Integration on manifold

Question

How can we generalised those integration by parts to arbitrary manifold? $$\int_{+\infty} ^{-\infty} f(x). g(x) \,dx$$ $$= f(x)\int_{+\infty}^{-\infty} g(x)-\int_{+\infty}^{-\infty} [f'(x)\int g(x)dx]\,dx$$

Answer 1

You use Stokes' theorem: $$ \int_\Omega d\omega= \int_{\partial \Omega} \omega $$ If you are not familiar with differential forms, you can also use the usual form of the divergence theorem $$ \int_M \nabla_\mu J^\mu \sqrt{g}d^nx = \int_{\partial M} J^\mu dS_\mu, $$ together with the fact that $\nabla_\mu$ is a derivation.