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:

(1) |

**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,

(2) |

Now, since the vector field is perpendicular to the path along
*C _{1}* and

(3) |

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*.)