Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1

Definition
Let $\mathbb X$ be an open subset of $\R^n$.

Let $f = \left({f_1, f_2, \ldots, f_m}\right)^\intercal: \mathbb X \to \R^m$ be a vector valued function.

$f$ is differentiable at $x \in \R^n$ there exists a linear transformation $T:\R^n \to \R^m$ and a mapping $r : U \to \R^m$ such that:


 * $(1):\quad$ $\displaystyle f \left({x + h}\right) = f \left({x}\right) + T(h) + r\left({h}\right)\cdot \Vert h\Vert$
 * $(2):\quad$ $\displaystyle\lim_{h\to 0} r(h) = 0$.