Mathematical Analysis II

Transkript

1 Universitext Mathematical Analysis II Translation by R. Cooke Bearbeitet von R. Cooke, Vladimir A. Zorich 1. Auflage Buch. xv, 688 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 2560 g Weitere Fachgebiete > Mathematik > Mathematische Analysis Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 9 *Continuous Mappings (General Theory) Metric Spaces Definition and Examples Open and Closed Subsets of a Metric Space Subspaces of a Metric space The Direct Product of Metric Spaces Problems and Exercises Topological Spaces Basic Definitions Subspaces of a Topological Space The Direct Product of Topological Spaces Problems and Exercises Compact Sets Definition and General Properties of Compact Sets Metric Compact Sets Problems and Exercises Connected Topological Spaces Problems and Exercises Complete Metric Spaces Basic Definitions and Examples The Completion of a Metric Space Problems and Exercises Continuous Mappings of Topological Spaces The Limit of a Mapping Continuous Mappings Problems and Exercises The Contraction Mapping Principle Problems and Exercises *Differential Calculus from a General Viewpoint Normed Vector Spaces Some Examples of Vector Spaces in Analysis Norms in Vector Spaces Inner Products in Vector Spaces... 45

3 X Problems and Exercises Linear and Multilinear Transformations Definitions and Examples The Norm of a Transformation The Space of Continuous Transformations Problems and Exercises The Differential of a Mapping Mappings Differentiable at a Point The General Rules for Differentiation Some Examples The Partial Derivatives of a Mapping Problems and Exercises The Finite-increment (Mean-value) Theorem The Finite-increment Theorem Some Applications of the Finite-increment Theorem Problems and Exercises Higher-order Derivatives Definition of the nth Differential Derivative with Respect to a Vector Symmetry of the Higher-order Differentials Some Remarks Problems and Exercises Taylor s Formula and the Study of Extrema Taylor s Formula for Mappings Methods of Studying Interior Extrema Some Examples Problems and Exercises The General Implicit Function Theorem Problems and Exercises Multiple Integrals The Riemann Integral over an n-dimensional Interval Definition of the Integral The Lebesgue Criterion for Riemann Integrability The Darboux Criterion Problems and Exercises The Integral over a Set Admissible Sets The Integral over a Set The Measure (Volume) of an Admissible Set Problems and Exercises General Properties of the Integral The Integral as a Linear Functional Additivity of the Integral Estimates for the Integral

4 XI Problems and Exercises Reduction of a Multiple Integral to an Iterated Integral Fubini s Theorem Some Corollaries Problems and Exercises Change of Variable in a Multiple Integral Statement of the Problem. Heuristic Considerations Measurable Sets and Smooth Mappings The One-dimensional Case The Case of an Elementary Diffeomorphism in R n Composite Mappings and Change of Variable Additivity of the Integral Generalizations of the Change of Variable Formula Problems and Exercises Improper Multiple Integrals Basic Definitions The Comparison Test Change of Variable in an Improper Integral Problems and Exercises Surfaces and Differential Forms in R n Surfaces in R n Problems and Exercises Orientation of a Surface Problems and Exercises The Boundary of a Surface and its Orientation Surfaces with Boundary The Induced Orientation on the Boundary Problems and Exercises The Area of a Surface in Euclidean Space Problems and Exercises Elementary Facts about Differential Forms Differential Forms: Definition and Examples Coordinate Expression of a Differential Form The Exterior Differential of a Form Transformation under Mappings Forms on Surfaces Problems and Exercises Line and Surface Integrals The Integral of a Differential Form The Original Problems. Examples Integral over an Oriented Surface Problems and Exercises The Volume Element. Integrals of First and Second Kind

5 XII The Mass of a Lamina The Area of a Surface as the Integral of a Form The Volume Element Cartesian Expression of the Volume Element Integrals of First and Second Kind Problems and Exercises The Fundamental Integral Formulas of Analysis Green s Theorem The Gauss Ostrogradskii Formula Stokes Formula in R The General Stokes Formula Problems and Exercises Elements of Vector Analysis and Field Theory The Differential Operations of Vector Analysis Scalar and Vector Fields Vector Fields and Forms in R The Differential Operators grad, curl, div, and Some Differential Formulas of Vector Analysis *Vector Operations in Curvilinear Coordinates Problems and Exercises The Integral Formulas of Field Theory The Classical Integral Formulas in Vector Notation The Physical Interpretation of div, curl, and grad Other Integral Formulas Problems and Exercises Potential Fields The Potential of a Vector Field Necessary Condition for Existence of a Potential Criterion for a Field to be Potential Topological Structure of a Domain and Potentials Vector Potential. Exact and Closed Forms Problems and Exercises Examples of Applications The Heat Equation The Equation of Continuity Equations of Dynamics of Continuous Media The Wave Equation Problems and Exercises *Integration of Differential Forms on Manifolds A Brief Review of Linear Algebra The Algebra of Forms The Algebra of Skew-symmetric Forms Linear Mappings and their Adjoints

6 XIII Problems and Exercises Manifolds Definition of a Manifold Smooth Manifolds and Smooth Mappings Orientation of a Manifold and its Boundary Partitions of Unity. Manifolds as Surfaces Problems and Exercises Differential Forms and Integration on Manifolds The Tangent Space to a Manifold at a Point Differential Forms on a Manifold The Exterior Derivative The Integral of a Form over a Manifold Stokes Formula Problems and Exercises Closed and Exact Forms on Manifolds Poincaré s Theorem Homology and Cohomology Problems and Exercises Uniform Convergence and Basic Operations of Analysis Pointwise and Uniform Convergence Pointwise Convergence Statement of the Fundamental Problems Convergence of a Family Depending on a Parameter The Cauchy Criterion for Uniform Convergence Problems and Exercises Uniform Convergence of Series of Functions Basic Definitions. Uniform Convergence of a Series Weierstrass M-test for Uniform Convergence The Abel Dirichlet Test Problems and Exercises Functional Properties of a Limit Function Specifics of the Problem Conditions for Two Limiting Passages to Commute Continuity and Passage to the Limit Integration and Passage to the Limit Differentiation and Passage to the Limit Problems and Exercises *Subsets of the Space of Continuous Functions The Arzelà Ascoli Theorem The Metric Space C(K, Y ) Stone s Theorem Problems and Exercises

7 XIV 17 Integrals Depending on a Parameter Proper Integrals Depending on a Parameter The Basic Concept Continuity of the Integral Differentiation of the Integral Integration of the Integral Problems and Exercises Improper Integrals Depending on a Parameter Uniform Convergence with Respect to a Parameter Continuity of an Integral Depending on a Parameter Differentiation with Respect to a Parameter Integration of an Improper Integral Problems and Exercises The Eulerian Integrals The Beta Function The Gamma Function Connection Between the Beta and Gamma Functions Examples Problems and Exercises Convolution and Generalized Functions Convolution in Physical Problems General Properties of Convolution Approximate Identities and Weierstrass Theorem *Elementary Concepts Involving Distributions Problems and Exercises Multiple Integrals Depending on a Parameter Proper Multiple Integrals Depending on a Parameter Improper Multiple Integrals Improper Integrals with a Variable Singularity *Convolution in the Multidimensional Case Problems and Exercises Fourier Series and the Fourier Transform Basic General Concepts Connected with Fourier Series Orthogonal Systems of Functions Fourier Coefficients and Fourier Series *A Source of Orthogonal Systems Problems and Exercises Trigonometric Fourier Series Basic Types of Convergence of Fourier Series Pointwise Convergence of a Fourier Series Smoothness and Decrease of Fourier Coefficients Completeness of the Trigonometric System Problems and Exercises The Fourier Transform

8 XV Fourier Integral Representation Differential Properties and the Fourier Transform Main Structural Properties Examples of Applications Problems and Exercises Asymptotic Expansions Asymptotic Formulas and Asymptotic Series Basic Definitions General Facts about Asymptotic Series Asymptotic Power Series Problems and Exercises The Asymptotics of Integrals (Laplace s Method) The Idea of Laplace s Method The Localization Principle for a Laplace Integral Canonical Integrals and their Asymptotics Asymptotics of a Laplace Integral *Asymptotic Expansions of Laplace Integrals Problems and Exercises Topics and Questions for Midterm Examinations Series and Integrals Depending on a Parameter Problems Recommended as Midterm Questions Integral Calculus (Several Variables) Problems Recommended for Studying the Midterm Topics Examination Topics Series and Integrals Depending on a Parameter Integral Calculus (Several Variables) References Classic Works Primary Sources Major Comprehensive Expository Works Classical courses of analysis from the first half of the twentieth century Textbooks Classroom Materials Further Reading Index of Basic Notation Subject Index Name Index