To state the answer to this question as easily as possible, circulation is the line integral of a vector field around a closed path. That's it. As such, it is another way to measure the amount of ``swirl'' in a vector field. Often, circulation is noted by a modified line integral:
Example 1. The circulation of the vector field around the path shown is zero. We can see this by first rewriting the circulation as a sum of line integrals along each separate piece of the curve. Thus,
Now, since the vector field is perpendicular to the path along C1 and C3, their contribution to the integral is zero. Since the magnitude of is the same along C2 and C4 but they are traversed in opposite directions, we have that
so that each of these integrals cancels the other and the total is zero.
Example 2. The circulation of around the curve C shown below. This curve is broken up into three segments:
We can break the integral up into three pieces: Calculating each piece separately, we find:
Thus, the total circulation of around this curve is the sum of these values, namely, .
Example 3. What about the circulation of a conservative vector field? This is obviously zero since so that and the curve C starts and ends at the same spot (P = Q.)