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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.

Mathematical Analysis I. Vol.1

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