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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.

Divergence Theorem Suppose that the components of have continuous partial
derivatives. If is a solid bounded by a surface oriented with the normal vectors
pointing outside, then:

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.

Let and compute:

To compute this integral, we’ll use the divergence theorem. Since
our region is a box, the limits of the triple integral will be easy to work with. Moreover,
computing the divergence of we see So by the divergence theorem, we have

Now let’s see another example:

Let

and compute:

To compute this integral, we’ll use the divergence theorem. This time
our region is a tetrahedron, we’ll work in cartesian coordinates. Moreover,
computing the divergence of we see So by the divergence theorem, we have

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

Let

and compute:

To compute this integral, we’ll use the divergence theorem.
Computing the divergence of we see So by the divergence theorem, we have

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.

Let be the cone whose base is a disk of radius in the plane and whose vertex is a
the origin. Compute the flux of across the boundary of .

Again, we will use the
divergence theorem. Computing the divergence of we see: Now write with me

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!