What is the relationship between k-forms and l-forms on an m-manifold?

[blendecho]

So I was wondering about this... if \omega is a k-form and \eta is a l-form, and m is a k+l+1 manifold in \mathbb{R}^n, what's the relationship between \int_M \omega\wedge d\eta and \int_M d\omega\wedge \eta
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 \int_{\partial M}\omega\wedge\eta=0?

 

[explain]

So I was wondering about this... if \omega is a k-form and \eta is a l-form, and m is a k+l+1 manifold in \mathbb{R}^n, what's the relationship between \int_M \omega\wedge d\eta and \int_M d\omega\wedge \eta
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 \int_{\partial M}\omega\wedge\eta=0?

I think it's basically integration by parts. You start with the identity
\textrm{d}(\omega\wedge \eta) = \textrm{d}\omega\wedge \eta+(-1)^{k}\omega\wedge\textrm{d}\eta. Then you integrate both sides over M, taking into account that \int_M \textrm{d}(...)=0 since \partial M=0.