This video extends the probability transformation rule from the scalar case to the vector case for vector functions of random vectors. The video discusses how the Jacobian determinant needs to be calculated instead of a simple gradient, and therefore this can influence whether the Jacobian or its inverse should be calculated depending on the ease of calculating the derivatives for the mapping or inverse mapping. The video provides a reminder of the physical interpretation of the Jacobian. Finally, the video considers the probability transformation for the Polar to Cartesian coordinate mappings.