Math 3593H Honors Mathematics II Spring Semester 2004
Assignment 8 - Due Thursday 3/18/2004
Read: Hubbard and Hubbard Sections 5.3, beginning of 6.1
Section 5.3 (pages 550-553): 1*, 2, 3, 4, 5, 6*, 7, 8*, 9a, 10, 11, 15, 16*, 18, 20*.
Section 6.1: 1, 2, 3*, 4a*.
I see now why they wanted to consider the volume of a k-parallelogram in Section 5.1. It is because they use it in Section 5.3, and this seems quite a good intuitive way to go about doing things. They do however introduce some more non-standard notation in Section 5.3 in equation 5.3.2, but we can live with it.
Having read through Section 5.2 myself, I think it is best if we miss it out. I find it to be rather inconclusive, fussy, complicated and hard to understand. They are dealing with the problem that the notion of parametrization, as introduced previously, is too restrictive. In that previous sense, most manifolds do not have a parametrization. The reason they are troubled by this is that the volume of a manifold depends on the way it sits inside its ambient space, and we have to get at it by means of parametrizations. In fact it is not necessary to deal with this matter in the way they suggest in Section 5.2. In the graduate course 'Manifolds and Topology' a different approach is adopted, but this is too much for us to take on board here. I think it is sufficient for us to deal with manifolds which, apart from a set of volume zero (measured in the dimension of the manifold), are made up of a finite number of pieces, each of which has a parametrization. To do an integral over the manifold, we do an integral on each piece, and add up the answers.