\begin{definition}[Definition:Differential of Mapping/Vector-Valued Function/Point]
Let $U \subset \R^n$ be an open set.
Let $f: U \to \R^m$ be a vector-valued function.
Let $f$ be differentiable at a point $x \in U$.
The '''differential of $f$ at $x$''' is the linear transformation $\d \map f x: \R^n \to \R^m$ defined as:
:$\map {\d \map f x} h = \map {J_f} x \cdot h$
where:
:$\map {J_f} x$ is the Jacobian matrix of $f$ at $x$.
Category:Definitions/Differentials
\end{definition}