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

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

and determines the linear tranformation

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

given by

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

For a vector function $F:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ 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

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