Math 3593H Honors Mathematics II Spring Semester 2004
Assignment 1 - Due Thursday 1/29/2004
Read: Hubbard and Hubbard Sections 3.1 and 3.2. I hope that we will cover 3.1 in class on Wednesday 21 and Friday 23 and do 3.2 on Monday 26.
Hand in only the exercises which have stars by them. I list all the questions which I think are reasonable for you to look at, and it is not my intention that you should actually do them all. Try some of them, skim others, and use your judgement.
Section 3.1 (pages 311-316): 1, 2, 3, 4, 5*, 7*, 8, 11*, 13, 14, 15, 16, 19*, 22*, 23, 24, 28
Seciont 3.2 (pages 321-323): 1*, 2, 3, 4, 5a, 6*
A. Find the equation for the tangent line at the point (2,4,8) to the curve parametrized in question 3.1.11a
B. Find the equation for the tangent plane at the point (cos(1), sin(1), 1) to the surface parametrized in question 3.1.12.
C*. Find the equation for the tangent plane at the point (sin(2)+1, 3, 2) to the surface parametrized in question 3.1.28.
The idea of a smooth manifold is intuitively not difficult, but making it precise in mathematical terms appears to be quite complicated, according to Section 3.1. Not only that, but although you might think Section 3.1 is giving you the complete picture, in fact it is not. It is one of those situations we have discussed a number of times, where you are told something but only discover later that there is more to the story than you were told the first time round.
The definition in 3.1 specifies that a manifold should be given as a subset of some ambient space. In general manifolds are defined more abstractly without specifying a particular embedding of the manifold into a larger space. Although the abstract definition is actually easier to work with in some ways, it does raise the question of whether every manifold can be embedded in R^n, and the definition of the tangent space becomes more difficult. The book avoids these issues, and probably rightly so. The course where this approach is taught is the graduate level course, 'Manifolds and Topology'.
As it is, we need to be able to work with manifolds given by means of parametrizations, and also as the set of solutions of an equation, and this is what most of the exercises to 3.1 are about.