Hessianizability of surface metrics

A symmetric quadratic form $g$ on a surface~$M$ is said to be locally Hessianizable if each $p\in M$ has an open neighborhood~$U$ on which there exists a local coordinate chart $(x^1,x^2):U\to\mathbb{R}^2$ and a function $f:U\to\mathbb{R}$ such that, on $U$, we have $$g = \frac{\partial^2 f}{\partial x^i\partial x^j}\,\mathrm{d} x^i\circ\mathrm{d} x^j.$$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable.