Having a directional derivative

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, let $v \in \mathbb{R}^n$, and let $L \in \mathbb{R}^m$. Let $g(t) = f(a + t v)$ for real $t$ near $0$. The function $f$ has the directional derivative $L$ at $a$ in the direction $v$ when $g$ is differentiable at $t = 0$ with $g'(0) = L$. Equivalently, $\dfrac{f(a + t v) - f(a)}{t} \to L$ as $t \to 0$.