Given the 2-form $ \omega= \frac{x}{r^3} \, dy \, dz + \frac{y}{r^3} \, dz \, dx + \frac{z}{r^3} \, dx \, dy$, defined on the spherical shell $U=\{(x,y,z) : a^2 < x^2 + y^2 +z^2<b^2 \}$ with $0<a<b$ , and $r=\sqrt{x^2+y^2+z^2}$, why is $\omega$ not exact.
I've been able to show that $\omega$ is closed. My idea was to use Stoke's theorem and find some manifold, $M$, and then show that $\int_{\partial M} w \neq 0$.
I've tried having $M$ be the area between two concentric spheres and then $\partial M$ just the inner and outer spheres but for those I got $\int _{\partial D} w = 0$. I've also tried some variations on those with $M$ being the area between two half spheres but again got $0$.
Any help on what $M$ I should try or some alternative approach would be greatly appreciated.