Our grader for the class is Chenzi Jin (cjin123@terpmail.umd.edu)
All homework refers to the textbook unless otherwise stated.
Homework 1 (Due 2/6):
1.1: 4, 12.
1.2: 3, 6, 12
Homework 2 (Due 2/13)
1.3: 1, 3, 4, 6, 9
Homework 3 (Due 2/20)
1.4: 1, 2 (As a hint for 2a: you can use the fact that the only nonempty set of a connected set that is both open and closed is the entire set.), 5, 6, 8
Homework 4 (Due 2/27)
1.4: 7
1.5: 1, 2, 4, 7
Homework 5 (Due 3/6)
1.6: 1 (Hint: You can use 1.1 #18 without proving it), 2, 5, 7
1.7: 1, 4 (Hint: You can use the fact that the rational numbers are countable, i.e. in bijection with the natural numbers)
Homework 6 (Due 3/13)
1.8 1, 2
Study for the midterm
Homework 7 (Due 3/27)
1.8 4, 6
2.1 1, 7, 8
For 2.1 1, here is a hint: one approach would be to use the fact that a smooth map is well-approximated by its derivative locally
Homework 8 (Due 4/3)
3.2 2, 5, 6
4.2 1, 5, 6
Homework 9 (Due 4/10)
4.3 Prove the properties of pull-back at the top of p. 164
4.4 3, 4, 5, 7, 8
Remarks/Hints: For 3, the bottom paragraph (starting "Many examples...") summarizes the d operation on functions. You should also use the equation half-way through p. 163, in the middle of the sentence "Here is an easy exercise..."; you do not have to do this exercise to use it. Finally, you should use the exercise at the very end of 4.3; you similarly do not have to do this exercise to use it. For 8b, the same equation half-way through p.163 is also very useful.
Homework 10 (Due 4/24)
4.4: 9, 10 (For problem 9, there appears to be a typo in the book. The references in exercises 6 and 5 should refer to exercises 7 and 6, in that order.)
4.5: 1
Prove the Multiplication Law (for open subsets of R^k) at the bottom of p. 174 and then also prove the Multiplication Law (for a general manifold) on p.176
4.7: 2
Homework 11 (Due 5/1)
4.4 11, 12
4.6 prove statements 1, 2 on p. 179
4.7 5, 7