Differential

In modern mathematics the differential of a function is the linear transformation associated to each point in the domain of the function. This linear tranformation is given by the derivative.

For example if $f:\mathbb{R} \to \mathbb{R}$ is given by $f(x)=x^2$ the the derivative is $f'(x)=2x$ . At $x=5$ the function value is $f(5)=25$ but $f'(5)=10$ is the linear transformation

$x\mapsto 10x$ .

Another if $F(x,y)=x^2+3y$ at $x=a,y=b$ it differential is the gradient

$\left[\frac{\part F(a,b)}{\part x},\frac{\part F (a,b)}{\part y}\right]$

and determines the linear tranformation

$\left[\frac{\part F(a,b)}{\part x},\frac{\part F(a,b)}{\part y}\right]:\R^2\to\R$

given by

$(x,y)\mapsto\frac{\part F(a,b)}{\part x}x+\frac{\part F(a,b)}{\part y}y$

For a vector function $F:\R^n\to\R^m$ let us ilustrate with another beispiel: Suppose that

$\begin{bmatrix}v\\w\end{bmatrix}\mapsto\begin{bmatrix}5v+w\\v^2\\-v+8w\end{bmatrix}$

then

$\begin{bmatrix}5&1\\2v&0\\-1&8\end{bmatrix}$

is the Jacobian. So at $v=2,w=3$ the differential is the map

$\begin{bmatrix}v\\w\end{bmatrix}\mapsto \begin{bmatrix}5&1\\4&0\\-1&8\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} =\begin{bmatrix}5v+w\\4v\\-v+8w\end{bmatrix}$