Integration by Parts

Recall the product rule \begin{equation} d(\alpha\wedge\beta) = d\alpha\wedge\beta + (-1)^p \alpha\wedge d\beta \end{equation} for a $p$-form $\alpha$ and $q$-form $\beta$. Rearranging and integrating this expression, we obtain \begin{align} \int_R d\alpha\wedge\beta &= \int_R d(\alpha\wedge\beta) - (-1)^p \int_R \alpha\wedge d\beta \nonumber\\ &= \int_{\partial R} \alpha\wedge\beta - (-1)^p \int_R \alpha\wedge d\beta \end{align}

A special case is of course the usual formula for integration by parts, namely \begin{equation} \int_C f\,d g = \int_C (dg\>f) = \int_{\partial C} g\,f - \int_C g\,df \end{equation} and a more interesting example is \begin{equation} \int_D f\,d\alpha = \int_{\partial D} f\,\alpha - \int_D df\wedge\alpha \end{equation} for any function $f$ and $p$-form $\alpha$.

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