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 $$g=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}\mathrm{d}{x}^i\circ \mathrm{d}{x}^j\mathrm{.}$$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable.