MATH 223 : Multivariable Calculus
Course
Description
Fall
Term 2004
Course Title: Multivariable
Calculus
Catalog Description: The
calculus of functions of more than one variable. Introductory vector
analysis, analytic geometry of
three dimensions, partial differentiation, multiple integration, line
integrals, elementary vector field theory, and applications (formerly MA 201).
Additional Description from Mathematics Department Webpage: All the functions you've studied in calculus so far live on a flat piece of paper. But you live in (at least!) three dimensions. Now you certainly know that calculus was invented to solve problems about the physical world, so we're going to have to move off that flat paper at some point. MATH 223 is where it happens. The key is the concept of a vector. If you've had a little bit of physics, you may have heard a vector is an object having direction and magnitude. In MATH 223, we'll tighten that definition up, and study functions whose domains and ranges consist of vectors. Can limit, derivative and integral make sense out here? The answer is yes, and when you're through you'll know how Newton's calculus Š the greatest intellectual achievement of humankind! Š made sense of Kepler's empirical observations about the motion of the planets Š the greatest scientific discovery of all time! Come to think of it, maybe this course should be required for graduationÉ
Instructor: Michael
Olinick, 314 Warner, Phone: 443-5559. Home telephone: 388-4290; email: molinick@middlebury.edu. Office Hours: Monday, Wednesday and Friday from 10 to 11 AM and
Noon to 1 PM, Tuesday from 9:30 AM
to 1 PM, Thursday from 11 to Noon.
I would be happy to make an appointment to see you at other mutually convenient times.
Meeting Times: MWF 11:15 AM -
12:05 PM (Warner 202) and computer laboratory on Mondays from
1:30 to 2:20 PM (MA 223Y) or 2:35
to 3:25 PM (MA 223Z) in Munroe 214. WeÕll use some of the Monday periods for
make-up classes since I will have to cancel several Friday meetings.
Prerequisites: Calculus
II (MA 113 or MATH 0122) and
Linear Algebra (MA 200) or permission.
Textbook: Richard
E. Williamson and Hale F. Trotter, Multivariable
Mathematics, Fourth Edition, New York: Prentice Hall, 2004. ISBN 0130672769
Your daily assignments will include a few pages of reading in the text.
Be certain to read the book carefully (with pencil and paper close by!) and
to complete the relevant reading before coming to class and before embarking on the homework problems.
Supplemental Book: James
A. Carlson and Jennifer M. Johnson, Multivariable Mathematics with Maple, (Englewood Cliffs: Prentice-Hall, 1997).
Portions will be distributed in class.
.
Computer Laboratories: There
will be a number of computer exploratory assignments using Maple. These assignments will give you an
opportunity to investigate the ideas of
multivariable calculus in ways not available to previous generations of
students. You will see how relatively simple commands can direct a computer to
carry out complex calculations rapidly and without error. More importantly, you
can create and carry out experiments with the computer to develop and test your own conjectures. MapleÕs very powerful graphics capabilities
provide you with strong tools to deepen your understanding of calculus through
visualization of curves and surfaces.
Requirements: There
will be three midterm examinations and a final examination in addition to
required daily homework assignments, weekly laboratory reports and, perhaps,
occasional very short papers. The
midterm examinations will be given in the evening to eliminate time pressure.
Tentative dates for these tests are:
Tuesday,
October 12
Monday,
November 8
Wednesday,
December 1
Final Exam: The
College's Scheduling Officer has set Thursday, December 16 from 7 to 10 PM as the time and date of the final exam.
Homework: Mathematics is not a spectator sport! You
must be a participant. The only effective way to learn mathematics is to do
mathematics. In your case, this includes
working out several hundred calculus problems.
There
will be daily written homework
assignments which you will be expected to complete and submit. They will be
corrected and assigned a numerical score, but I view these assignments
primarily as learning
rather than testing experiences. I will occasionally assign some challenging
problems which everyone may not be able to solve. You should, however, make an
honest attempt at every problem.
Each
homework assignment will probably take you between 2 and 3 hours to complete;
this includes the reading and problem solving. If you keep pace with the course by spending an hour or so
each day on it, then you will be quite successful. If you wait until the end of
the week and then try to spend one six hour block of time on the material, then
experience shows you face disaster!
Grades: Grades
in the course will be based primarily on the examinations and class
participation; effort and success on the homework will be considered in
borderline situations
Help: Please
see me immediately if you have any difficulties with this course. There are
ample resources on campus for assistance.
One of the essential
characteristics of college life that distinguishes it from secondary school is the increased
responsibility placed on you for your own education. Most of what you
will learn will not be told to you by a teacher inside a classroom.
Even
if our model of you were an empty vessel waiting passively to be filled with
information and wisdom, there wouldnÕt be time enough in our daily meetings to
present and explain it all.
We
see you, more appropriately, as an active learner ready to confront aggressively
the often times subtle and difficult ideas our courses contain. You will need
to listen and to read carefully, to master concepts by wrestling with numerous
examples and problems, and to ask thoughtful questions.
As
you progress through the undergraduate mathematics curriculum, emphasis changes
from mastering techniques to solve problems to learning the theory that
underlies the particular subject you are studying. Multivariable Calculus is a transitional
course. You will do plenty of calculations, find many derivatives and deal with
a full quota of integrals. You will also find more of your effort directed
toward understanding definitions, statements of theorems and their proofs. You
will even be expected to come up with some short proofs of your own.
One
of my goals for you this term is to develop your skills in reading mathematical
expositions. I will expect that you will heave read (perhaps more than once!)
in advance the sections of the text relevant to the topic we will be exploring
in class that day. I will not normally present a lecture which substitutes for
reading the text. I will more likely use time in class to give a broader
overview or alternative proofs
or interesting applications and
extensions of the material or previews of the next section.
MATH 223: Fall, 2004
Tentative Course Outline
(Times are approximate)
I. Review
(on your own as needed)
Vectors
Equations
and Matrices
Vector
Spaces and Linearity
II.
Derivatives (2 weeks)
Functions
of One Variable
Several
Independent Variables
Partial
Derivatives
Parametrized
Surfaces
III. Differentiability (1 week)
Limits
and Continuity
Real-Valued
Functions
Directional
Derivatives
Vector-Valued
Functions
IV. Vector Differential Calculus (2+ weeks)
Gradient
Fields
The
Chain Rule
Implicit
Differentiation
Extreme
Values
Curvilinear
Coordinates
V.
Multiple Integration (3 weeks)
Iterated
Integrals
Multiple
Integrals
Integration
Theorems
Change
of Variable
Centroids
and Moments
Improper
Integrals
VI. Integrals
and Derivatives on Curves (1 week)
Line
Integrals
Weighted
Curves and Surfaces of Revolution
Normal
Vectors and Curvature
Flow
Lines, Divergence, and Curl
VII. Vector Field
Theory (2+
weeks)
GreenÕs
Theorem
Conservative
Vector Fields
Surface
Integrals
GaussÕs
Theorem
StokesÕs
Theorem