Non-vanishing volume form on $S^2$ I have been given the form $\mu=x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy\in\Omega^2(S^2)$. I am asked to prove that this form is never vanishing. That is, $\nexists p\in S^2$ such that $\mu_p=0$. Intuitively, I would say that th eonly point where $\mu$ vanishes is $(0,0,0)$ which is not in $S^2$, so $\mu$ is nowhere 0. However, I know there are expressions of the form $xdx+ydy+zdz$ which are 0 on $S^2$. How could I prove that $\mu$ is nowhere vanishing?

HINT: What is the unit normal vector to the sphere at $p=(x,y,z)$? So, if $v,w$ give a basis for $T_pS^2$, what is $\mu(p)(v,w)$?
