Math 3593H Honors Mathematics II Spring Semester 2004

Assignment 2 - Due Thursday 2/5/2004

Read: Hubbard and Hubbard Sections 3.3, 3.4 and 3.5.

Exercises:
Section 3.3 (pages 333-335): 1, 3, 6, 9, 13, 14*
Question 6b should presumably say f(-x) = -f(x).
Seciont 3.4 (page 342): 1, 2*, 3, 4, 5*, 6, 7*, 9
Note that question 3.4.3 is the same as 3.3.13. I can only imagine that in questions 6 and 9 you are supposed to use a form of Taylor's theorem with remainder, which they don't do here.
Section 3.5 (pages 351-353): 1, 2, 3, 3c*, 3d*, 4, 4b*, 5*, 8, 9, 10, 11, 17, 17a*, 17b*, 17e*
3d is the same as 5b and 10d.

Peter's Paragraphs
I find the sections this week frustrating in that there is not that much in them, really, but there is a lot to read and the authors spend a long time agonizing over things which I think are clear. The most important things in Section 3.3 are Theorem 3.3.9 about equality of the mixed derivatives when second partials are continuous (but they only prove it in an appendix), the form of the multivariable Taylor polynomial, and how to compute the coefficients in 3.3.12. They spend a very long time introducing multi-exponent notation, as though it is difficult, and I am not even sure we need to trouble ourselves with it.

I imagine that Section 3.4 will contain no surprises, except Theorem 3.4.7 about implicit functions. That seems like an awful lot of computational work anyway.

Section 3.5 does contain material which is probably important for you. It tells us what quadratic forms look like, and this is something which comes up throughout mathematics and physics. However, I would have preferred it if they had given an exposition using bilinear forms (they introduce these in exercise 3.5.13) and used the Gram-Schmidt orthogonalization process. This comes to the same thing as completing the square, but I like it better. Last semester there were some days when the class before us always seemed to be learning about this process.

Joke of the Week
Rene Descartes went into a bar, and the bartender asked him, 'Would you like a beer?' He replied, 'I don't think ...' .... and disappeared.