Definition:Differential of Mapping/Vector-Valued Function

Definition
Let $O \subseteq \R^n$ be an open set.

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

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


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

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

That is, if $h = \left({h_1, \ldots, h_n}\right)$:


 * $\displaystyle \mathrm d f \left({x; h}\right) =

\begin{pmatrix} \frac {\partial f_1} {\partial x_1} \left({x}\right) & \cdots & \frac {\partial f_1} {\partial x_n} \left({x}\right) \\ \vdots & \ddots & \vdots \\ \frac {\partial f_m} {\partial x_1} \left({x}\right) & \cdots & \frac {\partial f_m} {\partial x_n} \left({x}\right) \end{pmatrix}

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