CiNii 図書 - Mathematical analysis

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.

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

巻冊次: 2 ISBN 9783540406334

内容説明

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.

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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Mathematical analysis

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