Mathematical Analysis II (Universitext)

Cover Universitext Title: Mathematical Analysis II Copyright Prefaces Preface to the English Edition Preface to the Fourth Russian Edition Preface to the Third Russian Edition Preface to the Second Russian Edition Preface to the First Russian Edition Table of Contents 9 *Continuous Mappings (General Theory) 9.1 Metric Spaces 9.1.1 Definition 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 more General Point of View 10.1 Normed Vector Spaces 10.1.1 Some Examples of Vector Spaces in Analysis 10.1.2 Norms in Vector Spaces 10.1.3 Inner Products in Vector Spaces 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 Derivatives of a Mapping 10.3.5 Problems and Exercises 10.4 The Finite-increment Theorem and some Examples of its Use 10.4.1 The Finite-increment Theorem 10.4.2 Some Applications of the Finite-increment Theorem 10.4.3 Problems and Exercises 10.5 Higher-order Derivatives 10.5.1 Definition of the nth Differential 10.5.2 Derivative with Respect to a Vector and 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 the Study of Extrema 10.6.1 Taylor's Formula for Mappings 10.6.2 Methods of Studying Interior Extrema 10.6.3 Some Examples 10.6.4 Problems and Exercises 10.7 The General Implicit Function Theorem 10.7.1 Problems and Exercises 11 Multiple Integrals 11.1 The Riemann Integral over an n-Dimensional Interval 11.1.1 Definition of the Integral 11.1.2 The Lebesgue Criterion for Riemann Integrability 11.1.3 The Darboux Criterion 11.1.4 Problems and Exercises 11.2 The Integral over a Set 11.2.1 Admissible Sets 11.2.2 The Integral over a Set 11.2.3 The Measure (Volume) of an Admissible Set 11.2.4 Problems and Exercises 11.3 General Properties of the Integral 11.3.1 The Integral as a Linear Functional 11.3.2 Additivity of the Integral 11.3.3 Estimates for the Integral 11.3.4 Problems and Exercises 11.4 Reduction of a Multiple Integral to an Iterated Integral 11.4.1 Fubini's Theorem 11.4.2 Some Corollaries 11.4.3 Problems and Exercises 11.5 Change of Variable in a Multiple Integral 11.5.1 Statement of the Problem and Heuristic Derivation of the Change of Variable Formula 11.5.2 Measurable Sets and Smooth Mappings 11.5.3 The One-dimensional Case 11.5.4 The Case of an Elementary Diffeomorphism in IR^n 11.5.5 Composite Mappings and the Formula for Change of Variable 11.5.6 Additivity of the Integral and Completion of the Proof of the Formula for Change of Variable in an Integral 11.5.7 Corollaries and Generalizations of the Formula for Change of Variable in a Multiple Integral 11.5.8 Problems and Exercises 11.6 Improper Multiple Integrals 11.6.1 Basic Definitions 11.6.2 The Comparison Test for Convergence of an Improper Integral 11.6.3 Change of Variable in an Improper Integral 11.6.4 Problems and Exercises 12 Surfaces and Differential Forms in IR^n 12.1 Surfaces in IR^n 12.1.1 Problems and Exercises 12.2 Orientation of a Surface 12.2.1 Problems and Exercises 12.3 The Boundary of a Surface and its Orientation 12.3.1 Surfaces with Boundary 12.3.2 Making the Orientations of a Surface and its Boundary Consistent 12.3.3 Problems and Exercises 12.4 The Area of a Surface in Euclidean Space 12.4.1 Problems and Exercises 12.5 Elementary Facts about Differential Forms 12.5.1 Differential Forms: Definition and Examples 12.5.2 Coordinate Expression of a Differential Form 12.5.3 The Exterior Differential of a Form 12.5.4 Transformation of Vectors and Forms under Mappings 12.5.5 Forms on Surfaces 12.5.6 Problems and Exercises 13 Line and Surface Integrals 13.1 The Integral of a Differential Form 13.1.1 The Original Problems, Suggestive Considerations, Examples 13.1.2 Definition of the Integral of a Form over an Oriented Surface 13.1.3 Problems and Exercises 13.2 The Volume Element. Integrals of First and Second Kind 13.2.1 The Mass of a Lamina 13.2.2 The Area of a Surface as the Integral of a Form 13.2.3 The Volume Element 13.2.4 Expression of the Volume Element in Cartesian Coordinates 13.2.5 Integrals of First and Second Kind 13.2.6 Problems and Exercises 13.3 The Fundamental Integral Formulas of Analysis 13.3.1 Green's Theorem 13.3.2 The Gauss—Ostrogradskii Formula 13.3.3 Stokes' Formula in IR^3 13.3.4 The General Stokes Formula 13.3.5 Problems and Exercises 14 Elements of Vector Analysis and Field Theory 14.1 The Differential Operations of Vector Analysis 14.1.1 Scalar and Vector Fields 14.1.2 Vector Fields and Forms in IR^3 14.1.3 The Differential Operators grad, curl, div, and abla 14.1.4 Some Differential Formulas of Vector Analysis 14.1.5 *Vector Operations in Curvilinear Coordinates 14.1.6 Problems and Exercises 14.2 The Integral Formulas of Field Theory 14.2.1 The Classical Integral Formulas in Vector Notation 14.2.2 The Physical Interpretation of div, curl, and grad 14.2.3 Other Integral Formulas 14.2.4 Problems and Exercises 14.3 Potential Fields 14.3.1 The Potential of a Vector Field 14.3.2 Necessary Condition for Existence of a Potential 14.3.3 Criterion for a Field to be Potential 14.3.4 Topological Structure of a Domain and Potentials 14.3.5 Vector Potential. Exact and Closed Forms 14.3.6 Problems and Exercises 14.4 Examples of Applications 14.4.1 The Heat Equation 14.4.2 The Equation of Continuity 14.4.3 The Basic Equations of the Dynamics of Continuous Media 14.4.4 The Wave Equation 14.4.5 Problems and Exercises 15 integration of Differential Forms on Manifolds 15.1 A Brief Review of Linear Algebra 15.1.1 The Algebra of Forms 15.1.2 The Algebra of Skew-symmetric Forms 15.1.3 Linear Mappings of Vector Spaces and the Adjoint Mappings of the Conjugate Spaces 15.1.4 Problems and Exercises 15.2 Manifolds 15.2.1 Definition of a Manifold 15.2.2 Smooth Manifolds and Smooth Mappings 15.2.3 Orientation of a Manifold and its Boundary 15.2.4 Partitions of Unity and the Realization of Manifolds as Surfaces in IR^n 15.2.5 Problems and Exercises 15.3 Differential Forms and Integration on Manifolds 15.3.1 The Tangent Space to a Manifold at a Point 15.3.2 Differential Forms on a Manifold 15.3.3 The Exterior Derivative 15.3.4 The Integral of a Form over a Manifold 15.3.5 Stokes' Formula 15.3.6 Problems and Exercises 15.4 Closed and Exact Forms on Manifolds 15.4.1 Poincare's Theorem 15.4.2 Homology and Cohomology 15.4.3 Problems and Exercises 16 Uniform Convergence and the Basic Operations of Analysis on Series and Families of Functions 16.1 Pointwise and Uniform Convergence 16.1.1 Pointwise Convergence 16.1.2 Statement of the Fundamental Problems 16.1.3 Convergence and Uniform Convergence of a Family of Functions Depending on a Parameter 16.1.4 The Cauchy Criterion for Uniform Convergence 16.1.5 Problems and Exercises 16.2 Uniform Convergence of Series of Functions 16.2.1 Basic Definitions and a Test for Uniform Convergence of a Series 16.2.3 The Abel-Dirichlet Test 16.2.4 Problems and Exercises 16.3 Functional Properties of a Limit Function 16.3.1 Specifics of the Problem 16.3.2 Conditions for Two Limiting Passages to Commute 16.3.3 Continuity and Passage to the Limit 16.3.4 Integration and Passage to the Limit 16.3.5 Differentiation and Passage to the Limit 16.3.6 Problems and Exercises 16.4 *Compact and Dense Subsets of the Space of Continuous Functions 16.4.1 The Arzelà—Ascoli Theorem 16.4.2 The Metric Space C(K, Y) 16.4.3 Stone's Theorem 16.4.4 Problems and Exercises 17 Integrals Depending on a Parameter 17.1 Proper Integrals Depending on a Parameter 17.1.1 The Concept of an Integral Depending on a Parameter 17.1.2 Continuity of an Integral Depending on a Parameter 17.1.3 Differentiation of an Integral Depending on a Parameter 17.1.4 Integration of an Integral Depending on a Parameter 17.1.5 Problems and Exercises 17.2 Improper Integrals Depending on a Parameter 17.2.1 Uniform Convergence of an Improper Integral with Respect to a Parameter 17.2.2 Limiting Passage under the Sign of an Improper Integral and Continuity of an Improper Integral Depending on a Parameter 17.2.3 Differentiation of an Improper Integral with Respect to a Parameter 17.2.4 Integration of an Improper Integral with Respect to a Parameter 17.2.5 Problems and Exercises 17.3 The Eulerian Integrals 17.3.1 The Beta Function 17.3.2 The Gamma Function 17.3.3 Connection Between the Beta and Gamma Functions 17.3.4 Examples 17.3.5 Problems and Exercises 17.4 Convolution of Functions and Elementary Facts about Generalized Functions 17.4.1 Convolution in Physical Problems (Introductory Considerations) 17.4.2 General Properties of Convolution 17.4.3 Approximate Identities and the Weierstrass Approximation Theorem 17.4.4 *Elementary Concepts Involving Distributions 17.4.5 Problems and Exercises 17.5 Multiple Integrals Depending on a Parameter 17.5.1 Proper Multiple Integrals Depending on a Parameter 17.5.2 Improper Multiple Integrals Depending on a Parameter 17.5.3 Improper Integrals with a Variable Singularity 17.5.4 *Convolution, the Fundamental Solution, and Generalized Functions in the Multidimensional Case 17.5.5 Problems and Exercises 18 Fourier Series and the Fourier Transform 18.1 Basic General Concepts Connected with Fourier Series 18.1.1 Orthogonal Systems of Functions 18.1.2 Fourier Coefficients and Fourier Series 18.1.3 *An Important Source of Orthogonal Systems of Functions in Analysis 18.1.4 Problems and Exercises 18.2 Trigonometric Fourier Series 18.2.1 Basic Types of Convergence of Classical Fourier Series 18.2.2 Investigation of Pointwise Convergence of a Trigonometric Fourier Series 18.2.3 Smoothness of a Function and the Rate of Decrease of the Fourier Coefficients 18.2.4 Completeness of the Trigonometric System 18.2.5 Problems and Exercises 18.3 The Fourier Transform 18.3.1 Representation of a Function by Means of a Fourier Integral 18.3.2 The Connection of the Differential and Asymptotic Properties of a Function and its Fourier Transform 18.3.3 The Main Structural Properties of the Fourier Transform 18.3.4 Examples of Applications 18.3.5 Problems and Exercises 19 Asymptotic Expansions 19.1 Asymptotic Formulas and Asymptotic Series 19.1.1 Basic Definitions 19.1.2 General Facts about Asymptotic Series 19.1.3 Asymptotic Power Series 19.1.4 Problems and Exercises 19.2 The Asymptotics of Integrals (Laplace's Method) 19.2.1 The Idea of Laplace's Method 19.2.2 The Localization Principle for a Laplace Integral 19.2.3 Canonical Integrals and their Asymptotics 19.2.4 The Principal Term of the Asymptotics of a Laplace Integral 19.2.5 *Asymptotic Expansions of Laplace Integrals 19.2.6 Problems and Exercises Topics and Questions for Midterm Examinations 1. Series and Integrals Depending on a Parameter 2. Problems Recommended as Midterm Questions 3. Integral Calculus (Several Variables) 4. Problems Recommended for Studying the Midterm Topics Examination Topics 1. Series and Integrals Depending on a Parameter 2. Integral Calculus (Several Variables) References 1. Classic Works 1.1. Primary Sources 1.2. Major Comprehensive Expository Works 1.3. Classical courses of analysis from the first half of the twentieth century 2. Textbooks 3. Classroom Materials 4. Further Reading Index of Basic Notation Subject Index A B C D E F G H I J K L M N O P Q R S T U V W Z Name Index Back Cover This two-volume work by V.A.Zorich on Mathematical Analysis constitutes a thorough first course in real analysis, leading from the most elementary facts about real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, and elliptic functions. With masterful exposition, the author provides a smooth, gradual transition from each topic to the next, so that the slope never feels too steep for the reader. Making use of Cartan's concept of a filter base, the author disperses the fog of epsilons and deltas that have always made the crucial subject of limits a barrier for the nonmathematical specialist. As a result, the major theorems of differentiation and integrationreveal their essential unity in a nearly painless manner. The clarity of the exposition is matched by a wealth of instructive exercises and fresh applications to areas seldom touched on in real analysis books, many of which are taken from physics and technology. TOC:Prefaces.- 9 Continuous Mappings (General Theory).- 10 Differential Calculus from a General Viewpoint.- 11 Multiple Integrals.- 12 Surfaces and Differential Forms in Rn.- 13 Line and Surface Integrals.- 14 The Elements of Vector Analysis and Field Theory.- 15 Integration of Differential Forms on Manifolds.- 16 Uniform Convergence and the Basic Operations of Analysis.- 17 Integrals Depending on a Parameter.- 18 Fourier Series and the Fourier Transform.- 19 Asymptotic Expansions.- Some Problems from the Midterm Examinations.- Examination Topics.- References.- Subject Index.- Name Index

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, integral transforms, 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 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.

Definition 1. A set X is said to be endowed with a metric or a metric space structure or to be a metric space if a function d : X x X R (9.1) is exhibited satisfying the following conditions: a) d(x1, x2) = 0 x1 = x2, b) d(x1, x2) = d(x2, x2) (symmetry), c) d(x1, x3) d(x1, x2) + d(x2, x3) (the triangle inequality), where x1, x2, x3 are arbitrary elements of X.