Author | Krantz, Steven G. author |
---|---|
Title | The Implicit Function Theorem [electronic resource] : History, Theory, and Applications / by Steven G. Krantz, Harold R. Parks |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2003 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0059-8 |
Descript | XI, 163 p. online resource |
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 Hadamardโs Global Inverse Function Theorem -- 6.3 The Implicit Function Theorem via the Newton-Raphson Method -- 6.4 The Nash-Moser Implicit Function Theorem