From Munkres "Analysis on Manifolds" Consider the form $ \omega = xydx + 3dy -yzdz $. Check by direct computation that $ d(d\omega) = 0 $. Can someone show me how to do it, because I don't seem to be getting how to compute these differentials...
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
Differentiating means exactly what you do with usual functions. The only additional rule is: $d^2x=d^2y=d^2z=0$. Therefore, $dx\wedge dy=-dy\wedge dx$, etc.
$$ d\omega = d(xy\,dx) + 3\,d^2y - d(yz\,dz) \\ = dx\wedge y\,dx +x\,dy\wedge dx + 0 - dy\wedge z\,dz - y\,dz^2 =\\ = 0 -x\,dx\wedge dy - z\,dy\wedge dz - 0=\\ = -x\,dx\wedge dy - z\,dy\wedge dz.$$
$$ d^2\omega = -dx\wedge dx\wedge dy - dz\wedge dy\wedge dz = 0+0=0.$$