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

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