suppose Fi (a1, a2, ... an) , 0 < i <= k. a1, ..., an are reals(adsbygoogle = window.adsbygoogle || []).push({});

Then the Frechet derivative DF is a k x n matrix. If rank(DF) = k , does it still suggest functional dependence amonst Fi 's ?

Also, when rank(DF) < n (number of independent variables) , what does it signify ?

The theorem I had in mind was :

let f = (f1, ..., fn) be a C1 map from a connected open set U in Rm into Rn.

Suppose Df has rank k at every x in U, where k < n.

Then every x in U has a neighborhood N s.t. f1, ..., fn are func.dep on N,

and f(N) is a smooth k-dim submanifold of Rn.

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# Homework Help: General theorem for Functional dependence

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