Jacobian matrix and determinant

The jacobian is the matrix of all first-order partial derivatives of a system of differential equations, or dynamical system.

Given two n-spaces: $[x_{1},x_{2},...x_{n}]$, and $[y_{1},y_{2},...y_{n}]$, related by the mapping function $F:X->Y$, such that $y_{i}=F_{i}(x_{1},x_{2},...x_{n})$

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

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

Example

The Jacobian 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:

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