Holy Cross Mathematics and Computer Science


Mathematics 241, section 1 -- Multivariable Calculus

Syllabus Fall 2013

Professor: John Little
Office: Swords 331
Office Phone: (508) 793-2274
email: little@mathcs.holycross.edu or jlittle@holycross.edu
Office Hours: M 10am - 12 noon, T 1 - 3pm, W 3 - 5pm, R 9 - 11am, F 10am - 12 noon
Course Homepage: http://mathcs.holycross.edu/~little/Mult13/Mult13.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 given 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. If we start at a particular point and move in different directions, the temperature would usually change in different ways. (For instance it might decrease if we move due North, it might increase if we move due West, and it might decrease if we moved higher above the surface of the Earth. In Multivariable Calculus we will learn how to determine the rate of change of a quantity like the temperature in this example if we move in different directions. We will also understand how to determine quantities like the average temperature or the total thermal energy of a volume within the atmosphere. Mathematically, we will give 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: Parametric curves and their geometry, vector fields and applications (about 8 class days)
Unit III: Differential calculus for functions of several variables -- partial derivatives, differentiability, directional derivatives, gradients (about 8 class days)
Unit IV: Optimization (unconstrained and constrained max/min problems) (about 6 class days)
Unit V: Integral calculus for functions of several variables: double, triple integrals (about 10 class days)
Unit VI: Vector calculus: line and surface integrals, Green's Theorem, Gauss's Theorem (aka the Divergence Theorem) (about 7 class days)

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

Text

The text book for the course is Multivariable Calculus (Jones and Bartlett International Series in Mathematics, ISBN: 978-0-7637-8247-4) by Professors David Damiano and Margaret Freije of the HC Mathematics and Computer Science Department. (Professor Freije is also currently serving as the acting Vice President for Academic Affairs and Dean of the College.) One of the distinctive features of this book is that it has many more (and much more realistic) applied examples than almost all of its ``competitors.'' I think you will find reading and studying it 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.

Approximately seven times during the semester, the class will meet in the Swords 219 Sunray Linux 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. 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!

Depending on the class composition and how things are going in general, we may also break the class down periodically into groups of 3 or 4 students. In these meetings, each group will work as a team 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.

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 27; Friday, November 1; Tuesday, November 26.
  2. A two-hour final exam -- 25%. The final exam for this course will be given at the scheduled hour for 9:00am MWF classes
  3. Weekly Quizzes -- 10%
  4. Written reports from computer labs and possible small group discussions -- 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. Watch for announcements of the final exam schedule from the Registrar's Office and please take the date of our 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 9 quizzes, the 6 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 lab assignment.
  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, but it might convert to a B+ in some cases.)

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

Advice On How To Succeed In This Class


A good "work ethic" is key. You do not need to be a "math genius" to master this material and do well. But you will need to put in a consistent effort and keep up with the course. Contrary to what you might believe, mathematics professors want you to learn and do well in their courses! But mathematics at this level is not an easy subject for anyone. So some hard work and dedication is necessary to learn it!

Come to class. Unless you are deathly ill, have a genuine family emergency, have to be away at a College athletic event, etc. plan on showing up here at 9:00am every Monday, Tuesday, Wednesday, and Friday this semester. If attending class wasn't important, all college courses would be by correspondence, and your tuition would be much lower!

Read the textbook. Don't just use it to look for worked problems similar to ones on the problem sets. You will find alternate explanations of concepts that may help you past a "block" in your understanding. Reading a math book is not like reading a novel, though. You will need to read very carefully, with pencil and paper in hand, working through examples in detail and taking notes. Make a list of questions to ask in office hours or at the next class. One thing to bear in mind while reading your text is that the result of an example is often secondary to the process used in obtaining the result.

Take notes and use them. This may seem obvious, but in my experience too many students diligently copy down everything on the board, and then never look at their notes again. Used intelligently, your notes can be a valuable resource as you work on problem sets and prepare for exams.

Set up a regular study schedule and work at a steady pace. It's not easy to play catch-up in a mathematics course, since every day builds on the previous one. You should expect to budget at least 6 hours in a typical week for work outside of class. The best way to use your time is to do a few problems, a little reading from the book, and reviewing of class notes every day.

Most importantly, if you are having difficulty learning something, get help as soon as possible. You can do this by asking questions during class (any time something isn't clear), or seeing me during office hours.

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.