Finding the expression of an implicit vector field

Question

Suppose that we have some smoth function $f: \mathcal{U} \subseteq \mathbb{R}^2 \to \mathbb{R}^3$ and I define a vector field $\mathbf{F}$ such that $$ \mathbf{F} (f) = \frac{ \frac{\partial f}{\partial x} \times \frac{\partial f}{\partial y} }{ \left\| \frac{\partial f}{\partial x} \times \frac{\partial f}{\partial y} \right\| } $$ Is there a way to express $\mathbf{F}$ explicitly? If not, can we express at least its divergence explicitly?