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Switching d

  1. May 5, 2007 #1
    So I was wondering about this... if [tex]\omega[/tex] is a [tex]k[/tex]-form and [tex]\eta[/tex] is a [tex]l[/tex]-form, and [tex]m[/tex] is a [tex]k+l+1[/tex] manifold in [tex]\mathbb{R}^n[/tex], what's the relationship between [tex]\int_M \omega\wedge d\eta[/tex] and [tex]\int_M d\omega\wedge \eta[/tex]
    given the usual niceness of things being defined where they should be, etc. etc. The manifold has no boundary, so am I correct in writing [tex]\int_{\partial M}\omega\wedge\eta=0[/tex]?
    Last edited: May 5, 2007
  2. jcsd
  3. May 8, 2007 #2
    I think it's basically integration by parts. You start with the identity
    [tex] \textrm{d}(\omega\wedge \eta) = \textrm{d}\omega\wedge \eta+(-1)^{k}\omega\wedge\textrm{d}\eta [/tex]. Then you integrate both sides over M, taking into account that [tex]\int_M \textrm{d}(...)=0[/tex] since [tex]\partial M=0[/tex].
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