  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.
