This is from Woll's "Functions of Several Variables," but there's no proof.
If $g$ is of class $C^k$ ($k \ge 2$) on a convex open set $U$ about $p$ in $\mathbb{R}^d$, then for each $q \in U$,
$ g(q) = g(p) + \sum_{i=1}^d \frac{\partial g}{\partial r_i} \bigg|_p (r_i(q) - r_i(p)) + \sum_{i,j} (r_i(q) - r_i(p)) (r_j(q) - r_j(p)) \int_0^1 (1-t) \frac{\partial^2g}{\partial r_i \partial r_j} \bigg|_{p + t(q - p)} dt. $
It looks like Taylor's or mean value theorem. I especially don't understand the integral part.