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

Definition

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

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

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

$(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \cdot \norm h$

$(2): \quad \ds \lim_{h \mathop \to 0} \map r h = 0$