Definition:Differential of Mapping/Vector-Valued Function

On an Open Set
Let $O \subseteq \R^n$ be an open set.

Let $f = \tuple {f_1, \ldots, f_m}^\intercal: O \to \R^m$ be a vector valued function, differentiable at $x \in O$.

The differential $\d f$ is a function of two variables, defined as:


 * $\map {\d f} {x; h} = \map {J_f} x \cdot h$

where $\map {J_f} x$ be the Jacobian matrix of $f$ at $x$.

That is, if $h = \tuple {h_1, \ldots, h_n}$:


 * $\map {\d f} {x; h} = \begin {pmatrix}

\map {\dfrac {\partial f_1} {\partial x_1} } x & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } x \\ \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_m} {\partial x_1} } x & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } x \end {pmatrix}

\begin {pmatrix} h_1 \\ \vdots \\ h_n \end {pmatrix}$