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1 Introduction to the Implicit Function Theorem1

1.1 Implicit Functions1

1.2 An Informal Version of the Implicit Function Theorem3

1.3 The Implicit Function Theorem Paradigm7

2 History13

2.1 Historical Introduction13

2.2 Newton15

2.3 Lagrange20

2.4 Cauchy27

3 Basic Ideas35

3.1 Introduction35

3.2 The Inductive Proof of the Implicit Function Theorem36

3.3 The Classical Approach to the Implicit Function Theorem41

3.4 The Contraction Mapping Fixed Point Principle48

3.5 The Rank Theorem and the Decomposition Theorem52

3.6 A Counterexample58

4 Applications61

4.1 Ordinary Differential Equations61

4.2 Numerical Homotopy Methods65

4.3 Equivalent Definitions of a Smooth Surface73

4.4 Smoothness of the Distance Function78

5 Variations and Generalizations93

5.1 The Weierstrass Preparation Theorem93

5.2 Implicit Function Theorems without Differentiability99

5.3 An Inverse Function Theorem for Continuous Mappings101

5.4 Some Singular Cases of the Implicit Function Theorem107

6 Advanced Implicit Function Theorems117

6.1 Analytic Implicit Function Theorems117

6.2 Hadamard's Global Inverse Function Theorem121

6.3 The Implicit Function Theorem via the Newton-Raphson Method129

6.4 The Nash-Moser Implicit Function Theorem134

6.4.1 Introductory Remarks134

6.4.2 Enunciation of the Nash-Moser Theorem135

6.4.3 First Step of the Proof of Nash-Moor136

6.4.4 The Crux of the Matter138

6.4.5 Construction of the Smoothing Operators141

6.4.6 A Useful Corollary144

Glossary145

Bibliography151

Index161