Named after German mathematician Carl Gustav Jacob Jacobi, the Jacobian determinant (or just the Jacobian) is required when evaluating multiple integrals after undergoing a change of variables.
To understand the Jacobian, it may help to consider the analogous case in single variable calculus where a change of variable is used. Consider we have the integral of some function f(x), and we wish to undergo the change of variables of x = g(u). Now, using the fact that x = g(u), we can get the following equation:
Which if rearranged will give dx = g’(u)du. Substituting this into the original integral, we will get the following: