I have an issue with a differential geometry task. Given is the solid angle form:
$$\omega = \frac{\epsilon_{ijk} x^i dx^j \wedge dx^k}{2 [ (x^1)^2 + (x^2)^2 +(x^3)^2]^{3/2}}$$
The aim of the task is to integrate this form over the unity sphere $S^2$. Now let $h_+^{-1}$ be the inverse stereographic projection map (without north): $ h_+^{-1}: \mathbb{R} \rightarrow S^2 \setminus N$, $h_+^{-1}=\left(\frac{2y^1}{||y||^2+1} , \frac{2y^2}{||y||^2+1}, \frac{||y||^2-1}{||y||^2+1} \right)$.
Task a) is to compute the pullback $h_+^{-1}* (\omega)$ and task b) is to integrate the form over the sphere.
Now: I know how to compute a pullback and I know that I can integrate the pulled-back form over the changed domain. But the calculation for the pullback seems to be much too long. I do not think that I'm doing it correctly by straight-forwardly calculating the pullback. I didn't finish the calculation yet, but the result seems to be in a very ugly form, which I don't know to integrate either. Can someone please help me and give me a hint how this should be solved?
Parametrize in spherical coordinates!