Mathematical Analysis I. Vol.1

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Beschreibung

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.



This two-volume work by V.A.Zorich on Mathematical Analysis constitutes a thorough first course in real analysis. The text leads from the most elementary facts about real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, and elliptic functions. With masterful exposition, the author provides a smooth, gradual transition from each topic to the next, so that the slope never feels too steep for the reader. The clarity of the exposition is matched by a wealth of instructive exercises and fresh applications to areas seldom touched on in real analysis books, many of which are taken from physics and technology.



Klappentext

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.



Zusammenfassung

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.



Inhalt
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 set of 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 &nbsp

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Produktinformationen

Titel
Mathematical Analysis I. Vol.1
Autor
Übersetzer
EAN
9783540874515
ISBN
978-3-540-87451-5
Format
Kartonierter Einband
Herausgeber
Springer, Berlin
Genre
Mathematik
Anzahl Seiten
574
Gewicht
882g
Größe
H235mm x B235mm
Jahr
2008
Untertitel
Englisch
Auflage
1st ed. 2004. 2nd printing 2008
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