Integrability of Pseudomonotone Differentiable Maps and the Revealed Preference Problem

The problem considered is as follows: given $C\subset \R^n$ and $F:C\rightarrow \R^n$ differentiable, find $f:C\rightarrow \R$ differentiable such that $\norm \nabla f(x)\norm^{-1} \nabla f(x) = \norm F(x) \norm^{-1}F(x)$ for all $x\in C$. Conditions for $f$ to be pseudoconvex or convex are given. The results are applied to the differentiable case of the revealed preference problem.