Holy Cross Mathematics and Computer Science


Mathematics 241 -- Multivariable Calculus

Syllabus Fall 2007

Professor: John Little
Office: Swords 339
Office Phone: (508) 793-2274
email: little@mathcs.holycross.edu (preferred) or jlittle@holycross.edu
Office Hours: MW 1 - 3pm, TR 11am - noon, F 10 - 11 am, and by appointment.
Course Homepage: http://mathcs.holycross.edu/~little/Mult07/Mult07.html

Course Description

Most phenomena in the real world involve functions that depend on more than one variable. For instance, we can think of the temperature in a certain region of the atmosphere as a function of latitude and longitude on the earth, plus height over the surface of the earth -- three variables in all. Multivariable Calculus is an introduction to the differential and integral calculus of functions of several variables, building on the foundation of one-variable calculus, but including also a number of new concepts from the geometry of higher dimensional spaces and the algebra of vectors and matrices.

Course Objectives

The major objectives of the course are:

Course organization

The course will cover the following specific topics. (Also see the detailed course schedule on the course homepage for a day-by-day plan for the course.)

Unit I: Vector algebra and geometry in R2 and R3 (about 9 class days)
Unit II: Differential calculus for functions of several variables -- partial derivatives, differentiability, directional derivatives, gradients (about 10 class days)
Unit III: Parametric curves and their geometry, vector fields and applications (about 8 class days)
Unit IV: Optimization (unconstrained and constrained max/min problems) (about 7 class days)
Unit V: Integral calculus for functions of several variables: double, triple integrals (about 8 class days)
Unit VI: Vector calculus: line and surface integrals, Theorems of Green, Gauss, Stokes (about 7 class days)

The remaining 3 class days will be devoted to in-class examinations.

Text

The text book for the course is Vector Calculus, 3rd edition by Professor Susan Jane Colley of the Oberlin College department of mathematics, Pearson/Prentice Hall, ISBN 0-13-185874-2. This book is available in the H.C. bookstore. I think you will find reading and studying this book to be challenging, but ultimately very rewarding.

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 as a team 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.

Approximately seven times during the semester, the class will meet in the Swords 219 Sunray Unix laboratory for "math computer 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 Unix network is very similar. 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!

Most of the other meetings of the class will be structured as lecture/discussion classes.

Grading Policy

Grading for the course will be based on

  1. Three in-class tests -- 40% of the course grade. Tentative dates: Friday, September 28; Friday, October 26; Friday, November 30.
  2. A two-hour final exam -- 25%. The final exam for this course will be given Tuesday, December 11 at 8:30am.
  3. Quizzes -- 10%
  4. Written reports from small group discussions and computer labs -- 15%
  5. Weekly individual problem sets -- 10%

Notes:

  1. Since everyone can have a bad day occasionally, your lowest in-class exam score will be weighted less than the other two (8%, 16%, 16%).
  2. Please take the date of the final into account when making your travel plans for the semester break(!)
  3. A 10-minute quiz will be given at the start of the class period most Fridays when there is not a full-period exam (see course schedule). Of the 8 quizzes, the 5 highest scores will be used for this component of the course average.
  4. Information regarding the expected format will be given out with the first assignment of this kind.
  5. All problem set assignments will be posted on the course homepage. No credit will be given for late homework, except in the case of an excused absence.

I will be keeping your course average in numerical form throughout the semester, and only converting to a letter for the final course grade. The course grade will be assigned according to the following conversion table (also see Note below):

Note: Depending on how the class as a whole is doing, some downward adjustments of the above letter grade boundaries may be made. No upward adjustments will be made, however. (This means, for instance, that an 85 course average would never convert to a letter grade of B- or below.)

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

Departmental Statement on Academic Integrity

Why is academic integrity important?

All education is a cooperative enterprise between teachers and students. This cooperation works well only when there is trust and mutual respect between everyone involved. One of our main aims as a department is to help students become knowledgeable and sophisticated learners, able to think and work both independently and in concert with their peers. Representing another person's work as your own in any form (plagiarism or ``cheating''), and providing or receiving unauthorized assistance on assignments (collusion) are lapses of academic integrity because they subvert the learning process and show a fundamental lack of respect for the educational enterprise.

How does this apply to our courses?

You will encounter a variety of types of assignments and examination formats in mathematics and computer science courses. For instance, many problem sets in mathematics classes and laboratory assignments in computer science courses are individual assignments. While some faculty members may allow or even encourage discussion among students during work on problem sets, it is the expectation that the solutions submitted by each student will be that student's own work, written up in that student's own words. When consultation with other students or sources other than the textbook occurs, students should identify their co-workers, and/or cite their sources as they would for other writing assignments. Some courses also make use of collaborative assignments; part of the evaluation in that case may be a rating of each individual's contribution to the group effort. Some advanced classes may use take-home examinations, in which case the ground rules will usually allow no collaboration or consultation. In many computer science classes, programming projects are strictly individual assignments; the ground rules do not allow any collaboration or consultation here either.

What are the responsibilities of faculty?

It is the responsibility of faculty in the department to lay out the guidelines to be followed for specific assignments in their classes as clearly and fully as possible, and to offer clarification and advice concerning those guidelines as needed as students work on those assignments. The Department of Mathematics and Computer Science upholds the College's policy on academic honesty. We advise all students taking mathematics or computer science courses to read the statement in the current College catalog carefully and to familiarize themselves with the procedures which may be applied when infractions are determined to have occurred.

What are the responsibilities of students?

A student's main responsibility is to follow the guidelines laid down by the instructor of the course. If there is some point about the expectations for an assignment that is not clear, the student is responsible for seeking clarification. If such clarification is not immediately available, students should err on the side of caution and follow the strictest possible interpretation of the guidelines they have been given. It is also a student's responsibility to protect his/her own work to prevent unauthorized use of exam papers, problem solutions, computer accounts and files, scratch paper, and any other materials used in carrying out an assignment. We expect students to have the integrity to say ``no'' to requests for assistance from other students when offering that assistance would violate the guidelines for an assignment.

Specific Guidelines for this Course

Examinations and quizzes will be given in scheduled, proctored sessions. No sharing of information in any form with other students will be permitted during exams or quizzes. The other assignments will be the weekly individual problem sets and group discussion and computer lab writeups. On the problem sets, discussion of the questions with other students in the class, and with me during office hours is allowed, even encouraged. Your final problem write-ups should be prepared individually, however, and the wording and organization of the writeup should be entirely your own work. If you take advantage of any of the options described above for consultation on the problems, you will be required to state that fact in a "footnote" accompanying the problem solution. Failure to follow this rule will be treated as a violation of the College's Academic Integrity policy. For the group discussions, you will be expected to work closely with your fellow team members.