Orientations

Abstract

When we introduced differential forms, we suggested that they should be objects that can be integrated over manifolds in an invariant way. Before we can pursue that, though, we need to address a serious issue that we have so far swept under the rug. This is the small matter of the positive and negative signs that arise when we try to interpret a k-covector as a machine for measuring k-dimensional volumes. In the previous chapter we brushed this aside by saying that the value of a k-covector applied to a k-tuple of vectors has to be interpreted as a “signed volume” of the parallelepiped spanned by the vectors. These signs will cause problems, however, when we try to integrate differential forms on manifolds, for the simple reason that the transformation law for an n-form under a change of coordinates involves the determinant of the Jacobian, while the change of variables formula for multiple integrals involves the absolute value of the determinant.