# Definition:Differential/Vector-Valued Function

## 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