Let $\omega = x \, dy \wedge dz + y \, dz \wedge dx + z \, dx \wedge dy$ be a $2-$form on $\mathbb{R}^3$.
I need to compute $\omega$ in Spherical coordinates. I have thus far
\begin{align} \omega &= x \, dy \wedge dz + y \, dz \wedge dx + z \, dx \wedge dy\\ &=(\rho \sin \phi \cos \theta)\,d(\rho \sin \phi \sin \theta) \, \wedge \, d(\rho \cos \phi)\\ &+(\rho \sin \phi \sin \theta) \, d(\rho \cos \theta) \, \wedge d(\rho \sin \phi \cos \theta)\\ &+(\rho \cos \phi) \, d(\rho \sin \phi \cos \theta) \, \wedge \, d(\rho \sin \phi \sin \theta) \\ &=(\rho \sin \phi \cos \theta)\, (\sin \phi \sin \theta\, d \rho \,+\rho \cos \phi \sin \theta \, d \phi +\rho \sin \phi \cos \theta d \theta) \, \wedge (\cos \phi \, d \rho - \rho \sin \phi \, d \phi)\\ &+(\rho \sin \phi \sin \theta) \,(\cos \phi \, d \rho - \rho \sin \phi \, d \phi) \, \wedge \, (\sin \phi \cos \theta \, d \rho+ \rho \cos \phi \cos \theta \, d \phi - \rho \sin \phi \sin \theta \, d \theta) \\ &+\rho \cos \phi (\sin \phi \cos \theta \, d \rho + \rho \cos \phi \cos \theta \, d \phi - \rho \sin \phi \sin \theta \, d \theta) \, \wedge (\cos \phi \, d \rho - \rho \sin \phi \, d \phi) \end{align}
My question is, anywhere I see three variables, do I need to take the partial with respect to each variable individually ? In that case, this will get very messy.
EDIT: I took partials, did I do okay?