Define a parameterization given a vector field on a smooth manifold

Question

I'm not aware of any techniques or theorems that would give an answer to the following question.

Suppose we have a smooth surface $\mathcal{S} \subset \mathbb{R}^3$, and also suppose that for each ${p} = (x,y,z)^T \in \mathcal{S}$ we have associate a vector $v \in \mathcal{T}_p$ (tangent space/plane at $p$). Is it possible to define a parameterization of the surface using such vector field?

I assume there're probably conditions I'm not mentioning in order to make this valid.

Does this problem have a name so I can look it up?

Thank you

Answer 1

Does this problem have a name so I can look it up?

What I was wondering is if such problem has a mathematical setting in general

One way to interrupt this question is "How do you find a surface parameterization whose gradient matches that of the given vector field?"

The answer would be to formulate the problem as a differential equation and try to find solutions.

If your desired parameterization has a two dimensional input $f(x,y)$, and you only have a single constraining vector field $v$ then you will need to find an $f$ such that either

$\frac{\partial f}{\partial x} = v$ or $\frac{\partial f}{\partial y} = v$