Exercises: Chapter 5, Section 3

  1. If ch5c1.png is an ch5c2.png -dimensional manifold (or manifold-with-boundary) in ch5c3.png , with the usual orientation, show that ch5c4.png , as defined in this section, is the same as ch5c5.png , as defined in Chapter 3.

    We can assume in the situation of Chapter 3 that ch5c6.png has the usual orientation. The singular ch5c7.png -cubes with ch5c8.png can be taken to be linear maps ch5c9.png where ch5c10.png and ch5c11.png are scalar constants. One has with ch5c12.png , that ch5c13.png . So, the two integrals give the same value.

    1. Show that Theorem 5-5 is false if ch5c14.png is not required to be compact.

      For example, if we let ch5c15.png be the open interval ch5c16.png , one has ch5c17.png but ch5c18.png . One can also let ch5c19.png and ch5c20.png .

    2. Show that Theorem 5-5 holds for noncompact ch5c21.png provided that ch5c22.png vanishes outside of a compact subset of ch5c23.png .

      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 ch5c24.png vanishes outside of a compact subset of ch5c25.png .

  3. If ch5c26.png is a ch5c27.png -form on a compact ch5c28.png -dimensional manifold ch5c29.png , prove that ch5c30.png . Give a counter-example if ch5c31.png is not compact.

    One has ch5c32.png as ch5c33.png is empty. With ch5c34.png the set of positive real numbers, one has with ch5c35.png that ch5c36.png .

  4. An absolute ch5c37.png -tensor on ch5c38.png is a function ch5c39.png of the form ch5c40.png for ch5c41.png . An absolute ch5c42.png -form on ch5c43.png is a function ch5c44.png such that ch5c45.png is an absolute ch5c46.png -tensor on ch5c47.png . Show that ch5c48.png can be defined, even if ch5c49.png 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 ch5c50.png -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 ch5c51.png with its absolute value so that Theorem 3-13 could be applied. In our case, this is automatic because Theorem 4-9 gives ch5c52.png .

  5. If ch5c53.png is an ch5c54.png -dimensional manifold-with-boundary and ch5c55.png is an ch5c56.png -dimensional manifold with boundary, and ch5c57.png are compact, prove that

    ch5c58.png

    where ch5c59.png is an ch5c60.png -form on ch5c61.png , and ch5c62.png and ch5c63.png have the orientations induced by the usual orieentations of ch5c64.png and ch5c65.png .

    Following the hint, let ch5c66.png . Then ch5c67.png is an ch5c68.png -dimensional manifold-with-boundary and its boundary is the union of ch5c69.png and ch5c70.png . Because the outward directed normals at points of ch5c71.png are in opposite directions for ch5c72.png and ch5c73.png , the orientation of ch5c74.png are opposite in the two cases. By Stokes' Theorem, we have ch5c75.png . So the result is equivalent to ch5c76.png . So, the result, as stated, is not correct; but, for example, it would be true if ch5c77.png were closed.