Jacobian matrix

The Jacobian matrix is the matrix of all first-order partial derivatives of scalar components of a vector-valued function F of a vector variable, with respect to the scalar-valued components of the argument to F. Thus we have

$(y_{1},\dots ,y_{m})=F(x_{1},\dots ,x_{n})$

and the scalar components of F are

$y_{i}=F_{i}(x_{1},\dots ,x_{n}).$

The Jacobian matrix ($J$) of F is:

${\begin{bmatrix}\partial y_{1}/\partial x_{1}&\cdots &\partial y_{1}/\partial x_{n}\\\vdots &\cdots &\vdots \\\partial y_{m}/\partial x_{1}&\cdots &\partial y_{m}/\partial x_{n}\end{bmatrix}}$

Example

The Jacobian matrix of the system:

$y_{1}=x_{1}$

$y_{2}=5x_{3}$

$y_{3}=4(x_{2})^{2}-2x_{4}$

$y_{4}=x_{3}\sin(x_{1})$

is:

${\begin{bmatrix}dx_{1}&0&0&0\\0&0&5\,dx_{3}&0\\0&8x_{2}\,dx_{2}&0&-2\,dx_{4}\\x_{3}\cos(x_{1})\,dx_{1}&0&\sin(x_{1})\,dx_{3}&0\end{bmatrix}}$