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The Implicit Function Theorem

Part of the Modern Birkhäuser Classics Book Series

DOI: 10.1007/978-1-4614-5981-1

ISBN: 978-1-4614-5980-4

Publisher: Springer New York - Springer Journals

Oct 21, 2012

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Chapter 1Introduction to the Implicit Function Theorem

Krantz, Steven G.; Parks, Harold R.

Pages: 1 –12

[To the beginning student of calculus, a function is given by an analytic expression such as 1.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
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Chapter 2History

Krantz, Steven G.; Parks, Harold R.

Pages: 13 –33

[The earliest works on algebra beginning with Al-jabr w’al muqâbala by Mohammed ben Musa AI-Khowârîzml (circa A.D. 825), from whence we get the word “algebra” (and the word “algorithm”), presented problems and solutions by numerical example.]
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Chapter 3Basic Ideas

Krantz, Steven G.; Parks, Harold R.

Pages: 35 –59

[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.]
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Chapter 4Applications

Krantz, Steven G.; Parks, Harold R.

Pages: 61 –91

[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 Picard's iterative proof of the existence theorem for ordinary differential equations inspired Goursat to give an iterative proof...
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Chapter 5Variations and Generalizations

Krantz, Steven G.; Parks, Harold R.

Pages: 93 –115

[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.]
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Chapter 6Advanced Implicit Function Theorems

Krantz, Steven G.; Parks, Harold R.

Pages: 117 –144

[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 xi with a zj) and thereby turned into a...

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