We now choose to pursue perhaps the obscurest question anywhere
in the book. This choice will be redeemed by what it enables us
to understand about Electromagnetism a little later. We suppose
that we have taken an arbitrary flow field and mapped out its
circulation, to create yet another field of little arrows. We
then ask, what is the *divergence* of this other field of
circulations?

To answer such a strange question, we first change the rules of the game slightly, and rather than using little circles to map out the circulation, we use little squares, adding the components of the flow along the little squares, as shown in fig. 7.6.

It is an exercise in the homework to check that this indeed gives us the same answer for the measured circulation, which is not hard to believe. Similarly, when we compute divergences, we will use closed surfaces which are not spheres, but rather cubes.

The virtue of using squares and cubes is that squares, which we
use to compute circulations, can readily be assembled into
cubes, which are closed surfaces and can be used to compute
divergences. Even better, we have *defined* the arrow that
represents a circulation to point straight outward from the face
of its little square, which means that the entire length of the
arrow comes from its outward component. To compute the divergence,
we simply add the lengths of all the arrows. This addition,
though, is trivial. The way to see this is shown in
fig. 7.6.

For small cubes, we only need to know the flow field along
the edges of the cubes to compute the circulations on each of
the faces. We simply add them up, positively if they point
along the direction of the square's perimeter, negatively if
they point against it. The resulting sum gives the length of
the circulation arrow. But every edge is shared by exactly two
squares, and because all of the squares are referred to an
outward-pointing direction, any direction along an edge looks
positive to one square and negative to the other. No matter
what the flow field does, whatever number we add to one square,
we have to subtract from the other one that shares that edge.
When we then add all of the circulation arrows, we are adding
all of these sums, in which every term that appeared added,
appeared somewhere else subtracted. So for any flow field, **the divergence of the cirulation is zero**.

When our flow field comes from the flow of water, this looks
like trivial statement number two. We have defined a long
complicated sequence of steps, computing first cirulations and
then divergences of them, and all the while we have been
secretly adding and subtracting all the same numbers, to produce
zero at the end. When we get to Electromagnetism, though, the
same result, which necessarily holds by definition, will not
have been so obvious, and will have useful implications. This
is because, for both the electric and the magnetic field, it
will be easier to discover that the electric field behaves like
a divergent flow field, and the magnetic like a circulation,
than to discover *what they are the flow fields and
circulations of*. Then, understanding properties of divergences
and cirulations in their own right will be invaluable.

Thu Aug 31 12:01:42 CDT 1995