Define a 1-form $ω$ on the punctured plane $R^2-\{0\}$ by
$ω(x,y)=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2 }dy $
a) Calculate $∫_Cω$ for any circle C of radius r around the origin
b) Prove that in the half plane $\{x>0\}$, ω is the differential of a function.
c)Why isn’t $ω$ the differential of a function globally on $R^2-{0}$?
for a), it's easy, just use the polar coordinate, and I got the result is $2\pi$
for c), it's also easy, because if $ω$ is the differential of a function globally on $R^2-{0}$ by Stoke's theorem
$$\int_{C=\partial (R^2-0)} \omega =\int_{R^2-0} d\omega$$
but $d\omega=0$ so
$$\int_{C=\partial (R^2-0)} \omega =\int_{R^2-0} d\omega=0$$
but from part a)
$$\int_{C=\partial (R^2-0)} \omega =2\pi \not =0$$
for part b), the book told me to use $arctan(\frac y x)$. But I'm not sure what they mean by that.