Math 395: Honors Analysis I

Professor: David E Speyer

Fall 2017

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 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 Sets

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