Mathematical analysis
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Bibliographic Information
Mathematical analysis
（Universitext）
Springer, 2009, c2004
- 1 : softcover
- 2 : softcover
- Other Title
-
Matematicheskij analiz
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"Title of original Russian edition: Matematicheskij analiz (Part I-II, 4th corrected edition, Moscow, 2002)" -- T.p. verso
Includes bibliographical references and indexes
Description and Table of Contents
- Volume
-
1 : softcover ISBN 9783540874515
Description
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.
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
- Volume
-
2 : softcover ISBN 9783540874539
Description
The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions.
Table of Contents
CONTENTS OF VOLUME II Prefaces
Preface to the fourth edition
Prefact to the third edition
Preface to the second edition
Preface to the first edition
9* Continuous Mappings (General Theory) 9.1 Metric spaces 9.1.1 Definitions and examples
9.1.2 Open and closed subsets of a metric space
9.1.3 Subspaces of a metric space
9.1.4 The direct product of metric spaces
9.1.5 Problems and exercises 9.2 Topological spaces 9.2.1 Basic definitions
9.2.2 Subspaces of a topological space
9.2.3 The direct product of topological spaces
9.2.4 Problems and exercises 9.3 Compact sets 9.3.1 Definition and general properties of compact sets
9.3.2 Metric compact sets
9.3.3 Problems and exercises 9.4 Connected topological spaces 9.4.1 Problems and exercises 9.5 Complete metric spaces 9.5.1 Basic definitions and examples
9.5.2 The completion of a metric space
9.5.3 Problems and exercises 9.6 Continuous mappings of topological spaces 9.6.1 The limit of a mapping
9.6.2 Continuous mappings
9.6.3 Problems and exercises 9.7 The contraction mapping principle
9.7.1 Problems and exercises
10 *Differential Calculus from a General Viewpoint 10.1 Normed vector spaces 10.1.1 Some examples of the vector spaces of analysis
10.1.2 Norms in vector spaces
10.1.3 Inner products in a vector space
10.1.4 Problems and exercises 10.2 Linear and multilinear transformations
10.2.1 Definitions and examples
10.2.2 The norm of a transformation
10.2.3 The space of continuous transformations
10.2.4 Problems and exercises 10.3 The differential of a mapping
10.3.1 Mappings differentiable at a point
10.3.2 The general rules for differentiation
10.3.3 Some examples
10.3.4 The partial deriatives of a mapping
10.3.5 Problems and exercises 10.4 The mean-value theorem and some examples of its use
10.4.1 The mean-value theorem
10.4.2 Some applications of the mean-value theorem
10.4.3 Problems and exercises 10.5 Higher-order derivatives
10.5.1 Definition of the nth differential
10.5.2 The derivative with respect to a vector and the computation of the values of the nth differential.
10.5.3 Symmetry of the higher-order differentials
10.5.4 Some remarks
10.5.5 Problems and exercises 10.6 Taylor's formula and methods of finding extrema
10.6.1 Taylor's formula for mappings
10.6.2 Methods of finding interior extrema
10.6.3 Some examples
10.6.4 Problems and e
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