MATH 223A :
Multivariable Calculus
Course Description/Syllabus
Fall Term 2022
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 – the greatest intellectual achievement of humankind! – made sense of Kepler's empirical observations about the motion of the planets – the greatest scientific discovery of all time! Come to think of it, maybe this course should be required for graduation…
Course Website: f22.middlebury.edu/MATH0223A
Instructor: Michael Olinick, Office: 202 Warner, Phone: 443-5559. Home telephone: 388-4290; email: molinick@middlebury.edu. Usual Office Hours: Monday, Wednesday and Friday from 9 to 10 AM and 12:15 to 1:45 PM. I would be happy to make an appointment to see you at other mutually convenient times.
Meeting Times: MATH
223A: MWF 10:10
AM – 11:00 AM (Warner 101)
Prerequisites: Calculus II (MATH 0122) and Linear Algebra (MATH 200) or permission.
Textbook: Michael Olinick, Multivariable Calculus: A Linear Algebra Based Approach, First Edition: Kendall-Hunt, 2021; ISBN 9781792437915 (https://he.kendallhunt.com/product/multivariable-calculus-linear-algebra-based-approach)
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, (Prentice-Hall, 1997). We will distribute portions in class.
Computer Algebra Systems: Mathematically oriented software such as Maple, MATLAB, and Mathematica give you an opportunity to investigate the ideas of multivariable calculus in ways not available to previous generations of students. 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 to develop and test your own conjectures. The very powerful graphics capabilities of these applications provide you with strong tools to deepen your understanding of multivariable calculus through visualization of curves and surfaces. I will feature Maple in my presentations but you should feel free to MATLAB or Mathematica or other computer algebra systems. I will try to schedule a few optional introductory Maple lab sessions.
Requirements: There will be three midterm examinations and a final examination in addition to required daily homework assignments and an extended independent project. The midterm examinations will be given in the evening to eliminate time pressure. Tentative dates for these tests are:
Monday, October 3
Wednesday,
November 2
Wednesday,
November 30
Final Exam: The registrar’s office has set Wednesday, December 14 from 9 AM to Noon as the date and time for the final exam.
Course Grades: Each of the midterm exams
will be worth approximately 20%, the final about 25% and the independent
project and homework roughly 15%. I will make adjustments
with later work counting more heavily if students show improvement over earlier
results.
Accommodations Students who have Letters of Accommodation in this class are
encouraged to contact me as early in the semester as possible to ensure that
such accommodations are implemented in a timely fashion. For those
without Letters of Accommodation, assistance is available to eligible students
through the Disability Resource Center (DRC). Please contact ADA
Coordinators Jodi Litchfield and Peter Ploegman of
the DRC at ada@middlebury.edu for more information.
All discussions will remain confidential.
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 many multivariable 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!
Our homework grader this term is Cody Mattice, ’24 (cmattice@middlebury.edu)
s
Grades: Grades in the course will be based primarily on the examinations, homework, projects, and class participation.
Help: Please
see me immediately if you have any difficulties with this course. There are
ample resources on campus for assistance. Our course tutor is Abby Truex, ’04. She will soon post hours when
she is available for help.
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 would not 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 have 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 223A: Fall, 20222
Tentative Course Outline
(Times are approximate)
Single Variable
Calculus
Vectors
Equations and
Matrices
Vector Spaces and
Linearity
Functions of One
Variable
Several
Independent Variables
Partial
Derivatives
Parametrized
Surfaces
Limits and
Continuity
Real-Valued
Functions
Directional
Derivatives
Vector-Valued
Functions
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
Improper
Integrals
Line Integrals
Weighted Curves
and Surfaces of Revolution
Normal Vectors
and Curvature
Flow Lines,
Divergence, and Curl
Green’s Theorem
Conservative
Vector Fields
Surface Integrals
Gauss’s Theorem
Stokes’s Theorem