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:
Steven G. Krantz is one of Springer's and Birkh?user's most prolific and popular authors in the field of functional analysis, geometric analysis, and partial differential equations. Krantz is the series editor of Birkh?user's ' Cornerstones' graduate text series and the founder and editor-in-chief of The Journal of Geometric Analysis, considered a society journal previously published by the AMS and often acts as an advisor to several senior editors at Springer/ Birkh?user . He is also editor-in-chief of the Journal of Mathematical Analysis and Applications. Professor Krantz is currently the editor-in-chief of the AMS Notices and also edits for The American Mathematical Monthly, Complex Analysis and Elliptical Equations, and The Bulletin of the American Mathematical Monthly. Krantz is also known for his wide breadth of expertise in several areas of mathematics such as harmonica analysis, differential geometry, and Lie groups, to name a few. Notable awards include Chauvenet Prize (1992), Beckenbach Book Award (1994), Kemper Prize (1994), Outstanding Academic Book Award (1998), Washington University Faculty Mentor Award (2007)....

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?