Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2
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Definition
Let $\mathbb X$ be an open subset of $\R^n$.
Let $f : \mathbb X \to \R^m$ be a vector valued function.
$f$ is differentiable at $x \in \mathbb X$ if and only if there exists a linear transformation $T: \R^n \to \R^m$ such that:
- $\ds \lim_{h \mathop \to \bszero} \frac {\norm {\map f {x + h} - \map f x - \map T h}} {\norm h} = 0$
Sources
- 1965: Michael Spivak: Calculus on Manifolds: 2. Differentiation: Basic Definitions