Let $U, V, W$ be scalar functions in $R^3$, that is $U(x,y,z), V(x,y,z), W(x,y,z) : R^3 \rightarrow R$. I would like to show that a sufficient and necessary condition for $U$ to be a function of V and W is for $\nabla U$ to be a linear combination of $\nabla V$ andd $\nabla W$, that is to say
$\nabla U = p(V,W)\nabla V + q(V,W)\nabla W$ iff $U = f(V, W)$ where $p(V,W), q(V,W)$ are scalar functions of V and W.
I know that in the backwards direction, it works, since by taking the gradient of $U$ i.e. $\nabla U = \nabla f(V,W)$, we can show (the work has been omitted) that $\nabla U = \frac{\partial f}{\partial V}\nabla V + \frac{\partial f}{\partial W}\nabla W$ which satisfies the backwards direction, but I am lost as in how to show the forward direction. I have found several qualitative arguments for this, considering constant curves of $U = k$ where $k$ is a constant and building a function in reverse, but is there any theorem for showing it with mathematics?