The Implicit Function Theorem - Harold R. Parks

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Résumé :
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 function, and (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 uncorrected reprint of this classic?monograph. ?? ? Originally published in 2002,?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.?

Biographie:
Peter V. Dovbush is an Associate Professor and Leading Researcher at the Institute of Mathematics and Computer Science of Moldova State University, Moldova. His research interests lie on the geometric theory of functions of several complex variables. Steven G. Krantz earned his B.A. from the University of California at Santa Cruz (1971) and his PhD from Princeton University, USA (1974). With teaching periods at UCLA, Princeton, Penn State, and Washington University in St. Louis, he chaired the latter's mathematics department for five years. Dr. Krantz has authored or co-authored over 160 books and 350 scholarly papers, and he has edited numerous journals. His contributions to mathematics have earned him awards such as the Chauvenet Prize, the Beckenbach Book Award, and the Kemper Prize....

Sommaire:
?Preface.- Introduction to the Implicit Function Theorem.- History.- Basic Ideas.- Applications.- Variations and Generalizations.- Advanced Implicit Function Theorems.- Glossary.- Bibliography.- Index.

The authors offer a useful and fascinating manuscript which should be of interest and useful to?graduate and postgraduate students, professional mathematicians, teachers as well as researchers. Most of them will be able to find all fundamental ideas in the field and simple and transparent proofs of all main implicit function theorems and theorems closely related to these.

-Zentralblatt MATH?