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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.

An exact differential is a one-form equal to $d\Phi$ for a single-valued scalar function $\Phi$ on the domain. In Euclidean coordinates this means its vector coefficients form $\nabla \Phi$. Integrating the first component of $F$ with respect to $x$, then matching the other components, givesDifferentiating checks all three components. Since the domain ${\mathbb{R}}^3$ is connected, two such potentials differ only by a constant, so this is the most general potential.

At both specified endpoints the nonconstant terms vanish. The fundamental theorem for line integrals gives zero for the line integral along every path joining the two points.

For the planar field, choose a continuous lift $\theta (t)$ of the polar angle along a curve avoiding the origin. Differentiating the angle givesOn the closed curve the angle increases by $2\pi$ times its winding number. ThusThe angle differential is locally exact but not globally exact on the punctured plane: the angle lift need not return to its initial value. This is the nonexact angular one-form, whose nonzero closed-path integral obstructs a global potential.