2017 June 6,

Math 211: Introduction to Multivariable Calculus

Carleton College, Spring 2017, Prof. Joshua R. Davis, , CMC 228, x4095


Calculus has been a major driver of scientific and technological advancement ever since it was invented/discovered in the 1600s. Unfortunately, the material that one learns in a first or second calculus course is rarely sufficient for real-world problems. Why? Because many problems involve multiple interacting 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 manufacturing 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 handle several variables.

That's the point of this course: to extend the geometric and algebraic concepts of introductory calculus, to functions that either output or input multiple variables. Topics include partial derivatives, multiple integrals, and some vector calculus. The course materials are

Our class meets in Weitz 230 during period 4A (MonWed 12:30-1:40, Fri 1:10-2:10) or 5A (MonWed 1:50-3:00, Fri 2:20-3:20). If you want to meet with me outside class, then try to make my office hours. If you cannot, then e-mail me, listing several possible times. My office hours are currently


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.

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

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 a violation of Carleton's 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. We will do some short exercises to help you with the latter.

Although homework is assigned every day, it is collected only once a week. When handing in a week's homework, staple your pages into a single packet, in the correct order. Multi-sheet 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, no questions asked. Simply hand in your late packet when the next packet is due, writing "Late Pass Used" prominently at the top of the late packet.

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 Disability Services.


To help you decode the schedule, here is an example. On Day 2 we discuss the dot product. Section 12.3 of the textbook covers that material; read it if you want another treatment. You have homework called "Day 2", which is due on Day 3.

M 3/27112.1, 12.2vectorsDay 11, 2, 3
W 3/29212.3dot productDay 23
F 3/31312.4cross productDay 34, 6
M 4/03411.3, 12.7polar, spherical, cylindrical coordinatesDay 45, 6faults, faults, with slip, with slip
W 4/05513.1, 13.2vector-valued functionsDay 56, 9
F 4/07613.3, 13.4arc length, curvatureDay 67, 9
M 4/10714.1multivariable functionsDay 78, 9
W 4/12814.2, 14.3limits, continuity, partial derivativesDay 89, 12
F 4/14914.4Mathematica lab (in CMC 201)Day 910, 12introduction.nb, derivatives.nb
M 4/171014.5gradient, directional derivativesDay 1011, 15
W 4/1911Review
F 4/2112Exam A
M 4/241314.6, 14.7chain rule, optimizationDay 1314, 15
W 4/261414.8Lagrange multipliersDay 1415, 17
F 4/281515.1double integralsDay 1516, 17
M 5/01Midterm Break
W 5/031615.2, 15.3double integrals, triple integralsDay 1617, 20integrals.nb
F 5/051715.3, 15.4triple integrals, coordinate changesDay 1718, 20
M 5/0818coordinate changes, applicationsDay 1819, 20
W 5/101916.1vector fields, grad, curl, divDay 1920, 23
F 5/1220Mathematica lab (in CMC 201)Q1 - Q723vectorfields.nb
M 5/152116.2line integralsDay 2124, 26
W 5/1722Review
F 5/1923Exam B
M 5/222416.3fundamental theoremDay 2425, 26
W 5/242517.1, 16.4Green's theorem, parametrized surfacesDay 2526, 28
F 5/262616.5integration on surfacesDay 2627, 28
M 5/292717.2Stokes' theoremDay 2728
W 5/3128Review
M 6/05Final Exam 8:30-11:00 or 3:30-6:00

We did not cover the divergence theorem (Section 17.3).