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 $\d \map f x: \R^n \to \R^m$ 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 $x$.