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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.
Hint. Let $\gamma \colon (0,1)\to \mathbf R^2$ be a $C^1$-parametrisation of a level curve of $f_1$. Then $$ (f_2 \circ \gamma)'(t) = (\nabla f_2 \circ \gamma)(t) \cdot \gamma'(t) $$ On the other hand, as $J_f(x) = 0$ for all $x \in U$, we have that $\nabla f_1(x)$ and $\nabla f_2(x)$ are linearly indepedent for all $x$, say $\nabla f_1(x) = \alpha(x)\nabla f_2(x)$. This gives above $$ (\nabla f_2 \circ \gamma)(t) = (\alpha\circ \gamma)(t)(\nabla f_1 \circ \gamma)(t) $$ Now use that $\gamma$ is a level curve of $f_1$, hence $f_1 \circ \gamma$ is constant.
