An infinitesimal proof of the implicit function theorem
Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 163-166
Voir la notice de l'article provenant de la source Cambridge University Press
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→ RmwithF(x, f(x)) = 0for all x ε U. Moreover, f is of class Cp.
Cutland, Nigel J.; Hanqiao, Feng. An infinitesimal proof of the implicit function theorem. Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 163-166. doi: 10.1017/S001708950000971X
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author = {Cutland, Nigel J. and Hanqiao, Feng},
title = {An infinitesimal proof of the implicit function theorem},
journal = {Glasgow mathematical journal},
pages = {163--166},
year = {1993},
volume = {35},
number = {2},
doi = {10.1017/S001708950000971X},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S001708950000971X/}
}
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