The problem is "Calculating $\oint_{L} \frac{xdy – ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \frac{y^2 – x^2}{(x^2 + y^2)^2} = \frac{\partial{Q}}{\partial{x}}$$
Acorrding to the Green formula:
$$\oint_{L} \frac{xdy – ydx}{x^2 + y^2} = \iint (\frac{\partial{Q}}{\partial{x}} – \frac{\partial{P}}{\partial{y}})dxdy = 0$$
What's wrong with my solution?
Best Answer
That is valid as long as $L$ is a simple closed curve and the origin is not "inside" $L$, i.e., in the bounded region determined by $L$. For example, this would work if $L$ is a unit circle centered at $(2,0)$, but not if it is the unit circle centered at the origin, and not if it is not a simple closed curve.
(As Timothy Wagner already pointed out before I finished writing, you need to check the hypotheses on the theorem you invoked to see when it works.)
If $L$ is closed, your integral should be $2\pi$ times the number of times $L$ winds around the origin.