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The Implicit Function Theorem in the Scalar Case*

Published online by Cambridge University Press:  20 November 2018

H. I. Freedman
H. I. Freedman*
Affiliation:
University of Alberta, Edmonton
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The implicit function theorem has applications at all levels of mathematics from elementary calculus (implicit differentiation) to finding periodic solutions of systems of differential equations ([1, Chapter 14] and [4], for example).

In 1961 W. S. Loud [3] studied the case of two equations in three unknowns. He considered only cases where up to third order derivatives were involved and only those cases where the derivative of the solutions at the critical point existed. Coddington and Levinson [1] consider a specific singular case involving n equations in n + m unknowns. In general the number of distinct critical cases involving up to third derivatives for such a general system is not known.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 6 , December 1969 , pp. 721 - 732
Copyright
Copyright © Canadian Mathematical Society 1969

Footnotes

*

The preparation of this paper was partially supported by the National Research Council of Canada Grant No. A-4823.

References

1. Coddington, and Levinson, . Theory of ordinary differential equations. (McGraw-Hill, 1955).Google Scholar
2. Goursat, E., Cours d′analyse mathematique, Vol. 1. (Gauthier-Villars, 1910).Google Scholar
3. Loud, W.S., Some singular cases of the implicit function theorem. Amer. Math. Monthly 68 (1961) 965–977.Google Scholar
4. Loud, W. S., Periodic solutions of perturbed second-order autonomous equations. Memoirs of the Amer. Math. Soc. 47 (1964).Google Scholar
5. Rudin, W., Principles of mathematical analysis. (McGraw-Hill, 1953).Google Scholar