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Differential

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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

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

and determines the linear tranformation

[\frac{\partial F (a,b)}{\partial x},\frac{\partial F (a,b)}{\partial y}]:\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}

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