General theorem for Functional dependence

[chy1013m1]

suppose Fi (a1, a2, ... an) , 0 < i <= k. a1, ..., an are reals
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.

 

[lippyka]

Yes, it still suggests functional dependence amongst the Fi's. When rank(DF) < n, it implies that some of the variables in the function are functionally dependent on the others, meaning that for any given set of values for some of the variables, there is a unique set of values for the remaining variables. This can be seen as a result of the theorem I mentioned above - it states that when the rank of the derivative is less than the number of independent variables, the function is functionally dependent.