Jacobian matrix

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

Suppose F : Rⁿ → R^m is a function. Such a function is given by m real-valued component functions, y₁(x₁,...,x_n), ..., y_m(x₁,...,x_n). The partial derivatives of all these functions (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix J_F of F, as follows:

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

Example

The Jacobian matrix of the function with components:

$y_{1}=x_{1}\,$

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

$y_{3}=4x_{2}^{2}-2x_{3}\,$

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

is:

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