Math 402/502: Advanced Calculus II - Spring 2017
- HOMEWORK PROBLEMS - Spring 2017
Homework 9 (due on Tuesday April 25, 2017)
Reading Assignment: Sections 17.6 (Contraction Mapping theorem), 17.7 (Inverse function theorem), 17.8 (Implicit function theorem).
- Exercise 17.4.3 (chain rule in several variables).
- Exercise 17.4.4 (state and prove a version of the quotient rule, read example 17.4.2 regarding a product rule!).
- Exercise 17.4.5 (application of the chain rule).
- Exercise 17.5.1 (an example of a continuous differentiable function whose second order mixed partial derivatives do not coincide).
- (Bonus Problem) Exercise 17.6.8 (nearby contractions have nearby fixed points).
- (Bonus Problem) Exercise 17.7.3 (application of the Inverse Function Theorem)..
Homework 8 (due Thursday April 6, 2017))
Reading assignment: Sections 6.1-6.5
- Exercise 4.6.14 (complex limit laws).
- Exercise 4.7.8 (tangent and arctangent).
- Exercise 6.2.2 (Uniqueness of derivatives).
- Exercise 6.3.3 (a function not differentiable at zero with directional derivatives in all directions).
Homework 7 (due on Thursday March 30, 2017)
Reading Assignment: Chapter 4 in third edition or Chapter 15 in second edition.
- Exercise 4.1.2 (examples of power series).
- Exercises 4.2.5 and 4.2.6 (every polynomial is real analytic in R).
- Exercise 4.5.1 (basic properties of the exponential).
- Exercise 4.5.5 (logarithm properties).
Homework 6 (due on Thursday March 2, 2017)
Reading Assignment: Sections 14.1-14.7.
- Exercises 13.4.4 and 13.4.5 (Continuity preserves connectedness and how it implies the Intermediate Value Theorem).
- Exercise 14.2.2(c) (partial sums of geometric series converges point-wise but not uniformly on the open interval of convergence).
- Exercise 14.3.1 (uniform limits preserve continuity).
- Exercise 14.4.2 (metric space of bounded functions).
- 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)
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).
- Exercises 12.2.1 and 12.5.12 (Discrete metric: interior/exterior points, identify compact and not compact sets, always complete).
- Exercise 12.2.3(b)(e)(f)(g) (basic properties of open and closed sets).
- Exercise 12.2.4 (the closure of an open ball is not necessarily the closed ball)
- Exercise 12.5.3 (Prove Heine-Borel Theorem: K is compact in R^n if and only if K is closed and bounded)
- (Bonus) Exercise 12.5.2 (Compact metric spaces are complete and bounded)
- (Bonus) Exercise 12.5.11 (Show that is a metric space has the open cover property then it is compact)
Reading Assignment: Sections 12.2, 12.3, 12.5, 13.3, 13.4
Homework 4 (Due on Thursday Feb 16, 2017)
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).
- Exercise 12.1.3 (examples of pairs (X,d) where d satisfies all but 1 one of the defining properties of a metric).
- Exercise 12.1.5 (Cauchy-Schwarz and triangle inequality in R^n with ell^2 metric).
- Exercise 12.4.4 (Cauchy sequence with a convergent subsequence must converge and to the same limit).
- Exercise 13.1.1 (three equivalent definitions of continuity at a point).
Homework 3 (due on Thursday Feb 9th, 2017)
Reading assignment: Sections 6, 9 and 10.
- Exercise 11.6.3 (integral test).
- Exercise 11.9.2 (two anti-derivatives for the same function differ by a constant).
- Exercise 11.10.1 (integration by parts formula).
Homework 2 (due on Thursday Feb 2, 2017)
Reading Assignment: Sections 11.4, 11.5, 11.6, 11.7.
- Exercise 11.2.4(h) (additivity on domain for p.c. integral, you can use prior properties).
- Exercise 11.3.5 (comparing upper and lower Riemann sums with upper and lower Riemann integrals).
- Exercise 11.4.1 (b)(d) (Riemann integration laws).
- 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]).
Homework 1 (due on Th 1/26/2017):
Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:
Reading Assignment Sections 11.1, 11.2 and 11.3
- Exercise 11.1.4 (P1#P2 is a partition and a common refinement for both P1 and P2)
- 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),
- Exercise 11.2.3 (Piecewise constant integral is independent of the partition)
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Last updated: February 2nd, 2017