Necessary and sufficient condition for $u(x,y,z)$ , $v(x,y,z)$ ,$w(x,y,z)$ are functionally dependent

Question

Show that necessary and sufficient condition for that $u(x,y,z)$ , $v(x,y,z)$ ,$w(x,y,z)$ are functionally dependent through equation $F(u,v,w)= 0$ is $\nabla u\cdot (\nabla v \times \nabla w )$.

Any clue on this?

What if there are $2$ variable instead of $3$ (say only $u(x,y,z)$ and $v(x,y,z)$ are functionally dependent)?

Answer 1

Hint. If $F(u(x,y,z),v(x,y,z))=0$ and $F$ is differentiable then $$\frac{\partial F}{\partial u}\nabla u+\frac{\partial F}{\partial v}\nabla v={\bf 0}.$$ Note that $\frac{\partial F}{\partial u}$ and $\frac{\partial F}{\partial v}$ are scalars.