Differentiable function of several variables

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$. The function $f$ is differentiable at $a$ when there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that $$\lim_{h \to 0} \frac{\|f(a + h) - f(a) - L h\|}{\|h\|} = 0.$$ Here the limit is over $h \ne 0$ with $a + h \in U$, and $\|\cdot\|$ is the Euclidean norm. The map $L$ is then unique. It is the derivative $Df(a)$, also called the total derivative of $f$ at $a$.