Visualizing Curl

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Section 12.13 Visualizing Curl

Recall that

\begin{equation} (\grad\times\FF) \cdot \nn \approx \frac{\oint \FF \cdot d\rr}{\textrm{area of loop}} = \frac{\textrm{(oriented) circulation}}{\textrm{unit area}}\tag{12.13.1} \end{equation}

so that the (orthogonal component of the) curl measures how much a vector field “goes around” a loop.

Figure 12.14. Two vector fields.

Figure 12.14. Two vector fields.

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

Activity 12.6. The Geometry of Curl.

Consider the two vector fields in Figure 12.14. In each case, can you find the (\(\kk\)-component of the) curl?

A natural place to start is at the origin. So draw a small box around the origin, as shown in Figure 12.15. Is there circulation around the loop?

Figure 12.15. The same two vector fields, with loops at the origin.

Figure 12.15. The same two vector fields, with loops at the origin.

The figures above help us determine the curl at the origin, but not elsewhere. The curl is a vector field, 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 (\(\kk\)-component of the) curl using the loops in Figure 12.16? Imagine trying to do the same thing with a rectangular loop, or even a circular loop.

Figure 12.16. The same two vector fields, with loops not at the origin.

Figure 12.16. The same two vector fields, with loops not at the origin.

Finally, it is important to realize that not all vector fields which go around the origin have curl. The example in Figure 12.17 demonstrate this important principle; it has no curl away from the origin. This figure represents a solution of Maxwell's equations for electromagnetism, and describes the magnetic field due to an infinite current-carrying wire (with current coming out of the page at the origin).

Figure 12.17. One more vector field.