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1 Introduction to the Implicit Function Theorem 1

1.1 Implicit Functions 1

1.2 An Informal Version of the Implicit Function Theorem 3

1.3 The Implicit Function Theorem Paradigm 7

2 History 13

2.1 Historical Introduction 13

2.2 Newton 15

2.3 Lagrange 20

2.4 Cauchy 27

3 Basic Ideas 35

3.1 Introduction 35

3.2 The Inductive Proof of the Implicit Function Theorem 36

3.3 The Classical Approach to the Implicit Function Theorem 41

3.4 The Contraction Mapping Fixed Point Principle 48

3.5 The Rank Theorem and the Decomposition Theorem 52

3.6 A Counterexample 58

4 Applications 61

4.1 Ordinary Differential Equations 61

4.2 Numerical Homotopy Methods 65

4.3 Equivalent Definitions of a Smooth Surface 73

4.4 Smoothness of the Distance Function 78

5 Variations and Generalizations 93

5.1 The Weierstrass Preparation Theorem 93

5.2 Implicit Function Theorems without Differentiability 99

5.3 An Inverse Function Theorem for Continuous Mappings 101

5.4 Some Singular Cases of the Implicit Function Theorem 107

6 Advanced Implicit Function Theorems 117

6.1 Analytic Implicit Function Theorems 117

6.2 Hadamard's Global Inverse Function Theorem 121

6.3 The Implicit Function Theorem via the Newton-Raphson Method 129

6.4 The Nash-Moser Implicit Function Theorem 134

6.4.1 Introductory Remarks 134

6.4.2 Enunciation of the Nash-Moser Theorem 135

6.4.3 First Step of the Proof of Nash-Moor 136

6.4.4 The Crux of the Matter 138

6.4.5 Construction of the Smoothing Operators 141

6.4.6 A Useful Corollary 144

Glossary 145

Bibliography 151

Index 161