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

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.

**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).

**Webpage:** `http://www.math.lsa.umich.edu/~speyer/395`

**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
document.

You can read the solutions, and see when you are scheduled to record,
here.

**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)

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

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.

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