A current runs through it

Consider a smooth simple closed curve $C$ in $\mathbb{R}^3$ that does not intersect the $z$ -axis. Let's parameterize this curve, $\vec{s}(t)$ , with $0\leq t \leq 1$ . We can write this parameterization in terms of polar coordinates, $x=r(t)\cos(\theta(t)), y=r(t)\sin(\theta(t))$ , where the phase angle $\theta$ is measured continuously along the curve, of course. A straightforward if slightly tedious computation shows that $\int_{C}\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy=\theta(1)-\theta(0)$ , the change of $\theta$ along the way. Since $\vec{s}(1)=\vec{s}(0)$ for a closed curve, these two polar angles will differ by a multiple of $2\pi$ ,showing that $\oint_C \vec{F}\cdot d\vec{s}$ is an integer multiple of $2\pi$ . Let's mention, in passing, that $\frac{\theta(1)-\theta(0)}{2\pi}$ is called the "winding number" of the curve $C$ .

It is easy to construct simple closed curves with any winding number we want by considering helices $\vec{s}(t)=(\cos t,\sin t,t)$ for $0\leq t\leq 2n\pi$ for any positive integer $n$ (reverse the orientation to obtain negative winding numbers). To complete those helices to simple closed curve, add an arc in the $xz$ -plane.