Smooth Functions and Maps

Preface to the English Translation
Preface
Contents
Basic notation
Sets
Maps and Functions
Smooth Functions
Sets of Functions
Measures
Introduction. Preliminaries
0.1 The Space Rm
0.2 Open and Closed Sets
0.3 The Limit of a Sequence in Rm
0.4 Lindelöf’s Lemma
0.5 Compact Sets
0.6 The Limit of a Map
0.7 Continuity
0.8 Continuity on a Compact Set
0.9 Uniform Continuity
0.10 The Continuity of the Inverse Map
0.11 Connectedness
0.12 Convex Sets in Rm
0.13 Linear Maps Between Euclidean Spaces
Exercises
Chapter I. Differentiable Functions
1 Partial Derivatives and Increments
1.1 Partial Derivatives
1.2 Lagrange’s Mean Value Theorem
1.3 The Criterion of Constancy
1.4 The Lipschitz Condition
Exercises
2 The Definition of a Differentiable Function
2.1 Differentiability
2.2 Differentiability: a Necessary Condition
2.3 Differentiability: a Sufficient Condition
2.4 Gradient
Exercises
3 Directional Derivatives
3.1 The Derivative Along a Vector
3.2 Computing Directional Derivatives
3.3 A Coordinate-Free Description of the Gradient
Exercises
4 The Tangent Plane to a Level Surface
4.1 Level Sets
4.2 The Graph as a Level Set
4.3 Tangent Plane
Exercises
5 Differentiable Maps
5.1 Definition of Differentiability
5.2 The Jacobian Matrix
5.3 Differentiation of Composite Functions
5.4 Lagrange’s Inequality
5.5 Differentiability of the Inverse Map
Exercises
6 Higher Derivatives
6.1 Definitions and Notation
6.2 The Symmetry of Second Derivatives
6.3 Smooth Functions and Maps
6.4 Equality of Mixed Partial Derivatives of Arbitrary Order
6.5 Algebraic Operations on Smooth Functions
6.6 Composition of Smooth Maps
6.7 Fractional Smoothness
Exercises
7 Polynomials in Several Variables and Higher Differentials
7.1 Polynomials in Several Variables
7.2 Higher Differentials
Exercises
8 Taylor’s Formula
8.1 Preliminaries
8.2 Taylor’s Formula for Functions of Several Variables
8.3 Another Estimate for the Remainder
8.4 A Characteristic Property of Taylor Polynomials
Exercises
9 Extrema of Functions of Several Variables
9.1 A Necessary Condition for an Extremum
9.2 A Refined Necessary Condition for an Extremum
9.3 Sufficient Conditions for an Extremum
9.4 The Absolute Maximum and Minimum Values of a Function of Several Variables
Exercises
10 Implicit Function
10.1 Statement of the Problem
10.2 The Existence and Uniqueness of an Implicit Function
10.3 The Smoothness of an Implicit Function
10.4 The Role of the Gradient of F
Exercises
11 *Whitney’s Extension Theorem
11.1 Statement of the Problem and a Preliminary Result
11.2 Whitney’s Extension Theorem
11.3 Proof of Whitney’s Theorem
11.4 Preserving the Degree of Smoothness of the Highest Derivatives
11.5 Extension from Totally Connected Sets
11.6 Sets with Minimally Smooth Boundaries
Exercises
Chapter II. Smooth Maps
1 The Open Mapping Theorem and the Diffeomorphism Theorem
1.1 The Open Mapping Theorem
1.2 The Open Mapping Theorem (Continued)
1.3 The Diffeomorphism Theorem
1.4 The Smoothness of the Inverse Map
Exercises
2 Local Invertibility Theorems and Dependence of Functions
2.1 Local Invertibility
2.2 The Partial Inversion Theorem
2.3 Extending a Smooth Map to a Diffeomorphism
2.4 Dependence and Independence of Functions
Exercise
3 Curvilinear Coordinates and Change of Variables
3.1 Curvilinear Coordinates
3.2 Examples of Curvilinear Coordinates
3.3 Partial Derivatives in Curvilinear Coordinates
Exercises
4 Classification of Smooth Maps
4.1 Equivalence of Smooth Maps
4.2 The Rank Theorem
5 *The Global Invertibility Theorem
5.1 Preliminary Remarks
5.2 Coverings and Their Properties
5.3 The Main Result
Exercise
6 *The Morse Lemma
6.1 Uniform Reduction of Close Quadratic Forms to Canonical Form
6.2 Leibniz’s Rule for Differentiation Under the Integral Sign
6.3 Hadamard’s Lemma
6.4 The Main Result
Chapter III. The Implicit Map Theorem and Its Applications
1 Implicit Maps
1.1 Statement of the Problem
1.2 The Inverse Mapping Theorem
1.3 Concluding Remarks
Exercise
2 Smooth Manifolds
2.1 Definitions and Notation
2.2 Equivalent Descriptions of a Smooth Manifold
2.3 The Tangent Subspace
2.4 Examples
Exercises
3 Constrained Extrema
3.1 Heuristic Arguments and Statement of the Problem
3.2 A Necessary Condition for a Constrained Extremum
3.3 The Lagrange Function and Lagrange Multipliers Method
3.4 Examples of Applying the Lagrange Multipliers Method
3.5 Sufficient Conditions for a Constrained Extremum
3.6 Conditions for the Absence of a Constrained Extremum
Exercises
Chapter IV. Critical Values of Smooth Maps
1 *Statement of the Problem and the Main Result
1.1 Critical Points and Critical Values
1.2 The Main Theorem
1.3 Critical Values of c1 -Maps
2 *Well-Positioned Manifolds
2.1 The Main Definition
2.2 Two Lemmas About Increments
3 *Morse’s Theorem on t-Representations
3.1 Preliminaries
3.2 t-Representation
3.3 The Existence of a t-Representation
4 *The Main Results
4.1 Proof of the Main Theorem
4.2 Generalization to Lipschitz Maps
5 *The Sharpness of the Conditions in the Main Theorem
5.1 Preliminaries
5.2 Cantor-Like Sets
5.3 Constructing an Auxiliary Function
5.4 The Auxiliary Function (Continued)
5.5 Counterexamples: Functions of Several Variables
5.6 Counterexamples: Maps in the Case n > m
5.7 Counterexamples: Maps in the Case m > n
6 *Whitney’s Example
6.1 Statement of the Problem and the Main Result
6.2 The Sets ca2
6.3 Constructing the Arc
6.4 Constructing the Function
6.5 Proving Inequality (4)
6.6 Generalization to the Case of Several Variables
Chapter V. Addenda
1 *Smooth Partitions of Unity
1.1 Auxiliary Inequalities
1.2 The Partition of Unity Theorem
2 *Covering Theorems
2.1 A Preliminary Theorem
2.2 The Vitali Covering Theorem
2.3 Density Points
3 *Hausdorff Measures and Hausdorff Dimension
3.1 Outer Measures
3.2 The Construction of the Hausdorff Measures
3.3 The Basic Properties of the Hausdorff Measures
3.4 Relationship With the Lebesgue Measure
3.5 The Hausdorff Dimension
3.6 The Hausdorff Measure and Hausdorff Dimension of Cantor-Like Sets
3.7 The Cantor Function Corresponding to the Set Ca
4 *Comparing the Measures up x uq and up+q
4.1 The Upper Bound
4.2 Estimating up x uq From Above
4.3 Estimating up x uq From Below
4.4 Standard Sets
4.5 The Hausdorff Dimension of a Cartesian Product
5 *Estimates for Smooth Maps Related to the Hausdorff and Lebesgue Measures
5.1 Estimating the Hausdorff Measure of an Image
5.2 On Images of Zero Measure
5.3 A Refined Estimate of the Increment
References
Index