# Problem Sets

## Homework Assignment 1.

**Problem 1**- Section 1.2, in the course notes, exercises 5,6,7,8,9 and 10.**Problem 2**- Section 1.3, exercise 7 and section 1.4, exercise 8.**Problem 3**- Section 1.5, exercises 1,2,3,4 and 10.**Problem 4**- Section 1.5, exercises 5,6,7,8 and 9.

Warning: In exercises 5 and 6 there is a small typographical error: The "T" should be an arbitrary k-tensor.

## Homework Assignment 2. Due March 9th

**Problem 1**- Section 1.9 exercises 11 and 12 a,b**Problem 2**- Section 2.1 exercises 5 and 6**Problem 3**- Section 2.2 exercises 7 and 8**Problem 4**- Section 2.4 exercises 5 and 6

## Homework Assignment 3. Due March 23rd

**Problem 1**- Section 2.5, exercise 8**Problem 2**- Section 3.2, exercises 4,5,6 and 7**Problem 3**- Section 3.4, exercise 2**Problem 4**- Section 3.4 exercise 3

Warning: In Problem 2, exercise 5, there is a typo: the form ω should be supported on UxA not on U.

## Homework Assignment 4. Due April 13th

## Homework Assignment 5. Due April 27th

**Problem 1**- Section 4.3, exercises 4 and 9**Problem 2**- Section 4.4, exercises 6 and 7**Problem 3**- Section 4.6, exercise 10**Problem 4**- Section 4.7, exercises 3,4,and 5

## Homework Assignment 6. Due May 11th

This homework assignment will consist of the following two "diagram chasing" problems.

**Problem 1**- Prove Mayer-Vietoris. (Hint: Exercises 1,2 and 3 in section 5.2.)**Problem 2**- Prove that if a manifold, X, admits a finite good cover, the DeRham cohomology groups of X are isomorphic to the Cech cohomology groups of the cover. (Hint:the diagram chasing argument at the end of section 5.8.)