Mathematical Analysis I [Paperback]

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 softcover edition of a very popular work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, Fourier, Laplace, and Legendre transforms, elliptic functions and distributions.

CONTENTS OF VOLUME I PrefacesPreface to the English editionPrefaces to the fourth and third editionsPreface to the second editionFrom the preface to the first edition 1. Some General Mathematical Concepts and Notation 1.1 Logical symbolism1.1.1 Connectives and brackets1.1.2 Remarks on proofs1.1.3 Some special notation1.1.4 Concluding remarks1.1.5 Exercises 1.2 Sets and elementary operations on them1.2.1 The concept of a set1.2.2 The inclusion relation1.2.3 Elementary operations on sets1.2.4 Exercises 1.3 Functions1.3.1 The concept of a function (mapping)1.3.2 Elementary classification of mappings1.3.3 Composition of functions. Inverse mappings1.3.4 Functions as relations. The graph of a function1.3.5 Exercises 1.4 Supplementary material1.4.1 The cardinality of a set (cardinal numbers)1.4.2 Axioms for set theorlc<