# Switching d

1. May 5, 2007

### 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 $$\int_{\partial M}\omega\wedge\eta=0$$?

Last edited: May 5, 2007
2. May 8, 2007

### explain

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 $$\partial M=0$$.