Definition:Differential/Vector-Valued Function

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Definition

At a Point

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 $x$.


On an Open Set

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 $\rd f$ is a function of two variables, defined as:

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

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


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

$\displaystyle \rd f \left({x; h}\right) = \begin{pmatrix} \frac {\partial f_1} {\partial x_1} \left({x}\right) & \cdots & \frac {\partial f_1} {\partial x_n} \left({x}\right) \\ \vdots & \ddots & \vdots \\ \frac {\partial f_m} {\partial x_1} \left({x}\right) & \cdots & \frac {\partial f_m} {\partial x_n} \left({x}\right) \end{pmatrix} \begin{pmatrix} h_1 \\ \vdots \\ h_n \end{pmatrix}$


Notation

There are various notations for the differential of a function $f$ at $x$:

  • $\d f \left({x}\right)$
  • $\d f_x$
  • $\d_x f$
  • $D f \left({x}\right)$
  • $D_x f$


Substituting $\d y$ for $\d f \left({x; h}\right)$ and $\d x$ for $h$, the following notation emerges:

$\d y = f' \left({x}\right) \rd x$

hence:

$\d y = \dfrac {\d y} {\d x} \rd x$


Notes


1. When the dimension of $W$ is $1$, the differential of a function is generalised by the notion of differential forms on manifolds. Indeed the differential of $f : V \to W$ is an exact form of degree $1$.

2. The above definition also furnishes differentials of differential functions between affine spaces. This is due to Affine Space with Origin has Vector Space Structure