# Integration by parts on manifold with corners

Suppose that $$M$$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e. \begin{align*} \int_M g(\nabla f, X) \,d\textrm{vol} = \int_{\partial M} f\left\langle X,N \right\rangle \,d\textrm{vol}_{\tilde{g}} - \int_M f\cdot (\operatorname{div} X ) \,d\textrm{vol} \end{align*} with $$f\in C^\infty(M), X\in \Gamma(M,TM)$$. I know that this identity hold for domains with lipschitz boundary, but it is not very clear to me if a domain with corners is a special case of a lipschitz domain.

• I think the second term on the RHS should read $\int_Mf\text{div}X d\text{vol}$. – S.Surace May 16 at 13:58
• Thanks, I fixed it. – Δημήτρης Ο May 16 at 14:02