U-Derivative from the Jacobian

We now have a clean computational workflow for directional derivatives. Given $f$ and a point $\mathbf{a}$: (1) compute all partial derivatives at $\mathbf{a}$, assembling them into the Jacobian row vector $Df(\mathbf{a}) = [\partial_1 f(\mathbf{a}), \ldots, \partial_n f(\mathbf{a})]$; (2) for any unit direction $\mathbf{u}$, compute $D_\mathbf{u} f(\mathbf{a}) = Df(\mathbf{a})\mathbf{u}$ (matrix-vector product).

The matrix-vector product interpretation is powerful: the Jacobian row vector "dot products" with any direction to give the directional derivative. This is exactly the action of a linear functional on a vector.

For a vector-valued function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$, the directional derivative in direction $\mathbf{u}$ is $D_\mathbf{u} \mathbf{f}(\mathbf{a}) = J\mathbf{f}(\mathbf{a})\mathbf{u} \in \mathbb{R}^m$ — a vector of directional derivatives, one for each component.