Math 3593H Honors Mathematics II Spring Semester 2004
Assignment 11 - Due Thursday 4/15/2004
Read: Hubbard and Hubbard Sections 6.5 and 6.6.
Section 6.5 (pages 611-614): 1, 2*, 3, 4, 5, 6*, 7*, 8, 9, 11, 12, 15*, 16, 17, 18*.
Section 6.6 (pages 626-627): 5, 7a*, 7c*, 7d*, 7e*, 8.
Section 6.6 is a complicated treatment of boundaries, of a degree of complication which we do not need in order to understand Stokes' theorem. Unfortunately we do need to know what a manifold with boundary is, because these are essential for the theorem, so we cannot skip Section 6.6. Aside from knowing what a manifold with boundary is, the other thing we need to know is that an orientation of the manifold determines an orientation of the boundary, obtained by standing on the boundary on the side away from the rest of the manifold, and requiring that a basis of the tangent space to the boundary is direct if the basis for the tangent space to the manifold obtained by inserting a vector pointing away from the manifold before the basis for the tangent space to the boundary is direct. The definition of a manifold with boundary we will use is that it is a subset of n-dimensional space which locally can either be parametrized in the usual way, or else locally is the restriction of a usual parametrization to a half space in which the first coordinate is non-negative. The boundary of a k-dimensional manifold is then a (k-1)-dimensional manifold, and we recognize things like a half-space, a solid ball and a solid torus as examples of this.
I feel rather safe in advising you not to read Section 6.6 at all. For the homework exercises I have excluded parts of questions which require you to know what they mean by a 'piece with boundary'.