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.