Math 2573H Honors Calculus III Fall Semester 2002
Assignment 9 - Due Thursday 11/7/2002
There is a mid-term exam on 11/7 and no quiz. The exam will be on the material we have covered in Chapters 5 and 6. Some of the material in these chapters we have not studied at all, and other results have sometimes been mentioned with the indication that we are looking at them for your general knowledge but you will not be tested on them. As a rough guide to this, we did not study Chapter 5 Section 4 or Chapter 6 Section 1 (which include material about directional derivatives and the gradient which you already know) or Chapter 6 Section 3 (which we are missing out altogether) or Chapter 6 Section 4F. Several of the theorems (for instance, the inverse function theorem) have only been mentioned, and in any case you will not be tested from a theoretical point of view on results stated as theorems. You do not need to know their proofs. What you do need to know is how to use these results and compute with them. For example, you should be able to identify the points at which a function is continuous, or differentiable, you should be able to find the matrix which represents a linear map when its effect on vectors other than the standard basis vectors is given, and so on.
You will be able to use calculators on this exam, but not books or notes.
Read: Williamson and Trotter, Chapter 7 Sections 3 and 4.
Section 3 contains a bunch of properties of these multiple integrals we are studying which are almost all completely expected generalizations to high dimensions of the properties we know for single-variable integrals. I find the most interesting property is Leibnitz' Rule (3.7). Section 4 is about how you make a substitution in high dimensions. The only result in this section is Jacobi's Theorem (4.4), which says that instead of multiplying by a derivative when making a substitution (as you do in the 1-variable case) you multiply by the determinant of the derivative matrix. I suppose this is the main reason we learnt about determinants.
Chap 7 Sec 3 pages 280-282: 1, 4, 5, 6, 9c*
Chap 7 Sec 4 pages 291-294: 2, 3, 5*, 6, 8, 10*, 14, 15