Computing the restriction of a differential form 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
 A: You need to write down the inclusion map $\iota: S^2\rightarrow\mathbb{R}^3$ which is given by 
$$\iota(\phi,\theta)=(\cos\phi\cos\theta,\cos\phi\sin\theta,\sin\phi).$$
Therefore, the differential of $\iota$ is given by
$$d\iota=\left[
                           \begin{array}{cc}
                            -\sin\phi\cos\theta  & -\cos\phi\sin\theta \\
                             -\sin\phi\sin\theta & \cos\phi\cos\theta \\
                             \cos\phi & 0 \\ 
                           \end{array}
                         \right],$$
which implies that
$$\iota_*\left(\frac{\partial}{\partial\phi}\right)=-\sin\phi\cos\theta\frac{\partial}{\partial x}-\sin\phi\sin\theta\frac{\partial}{\partial y}+\cos\phi\frac{\partial}{\partial z},$$
$$\iota_*\left(\frac{\partial}{\partial\theta}\right)=-\cos\phi\sin\theta\frac{\partial}{\partial x}+\cos\phi\cos\theta\frac{\partial}{\partial y}.$$
Therefore,
$$dy\wedge dz\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)=-\cos^2\phi\cos\theta$$
because 
\begin{align}
dy\wedge dz\left(X,Y\right)= \det \left[
                           \begin{array}{cc}
                            dy(X)  & dz(X) \\
                             dy(Y) & dz(Y) \\
                           \end{array}
                         \right].
\end{align}
Similarly, we have
\begin{align}
dz\wedge dx\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)&=-\cos^2\phi\sin\theta,\\
dx\wedge dy\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right) &=-\sin\phi\cos\phi.
\end{align}
Hence, 
$$(\iota^*\omega)_{(\phi,\theta)}\left(\frac{\partial}{\partial\phi},\frac{\partial}{\partial\theta}\right)=\omega_{\iota(\phi,\theta)}\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)$$
$$=\cos\phi\cos\theta dz\wedge dy\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)+\cos\phi\sin\theta dz\wedge dx\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)$$
$$+\sin\phi dx\wedge dy\left(\iota_*\left(\frac{\partial}{\partial\phi}\right),\iota_*\left(\frac{\partial}{\partial\theta}\right)\right)$$
$$=-\cos^3\phi\cos^2\theta-\cos^3\phi\sin^2\theta-\sin^2\phi\cos\phi=-\cos\phi.$$
