6 Integration

As promised, we will be able to integrate differential forms on manifolds. However,

there is a slight catch. We said that differential forms give us the signed volume of

an infinitesimal parallelepiped, and we can integrate these infinitesimal volumes

up to get the whole volume of the manifold. However, there is no canonical

choice of the sign of the volume, so we do not, in general, get a well-defined

volume.

In order to fix this issue, our manifold needs to have an orientation.