Definition:Differentiable Mapping/Vector-Valued Function/Point

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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.


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:

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


Definition 2

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

$\ds \lim_{h \mathop \to \bszero} \frac {\norm {\map f {x + h} - \map f x - \map T h}} {\norm h} = 0$


Definition 3

$f$ 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 $x$.


Also see


Sources


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