Define $\omega$ on $\mathbb{R}^3$ by $\omega = x\,dy\wedge dz + y\,dz\wedge dx + z\,dx\wedge dy$.
Thus far I have computed $\omega$ in spherical coordinates $(\rho,\phi,\theta)$, as well as computed $d\omega$ in both Cartesian and spherical coordinates. I found $\omega=\rho^3\sin\phi\,d\phi\wedge d\theta$.
But now I'm asked to compute the restriction $\omega|_{S^2} = \iota^*\omega$, where $\iota:S^2\to\mathbb{R}^3$ is the inclusion map, using coordinates $(\phi,\theta)$ on the open subset where they are defined.
So far, this is all I have:
Fix $p\in S^2$ and consider the basis $\left(\frac{\partial}{\partial\phi},\frac{\partial}{\partial\theta}\right)$ on $T_pS^2$. Then
$(\iota^*\omega)_{(\phi,\theta)}\left(\frac{\partial}{\partial\phi},\frac{\partial}{\partial\theta}\right)=\omega_{\iota(\phi,\theta)}\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)=\sin\phi\,d\phi\wedge d\theta\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)\,.$ And I suppose I also know that $-\frac{\pi}{2}<\phi<\frac{\pi}{2}$ and $0<\theta<2\pi$.
I've done so few examples, though, that it's unclear to me where to go from here.
Any help is appreciated