"Verify by direct computations".Topology exercise.

**vercammen** · December 5th 2012, 07:22 PM

Consider the forms

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

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

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

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

Probably doing a mistake in computations
Please, help

**GJA** · December 6th 2012, 07:00 PM

Hi vercammen,

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

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

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

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

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

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