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2003 | Buch

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The Implicit Function Theorem

History, Theory, and Applications

verfasst von: Steven G. Krantz, Harold R. Parks

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

The implicit function theorem is part of the bedrock of mathematical 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 differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for C^k functions, (ii) formulations in other function spaces, (iii) formulations for non- smooth functions, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash--Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve. "The Implicit Function Theorem" is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas.

Inhaltsverzeichnis

Frontmatter

1. Introduction to the Implicit Function Theorem

Abstract

To the beginning student of calculus, a function is given by an analytic expression such as

$$ f(x) = {x^3} + 2{x^2} - x - 3 $$

(1.1)

$$ g(y) = \sqrt {{{y^2} + 1}} $$

(1.2)

$$ h(t) = \cos \left( {2\pi t} \right) $$

(1.3)

Steven G. Krantz, Harold R. Parks

2. History

Abstract

The earliest works on algebra beginning with Al-jabr w’al muqâbala by Mohammed ben Musa Al-Khowârizmî (circa A.D. 825), from whence we get the word “algebra” (and the word “algorithm”), presented problems and solutions by numerical example. The notion of a “function,” whether explicit or implicit, would make no sense in such a context. It was not until about 1600 that the idea of using letters to denote both unknowns and coefficients was introduced by François Viéte (1540–1603). The algebraic methods of Viète were taken up by René Descartes (1596–1650) and combined with Descartes’s own coordinate system inspiration. That fundamental advance in 1637 finally brought mathematics to the point that the notion of a function could make sense.

Steven G. Krantz, Harold R. Parks

3. Basic Ideas

Abstract

In order to make this book a convenient reference, we shall endeavor to make it locally self-contained. With this thought in mind, we shall begin by presenting a very classical treatment of the implicit function theorem in Euclidean space.

Steven G. Krantz, Harold R. Parks

4. Applications

Abstract

There is a strong connection between the implicit function theorem and the theory of differential equations. This is true even from the historical point of view, for Pi-card’s iterative proof of the existence theorem for ordinary differential equations inspired Goursat to give an iterative proof of the implicit function theorem (see Goursat [Go 03]). In the mid-twentieth century, John Nash pioneered the use of a sophisticated form of the implicit function theorem in the study of partial differential equations. We will discuss Nash’s work in Section 6.4. In this section, we limit our attention to ordinary (rather than partial) differential equations because the technical details are then so much simpler. Our plan is first to show how a theorem on the existence of solutions to ordinary differential equations can be used to prove the implicit function theorem. Then we will go the other way by using a form of the implicit function theorem to prove an existence theorem for differential equations.

Steven G. Krantz, Harold R. Parks

5. Variations and Generalizations

Abstract

In Section 2.2, we described the method Newton devised for examining the local behavior of the locus of points satisfying a polynomial equation in two variables. It is easy to extend that method to the locus of an equation of the form

$$ {z^m} + {a_{{m - 1}}}(w){z^{{m - 1}}} + {a_{{m - 2}}}(w){z^{{m - 2}}} + \cdots + {a_0}(w) = 0 $$

(5.1)

, where each a_i(w) is a holomorphic function of $ w \in \mathbb{C} $ that vanishes at $ w = 0 $. Such an extension is significant, because the Weierstrass preparation theorem will show us that the behavior near (0, 0) of the locus of an equation of the form given in (5.1) is completely representative of the local behavior of the locus of points satisfying $ F\left( {w,z} \right) = 0 $, where F is holomorphic.

Steven G. Krantz, Harold R. Parks

6. Advanced Implicit Function Theorems

Abstract

We will now consider implicit function theorems in both the real analytic and the complex analytic (holomorphic) categories. These are obviously closely related, as the problem in the real analytic category can be complexified (by replacing every x^j with a z ^j) and thereby turned into a holomorphic problem. Conversely, any complex analytic implicit function theorem situation is a fortiori real analytic and can therefore be treated with real analytic techniques. And both categories are subcategories of the C^∞ category.

Steven G. Krantz, Harold R. Parks

Backmatter

Titel: The Implicit Function Theorem
verfasst von: Steven G. Krantz
Harold R. Parks
Verlag: Birkhäuser Boston
Electronic ISBN: 978-1-4612-0059-8
Print ISBN: 978-1-4612-6593-1
DOI: https://doi.org/10.1007/978-1-4612-0059-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction to the Implicit Function Theorem

2. History

3. Basic Ideas

4. Applications

5. Variations and Generalizations

6. Advanced Implicit Function Theorems

Backmatter