Chapter 5, #17

Chapter 6, #7

Chapter 7, # 6

Chapter 7, #11

Show that the distribution in R3 given by the equation:

d z - y d x = 0

is nowhere integrable.

## Syllabus

## Tuesday, October 17, 2006

## Wednesday, October 04, 2006

### HW 1

Here is the homework assignment. It is due next Wed.

Chapt 2 (Vol 1) of Spivak 3,11

Show that if M is a smooth n manifold, and N is a smooth k-manifold, and f:M->N is a surjective, smooth map then there a set of full measure U in N, such that for each x in U, f^-1(x) is a smooth n-k manifold.

Show that a bundle E is orientable if and only if /\^n(E) admits a no where vanishing section. (The crazy asci mess there denote the highest wedge power of E). Show that E is orientable if and only if its dual is.

Chapter 3, number 27

Chapter 4 numbers 1,8

Chapt 2 (Vol 1) of Spivak 3,11

Show that if M is a smooth n manifold, and N is a smooth k-manifold, and f:M->N is a surjective, smooth map then there a set of full measure U in N, such that for each x in U, f^-1(x) is a smooth n-k manifold.

Show that a bundle E is orientable if and only if /\^n(E) admits a no where vanishing section. (The crazy asci mess there denote the highest wedge power of E). Show that E is orientable if and only if its dual is.

Chapter 3, number 27

Chapter 4 numbers 1,8

## Friday, September 22, 2006

### Syllabus, course text etc...

The basic "reference" text for the course will be Spivak, volume 1 and 2. The first portion of this course is a discussion of the language of modern differential geometry. Since the first year sequence now includes some of this language we will proceed quite quickly through the initial definitions. Spivak proceeds quite leisurely, so do not be alarmed at the apparently breakneck pace through the book.

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