Definition:Differentiable Mapping/Vector-Valued Function

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Let $m,n\geq1$ be natural numbers.

At a Point

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.

Definition 1

$f$ 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:

$(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \cdot \norm h$
$(2): \quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$

Definition 2

$f$ is differentiable at $x \in \R^n$ if and only if for each real-valued function $f_j: j = 1, 2, \ldots, m$:

$f_j: \mathbb X \to \R$ is differentiable at $x$.

In an Open Set

Let $S \subseteq \mathbb X$.

Then $f$ is differentiable in the open set $S$ if and only if $f$ is differentiable at each $x$ in $S$.

This can be denoted $f \in \map {\CC^1} {S, \R^m}$.