InhaltsangabeCONTENTS 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 e

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 ...

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