One of the basic properties of the Laplacian is that given a compact Riemannian manifold-with-boundary (to which all this business applies equally), then for vanishing on the boundary, the inner product is fairly large relative to . As an immediate corollary, if satisfies the Laplace equation and vanishes on the boundary, then is identically zero.

It turns out that the proof of this will require the divergence theorem. This is a familiar fact from multivariable calculus, but it generalizes to -dimensions nicely as a corollary of Stokes theorem and some of the other machinery thus developed.

So, let’s choose an oriented Riemannian manifold of dimension with boundary . There is a volume form because of the choice of orientation globally defined. On , there is an induced Riemannian metric and an induced orientation, with a corresponding volume form on . If is a compactly supported vector field, the **divergence theorem** states that