Jacobian

The $m\times n$ Jacobian matrix of $f$ is given by

$$\left[\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}& \cdots & \frac{\partial f_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \cdots & \frac{\partial f_m}{\partial x_n}\end{array}\right]$$

where

• $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a real valued function
• The Jacobian can be written as $Df$, $J_f$, $\nabla f$ and $\frac{\partial (f_1,\cdots ,f_m)}{\partial (x_1,\cdots ,x_n)}$.