Math 103L (Calculus III)

Fall 1996

Plan for Weeks 14 and 15

Notes:

1. Your homework papers for Weeks 13 and 14 will be turned in on Monday, December 9. Those papers should include solutions to the problems whose numbers are boldfaced in the assignment below. (For your conveneince, the Week 13 assignments are repeated.) The assignment dates are start dates.
2. The rest of the semester is devoted to exploring extensions of the Fundamental Theorem of Calculus (FTC) in the context of vector and scalar functions of two variables. First we find direct analogues of both parts of the FTC for integration along curves in gradient fields. In particular, line integrals in gradient fields depend only on values of a potential function (antigradient) at the end points, but are independent of the path connecting the end points. It follows that integrals around closed curves in a gradient field must be zero. However, in non-gradient fields integrals around closed curves often are not zero -- in fact, their values are characterized by Green's Theorem, another extension of the FTC. In one form, Green's Theorem characterizes the tendency of a vector field to circulate around a curve as a double integral of the local circulation at all points inside the curve. In another form, Green's Theorem describes the net flow across a closed curve in terms of local "divergence" (introduction or removal of fluid) -- specifically, as a double integral of the local divergence throughout the region enclosed by the curve. Chapter 16 brings together all the important ideas from earlier chapters: scalar and vector functions, parametric representations, level curves, the gradient, partial derivatives, multiple integrals.

Date            Class/lab activity      Homework exercises         Reading for 
                                                                   next class


Mon., 11/25     Gradient fields         16.2, #1,2,4,11,12,15,17   16.3a
Wed., 11/27     Line integrals          16.3, #3,4,6,7,9,11        16.3b
(Thanksgiving Break)

Mon., 12/2      FTC for line integrals  16.3, #15,16,18,19,21,     16.4
                                               22,23
                Start Take-home part of Final Exam
Tues., 12/3     37: Vector Field Rotation
Wed., 12/4      Green's Theorem         16.4, #1,4,7,9,13,15,      16.5
                                               21,22,23,24
Fri., 12/6      Arc length integrals    16.5, #1,3,4,7,10,11,      Review
                                               13,16,19,20

Mon., 12/9      Review                  Ch.16, #7,9,10,25,29,      Review
                                                30,32,33,34
Tues., 12/10    No lab
Wed., 12/11     Evaluation, Take-Home Final due

Wed., 12/18     Rest of Final Exam, 9 am to noon

Last modified: December 1, 1996