The Jacobian… If you are familiar with any multivariable calculus you may have heard of this term. Honestly speaking, the first time I learned about the Jacobian I did not understand how it worked at all, but knowing that the Jacobian is heavily used throughout multivariable calculus and that there will be articles in the future I want to write that use it, in this article, I will finally attempt to give my best explanation of it.
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:
That extra g’(u) term that must be multiplied to account for scaling is essentially what the Jacobian is but in multivariable calculus. In fact, a double integral undergoing a change of variables will look something like this:
