INSTRUCTOR: Calculate the circulation of F of x comma y equal to angle bracket negative y, over the quantity x squared plus y squared, comma, the fraction x divided by x squared plus y squared, along a unit circle oriented counterclockwise. Now the unit circle oriented counterclockwise is important because that tells us our parameterization is r of t equal to angle bracket cosine of t, comma, sine of t, with t from 0 to 2 pi.

Now let's go ahead and find f of r of t, now that we have our parameterization, since we will need that. So F of r of t will be equal to-- well, let's see. Our numerator will be a negative y so that's negative sine t. And our denominator be x squared plus y squared. Well, those are cosine and sine, respectively. So that would just be equal to 1 actually. So this would be equal to negative sine t.

And our second component will be x over x squared plus y squared. Well, again, x squared plus y squared is going to be equal to 1. So this will be equal to our x component, which is cosine, of t. All right.

Now we also need to find r prime of t. I should have done that earlier, but r prime of t is equal to negative sine of t, comma, cosine of t. Now we need that because the way this is defined, we're going to take the integral to find the circulation of F of r of t, dotted with r prime of t, with respect to t.

So our circulation from 0 to 2 pi with respect to t will be angle bracket negative sine t, comma, cosine t, dotted with angle bracket negative sine t, comma, cosine t. Now the dot product of those two vectors is the circulation from 0 to 2 pi of sine squared t, plus cosine squared t, with respect to t, which will be equal to 2 pi.