Implicit function theorem : history, theory, and applications

✍ Scribed by Steven G Krantz; Harold R Parks

• Publisher: Birkhauser
• Year: 2013
• Tongue: English
• Leaves: 173
• Series: Modern Birkhäuser classics
• Category: Library

✦ Synopsis

The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for Ck functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth functions, and (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash-Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present uncorrected reprint of this classic monograph.Originally published in 2002, The Implicit Function Theorem is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas. Read more... Introduction to the implicit function theorem -- History -- Basic ideas -- Applications -- Variations and generalizations -- Advanced implicit function theorems

✦ Table of Contents

Cover......Page 1
The Implicit Function Theorem......Page 4
Contents......Page 10
Preface......Page 12
1.1 Implicit Functions......Page 16
Example 1.1.1......Page 17
1.2 An Informal Version of the Implicit Function Theorem......Page 18
Example 1.2.1......Page 20
Example 1.2.3......Page 21
1.3 The Implicit Function Theorem Paradigm......Page 22
Theorem 1.3.1......Page 23
Example 1.3.2......Page 24
Theorem 1.3.3......Page 25
Corollary 1.3.5......Page 26
Remark 1.3.7......Page 27
2.1 Historical Introduction......Page 28
Example 2.2.1......Page 30
Remark 2.2.2......Page 34
2.3 Lagrange......Page 35
Theorem 2.3.1......Page 37
Lemma 2.3.3......Page 38
Lemma 2.3.4......Page 39
Theorem 2.3.5......Page 40
Theorem 2.4.1......Page 42
Remark 2.4.2......Page 43
Theorem 2.4.4......Page 44
Theorem 2.4.6......Page 47
3.1 Introduction......Page 49
Theorem 3.2.1......Page 50
Notation 3.2.3......Page 52
Theorem 3.2.4......Page 53
3.3 The Classical Approach to the Implicit Function Theorem......Page 55
Theorem 3.3.2......Page 57
Proof that the Inverse Function Theorem Implies the Implicit Function Theorem......Page 58
Proof of the Inverse Function Theorem......Page 59
Theorem 3.4.1......Page 62
Proposition 3.4.2......Page 63
Theorem 3.4.6......Page 64
Theorem 3.4.10......Page 65
3.5 The Rank Theorem and the Decomposition Theorem......Page 66
Theorem 3.5.1......Page 67
Lemma 3.5.2......Page 68
Theorem 3.5.4......Page 70
Example 3.6.1......Page 72
Theorem 4.1.1......Page 74
Theorem 4.1.3......Page 75
Remark 4.1.4......Page 76
4.2 Numerical Homotopy Methods......Page 78
Theorem 4.2.1......Page 79
Example 4.2.2......Page 81
Theorem 4.2.3......Page 82
Remark 4.2.4......Page 83
Definition 4.2.6......Page 84
Theorem 4.2.8......Page 85
Theorem 4.3.1......Page 86
Definition 4.3.2......Page 87
4.4 Smoothness of the Distance Function......Page 91
Lemma 4.4.3......Page 92
Lemma 4.4.4......Page 93
The Distance to a Submanifold......Page 94
Lemma 4.4.7......Page 95
Definition 4.4.8......Page 96
Theorem 4.4.10......Page 97
Example 4.4.12......Page 98
Lemma 4.4.13......Page 99
Remark 4.4.16......Page 101
Theorem 4.4.17......Page 103
5.1 The Weierstrass Preparation Theorem......Page 105
Example 5.1.1......Page 106
Lemma 5.1.4......Page 108
Remark 5.1.5......Page 110
Theorem 5.2.2......Page 111
Theorem 5.2.4......Page 112
Remark 5.3.2......Page 113
Example 5.3.5......Page 114
Example 5.3.7......Page 115
Remark 5.3.9......Page 116
Example 5.3.12......Page 117
Example 5.3.13......Page 118
Preliminary Remarks......Page 119
The Case of Jacobian Matrix of Rank 1......Page 120
Case III......Page 123
Case IV......Page 124
Example 5.4.1......Page 125
Example 5.4.3......Page 126
6.1 Analytic Implicit Function Theorems......Page 128
Theorem 6.1.2......Page 129
6.2 Hadamard's Global Inverse Function Theorem......Page 132
Lemma 6.2.1......Page 133
Property 1......Page 134
Definition 6.2.2......Page 135
Theorem 6.2.4......Page 136
Remark 6.2.6......Page 137
Theorem 6.2.8......Page 138
Theorem 6.2.9......Page 139
Definition 6.3.2......Page 140
Theorem 6.3.3......Page 141
Theorem 6.3.5......Page 143
Theorem 6.3.6......Page 144
6.4.1 Introductory Remarks......Page 145
Theorem 6.4.2......Page 146
Proposition 6.4.3......Page 147
Remark 6.4.4......Page 148
6.4.4 The Crux of the Matter......Page 149
6.4.5 Construction ofthe Smoothing Operators......Page 152
Conclusion......Page 155
Glossary......Page 156
Bibliography......Page 162
Index......Page 171