Definition:Differential of Mapping/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 $\d \map f x: \R^n \to \R^m$ defined as:

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

where:

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

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}$

Notation

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

• $\map {\d f} x$
• $\d f_x$
• $\d_x f$
• $\map {D f} x$
• $D_x f$

Substituting $\d y$ for $\map {\d f} {x; h}$ and $\d x$ for $h$, the following notation emerges:

$\d y = \map {f'} x \rd x$

hence:

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

Notes

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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