Even in the presence of an orientation, there is in general no meaningful way to integrate -forms over subsets for because there is no consistent way to use the ambient orientation to orient -dimensional subsets. Geometrically, a -dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare the Gram determinant of a set of vectors in an -dimensional space, which, unlike the determinant of vectors, is always positive, corresponding to a squared number. An orientation of a -submanifold is therefore extra data not derivable from the ambient manifold.