"Verify by direct computations".Topology exercise.

Consider the forms

$\displaystyle \omega =xy dx +3dy - yz dz$

$\displaystyle \eta = xdx - yz^2 dy +2x dz$,

in $\displaystyle \mathbb{R}^3$. Verify by direct computations that

$\displaystyle d(d\omega) = 0$ and $\displaystyle d(\omega \wedge \eta) = (d\omega) \wedge \eta -\omega \wedge d\eta$

Probably doing a mistake in computations

Please, help

Re: "Verify by direct computations".Topology exercise.

Hi vercammen,

I'll do $\displaystyle d(d\omega)=0. $

$\displaystyle d\omega = d(xy)\wedge dx + d(3)\wedge dy - d(yz)\wedge dz $

$\displaystyle = x dy\wedge dx - z dy\wedge dz. $

Then

$\displaystyle d(d\omega) = dx\wedge dy\wedge dx - dz\wedge dy\wedge dz = 0 - 0 = 0. $

Proving the second one is similar to this, just more steps. Does this get things going in the right direction? Good luck!