Preface
1 IntroductIon to the Implicit Function Theorem
1.1 Implicit Functions
1.2 An Informal Version ofthe Implicit Function Theorem
1.3 Thelmplicit Function Theorem Paradigm

2 History
2.1 Historicallntroduction
2.2 Newton
2.3 Lagrange
2.4 Cauchy

3 Basfcldeas
3.1 Introduction
3.2 The Inductive Proof of the Implicit Function Theorem

Preface
1 IntroductIon to the Implicit Function Theorem
1.1 Implicit Functions
1.2 An Informal Version ofthe Implicit Function Theorem
1.3 Thelmplicit Function Theorem Paradigm

2 History
2.1 Historicallntroduction
2.2 Newton
2.3 Lagrange
2.4 Cauchy

3 Basfcldeas
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

4 Applications
4.1 Ordinary Differential Equations
4.2 Numerical Homotopy Methods
4.3 Equivalent Definitions of a Smooth Surface
4.4 Smoothncss ofthc Distance Function

5 VariatIons and Genera Hzations
5.1 The Weicrstrass Preparation Theorem
5.2 ImplicU Function Theorems without Differenriability
5.3 An Inverse Function Theorcm for Continuous Mappings
5.4 Some Singular Cases of the Implicit Function Theorem

6 Advanced Impllclt Functlon Theorems
6.1 Analyticlmplicit Function Theorems
6.2 Hadamard's Globallnverse Function Thecntm
6.3 The Implicit Function Theorem via the Newton-RaphsonMethod
6.4 The Nash-Moscrlmplicit Function Theorem
6.4.1 Introductory Remarks
6.4.2 Enunciation of the Nash-MoserThcorem
6.4.3 First Step of the ProofofNash-Moscr
6.4.4 The Crux ofthe Matter
6.4.5 Construction ofthe Smoothing Operators
6.4.6 A UsefulCorollary
Glossary
Bibliography
Index