The subject: Since Math 21a is a mathematics course, I will start with a digression to
introduce a definition for the science of mathematics so as to distinguish it from the other
Mathematics consists of the study of all imaginable worlds with the goal of
uncovering transcendent, universal relationships and underlying symmetries.
By way of contrast, fields such as physics, chemistry or biology, or even the more social
sciences such as economics are concerned, by definition, with the details of the particular
universe that we inhabit. This is to say that the charge for the other sciences can be
summarized as follows:
Provide a predictive understanding of the given universe.
The other sciences find mathematics useful, and often remarkably so, because the
underlying relationships which transcend our particular context yield predictions which
would be unfathomable with a focus that restricts solely to our own world.
With the digression now over and with the preceding remarks as background,
consider that Math 21a (and Math 21b as well) focuses on various concrete and abstract
properties of 2 and 3 dimensional space. Of particular concern here in Math 21a are
functions whose values vary over such spaces. This is to say that the subject is calculus for
functions of two or more variables. Such functions arise ubiquitously in the sciences.
Here are some examples: First, a biologist might be interested in the concentration
of a certain cellular protein as a function of the age of the cell and the distance from the
cell's nucleus. In this case, the function measures protein concentration and it depends on
the two variables, age and distance. Second, a physicist might be concerned with the
strength of a current in a superconductor as a function of the temperature, the strength of the
ambient magnetic field and the density of impurities. So here, the function in question
depends on the three variables, temperature, magnetic field strength and impurity density.
Third, an environmental scientist might be concerned with the concentration of mercury in
the Charles River as a function of distance from Boston Harbor and water flow rate. Here,
the interest is in a function of two variables, distance and flow rate. Meanwhile, a geologist
might worry about a function that gives the height above sea level of the Modoc Plateau in
California as a function of the three variables, latitude, longitude and time (measured in units
of millions of years). Finally, an economist might be concerned with the Dow Jones
average as a function of various other economic indicators.
In any event, our concern in Math 21a is not so much on examples, but on
developing various general techniques from calculus for the purpose of analyzing and
predicting the behavior of functions of two or more variables. None-the-less, concrete
examples abound in the course. However, there are discussions ahead that can seem quite
abstract; but be aware that even the most abstract subjects in this course have applications in
the other sciences, even if such applications are not mentioned here.
The sections: The precise subjects that are covered in Math 21a depend to
some extent on the particular type of section that you choose. In this regard, you should
know that Math 21a is taught in small sections of size 25 students, with sections labeled as "Regular",
"Physics" and "BioChem". A section without a special label is automatically a Regular
- 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
- The Regular and Physics sections also cover the following: Line and surface integrals,
Greens theorem, the divergence theorem and Stokes' theorem. These are the multi-
variable generalizations of the Fundamental Theorem of Calculus.
- 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
Detailed syllabi for the various types of sections are provided below.
Which kind of section should you choose?
- If you contemplate being a physics concentrator, you would benefit by being in the
- 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'.
Follow the online
instructions from here. If there is a problem with your section assignment, contact Susan
Milano in office 308 of the science center. .
Course Head: Clifford Taubes, Science Center 504,
email email@example.com .
Drop in office hours on Mondays 12-1:30 and Fridays 2-3:30.
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 respective three Harvard University Math Placement Tests.
All of the sections require Multivariable Calculus by Ostebee and Zorn with
the Student Solutions Manual, published by Saunders College Publishing. The BioChem
sections also require Fundamentals of Biostatistics by Rosner, published by Duxbury
Press. These books are available at the Harvard Coop.
Class meetings and problem sessions: The first class meeting, which everyone should
attend, is on Monday, September 18 at 8am in Science Center Lecture Hall B. Except for
this one meeting and for the course wide exams, you meet in your assigned section. The
section meets for a total of three hours per week, either one hour each on Mondays,
Wednesdays and Fridays, or for one and one half hours each on Tuesdays and Thursdays.
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 in your
section 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 once each week to be turned in as
instructed in the subsequent week. You are strongly encouraged to discuss the homework
with your fellow students and to form study groups to work these assignments. 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.) The lowest homework score will be disregarded when your average homework
grade is computed.
The weekly homework assignments will be posted on the Math 21a web site. The
answers to the homework assignments will appear after the due date on the web site as well.
Moreover, selected problems from the homework will be discussed in the problem sessions.
Homework assignments that are submitted after their assigned due date will be accepted at
my discretion. In any event, no more than two late homework assignments will be accepted
per student over the course of the semester.
In addition to the weekly homework assignment, various problems of a more routine
sort will be suggested for the subsequent class meeting. These are not to be turned as their
answers are in the Student Solutions Manual. However, you are strongly urged to work
them on your own or with others in the class because their purpose is to supply practice
with the techniques and ideas that are presented in the lectures. By the way, you are also
strongly encouraged to try on your own other problems from the text to hone your ability
with the concepts and techniques. Donıt feel that you should limit yourself to the
suggested problems. In this regard, note that the Student Solution Manual answers most of
the odd numbered problems in the book.
Computer assignments: There will be a few specially designated assignments during the
semester whose purpose is to introduce you to graphing and mathematical manipulation
computer programs. The use of computer technology to solve mathematical problems is
one of the great advances of our age, and so I want all of you to have at least a fleeting
introduction to this side of the subject. No prior knowledge of the relevant software
technology is required to work these assignments. More details will come later in the
Exams: There are two course-wide midterms and a final. The midterms will take place
on Wednesday, October 18 and Wednesday, November 15, both from 7:00-9:00pm in
Science Center C & D. Mark these dates on your calendar now, as no make-ups will be
given. The final exam is scheduled by the University for a date in mid January. According
to the Course Catalogue, the preliminary schedule has the final on Saturday, January 13 in
the morning. The University will confirm this date later in the semester.
Grading: Your final grade will be based on your performance on the homework (30%), the
computer assignments (3%), the two midterms (10% for the first and 15% for the second),
and the final (42%). 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. In as much as computerized graphing
programs aid you to develop this ability, you are encouraged to employ them as part of the
learning process. In this regard, the scheduled computer assignments are designed to
introduce you to the tool of computer graphing and mathematical manipulation.
However, be forewarned that for the purposes of this course, computers should be
considered solely as an aid to the development of geometric intuition. This course is
teaching various concepts whose applications may or may not be facilitated by a computer.
However, without a strong understanding of the underlying concepts, the computer wonıt be
much use. The point here is that computers can do many things, but they canıt yet think for
you. As powerful as todayıs computers are (and tomorrowıs will be), none of us will live
to see the day when they can turn pig feed into gold.
In any event, 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. With the exception of the specially designated computer assignments, 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 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 sectionıs
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 that differ significantly from
material discussed in class.)