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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# General theorem for Functional dependence

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**