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:

and compute:

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

and compute:

But is just the volume of the hemisphere of radius , which we know is . Hence:

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.

If you know the volume of the cone described above, you can be done! If not, do not despair, we’ll simply use cylindrical coordinates to compute it. Write with me:

Hence:

### 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!