Let $U \subset \mathbb R^n$ be open. A function $f: U\to \mathbb R^m$ is differentiable at $a \in U$ if there exists a linear transformation $A:\mathbb R^n \to \mathbb R^m$ such that \(\lim_{h\to 0} \frac{f(a+h)-f(a)-A(h)}{\vert \vert h\vert \vert } = 0.\) The transformation $A$ is denoted $Df(a)$ and is called the total derivative of $f$ at $a$.
Equivalently, let $V$ and $V$’ be vector spaces with norms $\vert \cdot\vert $ and $\vert \cdot\vert ‘$, respectively. A function $f:U\to V’$ from $U$ an open subset of $V$ is differentiable at $x\in U$ if there exists a linear transformation $d_x f :V\to V’$ called the differential of $f$ at $x$ and a small neighborhood $U_x\subseteq U$ of $x$ such that for all $v\in U$, we have that \(f(x+v) = f(x) + d_x(f) + o(v)\) where $o: U_x\to V’$ is a map such that \(\lim_{\vert v\vert \to 0} \frac{\vert o(v)\vert '}{\vert v\vert } = 0.\)
Wikidata ID: Q636889