Definition:Differentiable Mapping/Vector-Valued Function/Region
Jump to navigation
Jump to search
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.
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 $f \in \map {\CC^1} {S, \R^m}$.
Also see
- Definition:Differentiability Class for insight into the notation $\map {\CC^1} {S, \R^m}$.
Sources
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $13.4$