Preface
Introduction to the Implicit Function Theorem
1.1 Implicit Functions
1.2 An Informal Version of the Implicit Function Theorem
1.3 The Implicit Function Theorem Paradigm
History
2.1 Historical Introduction
2.2 Newton
2.3 Lagrange
2.4 Cauchy
Basic Ideas
3.1 Introduction
3.2 The Inductive Proof of the Implicit Function Theorem
3.3 The Classical Approach to the Implicit Function Theorem
3.4 The Contraction Mapping Fixed Point Principle
3.5 The Rank Theorem and the Decomposition Theorem
3.6 A Counterexample
Applications
4.1 Ordinary Differential Equations
4.2 Numerical Homotopy Methods
4.3 Equivalent Definitions of a Smooth Surface
4.4 Smoothness of the Distance Function
Variations and Generalizations
5.1 The Weierstrass Preparation Theorem
5.2 Implicit Function Theorems without Differentiability
5.3 An Inverse Function Theorem for Continuous Mappings
5.4 Some Singular Cases of the Implicit Function Theorem
Advanced Implicit Function Theorems
6.1 Analytic Implicit Function Theorems
6.2 Hadamard's Global Inverse Function Theorem
6.3 The Implicit Function Theorem via the Newton-Raphson Method
6.4 The Nash-Moser Implicit Function Theorem
6.4.1 Introductory Remarks
6.4.2 Enunciation of the Nash-Moser Theorem
6.4.3 First Step of the Proof of Nash-Moser
6.4.4 The Crux of the Matter
6.4.5 Construction of the Smoothing Operators
6.4.6 A Useful Corollary
Glossary
Bibliography
Index
· · · · · · (收起)