(Matematicheskij Analiz). Part I. 6th Edition, Moscow, Publisher MCCME 2012

ISBN-10: 3540874518
ISBN-13: 9783540874515
Edition: 2004
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Description: This work by Zorich on mathematical analysis constitutes a thorough first course in real analysis, leading from the most elementary facts about real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier,  More...

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Book details

Copyright year: 2004
Publisher: Springer
Publication date: 11/21/2008
Binding: Paperback
Pages: 574
Size: 6.00" wide x 9.25" long x 1.25" tall
Weight: 2.464
Language: English

This work by Zorich on mathematical analysis constitutes a thorough first course in real analysis, leading 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.

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
Some General Mathematical Concepts and Notation
Logical symbolism
Connectives and brackets
Remarks on proofs
Some special notation
Concluding remarks
Exercises
Sets and elementary operations on them
The concept of a set
The inclusion relation
Elementary operations on sets
Exercises
Functions
The concept of a function (mapping)
Elementary classification of mappings
Composition of functions. Inverse mappings
Functions as relations. The graph of a function
Exercises
Supplementary material
The cardinality of a set (cardinal numbers)
Axioms for set theory
Set-theoretic language for propositions
Exercises2. The Real Numbers
Axioms and properties of real numbers
Definition of the set of real numbers
Some general algebraic properties of real numbers
Consequences of the addition axioms
Consequences of the multiplication axioms
Consequences of the axiom connecting addition and multiplication
Consequences of the order axioms
Consequences of the axioms connecting order with addition and multiplication
The completeness axiom. Least upper bound
Classes of real numbers and computations
Definition of the set of natural numbers
The principle of mathematical induction
Rational and irrational numbers
The integers
The rational numbers
The irrational numbers
The principle of Archimedes Corollaries
Geometric interpretation. Computational aspects
The real line
Defining a number by successive approximations
The positional computation system
Problems and exercises
Basic lemmas on completeness
The nested interval lemma
The finite covering lemma
The limit point lemma
Problems and exercises
Countable and uncountable sets

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