Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a smooth function. Let's assume that $f$ has a local minimum at $p \in \mathbb{R}^2$ and hence $| \nabla f|\ = 0$ at $p.$
Intuitively, one should be able to define a coordinate system around $p$ in the following way:
-one coordinate function is induced by the flow of the gradient vector field towards the singularity (call it $r$).
-the second coordinate is induced by moving perpendicularly along the level curves (call it $\theta.$).
This coordinate system should, moreover, look more or less like the polar coordinates. In particular, I would expect that this should be true up to local diffeomorphism.
Question(s): is this intuition correct? And is it easy to make rigorous? If this can be done, then I assume that this is a standard procedure, so references would be appreciated.
EDIT: Actually, I've just realized that this has no hope of being true unless we impose some more conditions. I think it would be enough to assume the Hessian is positive definite. But it should still be possible to construct these coordinates in some cases even when the Hessian vanishes I think (consider for instance $f=x^4+y^4,$ for which the usual polar coordinates do the trick.