# Coordinates that map vertical planes to the planes of constant value of a smooth function

Suppose $$f: \mathbb{R}^3 \to \mathbb{R}$$ is a smooth function and $$V = \nabla f$$ is a smooth vector field. Let $$\theta : \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R}^3$$ be the integral curves of $$V$$. It would be very good for my problem if I could construct a chart $$\psi = (t,\theta,\varphi)$$ with the property that $$(f\circ \psi)^{-1}(a) = \{p\} \times [0,2\pi]\times [0,\pi]$$ is some vertical two dimensional plane in the domain of $$\psi$$ for each $$a$$ in the image of $$f$$. I would very much like to determine the Laplacian in these coordinates. I have taken an introductory course to smooth manifolds during my master studies, so I have a basic understanding of smooth manifolds. I feel this should be possible for the particular $$f$$ that I am using, but I am not sure how to do it. (Please let me know if the question is not well-posed, I am a beginner, so that would be helpful.) If you know how it is possible to solve my problem, it would be great if you can provide some details, such that I can read up on it and see if it works.

You can do it away from the zero of $$V$$ (i.e. the critical point of $$f$$).
Let $$q \in \mathbb R^3$$ be a point where $$V(q)\neq 0$$. By the Straightening theorem for vector fields, there is an open set $$U$$ with coordinates $$(t, x, y)$$ and a local coordinates $$\varphi : U \to \mathbb R^3$$ so that $$V = \frac{\partial }{\partial t}$$ in $$\varphi(U)$$. By shrinking $$U$$, we assume $$U = I \times V$$ for some interval $$I$$. Under this coordinates, the integral curve $$\theta$$ is given by
$$\theta (t_0, (t, x, y )) = (t_0+ t, x, y)$$
and after a scaling in the $$(x, y)$$ direction, this coordinates has the properties you need.
• Great news Arctic Char! Do you know offhand how to get a formula for the Laplacian of these coordinates, or must I go through the proof of the straightening theorem to see if I can get an expression for $\varphi$? – Mikkel Rev Nov 29 '19 at 23:30
• Well you can represent the Laplacian as the sum of Laplacian (of the level set $f=constant$) and the mean curvature of the level set (together with one more term). Is that what you want? @MikkelRev – Arctic Char Nov 30 '19 at 1:18