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 …… 12 Surfaces and Differential Forms in Rn 13 Line and Surface Integrals 14 Elements of Vector Analysis and Field Theory 15 *Integration of Differential Forms on Manifolds 16 Uniform Convergence and the Basic Operations of Analysis on Series and Families of Functions 17 Integrals Depending on a Parameter 18 Fourier Series and the Fourier Transform 19 Asymptotic Expansions Topics and Questions for Midterm Examinations Examination Topics Examination Problems (Series and Integrals Depending on a Parameter) Intermediate Problems (Integral Calculus of Several Variables) Appendix A Series as a Tool (Introductory Lecture) Appendix B Change of Variables in Multiple Integrals (Deduction and First Discussion of the Change of Variables Formula) Appendix C Multidimensional Geometry and Functions of a Very Large Number of Variables (Concentration of Measures and Laws of Large Numbers) Appendix D Operators of Field Theory in Curvilinear Coordinates Appendix E Modern Formula of Newton-Leibniz and the Unity of Mathematics (Final Survey) References Index of Basic Notation Subject Index Name Index
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