Course Outline

Calculus on Manifolds

Math 4B03/6B03, Winter 2012
Dr. Ben Mares

Logistics

Classroom: HH/217, Tu Th Fr 2:30-3:20pm.
Office hours: Tu Th 1-2pm or by appointment, HH/319
E-mail: last name @math.mcmas...
Webpage: http://tensorial.com/math/manifolds

References

Primary reference:

From Calculus to Cohomology by Masden and Tornehave

Secondary references:

Differentiable Manifolds notes by Hitchin
Calculus on Manifolds by Spivak

Topics

Review of multivariable calculus
Tensors and differential forms
Generalized Stokes' Theorem
De Rham cohomology
Manifolds

Roughly, we will follow §1-§14 of Masden and Tornehave, with additional material as needed.

Homework

There will be approximately one problem set per week. Problem sets will be due approximately one week after they are posted online. They should be handed in at the beginning of class on the indicated date. Graduate students may have additional required problems. Late work will be accepted either with prior permission of the instructor, or via the MSAF system.

Grading

Letter grades will be determined primarily based on satisfactory completion of the problem sets. However, passing is contingent on a final evaluation covering the problem sets handed in. For this reason, it's important to follow the collaboration policy. Students are welcome to inquire about the current status of their grades at anytime.

Marking scheme for problem sets

Each problem will be assigned a number from 0-3, which have the following meanings:

3	Correct, or correct up to minor technical details.
2	Mostly correct, but has significant errors.
1	Incorrect, but some progress towards a solution.
0	No.

You are doing well if you get mostly 3s with occasional 2s. If you get 1s and 0s, that means you still need to talk with your classmates and/or me to figure out the related concepts to avoid falling behind. Please take this seriously and seek help if you need it. I'm happy to answer questions and help you out!

Collaboration Policy

You are strongly encouraged to work through problems with your classmates, and to seek out your own references. However, you must write up answers based entirely on your own understanding. Specifically, when writing the answers you turn in, you should be alone and not looking at notes from discussions or unofficial external sources. The primary purpose of the final evaluation is to verify compliance with this collaboration policy.

Final evaluation

At the end of term, I want to provide you the opportunity to reflect on what you learned throughout the semester, without the stress and burden of a typical final exam. In late March, prior to the "Test and Examination Ban," I plan to schedule an individual meeting with each student, approximately twenty minutes in length. It will be a friendly, non-adversarial, open-note discussion about completed homework. (Incomplete problems will be excluded from discussion.) I expect that, after briefly rereading your solution, you should be able to explain your reasoning on any problem. Typical questions might include, "Which problem did you find the most challenging?" or "Could you elaborate a bit more on your solution to Problem X?" Briefly rereading (but not redoing) your answers beforehand should be sufficient preparation.

According to the policies regarding oral examinations, I may need to make an audio recording.

* I welcome any comments or requests for alternate accommodations, but please make them sooner rather than later.

Grading philosophy

Your final grade will be based on completed homework problems, i.e. problems for which you could comfortably explain your answers during the final evaluation. Answers marked with '3' are those I consider complete. You will still get full credit towards problems with lesser marks wherever you are comfortable with the correct answers.

If you are unsure of the correct answers to old problems, you can discuss them with me. To maximize your credit, I recommend maintaining a list of the answers you comfortably understand, and then making this list as large as possible.

The final evaluation will be a guided, open-note discussion where I select problems from your list, and you explain your answers. I will not adjust your grade during the final evaluation as long as you demonstrate a basic understanding of the correct solutions from your list. I will schedule appointments with everyone for late-March via a Doodle scheduler.

Not all problems are of equal weight, and to maximize flexibility, there is no predetermined formula for the final grade. An 'A' corresponds to a list containing most (but not necessarily all) problems. As of winter break, everyone still handing in assignments can earn an 'A' with the appropriate amount of review. (Those with lower marks will require heavier review.)

Important Message

The instructor and university reserve the right to modify elements of the course during the term. The university may change the dates and deadlines for any or all courses in extreme circumstances. If either type of modification becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.