Let $(M,g)$ be a compact, oriented Riemannian manifold with boundary. Let $E,F$ be two Hermitian vector bundles over $M$, and let $P:\Gamma (E) \to \Gamma (F) $ be a linear differential operator of order $k$. Then there is a sesquilinear map $B_L : \Gamma (E) \times \Gamma (F) \to C^\infty(M,\mathbb C)$ such that, for every pair of sections $u \in \Gamma (E)$ and $v \in \Gamma (F)$, the following integration by parts formula holds:
$$\int _M \langle Pu,v \rangle = \int_M \langle u,P^*v \rangle + \int_{\partial M} B_L (u,v)$$
Do you have a reference in which this theorem is proved? Thank you.