The Implicit Function Theorem: History, Theory, and Applications

by Steven G. Krantz, Harold R. Parks

1 Introduction to the Implicit Function Theorem.- 1.1 Implicit Functions.- 1.2 An Informal Version of the Implicit Function Theorem.- 1.3 The Implicit Function Theorem Paradigm.- 2 History.- 2.1 Historical Introduction.- 2.2 Newton.- 2.3 Lagrange.- 2.4 Cauchy.- 3 Basic Ideas.- 3.1 Introduction.- 3.2 The Inductive Proof of the Implicit Function Theorem.- 3.3 The Classical Approach to the Implicit Function Theorem.- 3.4 The Contraction Mapping Fixed Point Principle.- 3.5 The Rank Theorem and the Decomposition Theorem.- 3.6 A Counterexample.- 4 Applications.- 4.1 Ordinary Differential Equations.- 4.2 Numerical Homotopy Methods.- 4.3 Equivalent Definitions of a Smooth Surface.- 4.4 Smoothness of the Distance Function.- 5 Variations and Generalizations.- 5.1 The Weierstrass Preparation Theorem.- 5.2 Implicit Function Theorems without Differentiability.- 5.3 An Inverse Function Theorem for Continuous Mappings.- 5.4 Some Singular Cases of the Implicit Function Theorem.- 6 Advanced Implicit Function Theorems.- 6.1 Analytic Implicit Function Theorems.- 6.2 Hadamard's Global Inverse Function Theorem.- 6.3 The Implicit Function Theorem via the Newton-Raphson Method.- 6.4 The Nash-Moser Implicit Function Theorem.- 6.4.1 Introductory Remarks.- 6.4.2 Enunciation of the Nash-Moser Theorem.- 6.4.3 First Step of the Proof of Nash-Moser.- 6.4.4 The Crux of the Matter.- 6.4.5 Construction of the Smoothing Operators.- 6.4.6 A Useful Corollary.

Genres: Mathematics

180 pages, Paperback

First published April 5, 2002