# Math 402/502: Advanced Calculus II - Spring 2017

## Homework Problems

### MATH 402/502 - HOMEWORK PROBLEMS - Spring 2017

Homework 9 (due on Tuesday April 25, 2017)

1. Exercise 17.4.3 (chain rule in several variables).
2. Exercise 17.4.4 (state and prove a version of the quotient rule, read example 17.4.2 regarding a product rule!).
3. Exercise 17.4.5 (application of the chain rule).
4. Exercise 17.5.1 (an example of a continuous differentiable function whose second order mixed partial derivatives do not coincide).
5. (Bonus Problem) Exercise 17.6.8 (nearby contractions have nearby fixed points).
6. (Bonus Problem) Exercise 17.7.3 (application of the Inverse Function Theorem)..
Reading Assignment: Sections 17.6 (Contraction Mapping theorem), 17.7 (Inverse function theorem), 17.8 (Implicit function theorem).

Homework 8 (due Thursday April 6, 2017))

1. Exercise 4.6.14 (complex limit laws).
2. Exercise 4.7.8 (tangent and arctangent).
3. Exercise 6.2.2 (Uniqueness of derivatives).
4. Exercise 6.3.3 (a function not differentiable at zero with directional derivatives in all directions).

Homework 7 (due on Thursday March 30, 2017)

1. Exercise 4.1.2 (examples of power series).
2. Exercises 4.2.5 and 4.2.6 (every polynomial is real analytic in R).
3. Exercise 4.5.1 (basic properties of the exponential).
4. Exercise 4.5.5 (logarithm properties).
Reading Assignment: Chapter 4 in third edition or Chapter 15 in second edition.

Homework 6 (due on Thursday March 2, 2017)

1. Exercises 13.4.4 and 13.4.5 (Continuity preserves connectedness and how it implies the Intermediate Value Theorem).
2. Exercise 14.2.2(c) (partial sums of geometric series converges point-wise but not uniformly on the open interval of convergence).
3. Exercise 14.3.1 (uniform limits preserve continuity).
4. Exercise 14.4.2 (metric space of bounded functions).
5. Exercise 14.6.1 (if a series of integrable converges uniformly on [a,b] then can interchange integration and series).

Homework 5 (due on Thursday Feb 23, 2017)

1. Exercises 12.2.1 and 12.5.12 (Discrete metric: interior/exterior points, identify compact and not compact sets, always complete).
2. Exercise 12.2.3(b)(e)(f)(g) (basic properties of open and closed sets).
3. Exercise 12.2.4 (the closure of an open ball is not necessarily the closed ball)
4. Exercise 12.5.3 (Prove Heine-Borel Theorem: K is compact in R^n if and only if K is closed and bounded)
5. (Bonus) Exercise 12.5.2 (Compact metric spaces are complete and bounded)
6. (Bonus) Exercise 12.5.11 (Show that is a metric space has the open cover property then it is compact)
Please write the bonus problem(s) in a separate page with your name, I will grade these bonus problem for those who choose to turn it in).

Reading Assignment: Sections 12.2, 12.3, 12.5, 13.3, 13.4

Homework 4 (Due on Thursday Feb 16, 2017)

1. Exercise 12.1.3 (examples of pairs (X,d) where d satisfies all but 1 one of the defining properties of a metric).
2. Exercise 12.1.5 (Cauchy-Schwarz and triangle inequality in R^n with ell^2 metric).
3. Exercise 12.4.4 (Cauchy sequence with a convergent subsequence must converge and to the same limit).
4. Exercise 13.1.1 (three equivalent definitions of continuity at a point).
Reading Assignment: Sections 12.1, 12.4, 13.1 (remember that Chapter 12 is the first chapter in book 2 in 2nd edition, renamed Chapter 1 in the third edition).

Homework 3 (due on Thursday Feb 9th, 2017)

1. Exercise 11.6.3 (integral test).
2. Exercise 11.9.2 (two anti-derivatives for the same function differ by a constant).
3. Exercise 11.10.1 (integration by parts formula).
Reading assignment: Sections 6, 9 and 10.

Homework 2 (due on Thursday Feb 2, 2017)

1. Exercise 11.2.4(h) (additivity on domain for p.c. integral, you can use prior properties).
2. Exercise 11.3.5 (comparing upper and lower Riemann sums with upper and lower Riemann integrals).
3. Exercise 11.4.1 (b)(d) (Riemann integration laws).
4. Exercise 11.4.2 (if a positive, continuous function on [a,b] has Riemann integral equal to zero, then the function must be identically equal to zero on [a,b]).
Reading Assignment: Sections 11.4, 11.5, 11.6, 11.7.

Homework 1 (due on Th 1/26/2017): Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:

1. Exercise 11.1.4 (P1#P2 is a partition and a common refinement for both P1 and P2)
2. Exercise 11.2.2 (only show for the product: if f,g are piecewise constant (p.c.) on I then fg is p.c. on I),
3. Exercise 11.2.3 (Piecewise constant integral is independent of the partition)
Reading Assignment Sections 11.1, 11.2 and 11.3