Find a function $f$ on $\mathbb{R}^3$ such that $A^*df=0$

Question

Let $A : \mathbb{R}^2 = \{(u,v)\} \rightarrow \mathbb{R}^3=\{(x,y,z)\}$ be given by

$A(u,v)=(u,v,F(u,v))$

Find a function $f$ on $\mathbb{R}^3$ such that $A^*df=0$.

I have tried $A^*df=d(A^*f)=d(f \circ A)=df(u,v,F(u,v))$, which gives that

$\frac{\partial f}{\partial u}du + \frac{\partial f}{\partial v}dv+\frac{\partial f}{\partial z}dF=0$.

I cannot see how to use this to now to find an $f$.

I would like to look for a sufficient and or neceesary condition for a solution and not just find some semi trivial solution.

Answer 1

HINT: The image of $A$ is a level surface of the function $f(x,y,z)=z-F(x,y)$.