How do I have to approach this problem? Intuitionally I can imagine the situation, but I have no idea how to prove this.
Problem : Let $f=(f_1, f_2)$ be a continuously differentiable function defined on an open set $U$ in $R^2$ such that $\nabla f_1$ and $\nabla f_2$ do not vanish at any point of $U$.
a) Suppose that $J_f($x$)=0$ for all x in $U$. Prove that a curve $C$ in $U$ is a level curve of $f_1$ if and only if it is also a level curve of $f_2$.
b) Suppose that $f_1$ and $f_2$ have the same level curves on $U$. Prove that $J_f($x$)=0$ for all x in $U$.
We are given two $C^1$-functions $f_1$, $f_2$ with a common domain $U\subset{\mathbb R}^2$, whereby both $\nabla f_1$ and $\nabla f_2$ do not vanish in $U$.
(a) The condition $$J_f({\bf x})=\nabla f_1({\bf x})\wedge\nabla f_2({\bf x})=0\qquad({\bf x}\in U)$$ means that $\nabla f_1({\bf x})$ and $\nabla f_2({\bf x})$ are parallel at all points ${\bf x}\in U$. It follows that the orthogonal trajectories of the two gradient fields are the same. But these orthogonal trajectories are just the level lines of $f_1$, resp., $f_2$.
(b) Same thing: Since $\nabla f_i\ne{\bf 0}$ in $U$ the level lines of both $f_1$ and $f_2$ are smooth curves, whereby through each point ${\bf x}\in U$ passes exactly one curve for each of $f_1$ and $f_2$. If these curves do coincide then their tangents, and hence their normals $\nabla f_i({\bf x})$, have to be parallel at each point ${\bf x}\in U$. This implies $J_f({\bf x})=0$ for all ${\bf x}\in U$.