The Implicit Function Theorem History, Theory, and Applications
The implicit function theorem is part of the bedrock of mathematics 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 differen...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Boston, MA
Birkhäuser Boston
2003, 2003
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| Edition: | 1st ed. 2003 |
| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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