Paperback(Softcover reprint of the original 2nd ed. 2015)

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Product Details

ISBN-13: 9783662569559
Publisher: Springer Berlin Heidelberg
Publication date: 04/26/2018
Series: Universitext
Edition description: Softcover reprint of the original 2nd ed. 2015
Pages: 616
Product dimensions: 6.10(w) x 9.25(h) x (d)

About the Author

VLADIMIR A. ZORICH is professor of mathematics at Moscow State University. His areas of specialization are analysis, conformal geometry, quasiconformal mappings, and mathematical aspects of thermodynamics. He solved the problem of global homeomorphism for space quasiconformal mappings. He holds a patent in the technology of mechanical engineering, and he is also known by his book “Mathematical Analysis of Problems in the Natural Sciences”.

Table of Contents

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
 

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