Definition:Differential of Mapping/Vector-Valued Function

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

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

The differential $\mathrm d f$ is a function of two variables, defined as:
 * $\mathrm d f \left({x; h}\right) = J_f(x) \cdot h$

where $J_f(x)$ be the Jacobian of $f$ at $x$.

That is, if $h = (h_1,\ldots,h_n)$,
 * $\displaystyle \mathrm d f \left({x; h}\right) =

\begin{pmatrix} \frac{\partial f_1}{\partial x_1}(x) & \cdots & \frac{\partial f_1}{\partial x_n}(x) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(x) & \cdots & \frac{\partial f_m}{\partial x_n}(x) \end{pmatrix} \begin{pmatrix} h_1\\ \vdots\\ h_n \end{pmatrix}$