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First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see page 105 of these notes.

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First of all, the things that you actually integrate are densities, which (this is are the differential geometric equivalent counterparts of a measuremeasures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see page 105 of these notes.

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First of all, the things that you actually integrate are densities, which (this is the differential geometric equivalent of a measure. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see page 105 of these notes.