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.