Math 395: Honors Analysis I
Professor: David E Speyer
Overview: This is a rigorous course in
multivariable calculus, including the necessary constructions from
linear algebra and many of the advanced topics which usually are
considered too challenging to fit into a first course. Students should
expect a heavy workload and a rapid pace.
For the first half of the course, we will work with
differentiation. The big questions are "How should we define the
derivative of a multivariable function?" "How do we shift between
equations like x2 + y2=1 and
parametrizations like (x,y) = (cos t, sin t)?" "How do we do optimization in many variables?"
In the second half of the course, we will work with integration. Here
the big questions are "How do we define the multivariate integral?"
"How can we manipulate such integrals?" "What is the analogue of the
fundamental theorem of calculus in many variables?"
My bias is toward the geometrical and computational, and away from
analysis. I would rather prove beautiful properties of nice functions
than study the zoo of bizarre functions.
Course meets: Monday, Wednesday, Friday 2:30-4:00; 1449 Mason Hall
Office Hours: Wednesday 9-12, Thursday 1-4 in 2844 East
Hall. TA's office hours Thursday 5:30-6:30 in 2851 East Hall (Nesbitt Room).
Textbook: Analysis on Manifolds, by Munkres. The book
will be only loosely connected to class; I hope it will primarily
serve as a reference when my lectures are confusing.
Teaching Assistants: Noah Luntzlara (nluntzla@umich)
and Wenyu Jin (wyjin@umich))
Intended Level: Undergraduate math majors who are comfortable with
rigorous proofs and have already taken proof-based courses on single
variable calculus and linear algebra. Most students will have taken
the 295-296 sequence.
Lie groups on Fridays Each Friday, I will bring a collection of problems. These
problems will start with developing the properties of the matrix
exponential and its applications and then pivot toward understanding the structure of Lie
groups. I will assign students to take
turns recording solutions to these problems in a shared
You can read the solutions, and see when you are scheduled to record,
Sections: Fridays 4-5 in 2866 East Hall
Student work expected: I will assign weekly problem sets, due
on Fridays in class. No late homeworks will be accepted, but I will
drop the lowest two homework grades. See below for policy on collaboration.
Students will be expected to take turns recording solutions to the
problems which are discussed on Fridays.
There will be one in class exam on October 20 and one final on Tuesday,
December 19, 1:30-3:30 PM. The problems on the exams will be similar to the
homework, and may be distributed in advance.
Grading: I will combine your grades into a numerical score,
which will then be turned into letter grades by a curve which I expect will
be quite generous. The numerical score will be computed as follows:
60% homeworks, 10% for the midterms, 20% for the final, 10%
for participation in the Friday discussions and in recording the
results of them.
One of the things which I think is wrong with
typical American education is that we teach people that 60% success is
a failure; if 60% of my research projects succeeded, I would consider
it great. I intend to assign problems difficult enough that you will
not solve all of them.
- Problem Set 0, Due September 15 (TeX)
- Problem Set 1, Due September 22 (TeX)
- Problem Set 2, Due September 29 (TeX)
- Problem Set 3, Due October 6 (TeX)
- Problem Set 4, Due October 13 (TeX)
- No problem set for October 20, due to midterm.
- Problem Set 5, Due October 27 (TeX)
- Problem Set 6, Due November 3 (TeX)
- Problem Set 7, Due November 10 (TeX)
- Problem Set 8, Due November 17 (TeX)
- No problem set due for Thanksgiving.
- Problem Set 9, Due December 1 (TeX)
- Problem Set 10, Due December 8 (TeX)
Homework Policy: You are welcome to consult each other
provided (1) you list all people and sources who aided you, or
whom you aided and (2) you write-up the solutions independently, in
your own language. If you seek help from mathematicians/math
students outside the course, you should be seeking general advice, not
specific solutions, and must disclose this help. I am, of course, glad to
provide help, as are your TA's Noah Luntzlara and Wenyu Jin.
I don't intend for you to need to consult books and papers outside
your notes. If you do consult such, you should be looking for
better/other understanding of the definitions and concepts, not
solutions to the problems.
You MAY NOT post homework problems to internet fora seeking
solutions. Although I participate in some such fora, I feel that they have a major tendency to be
too explicit in their help; you can read further thoughts of mine here. You may post questions asking for clarifications
and alternate perspectives on concepts and results we have covered.
Seeking problem solutions from people and sources outside this course,
or directly copying solutions from your fellow students, is
plagiarism and will meet severe consequences.
Integration of vector valued functions of a real variable
Four different ways we can define
a manifold. Inspired by this excellent remark by Tom Goodwillie.
The hardest implication between the
definitions of manifold
Coordinate patches on manifolds have left inverses
Every open set has an exhaustion