Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$ \begin{equation} (\partial_i f) (\partial_j g) - (\partial_j f) (\partial_i g)=0 \end{equation} Denote by $L_c=\left\{\mathbf{r} \in U \left.\right| f(\mathbf{r})=c\right\}$ a "level set" of $f$. Let $C$ be a connected component of $\partial L_c$. Is $g$ constant on $C$?

This sounds like a standard question to me, so a simple reference will do as an answer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.