Visualizing Divergence

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Section 12.6 Visualizing Divergence

Recall that

\begin{equation} \grad\cdot\FF \approx \frac{\Int \FF \cdot d\AA}{\textrm{volume of box}} = \frac{\textrm{flux}}{\textrm{unit volume}}\tag{12.6.1} \end{equation}

so that the divergence measures how much a vector field “points out” of a box.

Figure 12.6. Two vector fields.

Figure 12.6. Two vector fields.

Can we use these ideas to investigate graphically the divergence of a given vector field?

Activity 12.3. The Geometry of Divergence.

Consider the two vector fields in Figure 12.6. In each case, can you find the divergence?

A natural place to start is at the origin. So draw a small box around the origin, as shown in Figure 12.7. ¹ Is there flux across the loop?

Figure 12.7. The same two vector fields, with loops at the origin.

Figure 12.7. The same two vector fields, with loops at the origin.

The figures above help us determine the divergence at the origin, but not elsewhere. The divergence is a function, which can vary from point to point. We therefore need to examine loops which are not at the origin. It is useful to adapt the shape of our loop to the vector field under consideration. Both of our vector fields are better adapted to polar coordinates than to rectangular coordinates, so we use polar boxes. Can you determine the divergence using the loops in Figure 12.8? Imagine trying to do the same thing with a rectangular loop, or even a circular loop.

Figure 12.8. The same two vector fields, with loops not at the origin.

Figure 12.8. The same two vector fields, with loops not at the origin.

Finally, it is important to realize that not all vector fields which point away from the origin have divergence. The example in Figure 12.9 demonstrates this important principle; it has no divergence away from the origin. This figure represents a solution of Maxwell's equations for electromagnetism, and describes the electric field of an infinite charged wire

Figure 12.9. One more vector field.

In two dimensions, both “boxes” are loops. In three dimensions, the oriented box used to measure circulation is still a loop, but the box used to measure flux is now an ordinary, 3-dimensional box.