TITLE: Local expression for a 1-form on a surface
QUESTION [5 upvotes]: Suppose that $\alpha$ is a non-vanishing 1-form on a 2-dimensional manifold. Why can $\alpha$ locally be written as $\alpha = f \ dg$ for some smooth functions $f$ and $g$?

REPLY [12 votes]: Given a point $p$ in the manifold, choose a smooth nonvanishing vector field $V$ in a neighborhood of $p$ such that $\alpha(V)\equiv 0$. The straightening lemma implies that there are smooth coordinates $(x,y)$ in a neighborhood of $p$ such that $V = \partial/\partial x$ in these coordinates. In these coordinates, $\alpha = f\,dy$ for some smooth function $f$.