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# Mathematical Analysis II / Edition 1

Mathematical Analysis II / Edition 1 available in Hardcover

- ISBN-10:
- 3540406336
- ISBN-13:
- 9783540406334
- Pub. Date:
- 01/22/2004
- Publisher:
- Springer Berlin Heidelberg

## Hardcover

###### ADVERTISEMENT

## Overview

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.

## Product Details

ISBN-13: | 9783540406334 |
---|---|

Publisher: | Springer Berlin Heidelberg |

Publication date: | 01/22/2004 |

Series: | Universitext Series |

Edition description: | 2004 |

Pages: | 688 |

Product dimensions: | 1.56(w) x 6.14(h) x 9.21(d) |

## About the Author

## Table of Contents

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 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 (content) 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 Rn 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 Some 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 exercise

12 Surfaces and Differential Forms in Rn

12.1 Surfaces in Rn 12.1.1 Problems and exercises

12.2 The 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 expressions of a differential form 12.5.3 Exterior differential forms 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 material surface 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 R3 13.3.4 The general Stokes formula 13.3.5 Problems and exercises

14 The 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 R3 14.1.3 The differential operators grad, curl, div, and nabla 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 Some 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 A necessary condition for the existence of a potential 14.3.3 A criterion for a field to be a potential field 14.3.4 The topological structure of a domain and the potential 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 Definintion 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 Rn 15.2.5 Problems and exercises

15.3 Differential forms and their integrals over 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 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 16.2.2 The Weierstrass M-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 the 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 *Subsets of the space of continuous functions 16.4.1 The Arzela—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 the 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 over the 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 The connection between the beta and gamma functions 17.3.4 Some examples 17.3.5 Problems and exercises

17.4 Convolution and generalized functions 17.4.1 Convolution in physical problems (introductory considerations) 17.4.2 Some general properties of the 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 The basic kinds of convergence of the classical Fourier series 18.2.2 Investigation of the 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 technical 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 the 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 Some Problems from the Midterm Examinations

1. Introduction to analysis (numbers, functions, limits)

2. One-variable differential calculus

3. Integration. Introduction to several variables

4. Differential calculus of several variables Examination Topics

1. First semester 1.1. Introduction and one-variable differential calculus

2. Second semester 2.1. Integration. Multivariable differential calculus References

1. Classic works 1.1 Primary sources 1.2. Major comprehensive expository works

2. Textbooks

3. Classroom materials

4. Further reading Subject Index Name Index