We can assume in the situation of Chapter 3 that has the usual orientation. The singular -cubes with can be taken to be linear maps where and are scalar constants. One has with , that . So, the two integrals give the same value.
For example, if we let be the open interval , one has but . One can also let and .
The compactness was used to guarantee that the sums in the proof were finite; it also works under this assumption because all but finitely many summands are zero if vanishes outside of a compact subset of .
One has as is empty. With the set of positive real numbers, one has with that .
Make the definition the same as done in the section, except don't require the manifold be orientable, nor that the singular -cubes be orientation preserving. In order for this to work, we need to have the argument of Theorem 5-4 work, and there the crucial step was to replace with its absolute value so that Theorem 3-13 could be applied. In our case, this is automatic because Theorem 4-9 gives .
where is an -form on , and and have the orientations induced by the usual orieentations of and .
Following the hint, let . Then is an -dimensional manifold-with-boundary and its boundary is the union of and . Because the outward directed normals at points of are in opposite directions for and , the orientation of are opposite in the two cases. By Stokes' Theorem, we have . So the result is equivalent to . So, the result, as stated, is not correct; but, for example, it would be true if were closed.