionen, trotz der Überfülle an Einführungsbüchern. Eine genaue, bewußte Lektüre dieses profunden Werks könnte mögliche künftige Autoren mittelmäßiger Analysisbücher vielleicht abschrecken.

[...]Meisterhaft wird hier intuitives Verstehen gefördert, vermittelt durch anschauliche geometrische Denkweisen, heuristische Ideen und induktive Vorgangsweisen, ohne Exaktheitsansprüche hintanzustellen oder konkrete Details oder Anwendungen auch nur ansatzweise zu vernachlässigen. Der Aufbau ist in vieler Hinsicht ungewöhnlich, eröffnet frühe Einblicke und Weitblicke und regt zum Denken an [...], ist auch der historischen Entwicklung angemessen und bietet eine wichtige Alternative zu den vielen "eleganten" Zugängen, bei denen die Vermittlung wichtiger nötiger Entwicklungsschritte für ein aktives Verständnis zu kurz kommt.

Der umfassende, Nachbardisziplinen laufend berührende Zugang trägt reiche Früchte, ebenso die facettenreiche Fülle an Erklärungen der Wurzeln und Essenz der grundlegenden Konzepte und Resultate, die Beschreibungen von Zusammenhängen und Ausblicke auf weitere Entwicklungen mit vielen in Einführungsbüchern leider eher unüblichen Anwendungen und Querbezügen [...]. Man erwirbt mit diesem Werk zusätzlich ein vollständiges, umfangreiches und wertvolles "Problem-Buch". Bei aller reichhaltiger Fülle stellt sich die Mathematik hier aber immer als eine Einheit dar, in ihrer auf den heutigen Stellenwert Bezug nehmenden historischen und philosophischen Entwicklung, geprägt durch, an passender Stelle kompetent gewürdigte, bedeutende große schöpferische Persönlichkeiten. [...] Dieses vorzügliche Werk atmet den Geist einer bewunderungswürdigen, vielschichtigen Forscher- und Lehrerpersönlichkeit."

H.Rindler, Monatshefte für Mathematik 146, Issue 4, 2005 "Die vorliegenden zwei Bände sind die englische Übersetzung eines russischen Werkes, das bereits Anfang der achtziger Jahre erschienen ist und inzwischen bereits zum vierten Mal aufgelegt wurde. Die Bücher beinhalten auf über 1200 Seiten die klassische Analysis in einer zeitgemäßen 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 Studienanfängern und fortgeschrittenen Studierenden uneingeschränkt empfohlen werden, aber auch Lehrende werden viele Anregungen darin finden."

M.Kronfellner (Wien), IMN - Internationale Mathematische Nachrichten 59, Issue 198, 2005, S. 36-37
…mehr
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…mehr