Let $f = (f_1,f_2)$ (with each $f_1,f_2$ a real valued function) be a continuously differentiable function defined on an open set $U$ in $\mathbb{R^2}$ such that $\nabla f_1$ and $\nabla f_2$ do not vanish at any point of $U$. Suppose that $J_f (x) = 0$ for all $x$ in $U$.
With the given conditions, is a curve $C$ in $U$ a level curve of $f_1$ iff it is also a level curve of $f_2$?
Any pointers are appreciated.