# math 439

Course Syllabus: https://drive.google.com/open?id=1OOAq2im0JnU3ajZGS9xQ4xdUFA4AVWPd

Isoperimetric inequality supplement: https://drive.google.com/file/d/1CGOV2l4X5PlXHhkfrxmUlzMu8uP0cOZ1/view?usp=sharing (now with rigidity case)

(please let me know if you find any typos)

Homework assignments:

**Homework 1 (assigned 9/3 due 9/10):** out of Do Carmo do section 1-2 problems 2, 3, section 1-3 problems 1, 7, 8, 10

**Homework 2 (assigned 9/10 due 9/17): **out of Do Carmo do section 1-4 problems 12, 13, 14 and section 1-5 problems 2, 4, 5

**Homework 3 (assigned 9/17, due 9/24): **out of the isoperimetric inequality handout, do exercises 2,3, 4,6. Out of Do Carmo do section 1-5 problems 10, 13, 14

**Homework 4 (assigned 9/24, due 10/1):**** **There is no homework this week!! Study for the test instead, which will be 10/3 (Thursday) in class.

What you need to know for midterm 1 (I won't include anything not discussed below): You should be familiar with all the definitions and theorems (including proofs) in sections 1-1 to 1-5, and sections 2-2 and 2-3. I wouldn't ask any harder proofs without breaking it up into some easier parts (and in doing so give hints).

You should know (this is something somewhat hidden in the book) how to show every regular parameterized curve admits a unit speed parameterization.

You'll also need to know a couple things from the appendix to chapter 2. In particular, you should know the statement of the inverse function theorem, and you should know the "original" definition of differential of a function F (definition 1 in the appendix), how to show that definition is well defined, and how to compute the differential "in coordinates" from it (basically, know how to show proposition 7 in the appendix).

As for problems, all the homework problems assigned from the book are fair game and may possibly appear. Out of the isoperimetric supplement, I may or may not ask you to prove the isoperimetric inequality using the Brunn-Minkowski inequality (if I choose to ask, I'll give you the inequality). Besides those, I might also ask for solutions of the following problems out of Do Carmo:

section 1-2: problems 1, 4

section 1-4: problems 1, 7

section 1-5, problems 1, 6, 7, 13, 14, 15

section 2-2, problems 1, 2, 3, 4, 7, 8, 9

section 2-3, problems 1, 2, 3, 4, 6, 7, 9, 11

Of course, you don't need to turn your solutions to these problems in, but thinking about all of them is obviously encouraged. I'd be more than happy to answer any questions over email or during office hours.

**Homework 4 (assigned 10/8 due 1017)**: Out of Do Carmo do section 2.2: problems 17, 19, section 2.3: problems 5, 13 and fill in the details to example 3 on page 77, section 2.4: 15, 17, 18, 28, section 2.5: problems 1,5.

**Homework 5 (assigned 10/18 due 10/29): **Out of Do Carmo do section 3.2: problems 2, 3, 5, 8, section 3.3 problems 1, 22, 24

**Homework 6 (assigned 10/29 due 11/7): **Out of Do Carmo do section 4-2 problems 2, 3, 10, 18

**Second midterm preparation: **The date of the midterm has been moved to be on 11/21. It is not cumulative and will cover material after midterm 1. I won't ask you anything that is not discussed below:

As before, you should know how to do the homework problems. You should also be able to do most of the proofs - if the calculation is very long (e.g. Gauss formula) I probably won't ask you to do it. On that note I won't ask you to remember any particularly long formulas, but I will probably give you some formulas and ask you to derive some other formula.

Besides the homework problems (namely, those in homeworks 4-6) above, I might ask you the following questions:

Section 3.3: problem 13. Section 4.3: 1, 3,4, 6, 7, 8,9. Section 4.4 problems 3, 12,13. Section 4.6 problems 5, 9, 10