6Integration

III Differential Geometry



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.

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