ma122 — Series and Multivariable Calculus by lzg15357


									         ma122 — Series and Multivariable Calculus
                                                  Fall, 2009
                                     Fernando Q. Gouvêa

    As our official title suggests, this course focuses on two important topics within
Calculus. But they will have unequal status in the course. Our dominant concern
will be to extend the ideas you learned in your first calculus course to the multi-
variable setting. In order to do that, however, we will frequently have to rethink
the ideas in single-variable calculus, finding the right way to understand them, the
point of view that can be generalized to several variables.
    Along the way, we will have the chance to add to the single-variable theory as
well, by exploring the theory of series. The most important aspect if this is the
representation of functions as infinite sums of simpler functions. This is one of the
fundamental methods of applied mathematics, and we will focus mostly on those
aspects that are most useful.
    In fact, all the while we will be concentrating on those ideas that have actually
been useful in the sciences. We won’t, however, merely focus on recipes for doing
things: we want to understand the Calculus, and not just to know how to use it.
In particular, I expect you to develop the ability to use your understanding of the
theory to apply it in new practical situations.

   Enough chatter; here’s the practical stuff:

Where to find me: My office is Mudd 412; my phone extension is 5836; my email
address is If you need to reach me when I’m not in my office,
email is the best method. If you prefer, feel free to call and leave a message — but
sometimes I take a while to notice the little red light. In any case, see the note on
email below.

Office hours: I’ve separated the following times as office hours:
   • Mondays, 11:00–12:00

   • Tuesdays, 1:00-2:00 and 3:00–4:00

   • Fridays, 11:00-12:00 and 3:00–4:00
Since I am teaching three courses this semester, you may arrive at my office to find
me busy with another student. Please be patient; I’ll try to make sure that everyone
gets a chance to see me. If you can’t come at any of these times, please call and
make an appointment. You may find me in my office at other times, but it is only
at the times above that I can guarantee that I’ll be available. I am usually not on
campus on Thursdays.

                   Living on Earth may be expensive, but it includes an annual free trip around the Sun.
2                                                                            ma122, Gouvêa, Fall 2005

   You are encouraged to come see me. It is part of your education, and one of
your privileges as a Colby student.

How the class will be organized: I will be trying to run the course using a mixture
of lectures and discussion and in-class activity. You will often be asked to work in
class in a group with others. At any time (and especially in classes where I’m doing
most of the talking) you should feel free to interrupt with questions and comments.
I will try to prompt this participation by asking you questions too!

Text: Our basic texts will be

    1. Chapters 9 and 10 of Calculus, by Hughes-Hallett, Gleason, et. al. (third

    2. Chapters 12–16 of Multivariable Calculus, by McCallum, Hughes-Hallett,
       Gleason, et. al. (fourth edition)

These books belong to a new generation of calculus texts which are designed to
be read, and which put an emphasis on understanding rather than on mechanical
proficiency. I will be asking you to read sections of the book as we go along, and I
will expect you to gain some understanding from this reading.
    But don’t stop there. Colby has a library, and it contains a great many books
about calculus. Some of them will be textbooks, and others are supplementary
books of various kinds. Do check them out!
    Some of you may feel the need to get a book to help you review material from
your first Calculus course. If you own a copy of your Calculus I text, that’s the
obvious thing to use. If not, I suggest that you get a copy of a book called How
to Ace Calculus, by Adams, Hass, and Thompson; it’s short, practical, and will
probably get the job done. (There’s also a How to Ace the Rest of Calculus, which
covers the material in this course. It’s worth a look too.)
    Some students find that Student Solution Guides and Study Guides are helpful.
These do exist for our textbook. The Student Solution Guide, for example, includes
full solutions (not just answers) for the odd-numbered problems in the book. If you
think something like this would help, check with the bookstore.

Technology: Calculators and computers are becoming ever more useful to people
who need to use mathematics, and it is important to learn how to use these tools.
For this class, I will often be using a calculator and will also often ask you to
do computations and/or to graph functions on a calculator. If you do not own a
graphing calculator, you might want to consider getting one. If you do own one,
learn to use it!
    In addition to using a graphing calculator, I will occasionally make use of a very
powerful computer program called Mathematica. This is rather hard to use, but if
you are interested in computing and are willing to put in the effort you might want
to investigate this program on the computers in the Math&CS Computer Lab and

                          We have enough youth — how about a fountain of SMART?
3                                                                                      ma122, Gouvêa, Fall 2005

in the Olin Lab. People that use mathematics in their work in the sciences or in
economics are more likely to use computer tools than calculators, so it makes sense
to begin to learn how to use these tools.

email: Email has become a fundamental communication tool, and we will be using
it in our course. Your first assignment will involve sending me an email message,
and I will occasionally use email to communicate with you either individually or as
a class.

Prerequisites: This is nominally a “second-semester calculus” class, some of you
may be worried about what I will be assuming you already know. (Of course, if
you’ve taken ma121 at Colby, you should be OK.) I will assume that you know
the basics about limits, continuity, derivatives, and integrals of functions of one
variable. I will expect you to be familiar with the logarithm and the exponential
function in addition to the basic trigonometric functions. As topics arise during
the semester, I can offer occasional quick reviews of the necessary background
material, but if you need extensive review you’ll need to work on your own. If you
have doubts about your background please come talk to me.

Quizzes: Approximately every other Friday I will give you a short in-class quiz.
These are intended simply to give you (and me) a reference point about how well
you are absorbing the material. These will not be announced. Since these are
partially a diagnostic tool, I will discard the worst of your quiz scores when I work
out your grade.

Homework: One can’t learn mathematics without doing mathematics, and doing
mathematics, in the context of the calculus, consists in solving non-trivial problems.
On the other hand, learning a new subject often requires doing a number of “five
finger exercises”: relatively simple problems that basically drill you on what you
have just learned. The odd-numbered problems in the book, which have answers
in the back, are a good way to provide yourself with this kind of practice.
    You will receive a weekly homework assignment every Tuesday, and it will be
due the following Tuesday. These assignments will often contain problems that
require a little creativity to solve. It is by solving these types of problems that you
will really begin to understand what the mathematics is about, and it is also this
kind of problem that you will meet in the “real world” when it comes time to use
your knowledge. So, while these problems will be difficult, solving them will be
worth the effort.
    Since these assignments are difficult, I will separate some class time to discuss
homework problems. We will do this on Fridays. This means you should make
sure to take a first stab at the problems before the Friday class so that you know
what questions you need to ask.
    Please don’t leave your assignments for the last day, because you will probably
not be able to do them in one day. The problems I’ll assign will require time for

                            Thesaurus: ancient reptile with an excellent vocabulary.
4                                                                                     ma122, Gouvêa, Fall 2005

their solution, and you should plan to put in that time. (Read this paragraph again.
You have been warned!)
    Since homework is so important, it will have a relatively heavy weight in your
grade. On the other hand, since homework is also the place to make mistakes and
learn from them, I will discard the worst homework score from each half of the

Outside of class: You should expect to have to study two to three hours outside
of class for every hour you spend in class. The best use of this time is to spend a
part of it working with a group of friends. This will add a social dimension to your
study, and will also help you resolve difficulties by using the differing strengths of
people in your group. The ideal sequence seems to be to work the problems on
your own first, then to work in a group to resolve any difficulties and reconcile any
differences (when two people in the group get different answers, it can be a great
learning opportunity).
    An important resource to help you with studying and homework is Calculus
After Hours, a Calculus lab where you can ask questions, work on homework, and
generally learn with others. Watch for the handout with more information about
this program.
    While you are free to work on homework with a group, you should write your
final draft by yourself, so that it reflects your state of understanding of the mate-
rial. In other words, getting help from others to understand, then writing a solu-
tion that reflects that understanding is completely acceptable, and even desirable.
Copying someone else’s solution without understanding is not acceptable, and will
be treated as a form of academic dishonesty. When you write up your solutions, be
careful: use good grammar and full sentences, explain your reasoning, justify your
approach, make it readable. Writing well requires real understanding; by requiring
yourself to write well you will make sure that you really do understand what is go-
ing on. If you have questions about what kind of writing I’ll be looking for, please
feel free to ask.

Writing Assignments: There will be one or two writing assignments during the
semester. These will probably involve working with a group of others to find some-
thing out, then writing up the results. I will provide more information on these

Tests: We will have three tests: two midterm tests and a final exam. The midterms
will be on October 7 and November 17. In order to allow you more time and to
let two of my courses take them together, the midterms will be in the evening: from
7:00 to 8:30 pm, in Olin 1. Please mark your calendars.
    All of the tests will consist of problems similar in style (but not in difficulty!) to
the problems assigned for homework.

Cheating: While you are encouraged to cooperate with others in class and in your

                     If bankers can count, how come they have eight windows and only four tellers?
5                                                                                       ma122, Gouvêa, Fall 2005

homework assignments, your other work is expected to be your own. In particular,
you are forbidden to get or give help during quizzes and tests. Please refer to the
Colby policy on academic honesty as stated in the College Catalogue.

Attendance: Class attendance is required. Should you need to miss a class, please
talk to me in advance to see if your absence can be excused. Missing too many
classes will result in academic warnings, grade penalties, and eventually dismissal
from the class. Please note: If you miss a Friday class during which a quiz was
given, you will not be offered an opportunity to make up that grade.

Grading: Your grade will be computed as follows:

                        quizzes                                10%
                        writing assignments                    15%
                        homework                               20%
                        midterm tests                          25%
                        final exam                              30%

An outline: Rather than giving you a week-by-week schedule (which I’d end up not
following), here’s an outline of what I’m planning to do, in roughly the order we’ll
want to do it. As we go along, I’ll point you to the relevant bits in the textbook.

     1. Functions of one and several variables.

     2. Linear functions, linear approximations.

     3. Derivatives as linear approximations, differential.

     4. Geometry of R2 and R3 ; vectors.

     5. Differentials and directional derivatives, the gradient.

     6. Improving the approximation: Taylor polynomials in one and several vari-

     7. Optimization, constrained optimization.

     8. Sums and integrals, infinite series.

     9. Integration of functions of several variables.

    10. If time allows, other topics.

                           The careful application of terror is also a form of communication.
6                                                                          ma122, Gouvêa, Fall 2005

About Me

   Students often wonder about their professors. You can find out more about me
by looking at my home page, at


But, just for fun, here are some factoids:

    • I was born in Brazil.

    • I have been at Colby since 1991.

    • I have two sons. One has a PhD in Political Science and works at a non-profit
      consulting firm. The other is a PhD student in Classics at the University of

    • My main research interests are in number theory and in the history of math-

    • I’m also interested in lots of other things. (The web site has a partial list.)

    • I’m a Christian; currently I am a member of the Lutheran Church of the
      Resurrection in Waterville.

    • I sing, but only in church.

    • According to philosopher Richard Rorty, I am “frightening, dangerous, vi-
      cious. . . ”

    • I have had a beard since 1980, but it has only been gray for the last few years.

    • I vote.

    • My house is the best place to eat in Waterville, but it’s only open by invitation.
      If you want to have a taste, come to the Mathematics Colloquia.

    • I own a rather excitable dog called Jelly Bean. She’s about one year old and
      is lots of fun.

    • On my bookshelves, you’ll find many books by (among others): Gene Wolfe,
      D. J. Enright, Dorothy L. Sayers, Jaroslav Pelikan, Reviel Netz, Barry Mazur,
      Gordon Fee, Robert Jenson, G. K. Chesterton, Walt Kelly, Mark Helprin,
      David B. Hart, Niccolò Tartaglia.

                                    93.5% of all statistics are made up.

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