Mathematics 21a - Spring 2000

Subject matter: There are three distinct types of sections of Math 21a from which to choose: Regular, Physics and BioChem.

All sections cover at least the following topics: Functions of several variables, differentiation and integration of functions of several variables, parametric curves and surfaces, optimization, vector fields, linear approximations and various topics in partial differential equations.

The Regular and Physics sections also cover: Line and surface integrals, Greens’ Theorem, the Divergence Theorem and Stoke’s Theorem.

The Physics sections will use examples drawn from physics to illustrate some of the covered topics. Certain of the homework problems assigned to this section will also come from physics.

Meanwhile, the BioChem sections covers, in addition to the material in the first point, various introductory topics in statistics and probability. Note that there are no specific references to either biology or chemistry in this section. Thus, the material is accessible to all.

Detailed syllabi for the various types of sections will be provided elsewhere on this website.

With regard to these sections, you should know that Math 21a is taught in small sections of size ~25 students, with some sections specially labled ‘Physics’ and ‘BioChem’. A section without a special label is a Regular section.

Which kind of section should you choose?

If you contemplate being a physics concentrator, you would benefit by being in the Physics section.

In any event, if you are planning to take either the Physics 15-16 or Physics 11 sequences, you should enroll in either a Regular or a Physics section.

If you are planning to concentrate in BioChemical Sciences, then you are strongly urged by that department to enroll in a BioChem section.

Anyone who will not, at some point, enroll in Physics 11 or 15-16 might consider a BioChem section as well.

To section: If you have an email account, log on to the Harvard computer system, then type 'section' instead of 'pine. If not, use any Harvard computer, telnet to 'fas' and when prompted to 'login', type 'section'. At the password prompt, press 'enter'. If there is a problem with your section assignment, contact Susan Milano in Sci Ctr 308.

Course Head: Robert Winters, Science Center 435, rwinters@math.harvard.edu. Office hours are tentatively scheduled for Monday, Wednesday, Friday - noon to 1pm.

Prerequisites: Math 1b with a satisfactory grade, or AB-BC score of at least 4, or scores of at least (20, 8, 4) on the Harvard Math Placement Test (HMPT).

Textbooks: All of the sections require Multivariable Calculus by Ostebee and Zorn, published by Saunders College Publishing. The BioChem sections also require Study Guide for Fundamentals of Biostatistics by Rosner, published by Duxbury Press. These books are available at the Harvard Coop.

Classes, problem sessions and labs: As remarked above, this course is taught in sections and each student is assigned to a section which meets three hours per week. Each student is also assigned to a 1-hour math problem session, conducted weekly by a Course Assistant. The meeting time for the problem session will be arranged during the first week of classes. You may attend more than one problem session per week; and the schedule of all problem sessions will be posted on the Calculus Office bulletin board outside of Science Center 308.

Homework: A substantial problem set will be assigned during every section meeting and will be due at the following meeting. You are strongly encouraged to discuss the homework with your fellow students and to form study groups. However, you must write up the solutions by yourself, and you must note the names of your coworkers somewhere on the homework. (This last point is simply a matter of professional ethics.) For Monday-Wednesday-Friday sections, the three lowest and for Tuesday-Thursday sections, the two lowest homework scores will be disregarded when your average homework grade is computed.

Several of the homework assignments will be required to be produced using word processing software and mathematics software such as Mathematica, Maple, or Matlab to produce graphs and more. These homework assignments will be mandatory.

Answers to the homework problems will be made available at the Math 21a course website and selected problems will be discussed in the problem sessions.

Homeworks submitted after the day of the subsequent section meeting will be accepted only by the discretion of the teaching staff. In any event, no more than three late homeworks will be accepted per student over the course of the semester.

Exams: There are two course-wide midterms and a final. The midterms will take place as follows:

Midterm Exam 1: Tuesday, February 29, 7:00pm to 8:30pm in Sci Ctr Hall C
Midterm Exam 2: Tuesday, April 4, 7:00pm to 8:30pm in Sci Ctr Hall C

Grading: Your final grade will be based on your performance on the homework (20%), the two midterms (20% each), and the final exam (40%). A small upward adjustment in the grade is possible when the final is dramatically better than the average of the midterms and the homework.

Computers and calculators: The visualization of surfaces and other geometric phenomena is an important aspect of this course. Because computerized graphing programs will aid you in developing this ability, you are encouraged to employ them as part of the learning process. In this regard, we are planning to have one or more resource people available to answer questions about Mathematica and possibly other software tools. (No prior experience with Mathematica is required.)

However, computers should be considered solely as an aid to the development of geometric intuition. In particular, the use of computers and other electronic aids will not be permitted during exams. (Bring only your brain and some pencils.) With this in mind, note that various homework problems ask you to sketch or otherwise describe various geometric objects. You are strongly advised to struggle with these first without electronic aids, as they may be quite trivial with a graphing program.

Words of Caution and Advice: This course will be more demanding then your previous mathematics courses at Harvard and elsewhere. In particular, the daily assignments will be time consuming and you should plan now to set aside regular hours to wrestle with them. It is virtually impossible to do well in this course without working the homework assignments in a timely fashion. Note also that this course is fast paced, and new material builds on old. Thus, do not fall behind. If you find yourself falling behind, please contact your section teacher immediately to discuss options for personal help. Indeed, Harvard provides many services along these lines for its students, and your section teacher can help you find them.

When you are working your assignments, keep in mind that your success in this course will require more than just memorizing formulas and "plugging in values". Numerical calculations are still important, but play a smaller role than in 1-variable calculus. Here is the key to success: Understand the underlying concepts and then work enough problems so that you can employ them in any example thrown at you. (In this regard, you will consistently battle with homework and exam problems which differ significantly from material discussed in class.)