# Definition:Differential/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 $\d f \left({x}\right): \R^n \to \R^m$ defined as:

$\d f \left({x}\right) \left({h}\right) = J_f \left({x}\right) \cdot h$

where:

$J_f \left({x}\right)$ is the Jacobian matrix of $f$ at $x$.