General Relativity: An Introduction for Physicists · Problem 3.12

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · divergence-theorem-manifold

Problem

If a vector field

v^a

vanishes on a hypersurface

S

that bounds a region

V

of an

N

-dimensional manifold, show that

\int_V (\nabla_a v^a) \sqrt{-g}\, d^N x = 0.