Math 2573H Honors Calculus III Fall Semester 2002

Assignment 6 - Due Thursday 10/17/2002

Read: Williamson and Trotter, 6.1ABC, 6.2
We have already studied the gradient of a function and directional derivatives last year, so most of section 6.1 should be familiar. The thing we did not do last year is the chain rule, and that is the most important thing for us in these sections. I would as soon forget about the gradient right now, but the proof of the chain rule given in section 6.2 relies on the particular case given in Theorem 1.3 on page 187, which in turn relies on Theorem 4.3 of Chapter 5 in a section I told you to skim over last week. I am not particularly happy about the treatment of this in the book. First of all, we don't have to prove the chain rule in this way (you may wonder if we need to prove it at all ...). Secondly the approach to gradient and directional derivatives is not that easy to understand the way it is written in this book, in my view. You may do better to read this stuff from last year's book, by Simmons. In Trotter and Williamson they introduce a special notation for the directional derivative which we really don't need to know about, and so I am going to try to avoid it.

You may sense my frustration over the treatment of this material! Please bear with me as I try to steer us through it.

Chap 6 Sec 1 pages 190-192: 2, 4, 8*, 14*
Chap 6 Sec 2 pages 204-207: 2*, 3, 4, 5*, 9, 10, 13, 15*