Does the lemma at the bottom of this page have a name? Or could someone give me an idea of where I can find a proof?
In case you can't access the link:
Lemma $\ \ $ If $g$ is of class $C^k(k\geqslant2)$ on a convex open set $U$ about $p$ in $\Bbb R^d$, then for each $q\in U$,
$\displaystyle\begin{array}{rl}(1) & g(q)=g(p)+\sum_{i=1}^d\dfrac{\partial g}{\partial r_i}\left|\right._p\big(r_i(q)-r_i(p)\big)\\&\qquad\ +\sum_{i,j}\big(r_i(q)-r_i(p)\big)\big(r_j(q)-r_j(p)\big)\int_0^1(1-t)\dfrac{\partial^2g}{\partial r_i\partial r_j}\left|\right._{(p+t(q-p))}dt. \end{array}$
In particular, if $g\in C^\infty$, then the second summation in $(1)$ determines an element of $F_p^2$ since the integral as a function of $q$ is of class $C^\infty$.