The College of the Holy Cross

Mathematics 41, section 1 -- Multivariable Calculus

Syllabus, Fall 1999

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@math.holycross.edu or jlittle@holycross.edu
Office Hours: MWF 10-12, TR 1-3, and by appointment.
Course Homepage: http://math.holycross.edu/~little/Mult99/Mult99.html

Course Description

Multivariable Calculus is an introduction to the differential and integral calculus of functions of several variables. The course will cover the following topics. (Also see the detailed course schedule at the end of this syllabus.)

Unit I: Vectors in R² and R³
Unit II: Vector fields, critical points
Unit III: Differential calculus for functions of several variables -- directional derivatives, partial derivatives, gradients
Unit IV: Optimization
Unit V: Double, triple integrals
Unit VI: Line integrals and Green's Theorem

Text

The text book for the course is Vector Calculus, A Collaborative Approach by Professors Damiano and Freije of the Holy Cross department of mathematics, available in the H.C. bookstore. We think you will find reading and studying this book to be challenging, but ultimately very rewarding. It is definitely not a standard calculus book --

it has more, and more realistic, applications from the sciences than most books,
it consistently stresses qualitative understanding of the meaning and purpose of the mathematics, rather than mere proficiency at calculations,
(along the same lines) many of the problems are non-routine (they require thought, not just calculation!) and frequently open-ended -- there can be several different "right answers" for many of them,
it is designed for use with mathematical software that can perform many of the routine tasks that used to be the focus of calculus, freeing your time and effort for more interesting things!
it is designed for a collaborative teaching strategy; in addition to regular text sections and problem sets, the text includes discussion questions designed for in-class group work. We will be using these on a regular basis in class.

Note: Because this text is still a work in progress, we are especially interested in your comments throughout the semester. Are there things that "work" especially well? Are there things that need to be explained in more detail, or through different examples? Feel free to address your comments to me, or directly to Professor Damiano or Professor Freije if you prefer!

Course Format

In order for a student to get as much as possible out of this or any course, regular active participation and engagement with the ideas we discuss are necessary. To get you more directly involved in the subject matter of this course, regularly throughout the semester the class will break down into groups of 3 or 4 students for one or more days, and each group will work individually for (a portion of those) class periods on a group discussion exercise. I will be responsible for designing and preparing these exercises, and I will be available for questions and other help during these periods. Each group will keep a written record of their observations, results, questions, etc. which will be handed in.

Some of the other meetings of the class will be structured as lectures when that seems appropriate.

Approximately eight times during the semester, the class will meet in the Swords 219 Sun workstation laboratory for "math lab" classes. Each of these sessions will lead to a lab writeup assignment. (The due date will be announced when the assignment is given out.) We will be using the excellent, extremely powerful, general-purpose mathematical software package Maple that some of you used last year on the PC's in HA 408. The version of Maple that runs on the Sun workstations is very similar, and the Suns themselves are more powerful computers, so we'll be able to do even more. We will use Maple to produce graphs of functions, calculate numerical approximations to integrals, and compute symbolic derivatives and integrals of functions. But in fact Maple is so powerful that we will be using only a small fraction of what it can do. If you take additional mathematics courses at Holy Cross, you will probably use many other features of this same program. And programs like this are becoming a standard tool in many areas of science, engineering, and even finance. Being able to use them effectively is a valuable skill to have!

Unlike the situation in the PC lab, you will need a username and password to log onto the Sun network. I will distribute your usernames and passwords in our first lab session, which will be devoted to getting acquainted with the Sun network, running Maple, etc.

Grading Policy

Grading for the course will be based on

Three in-class tests, together worth 45% of the course grade,
A two-hour final exam, worth 30% of the course grade,
Written reports from small group discussions and computer labs -- one report from each group. Information regarding the expected format will be given out with the first assignment of this kind. Together, worth 15% of the course grade.
Individual homework assignments, given out in class. The homework will count as 10% of your course grade, if it is to your benefit, that is, if your homework average is greater than or equal to the average of your other grades. Otherwise, it will be discounted, and your average will be the average of your other grades. No credit will be given for late homework, except in the case of an excused absence, or with my permission.

A 5-page paper may be substituted for one of the in-class exams. I will prepare a list of possible topics and distribute it later in the semester.

If you ever have a question about the grading policy or your standing in the course, don't hesitate to ask me.

Course Schedule

The following is an approximate schedule for the course. Some rearrangement, expansion, or contraction of topics may become necessary. I will announce any changes in class.

Date	Class Topic	Reading (Damiano and Freije)

9/1	Course introduction, coordinates in 3D space	1.1
9/2	Vectors, geometry and algebra	1.2
9/3	More on vectors and geometry	1.2
9/6	The ``dot'' product of vectors	1.3
9/8	The ``cross'' product	1.3
9/9	Lines in R³	1.4
9/10	Lines and Planes	1.4
9/13	Lab Day 1: Maple on the Sun network
9/15	Review of parametric curves, tangent lines	2.1, 2.2
9/16	Vector Fields	2.4
9/17	Vector Fields and Differential Equations	2.4
9/20	Lab Day 2: Critical points of vector fields	2.4
9/22	Functions of two variables	3.1
9/23	Graphs and level curves	3.1
9/24	Exam I	(covers material from 9/1 through 9/20)
9/27	More on graphs and level curves	3.1
9/29	Lab Day 3: Maple surface plotting
9/30	Rates of Change	3.2
10/1	Directional derivatives	3.2
10/4	Partial derivatives	3.3
10/6	Lab Day 4: Applications	3.3
10/7	Continuity, Differentiabilityfor f(x,y)	3.4,3.5
10/8	No Class -- I'll be away at a conference	3.5
10/11	No Class -- Columbus Day
10/13	Lab Day 5: Local Linearity	3.5
10/14	Gradients and tangent planes	3.5
10/15	Critical points	4.2
10/18	The gradient vector field grad f(x,y)	4.1
10/20	Algebraic classification of critical points	4.2
10/21	Second derivative test	4.2
10/22	Exam II	(covers material from 9/22 through 10/18)
10/25	More on the second derivative test	4.2
10/27	Geometry of constrained optimization	4.3
10/28	Method of Lagrange Multipliers	4.3
10/29	Lagrange Multipliers, continued	4.3
11/1	Lab Day 6: Introduce sums	5.1
11/3	Riemann Sums -- rectangular regions	5.2
11/4	Double integrals	5.2
11/5	Double integrals over non-rectangular regions	5.2
11/8	Lab Day 7: Polar coordinates in R²	5.3
11/10	Polar double integrals	5.3
11/11	Riemann sums over 3D regions	5.4
11/12	Triple integrals	5.4
11/15	Cylindrical and spherical Coordinates	5.5
11/17	Lab Day 8: Cylindrical and spherical coordinates	5.5
11/18	Cylindrical and spherical triple integrals	5.6
11/19	Exam III	(covers material from 10/20 through 11/15)
11/22	Path Integrals	6.1
11/24,25,26	No Class -- Thanksgiving Break
11/29	Path and line integrals	6.1
12/1	Path and line integrals	6.1
12/2	More on line integrals	6.2
12/3	SAC, Green's Theorem	6.3
12/6	Finish Green's Theorem, course wrap-up	6.3

The final exam for this course will be given at 2:30 pm on Wednesday, December 15.