Integration by parts for Riemannian divergence
= Integration by parts for Riemannian divergence
{title2=$\int_M X(f)\omega_g=-\int_M f\operatorname{div}X\,\omega_g$}
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 $d(f\iota_X\omega_g)=[X(f)+f\operatorname{div}X]\omega_g$. With boundary, the omitted boundary integral is $\int_{\partial M}f\iota_X\omega_g$.