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This is a question found in Theodore Gamelin's Complex Analysis, Chapter 3, Section 2.

We are given the differential $$\frac{-ydx+xdy}{x^2+y^2}\quad \text{where }\ (x,y)\neq(0,0)$$ The first part says to show that the differential is closed, which I've already done.

The second part says to show that the line integral in any annulus centered at $0$ is not independent of path. I need help for this second part.

I tried evaluating the line integral over the piecewise boundaries $|z|=R$ and $|z|=r$. When calculating the latter, I put a negative sign since the inner circle should have the opposite orientation. But both integrals came out to be $2\pi$ and hence the sum equaled $0$.

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    The obvious thing to do is to try evaluating the integral along a variety of paths. So which have you tried?2012-10-22
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    I tried evaluating the line integral over the piecewise boundaries $|z| = R$ and $|z| = r$. When calculating the latter, I put a negative sign since the inner circle should have the opposite orientation. But both integrals came out to be $2\pi$ and hence the sum equaled 0.2012-10-23
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    You already solved it. You showed that the integral on a circle around the origin is not zero, so it's not independent of path. (If it were path independent, then integrating along a circle should yield zero, since you can pick any two points on the circle and view the circle as the union of two paths between those two points.)2012-10-23
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    Yeah, I actually got it thanks :) I made a mistake with the definition of independence of path. What I did was find the line integral over the boundary of the annulus. To find out that the line integral is independent of path, you just need to show that the line integral over any closed path in D (annulus in this case) is equal to 0. So I needed to find some line integral over a path that does not equal to 0. And I already did that as Alan have explained. Anyways, thank you everyone.2012-10-23
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    If you get familiar to the complex analysis, you will soon recognize the given differential as $dz/iz$. This special differential measures the infinitesimal change in angle, or in other words, the *winding*. This is undisputedly one of the most important object in complex analysis.2012-10-23
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    Just 'wiki'ed about winding numbers. Fascinating. Put a note next to the question, thank you sos440 :)2012-10-23

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