Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be twice Frechet differentiable at $a$ in the sense that $f$ is Frechet differentiable on an open neighbourhood of $a$ and the derivative map
is Frechet differentiable at $a$ for some open neighbourhood $U_0 \subset U$ of $a$. Let $u,v \in \mathbb{R}^m$. Then the [iterated directional derivative](/page/Iterated%20Directional%20Derivative) $D_uD_v f(a)$ exists and