隐函数定理

《隐函数定理(英文)》内容包括：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 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 History 2.1 Historical Introduction 2.2 Newton 2.3 Lagrange 2.4 Cauchy 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 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 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 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 Glossary Bibliography Index