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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.
Reviews / Votes
"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
"This unique and worthwhile book deserves an audience ranging from lower-division undergraduate calculus students through graduate students and faculty. "
-CHOICE
"For the analyst who uses the Implicit Function Theorem or the instructor who teaches elementary or more advanced variants of it, this book is extremely useful.. The presentation is nice and fluent, and the book is accessible even to undergraduate students with a minimum of background in calculus."
-JOURNAL OF OPERATOR THEORY
"This is an excellent book devoted to the implicit function theorem and related results (like the inverse function theorem) that play one of the most important roles in modern mathematics...The book is mainly self-contained and undoubtedly will serve as a useful resource for advanced undergraduates, graduate students, professional mathematicians, and scientists of other types. The bibiography is extensive, including references to various topics related to the implicit function theorem and its generalizations, both those which are considered and those which are not considered in the book."
---MathSciNet
"The authors collect in this book many variants of the Implicit Function Theorem and various methods of the proof. They emphasize the IFT as a powerful tool in many branches of mathematics."
---Applications of Mathematics
"This small book consists of six chapters and is entirely devoted to one of the most important resultsin analysis - the implicit function theorem and its variations. The book starts with historical comments on the evolution of ideas leading to the theorem. . .The book will appeal to a large part of the mathematical community since everybody will find there some far going generalizations of classical results in a very readable form."
---EMS Newsletter
Language
Place of publication
Publishing group
Illustrations
File size
ISBN-13
978-1-4612-0059-8 (9781461200598)
DOI
10.1007/978-1-4612-0059-8
Schweitzer Classification
1 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.- 2 History.- 2.1 Historical Introduction.- 2.2 Newton.- 2.3 Lagrange.- 2.4 Cauchy.- 3 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.- 4 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.- 5 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.- 6 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.
