TITLE: integrating by parts on a manifold
QUESTION [5 upvotes]: Suppose $M$ is compact. Let $\phi$ be some smooth function, and $\beta$ an $n-1$-form. Then does integration by parts say that $$\int_M\phi d\beta=\int_Md\phi\wedge\beta?$$
If not, how does integration by parts lets you rewrite the integral $\int_M\phi d\beta$?

REPLY [6 votes]: Integration by parts follows from Stokes' Theorem (Stokes). If $M$ is compact and closed, i.e. with empty boundary, you have that
$$\int_{M}d\omega=0$$
for every $n-1$-form. Let $\omega=\phi\beta$, then you have
$$0=\int_{M}d(\phi\beta)=\int_M\phi d\beta+\int_Md\phi\wedge \beta$$
hence
$$\int_M\phi d\beta=-\int_Md\phi\wedge\beta\;.$$