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Math Help - "Verify by direct computations".Topology exercise.

  1. #1
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    "Verify by direct computations".Topology exercise.

    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
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  2. #2
    GJA
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    Re: "Verify by direct computations".Topology exercise.

    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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