2021 November 30,
Carleton College, Fall 2021, Joshua R. Davis, , CMC 324, x4095
Calculus has been a major driver of science and technology over the past few centuries. Unfortunately, the material that you learned in your first two calculus courses is rarely sufficient for real-world problems. Why? Because real problems usually involve multiple variables. For example, a satellite orbiting Earth is not moving in a one-dimensional space with coordinate x but rather a three-dimensional space with coordinates x, y, z. A manufacturer trying to maximize its profit must consider not just one raw material but many interacting goods and services. Therefore, our ability to apply calculus is greatly improved if we can handle several variables.
That's the point of this course: to extend the geometric and algebraic concepts of introductory calculus to functions that input or output multiple variables. Topics include partial derivatives, multiple integrals, and some vector calculus. This material finds application in physics, engineering, economics, geology, computer science, and many other fields. We explore as many of these applications as we can.
The prerequisites are two Advanced Placement calculus courses. (Students who take Calculus 2 at Carleton should instead take Math 210.) We sometimes use Mathematica to visualize shapes and expedite calculations. Carleton gives you this software, and I give you tutorials and instruction. No prior experience is expected, you are not required to write programs, and there are no exam questions about Mathematica. Talk to me if you are concerned about your background.
I am teaching two sections of this course. Section 01 meets in CMC 210 during period 2A (MonWed 9:50-11:00, Fri 9:40-10:40) and takes its final exam Monday November 22 7:00-9:30 PM. Section 03 meets in CMC 301 during period 5A (MonWed 1:50-3:00, Fri 2:20-3:20) and takes its final exam Tuesday November 23 8:30-11:00 AM.
The course materials are
My office hours are Mon 3:10-4:20, Wed 11:10-12:20, Thu 1:30-2:30, Fri 1:10-2:10. As of Thursday October 21, office hours are in my office (CMC 324) with masks please. No appointment is needed for office hours. Just drop in! Going to office hours is a normal part of college. As you advance through your education, office hours become increasingly important, because the material gets harder, your interests become more specific, and you need advice and recommendations from faculty. If you want to schedule an appointment outside office hours, then consult my weekly schedule and e-mail me with several times, at which we are both available.
Here are some other resources.
The following elements contribute to your numerical grade. The percentages are inspired by recent scholarship on equity (racial, economic, etc.) in education. They are intended to reward competence much more than effort, because perceived effort is distorted by instructor bias. They express the view that mistakes on homework are educational and should not be punished.
Numerical grades are converted to letter grades only at the end of the term. There are no predetermined percentages (90%, 80%, 70%, etc.) required for specific grades (A, B, C, etc.), because I cannot write problems that are so precisely and reliably tuned. Instead, I assign letter grades by comparing students' scores to the course goals. Roughly speaking, a student who meets most of the goals earns a B. A student who meets almost all goals — and sometimes exceeds them — earns an A. A student who demonstrates effort but meets only a few of the goals earns a C. Students, for whom I have insufficient evidence of learning and effort, might earn grades below C.
An advantage of this system is that students are not in competition with each other. Also, students don't suffer when I accidentally write a difficult exam. The disadvantage is that you cannot compute your own grade. Send me e-mail, if you want me to estimate your current grade for you.
On homework, you are encouraged to figure out the problems with other students. However, you should always write/type your solutions individually, in your own words. You may not copy someone else's work or allow them to copy yours. Presenting someone else's work as your own is an act of academic dishonesty. The College requires me to report you, if I suspect that you have not upheld its Academic Integrity standards.
Writing is not just for English and history majors. Written and oral communication skills are essential to every academic discipline and are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation. (To help you, I might assign writing exercises. I haven't decided yet.)
Homework is assigned nearly every day, and you are expected to attempt the problems immediately. However, to give you some flexibility, homework is usually due two meetings after it was assigned. When handing in a multi-page homework assignment, please staple your pages into a single packet, in the correct order.
Depending on time constraints in any given week, perhaps not all of your homework will be graded. In order to ensure full credit, do all of the assigned problems.
The College's accreditation says that a 6-credit course is 150 hours of work. That's about 15 hours per week or 5 hours per class meeting. Those 5 hours break down into about 1 hour for class itself and 4 hours for homework, reading, studying, etc. If you find yourself spending much more time than this, then talk to me.
Usually I have a very simple, clear policy on late homework. But it's not flexible enough to handle a pandemic. So let's use the following late policy instead. If you need to hand in homework late, then do so. Put it in a separate pile from the on-time homework. Depending on the grader's schedule, it might be graded for full credit, or it might not be graded at all.
If some medical condition affects your participation in class or your taking of exams, let me know in the first week of class. You may need to make official arrangements with Accessibility Resources.
Because the pandemic makes everything unpredictable, we all need to be empathetic and flexible. Some of the course policy details might need to change.
To help you decode the schedule, here is an example. Day 01 is Wednesday September 15. We discuss coordinate systems and vectors. Section 12.1 of the textbook covers that material; read it before or after class, to get another treatment. You have seven homework problems, which are due at the start of class on Day 02. If you wish, you can review the worksheet fourTheorems.pdf that we used in class.
| Date | Day | Topic | Homework | Due | Reading | Notes |
|---|---|---|---|---|---|---|
| W 09/15 | 01 | coordinate systems | 12.1 #4, 8, 9, 15, 17, 40, 52 | 02 | 12.1 | fourTheorems.pdf |
| F 09/17 | 02 | vectors | 2021f211day02.pdf | 04 | 12.2 | |
| M 09/20 | 03 | dot product | 2021f211day03.pdf | 03, 05 | 12.3 | |
| W 09/22 | 04 | cross product, lines, planes | 2021f211day04.pdf | 06 | 12.4, 12.5 | |
| F 09/24 | 05 | parametrized curves, polar coordinates | 2021f211day05.pdf | 07 | 10.1, 10.3 | |
| M 09/27 | 06 | vector functions | 2021f211day06.pdf | 08 | 13.1, 13.2 | parametrizedCurves.pdf |
| W 09/29 | 07 | arc length, motion in space | 2021f211day07.pdf | 09 | 13.3, 13.4 | calculusCurves.pdf |
| F 10/01 | 08 | functions of several variables | 2021f211day08.pdf | 10 | 14.1, 14.2 | |
| M 10/04 | 09 | Exam A | ||||
| W 10/06 | 10 | partial derivatives | 2021f211day10.pdf | 12 | 14.3 | |
| F 10/08 | 11 | tangent plane, chain rule | 2021f211day11.pdf | 13 | 14.4, 14.5 | chainRule.pdf |
| M 10/11 | 12 | directional derivatives, gradient | 2021f211day12.pdf | 14 | 14.6 | |
| W 10/13 | 13 | optimization | 2021f211day13.pdf | 15 | 14.7 | optimization.pdf |
| F 10/15 | 14 | Lagrange multipliers | 2021f211day14.pdf | 16 | 14.8 | multipliers.pdf |
| M 10/18 | Midterm Break | |||||
| W 10/20 | 15 | double integrals | 15.1 #6, 12, 15, 19, 32, 34; 15.2 #1, 5, 7, 11 | 17 | 15.1, 15.2 | |
| F 10/22 | 16 | Exam B | ||||
| M 10/25 | 17 | polar integrals | 15.2 #9, 20, 25, 36, 65; 15.3 #7, 11, 18, 23, 38, 46 | 19 | 15.3 | |
| W 10/27 | 18 | double, triple integrals | 2021f211day18.pdf | 20 | 15.4 | accelerometer.pdf |
| F 10/29 | 19 | triple integrals | 15.6 #3, 9, 15, 19, 37; 15.7 #9, 13, 15 | 21 | 15.6, 15.7 | |
| M 11/01 | 20 | cylindrical, spherical integrals | 2021f211day20.pdf | 22 | 15.7, 15.8 | cylindricalSpherical.pdf |
| W 11/03 | 21 | spherical more, vector fields | 2021f211day21.pdf | 23 | 15.8, 16.1 | |
| F 11/05 | 22 | potential functions, line integrals | 2021f211day22.pdf | 24 | 16.3, 16.2 | |
| M 11/08 | 23 | Exam C | ||||
| W 11/10 | 24 | line integrals, fundamental theorem | 2021f211day24.pdf | 26 | 16.2, 16.3 | |
| F 11/12 | 25 | fundamental theorem, Green's theorem | 16.3 #35, 42c; 16.4 #1, 2, 3 | 27 | 16.3, 16.4 | |
| M 11/15 | 26 | Green's theorem, curl, divergence | 2021f211day26.pdf | 28 | 16.4, 16.5 | gradCurlDiv.pdf |
| W 11/17 | 27 | curl, divergence | 16.5 #1, 8, 9-12, 14, 21, 23, 24, 29, 40a | no | 16.5 | |
| F 11/19 | 28 | review | review.pdf |