Integration by parts for Riemannian divergence (${\int }_{M}X(f){\omega }_g=-{\int }_{M}f\mathrm{div}X{\omega }_g$)

On a compact oriented Riemannian manifold without boundary, the Generalized Stokes theorem applied to $f{\iota }_X{\omega }_g$ gives the displayed identity. The product expansion is $d(f{\iota }_X{\omega }_g)=[X(f)+f\mathrm{div}X]{\omega }_g$. With boundary, the omitted boundary integral is ${\int }_{\partial M}f{\iota }_X{\omega }_g$.