Beschreibung
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.
Inhaltsverzeichnis
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 2.3.1 The nested interval lemma 2.3.2 The finite covering lemma 2.3.3 The limit point lemma 2.3.4 Problems and exercises 2.4 Countable and uncountable sets 2.4.1 Countable sets Corollaries 2.4.2 The cardinality of the continuum Corollaries 2.4.3 Problems and exercises 3. Limits 3.1 The limit of a sequence 3.1.1 Definitions and examples 3.1.2 Properties of the limit of a sequence a. General properties b. Passage to the limit and the arithmetic operations c. Passage to the limit and inequalities 3.1.3 Existence of the limit of a sequence a. The Cauchy criterion b. A criterion for the existence of the limit of a monotonic sequence c. The number e d. Subsequences and partial limits of a sequence Concluding remarks 3.1.4 Elementary facts about series a. The sum of a series and the CauchyCauchy, A. criterion for convergence of a series b. Absolute convergence; the comparison theorem and its consequences c. The number e as the sum of a series 3.1.5 Problems and exercises 3.2 The limit of a function 3.2.1 Definitions and examples 3.2.2 Properties of the limit of a function a. General properties of the limit of a function b. Passage to the limit and arithmetic operations c. Passage to the limit and inequalities d. Two important examples 3.2.3 Limits over a base a. Bases; definition and elementary properties b. The limit of a function over a base 3.2.4 Existence of the limit of a function a. The Cauchy criterion b. The limit of a composite function c. The limit of a monotonic function d. Comparison of the asymptotic behavior of functions 3.2.5 Problems and exercises 4. Continuous Functions 4.1 Basic definitions and examples 4.1.1 Continuity of a function at a point 4.1.2 Points of discontinuity 4.2 Properties of continuous functions 4.2.1 Local properties 4.2.2 Global properties of continuous functions Remarks to Theorem 2 4.2.3 Problems and exercises 5. Differential Calculus 5.1 Differentiable functions 5.1.1 Statement of the problem 5.1.2 Functions differ ...
