Nonexact angular one-form ($\omega =(-ydx+xdy)/(x^2+y^2)$)

On the punctured plane, this one-form is closed and locally the differential of the polar angle. Its line integral around a closed curve is $2\pi$ times the winding number. A once-winding circle therefore has nonzero integral, proving that no global single-valued angle potential exists and the form is not an exact differential form.

For a counterclockwise circle of radius $R>0$, substitute $x=R\cos t$, $y=R\sin t$. The pullback of a differential form isThe radius cancels. In coordinates, $P=-y/(x^2+y^2)$ and $Q=x/(x^2+y^2)$ satisfy ${\partial }_{x}Q={\partial }_{y}P$ away from the origin, so $d\omega =0$ there. The nonzero integral is a global obstruction to being an exact differential form, not a nonzero local exterior derivative.

Compare this with the smooth one-form $\eta =(-ydx+xdy)/2$ on the whole plane. Here $d\eta =dx\wedge dy$, and its integral around a radius-$R$ circle is $\pi R^2$, the enclosed area. The Generalized Stokes theorem applies to the full disc for $\eta$. For $\omega$, that disc crosses the excluded origin, where the one-form is undefined. On an annulus the theorem applies, but its induced inner boundary is clockwise: the two integrals $2\pi$ and $-2\pi$ cancel. This explains exactly why closedness and nonzero circulation can coexist.

For real $a,b\ne 0$, this differential one-form on the punctured plane is the pullback of a differential form of the nonexact angular one-form under the invertible linear transformation $(X,Y)=(ax,by)$. It is a closed differential form for every such pair. On a region admitting a continuous complex argument, it has the local potential $f=\mathrm{Arg}(ax+iby)$. In a chart with $x\ne 0$, this can be written $f=\arctan (by/(ax))+C$.

There is no globally single-valued potential on the whole punctured plane. On the closed path $x=\cos t/a$, $y=\sin t/b$, $0\le t\le 2\pi$, substitution gives ${\omega }_{a,b}=dt$, hence its line integral is $2\pi \ne 0$. This contradicts the zero integral around every closed path required of an exact differential form. Around a positively oriented once-winding path the integral is $2\pi \mathrm{sgn}(ab)$. The local equation ${\omega }_{a,b}=0$ gives radial rays; away from a vertical ray its solutions as graphs are $y=Cx$ on intervals not crossing the excluded origin.