[Math] integrating by parts on a manifold

differential-geometryintegration

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 $\int_M\phi d\beta$?

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 $$\int_M\phi d\beta=-\int_Md\phi\wedge\beta\;.$$