Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, differentiable,...) functions $f:\mathbb R^n \to \mathbb R^m$ such that there exist $h:\mathbb R\to\mathbb R$ and a linear map $A:\mathbb R^n\to \mathbb R^m$ with $f(x_1,\ldots,x_n)= A(h(x_1),\ldots,h(x_n))$ for all $x=(x_1,\ldots,x_n)\in \mathbb R^n$?
This smells like a functional equation but knowing close to nothing about this I dare to ask this question here.
It seems to me that you are willing for the implicit function theorem and the Weiestrass preparation lemma
See
http://en.wikipedia.org/wiki/Implicit_function_theorem
http://en.wikipedia.org/wiki/Weierstrass_preparation_theorem