Hessian matrix defines bilinear form that outputs second-order directional derivatives

Suppose $f$ is a function of $n$ variables $x_1,x_2,\dots,x_n$, which we think of as a vector variable $\overline{x}$. Suppose $\overline{u},\overline{v}$ are unit vectors in $n$-space. Then, we have the following:
$D_{\overline{v}}(D_{\overline{u}}(f)) = \overline{u}^TH(f)\overline{v}$
where $\overline{u},\overline{v}$ are treated as column vectors, so $\overline{u}^T$ is $\overline{u}$ as a row vector, and $\overline{v}$ is $\overline{v}$ as a column vector. The multiplication on the right side is matrix multiplication. Note that this tells us that the bilinear form corresponding to the Hessian matrix outputs second-order directional derivatives.
$D_{\overline{v}}(D_{\overline{u}}(f)) = D_{\overline{u}}(D_{\overline{v}}(f))$