Does $\nabla A \times \nabla B = 0$ imply that A is a function of B?

Question

As stated in the title. Assuming that A and B are both non-constant functions of $x,y,z$, does $\nabla A \times \nabla B = 0$ imply that A is a function of B, $A=A(B)$ in the sense that there exists a function $f$ such that $A = f \circ B$ ? If so, how could I go about proving it?

Answer 1

No, If $A=\sin(x)$ and $B=0$ then of course $A$ is not a function of $B$ despite $\nabla A\times\nabla B = (\sin(x)e_x)\times 0 = 0$.

Actually you can have $A$ and $B$ be almost any function of $x$, I could have chosen $B=1$ or even $B=x^2$.