Jacobian Matrix

Used for functions with multiple inputs and multiple outputs. It is the Matrix of all possible partial derivatives of a function $f:{\mathbb{F}}_1^n\mapsto {\mathbb{F}}_2^n$ given by:

$$\frac{dy}{dx}=\left[\begin{array}{cccc}\frac{\partial y_1}{x_1}& \frac{\partial y_1}{x_2}& \dots & \frac{\partial y_1}{x_n}\\ & & & \\ ⋮& ⋮& ⋮& ⋮\\ \frac{\partial y_n}{x_1}& \frac{\partial y_n}{x_2}& \dots & \frac{\partial y_n}{x_n}\end{array}\right]$$

2-Variable Jacobian

$$|J(x,y)|=\left|\begin{array}{cc}\frac{\partial h_1(x,y)}{\partial x}& \frac{\partial h_1(x,y)}{\partial y}\\ \frac{\partial h_2(x,y)}{\partial x}& \frac{\partial h_2(x,y)}{\partial y}\end{array}\right|=|\frac{\partial h_1(x,y)}{\partial x}\frac{\partial h_2(x,y)}{\partial y}-\frac{\partial h_1(x,y)}{\partial y}\frac{\partial h_2(x,y)}{\partial x}|$$