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This problem is from 'Introduction to Mathematical Analysis' by Douglass.

Problem 9.15 Let $f=(f_1,f_2)$ 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 jacobian of $f$ is zero $\forall x\in U$. 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 $f_2$.

I don't know how to use jacobian condition to prove statement.

Emmy N.
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Lim do
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  • I have edited the question. You should learn how to use LaTex, since questions with a wrong math stlye are usually refused. – Emmy N. Oct 12 '23 at 12:42

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