Math 3592H Honors Mathematics I Fall Semester 2005

Assignment 12 - Due Tuesday 12/6/2005. There will be no quiz on this day. The third midterm exam is on December 8. You will be tested on the material in Sections 2.1 - 2.7 which we have studied.

Read: Hubbard and Hubbard Section 2.6, 2.7 (pages 237-242 and 250-253 only), then (if we get there) we will start on 3.3.

Exercises (Due Tuesday 12/6/2005):
Hand in only the exercises which have stars by them.

Section 2.6 (pages 235-237): 1, 2, 3, 4*, 5*, 6, 7*, 8*, 11*
(We have really done question 9 in class with subspaces of R^n, and the argument in abstract is the same. So far as I can see question 10 is pretty much the same as question 5, but maybe I am missing something.)
2.7 (pages 255-257): 2*, 4, 7a*, 10, 12a

The following exercises are relevant for the material we do this week. I list them because they may be useful practice for the mid-term exam.
2.10 (pages 285-290); 8, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22,

Comments:
The culmination of Chapter 2 is the discussion of the implicit and inverse function theorems in Section 2.9. These theorems have theoretical importance in that they allow us to deduce the equivalence of two different definitions of a manifold, and they appear here in the book because at the start of Chapter 3 we define manifolds. Usually in a course at this level these theorems are stated but not proved. Our book does provide a proof, but sections of it are relegated to the Appendix, and in any case I think a more streamlined proof could be given.

I think we should not study the theorems in Section 2.9 this semester, and in any case I do not think we should spend any time with the technical difficulties of proving them, and so I am going to be very selective in the material we study. We will not do Kantorovich's theorem in Section 2.7. That theorem does use the important notion of a Lipschitz condition, and it is a pity not to know about that, but I think we should move on. Section 2.8 on superconvergence we will not do at all.