We introduce the divergence theorem.

The divergence theorem

The divergence theorem states that certain volume integrals are equal to certain surface integrals. Let’s see the statement.

Integrals of the type above arise any time we wish to understand “fluid flow” through a surface. The “fluid” in question could be a real fluid like air or water, or it could be an electromagnetic field, or something else entirely. Unfortunately, many of the “real” applications of the divergence theorem require a deeper understanding of the specific context where the integral arises. For our part, we will focus on using the divergence theorem as a tool for transforming one integral into another (hopefully easier!) integral.

Now let’s see another example:

With our next two examples, we cannot help but flex our mathematical muscles a bit:

Above, we used the fact that we know that the volume of a sphere is . When you know the volume that’s great! If not you have to compute the integral.

A new fundamental theorem of calculus

How is the divergence theorem a fundamental theorem of calculus? Well consider this:

Are there more fundamental theorems of calculus? Absolutely, and we’re ready for the last one of this course. Read on young mathematician!