The Implicit Function Theorem

by Krantz, Steven G.; Parks, Harold R.

• ISBN13: 9780817642853
• ISBN10: 0817642854
• Format: Hardcover
• Copyright: 2002-06-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	ix
Introduction to the Implicit Function Theorem	1	(12)
Implicit Functions	1	(2)
An Informal Version of the Implicit Function Theorem	3	(4)
The Implicit Function Theorem Paradigm	7	(6)
History	13	(22)
Historical Introduction	13	(2)
Newton	15	(5)
Lagrange	20	(7)
Cauchy	27	(8)
Basic Ideas	35	(26)
Introduction	35	(1)
The Inductive Proof of the Implicit Function Theorem	36	(5)
The Classical Approach to the Implicit Function Theorem	41	(7)
The Contraction Mapping Fixed Point Principle	48	(4)
The Rank Theorem and the Decomposition Theorem	52	(6)
A Counterexample	58	(3)
Applications	61	(32)
Ordinary Differential Equations	61	(4)
Numerical Homotopy Methods	65	(8)
Equivalent Definitions of a Smooth Surface	73	(5)
Smoothness of the Distance Function	78	(15)
Variations and Generalizations	93	(24)
The Weierstriss Preparation Theorem	93	(6)
Implicit Function Theorems without Differentiability	99	(2)
An Inverse Function Theorem for Continuous Mappings	101	(6)
Some Singular Cases of the Implicit Function Theorem	107	(10)
Advanced Implicit Function Theorems	117	(28)
Analytic Implicit Function Theorems	117	(4)
Hadamard's Global Inverse Function Theorem	121	(8)
The Implicit Function Theorem via the Newton-Raphson Method	129	(5)
The Nash-Moser Implicit Function Theorem	134	(11)
Introductory Remarks	134	(1)
Enunciation of the Nash-Moser Theorem	135	(1)
First Step of the Proof of Nash-Moser	136	(2)
The Crux of the Matter	138	(3)
Construction of the Smoothing Operators	141	(3)
A Useful Corollary	144	(1)
Glossary	145	(6)
Bibliography	151	(10)
Index	161