Orthogonality of Eigenfunctions of Compact Riemannian Manifold without Boundary

Theorem
Let $\struct {M, g}$ be a compact Riemannian manifold without boundary.

Let $u, v \in \map {C^\infty} M : M \to \R$ be smooth mappings on $M$.

Suppose $u$ and $v$ are eigenfunctions of $M$ with distinct eigenvalues.

Then:


 * $\ds \int_M u v \rd V_g = 0$

where $\rd V_g$ is the Riemannian volume form.