Definition:Differential of Mapping/Vector-Valued Function/Point

Definition
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 $df(x) : \R^n \to \R^m$ defined as:
 * $df(x)(h) = J_f(x) \cdot h$

where:
 * $J_f(x)$ is the Jacobian matrix of $f$ at $x$.