目錄
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
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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 O
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