隐函数定理

　　The implicit function theorem is. along with its close cousin
the inverse func- tion theorem， one of the most important， and one
of the oldest， paradigms in modcrn mathemarics. One can see the
germ of the idea for the implicir func tion theorem in the writings
of Isaac Newton （1642-1727）， and Gottfried Leib-niz’s （1646-1716）
work cxplicitty contains an instance of implicit
differentiation.
　　Whilc Joseph Louis Lagrange （1736-1813） found a theorcm that is
essentially a version of the inverse function theorem， ic was
Augustin-Louis Cauchy （1789-1857） who approached the implicit
function theorem with mathematical rigor and it is he who is
gencrally acknowledgcd as the discovcrer of the theorem. In
Chap-ter 2， we will give details of the contributions of Newton，
Lagrange， and Cauchy to the development of the implicit function
theorem.

Preface
1 IntroductIon to the Implicit Function Theorem
1.1 Implicit Functions
1.2 An Informal Version ofthe Implicit Function Theorem
1.3 Thelmplicit Function Theorem Paradigm

2 History
2.1 Historicallntroduction
2.2 Newton
2.3 Lagrange
2.4 Cauchy

3 Basfcldeas
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 Smoothncss ofthc Distance Function

5 VariatIons and Genera Hzations
5.1 The Weicrstrass Preparation Theorem
5.2 ImplicU Function Theorems without Differenriability
5.3 An Inverse Function Theorcm for Continuous Mappings
5.4 Some Singular Cases of the Implicit Function Theorem

6 Advanced Impllclt Functlon Theorems
6.1 Analyticlmplicit Function Theorems
6.2 Hadamard’s Globallnverse Function Thecntm
6.3 The Implicit Function Theorem via the Newton-Raphson
Method
6.4 The Nash-Moscrlmplicit Function Theorem
6.4.1 Introductory Remarks
6.4.2 Enunciation of the Nash-MoserThcorem
6.4.3 First Step of the ProofofNash-Moscr
6.4.4 The Crux ofthe Matter
6.4.5 Construction ofthe Smoothing Operators
6.4.6 A UsefulCorollary
Glossary
Bibliography
Index