隐函数定理
2012-9
世界图书出版公司
（美）克朗兹 著
163
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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.
作者：（美国）克朗兹（Steven G.Krantz） （美国）Harold R.Parks
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
版权页： 插图： The picture we would like to see for the curve (t(s), x(s)) along which (4.10) holds should resemble that in Figure 4. l (a). It would be even better if the curve resembled that in Figure 4. I(b), because in that case we could parameterize the curve by t itself. On the other hand, it is conceivable that the solution set of H(t, x) = 0 might look like that in Figure 4.2 where, starting from a zero of the form H (0, x0),we can never arrive at a zero of the form H(I, xl ). Notice that there are four types of bad behavior for {(t, x) : H(t, x) = 0} in Figure 4.2: (I) A curve starts at t=0, but doubles back without ever getting to t = I, (2) a curve becomes unbounded in x, (3) a curve reaches a bifurcation point where curves cross, and (4) a curve comes to a dead end where it cannot be continued. All of these instances of bad behavior are possible; nonetheless they all can be ruled out by imposing some simple hypotheses and applying the implicit function theorem. To illustrate the ideas, we first state a theorem in which we can show that the curve H(t(s), x(s)) = 0 has the nice form shown in Figure 4. 1(b). Theorem 4.2,1 Let U be an open subset of RN. Suppose that H is continuously differentiable in an open set containing [0, I] x U, that the function Fo given by F0(x)=H(0, x) ning with the 1764 award given by the Paris Academy of Sciences for his paper on the libration of the moon.4A basic result in celestial mechanics is Kepler's equation E = M + esin(E), (2.15) where M is the mean anomaly,5 E is the eccentric anomaly, and e is the eccen-tricity of the orbit. We will describe these quantities in more detail later. For the moment, we note that M and e should be considered to be the quantities that can be measured and that e is assumed to be small. One of Lagrange's theorems, now called the Lagrange Inversion Theorem, gave a formula for the correction that must be made when, for some function ψ(.), ψ(M) is replaced by ψ(E).
《隐函数定理(英文)》介绍了隐函数定理的基本知识，是全英文版。在数学中，隐函数定理是一个描述关系以隐函数表示的某些变量之间是否存在显式关系的定理。
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隐函数定理在数学各个分支中还是很重要的, 值得拿来研读下