2014 November 29,

Math 211: Introduction to Multivariable Calculus

Carleton College, Fall 2014, Prof. Joshua R. Davis, , CMC 228, x4362


Calculus has been a major driver of scientific and technological advancement ever since it was invented/discovered in the mid 1600s. Many of the applications that make calculus so important involve multiple interacting variables. For example, a satellite entering orbit around Earth is not moving in a one-dimensional space with coordinate x but rather a three-dimensional space with coordinates x, y, z. A company trying to maximize its profit often must consider not just the price of one raw material, but prices of several materials and services.

Therefore, our ability to apply calculus to real-world problems is greatly improved if we are able to analyze functions of more than one variable. That's the point of this course: to extend the geometric and algebraic concepts of introductory calculus to those functions. Topics include partial derivatives, multiple integrals, and some vector calculus.

Our class meets in CMC 301 during period 5A (MonWed 1:50-3:00, Fri 2:20-3:20). My office hours are

If you want to meet with me, then try to make my office hours. If you cannot, then e-mail me, listing several possible times. The course materials are


Final grades (A, B, C, etc.) are assigned according to an approximate curving process. By this I mean that there are no predetermined percentages (90%, 80%, 70%, etc.) required for specific grades. The advantage of this system is that student grades don't suffer when I write a difficult exam. The disadvantage is that you cannot compute your own grade. Visit me in my office, if you want me to estimate your current grade for you. The following elements contribute to your final grade.

You are expected to spend about 10 hours per week on this course outside class. Some students need to spend more than 10 hours. If you find yourself spending more than 15 hours, then talk to me.

Standards for Work

You are encouraged to work with others on all assignments (but not exams). Work together to figure out the problems/programs, but write/type them up separately, 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. If Carleton College finds you to not to have upheld its Academic Honesty standards in this course, then you will receive an F for this course.

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. You are expected to write in complete sentences with punctuation. Your solutions should be self-explanatory. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that your grader can ascertain whether you yourself understand the material.

Although homework is assigned every day, it is collected only once a week (usually Fridays). When handing in a week's homework, staple your pages into a single packet, in the correct order. Packets that are not stapled are unacceptable. I will not accept packets that are not stapled. Is there a stapler in the classroom? Often not, so staple ahead of time. Is a paper clip okay? No.

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.

Special Accommodations

During the term, you have one free pass to hand in a week's homework packet late. Instead of handing in the packet, send me e-mail, by the due date, declaring that you are using your late pass. No explanation is necessary. Your packet is now due when the next packet is due. When you submit your late packet, write "Late Pass Used" prominently at the top. Once you have used your late pass, no late assignments are accepted, except in extreme circumstances that typically involve interventions by physicians or deans.

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 the Office of Disability Services.


This schedule is tentative. It will be adjusted as we go along. To help you decode the schedule, here is an example. On Day 2 we discuss vectors and the dot product. Before class (or maybe after), you should read sections 12.2 and 12.3 in the textbook. You have homework called "Day 2", which you hand in on Day 5.

M 9/15112.1, 12.2vectorsDay 11
W 9/17212.3dot productDay 25
F 9/19312.4cross productDay 35
M 9/22411.3, 12.7polar, spherical, cylindrical coordinatesDay 48
W 9/24513.1, 13.2vector-valued functionsDay 58
F 9/26613.3, 13.4arc length, curvatureDay 68
M 9/29714.1multivariable functionsDay 711
W 10/1814.2, 14.3limits, continuity, partial derivativesDay 811introduction.nb
F 10/3914.4Mathematica labDay 911derivatives.nb
M 10/61014.5gradient, directional derivativesDay 1014
W 10/81114.6, 14.7optimizationDay 1114
F 10/1012Exam A
M 10/131314.8Lagrange multipliersDay 1316
W 10/151415.1double integralsDay 1416integrals.nb
F 10/171515.2, 15.3triple integralsDay 1516
M 10/20Midterm Break
W 10/221615.4coordinate changesDay 1619
F 10/241716.1vector fieldsDay 1719vectorfields.nb
M 10/271816.2line integralsDay 1822
W 10/291916.3fundamental theorem for line integralsDay 1922
F 10/312017.1Green's theoremDay 2022
M 11/321grad, curl, div, LaplacianDay 2125
W 11/52216.4, 16.5integration on surfacesDay 2225
F 11/723Exam B
M 11/102417.2Stokes' theoremDay 2428
W 11/122517.3divergence theorem, Mandelbrot setDay 2528mandelbrot.nb
F 11/1426Review26
M 11/1727fluid dynamicsDay 2728fluids.pdf
W 11/1928fluids, finite differences
Sa 11/22Final Exam 8:30AM-11:00AM