Consider the 1-form $\alpha = xdz + ydw -(x^2 + y^2 + z^2 + w^2)dt$ on $\mathbb{R}^5$.
I'm trying to find $\int_S d\alpha \wedge d\alpha$, where $S \subset \mathbb{R}^5$ is given by $x^2 + y^2 + z^2 + w^2 =1$ and $0\leq t \leq 1$.
Restricting to $S$,we get $\alpha = xdz + ydw -dt$, so $d\alpha = dx\wedge dz + dy \wedge dw$.
Now $d\alpha \wedge d\alpha$ = $d(\alpha \wedge d\alpha)$, so by Stokes' Theorem, $\int_S d\alpha \wedge d\alpha = \int_{\partial S} \alpha \wedge d\alpha$.
I found that $\alpha \wedge d\alpha = xdz\wedge dy \wedge dw + ydw\wedge dx \wedge dz - dt \wedge dx \wedge dz - dt \wedge dy \wedge dw$.
The boundary $\partial S$ of $S$ is the disjoint union $S^3 \times \{0\} \cup S^3 \times \{1\}$, where $S^3$ is the unit sphere in the $x,y,z,w$-subspace of $\mathbb{R}^5 = \{(x, y, z, w, t)\}$.
How do I integrate this 3-form $\alpha \wedge d\alpha$ over a boundary sphere of $S$?
(This isn't homework! I've never seen explicitly how to integrate differential forms over a manifold, so a worked example would be really helpful.)