# The Implicit Function Theorem: History, Theory, and Applications

Springer Science & Business Media, 2002 - 163 Seiten
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 C^k functions, (ii) formulations in other function spaces, (iii) formulations for non- smooth functions, (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 volume. The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve. "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.

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### Inhalt

 Preface ix Introduction to the Implicit Function Theorem 1 12 An Informal Version of the Implicit Function Theorem 3 13 The Implicit Function Theorem Paradigm 7 History 13 22 Newton 15 23 Lagrange 20 24 Cauchy 27
 Variations and Generalizations 93 52 Implicit Function Theorems without Differentiability 99 53 An Inverse Function Theorem for Continuous Mappings 101 54 Some Singular Cases of the Implicit Function Theorem 107 Advanced Implicit Function Theorems 117 62 Hadamards Global Inverse Function Theorem 121 63 The Implicit Function Theorem via the NewtonRaphson Method 129 64 The NashMoser Implicit Function Theorem 134

 Basic Ideas 35 32 The Inductive Proof of the Implicit Function Theorem 36 33 The Classical Approach to the Implicit Function Theorem 41 34 The Contraction Mapping Fixed Point Principle 48 35 The Rank Theorem and the Decomposition Theorem 52 36 A Counterexample 58 Applications 61 42 Numerical Homotopy Methods 65 43 Equivalent Definitions of a Smooth Surface 73 44 Smoothness of the Distance Function 78
 642 Enunciation of the NashMoser Theorem 135 643 First Step of the Proof of NashMoser 136 644 The Crux of the Matter 138 645 Construction of the Smoothing Operators 141 646 A Useful Corollary 144 Glossary 145 Bibliography 151 Index 161 Urheberrecht

### Beliebte Passagen

Seite 155 - Graves, Implicit functions and their differentials in general analysis. Transactions of the American Mathematical Society, vol.
Seite 157 - WF Osgood, A Jordan curve of positive area, Transactions of the American Mathematical Society, Vol. 4 1903, S.

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