The Divergence Theorem and Stokes’ Theorem

When preparing the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a new homework problem to Chapter 4.

Problem 4. Integrate Eq. 4.8 over a volume and subtract the result from Eq. 4.4. The resulting relationship is called the divergence theorem.

For those of you who don’t keep a copy of IPMB always close at hand (what’s the matter with you?), Eq. 4.8 is

where “div j” is the divergence of the vector j, and Eq. 4.4 is

When you put the two equations together, you get (spoiler alert) ,

also known as the divergence theorem. It is one of the fundamental results of vector calculus.

Div, Grad, Curl, and All That,
by H. M. Schey.

If you want to learn more about the divergence theorem, I recommend H. M. Schey’s book Div, Grad, Curl and All That. He writes

For the remainder of this chapter we digress from the mainstream of our narrative to discuss a famous theorem that asserts a remarkable connection between surface integrals and volume integrals. Although this relation may be suggested by the work we have done in electrostatics, the theorem is a mathematical statement holding under quite general circumstances. It is independent of any physics and is applicable in many different places. It is called the divergence theorem, and sometimes Gauss’ theorem… It says that the flux of a vector function through some closed surface equals the triple integral of the divergence of that function over the volume enclosed by the surface.

Besides the divergence theorem, another basic tenet of vector calculus is Stokes’ theorem. Can we make a similar homework problem demonstrating that? Yes! Here is a new problem for Chapter 8.

Problem 23 ½. Integrate Eq. 8.22 over a surface and subtract the result from Eq. 8.21. The resulting relationship is called Stokes’ theorem.

If you don’t have IPMB handy, Eq. 8.21 is

and Eq. 8.22 is

where “curl E” is the curl of the vector E. When you put the two equations together, you get

 Schey writes

We [now] discuss another famous theorem, one strongly reminiscent of the divergence theorem and yet, as we’ll see, quite different from it. This theorem, named for the mathematician Stokes, relates a line integral around a closed path to a surface integral over what is called a capping surface of the path…In words, Stokes’ theorem says that the line integral of the tangential component of a vector function over some closed path equals the surface integral of the normal component of the curl of that function integrated over any capping surface of the path.

The divergence theorem and Stokes’ theorem are a bit too mathematical to develop in IPMB, with its emphasis on biological and medical applications. Yet there they are, implicit in our discussions of diffusion and of transcranial magnetic stimulation. If you want to learn more, start with Schey’s wonderful (and relatively inexpensive) book Div, Grad, Curl.