The implicit function theorem, part of the bedrock of mathematical analysis and geometry, has important implications in the theories of partial differential equations, differential geometry, and geometric analysis. Its history is lively and complex, 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 will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics.

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.

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.- 6.4.1 Introductory Remarks.- 6.4.2 Enunciation of the Nash-Moser Theorem.- 6.4.3 First Step of the Proof of Nash-Moser.- 6.4.4 The Crux of the Matter.- 6.4.5 Construction of the Smoothing Operators.- 6.4.6 A Useful Corollary.