# Math 250A

So far two misprints on the exam:
In Problem 4 a) let X \in Vect(X) should say let X \in Vect(M)
At the end of 4 c) it should say: \phi(x)=x for x not in U then \phi is
a diffeomorphism of M)

**Take home final exam **

Below is a proof of the inverse function theorem that I took from
the Harvard web page. It in turn is basically the proof in Spivak's
Calculus on Manifolds.

**The inverse function theorem **

Below is a nice discussion of the long line by Richard Koch
(U. Oregon).

**The long line **

The next file is chapter 1 in Warner's Differential Geometry:

**Differential Geometry Chapter 1 **

Homework Problems

pp.5-7 4,5,7,12,16,17

p. 12 4,5

p. 18 3,5

p. 25 1,2,11,13 Read section 4 of chapter 1 in text.

p.50 Warner problems 6,9,10,14 (In Warner's book the second axiom
is a part of the definition of a manifold.
Additional Exercises:

1. Prove that if M and N are connected, compact manifolds and f: M -> N is
smooth and has a bijective differential at every point then f defines
a covering of N. (In particular f is surjective.)

2. Prove that if M is a connected one dimensional manifold with an
everywhere non-vanishing vector field that M satsifies the second axiom of countability.

Below is a problem set that leads to a completion of our proof
that a 1-dimensioal cnnected compact manifold is diffeomorphic with
S^1.

**More problems 11/20 **

Problems from the text.

p.55-56 1,3,9,10,11

### Hand written lectures:

**Lecture 2 **

**Lecture 3 **

**Lecture 4 **

**Lecture 5 **

**Lectures 6&7 **

**Lecture 8 **

**Lecture 9 **

**Lecture 10 **

**Lecture 11 **

**Lecture 12 **

**Lecture 13 **

**Lecture 14 **

**Lecture 15 **

**Lecture 16 **

**Lecture 17 **

**Lecture 18 **

The following is to expand on the argument in Lecture 18 that came
out a bit scrambled.

**An additional explanation for
material in Lecture 18 **

**Lecture 19 **

**Lecture 20 **

**Lecture 21 **

**Lecture 22 **

**Lecture 23 **

**Lecture 24 **

**Lecture 25 **

**Lecture 26 **

**Lecture 27 **

**Lecture 28 **

**Lecture 29 **

**Lecture 30 **