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An infinitesimal proof of the implicit function theorem

Published online by Cambridge University Press:  18 May 2009

Nigel J. Cutland  and
Feng Hanqiao
Nigel J. Cutland
Affiliation:
Department of Pure MathematicsUniversity of HullHull HU6 7RXEngland
Feng Hanqiao
Affiliation:
Department of Computer ScienceShaanxi UniversityXian, 710062China
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We give a short and constructive proof of the general (multi-dimensional) Implicit Function Theorem (IFT), using infinitesimal (i.e. nonstandard) methods to implement our basic intuition about the result. Here is the statement of the IFT, quoted from [4];

Theorem. Let A ⊂ ℝn × ℝmbe an open set and let F:A → ℝ be a function of class Cp (p≥1). Suppose that (xO, yO) ε A with F(xO, yO) = 0 (xO ε ℝn, yO ε ℝm) and that the Jacobian determinantis not zero at (xO, yO). Then there is an open neighbourhood U of xO and a unique function f:U→ ℝmwith

F(x, f(x)) = 0

for all x ε U. Moreover, f is of class Cp.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 163 - 166
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

REFERENCES

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