# Exercises: Chapter 5, Section 3

1. If is an -dimensional manifold (or manifold-with-boundary) in , with the usual orientation, show that , as defined in this section, is the same as , as defined in Chapter 3.

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.

1. Show that Theorem 5-5 is false if is not required to be compact.

For example, if we let be the open interval , one has but . One can also let and .

2. Show that Theorem 5-5 holds for noncompact provided that vanishes outside of a compact subset of .

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 .

2. If is a -form on a compact -dimensional manifold , prove that . Give a counter-example if is not compact.

One has as is empty. With the set of positive real numbers, one has with that .

3. An absolute -tensor on is a function of the form for . An absolute -form on is a function such that is an absolute -tensor on . Show that can be defined, even if is not orientable.

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 .

4. If is an -dimensional manifold-with-boundary and is an -dimensional manifold with boundary, and are compact, prove that

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.