3.7 Jacobian matrix

Differentiation of vector-valued function with vector

The matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector

\(J = \begin{bmatrix} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} & \dots & \frac{\partial F_1}{\partial x_n} \\[0.3em] \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} & \dots & \frac{\partial F_2}{\partial x_n} \\[0.3em] \vdots & \vdots & \ddots & \vdots \\[0.3em] \frac{\partial F_m}{\partial x_1} & \frac{\partial F_m}{\partial x_2} & \dots & \frac{\partial F_m}{\partial x_n} \end{bmatrix}\)

Determinant of Jacobian matrix shows the area (integration value) ratio of coordinate transformation. If the random variables are transformed, this area ratio (determinant of Jacobian) should be multiplied to the distribution function to adjust the area to be 1, because total area of any probability function should be 1.