Jacobian

The Jacobian of a function which maps $(x_1,\dots ,x_n)$ to $(y_1,\dots ,y_n)$ is the $n$ by $n$ matrix of all possible partial derivatives given by

$$J=\frac{d\vec{y}}{d\vec{x}}=\left(\begin{array}{cccc}\frac{\partial y_1}{\partial x_1}& \frac{\partial y_1}{\partial x_2}& \cdots & \frac{\partial y_1}{\partial x_n}\\ \frac{\partial y_2}{\partial x_1}& \frac{\partial y_2}{\partial x_2}& \cdots & \frac{\partial y_2}{\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial y_n}{\partial x_1}& \frac{\partial y_n}{\partial x_2}& \cdots & \frac{\partial y_n}{\partial x_n}\end{array}\right)$$