Math 251:14--15 Spring 2005 Homework

Here is a partial list of lecture topics and homework problems for these sections. At any given time, it should show the schedule through the next exam. The recitation on Thursday will be devoted to topics discussed through the lecture of the previous Monday. The recitation instructor will announce any work that he wants submitted for grading. There may also be quizzes in recitation. The problems listed here do not have answers in the book. Any differences in your solutions will lead to a lively discussion.

There is another page prepared by the department giving some odd numbered problems that are suitable for practice. However, only this page reflects the problems that the lecturer plans to emphasize on exams.

Notes

1. Since the first part of the course deals only with curves, section 12.4, the cross product, and the second half of section 12.5, equations of planes,will be deferred until the second part of the course.
2. Exercise 32 in section 12.2 contains an additional complication not present in the example on page 804 because the object in the exercise is a mass instead of a weight. The textbook presents the author's description of the difference in section 6.4, p. 460. On an exam, this type of problem will be expressed in terms of weight to avoid the need for special conversion factors.
3. The only integrals on curves that will be considered in this course are the line integrals of vector fields introduced at the end of section 16.2, so the only topics from chapter 13 appearing will be those involving the differential calculus of space curves.
4. New slides are being developed, but the pages for Fall 2001 and Spring 2003 contain links to slides that are still useful and were shown in lecture.
5. Topics from 14.1 and 14.2 were included but downplayed. In Calculus, you work only with familiar functions that are usually defined and continuous everywhere. There are some exceptions that have an obvious restriction on the domain. Similarly, it is important to note that partial derivatives are a special case of the equally natural setting that leads to the chain rule. This viewpoint motived the slides an Maple worksheets shown in lecture that will be posted soon (along with an outline of the reminder of this segment of the course).
6. Our emphasis in section 14.7 will be on global extrema, so the second derivative test for characterizing local extrema isn't relevant. One of the Maple labs will investigate the classification of local extrema through a second derivative test, so examples that are simple enough to appear on an exam won't add much to your work on this topic. Also, the constrained extrema of section 14.8 are usually treated only in the case of global extrema since there is no easy version of a second derivative test in that case.
7. Sections 15.1 and 15.2 should be ignored. A theoretical treatment of multiple integrals using integrals of more general functions over rectangles has some value, but the description of the region of integration is the primary feature of the calculus of multiple integrals. It is important to get to that part of the subject as quickly as possible. Since Maple uses similar descriptions in both graphs and integrals, the use of Maple to visualize multiple integral problems is encouraged. The use of Green's theorem to relate double integrals over a region with line integrals around the boundary of the region requires that the geometry of the integral be stressed.
8. Identifying double integrals with the volume under a surface in section 15.3 is a misleading application. It will be an immediate consequence of the more natural definition of volume as a triple integral that will appear later in the course. Similarly, changing order of integration is an extreme special case of Green's theorem, so problems based on this technique will also be skipped in section 15.3.
9. A different point of view will be taken in section 15.5. Instead of starting with the application, formulating it as an integral, and then finding a way to evaluate the integral that probably will not be done correctly, the process will be reversed. That is, if a physical quantity like area or a moment is known (possibly by having done the calculation of an integral in an advantageous setting), that value can be used to evaluate any integral that is recognized as giving that quantity. When combined with Green's theorem, some line integrals can be evaluated with very little computation.
10. The formula for directly converting between rectangular and spherical coordinates is difficult to remember and not particularly useful. In almost all cases, it will be much easier to go by way of cylindrical coordinates when making this conversion.
11. There is a method to permuting the order of sections. Just as Green's theorem makes more sense when the proof, based on the method of calculating iterated integrals, is given before the theorem is stated, Stokes' Theorem should be proved first in order to see the need to introduce the curl of a vector field. Since the various derivatives are treated as requirements of a form of the fundamental theorem, exercises involving the divergence will not appear until that operation is needed.