Math 402/502: Advanced Calculus II - Spring 2019

Instructor: Cristina Pereyra

Homework Problems

MATH 402/502 - HOMEWORK PROBLEMS - Spring 2019

Homework 9 (due on Thursday April 11, 2019) This is the last homework. Choose 4 of the 6 problems to submit. You can do the other two problem for bonus points, please submit them on a separate page for me to grade.

  1. Exercise 6.5.1 (an example of a continuous differentiable function whose second order mixed partial derivatives do not coincide).
  2. Exercise 6.6.5 (some examples of contractions and strict contractions).
  3. Exercise 6.6.8 (nearby contractions have nearby fixed points).
  4. (Application of the fixed point theorem) Let f,g be real-valued continuous functions defined on the interval [0,1], that is f,g are in C([0,1]). Consider the uniform metric on C([0, 1]) given by d(f,g):= sup{|f(t)-g(t)|: t in [0,1]}. For f in C([0,1]), let F(f) be the continuous function defined for each t in [0, 1] by F(f)(t) = int_0^t uf(u)du. Show that F : C([0,1])->C([0,1]) is a contraction, i.e. d(F(f),F(g)) <= a d(f,g), f,g in C([0,1]) for some a in (0,1). Explain why this implies that the equation f(t) = int_0^t uf(u)du for t in [0,1] has a unique solution f in C[0, 1].
  5. Exercise 6.7.3 (application of the Inverse Function Theorem).
  6. (Application of the implicit function theorem) Prove that in some neighborhood of (0,0) of R^2 there exists a unique continuously differentiable function f:R^2->R such that in this neighborhood x+y+f(x,y)-sin(xyf(x,y))=0. Find the partial derivatives of the function f at (0,0).
Reading Assignment: Sections 6.6 (Contraction Mapping theorem), 6.7 (Inverse function theorem), 6.8 (Implicit function theorem).

Homework 8 (due Thursday April 4, 2019))

  1. Exercise 6.1.4 (show that linear transformations are bounded operators)
  2. Exercise 6.2.2 (Uniqueness of derivatives).
  3. Exercise 6.3.3 (a function not differentiable at zero with directional derivatives in all directions).
  4. Bonus: Exercise 6.4.3 (chain rule in several variables).
  5. Exercise 6.4.5 (application of the chain rule).
Reading assignment: Sections 6.1-6.5

Homework 7 (due on Thursday March 29, 2019)

  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 (definition and basic properties of the logarithm. All references in the Hints except for the last one are to Tao's Analysis I.).
Reading Assignment: Chapter 4.

Homework 6 (due on Thursday Feb 28, 2019)

  1. Exercise 3.2.2(c) (partial sums of geometric series converges point-wise but not uniformly on the open interval of convergence).
  2. Exercise 3.3.1 (uniform limits preserve continuity).
  3. Exercise 3.3.4 (uniformly convergent sequence of continuous functions then f_n(x_n) converges to f(x) if x_n converges to x)
  4. Exercise 3.4.2 (metric space of bounded functions).
  5. Exercise 3.6.1 (if a series of integrable functions converges uniformly on [a,b] then can interchange integration and series).
Reading Assignment: Sections 3.1-3.7.

Homework 5 (due on Thursday Feb 21, 2019)

  1. Exercise 1.5.3 (Prove Heine-Borel Theorem: K is compact in R^n if and only if K is closed and bounded. One implication is Corollary 1.5.6, use it)
  2. Exercise 2.2.3 (absolute value preserves continuity)
  3. Exercise 2.2.11 (continuous in each variable does not imply jointly continuous)
  4. Exercise 2.3.4 (composition preserves uniform continuity)
  5. Exercise 2.4.4 (continuity preserves connectedness, see definitions in Section 2.4)
Reading Assignment: Sections 2.2, 2.3, 2.4 (2.5 is about Topological spaces you are welcome to read the section, I won't lecture on it)

Homework 4 (Due on Thursday Feb 14, 2019) Exercises come from Tao's Analysis II book, 3rd edition. Note that in the first and second editions Chapters are enumerated starting at Chapter 12 instead of Chapter 1, so Exercise 12.1.3 in the second edition corresponds to Exercise 1.1.3 in the third edition. Please write the bonus problem in a separate piece of paper for me to grade.

  1. Exercise 1.1.3 (b)(d) (examples of pairs (X,d) where d satisfies all but 1 one of the defining properties of a metric).
  2. Exercise 1.1.5 (Cauchy-Schwarz and triangle inequality in R^n with Euclidean or ell^2 metric).
  3. Exercises 1.2.1 and 1.5.12 (Discrete metric: interior/exterior points, identify compact and not compact sets, always complete).
  4. Exercise 1.2.3(b)(e)(f)(g) (basic properties of open and closed sets).
  5. Exercise 1.4.4 (Cauchy sequence with a convergent subsequence must converge and to the same limit).
  6. Exercise 2.1.1 (three equivalent definitions of continuity at a point).
  7. Bonus Exercise 1.1.15 (l^1 and sup metric on the space of absolutely convergent sequences are not equivalent)
Reading Assignment: Sections 1.1, 1.4, 2.1 in Tao's Analysis II book.

Homework 3 (due on Thursday Feb 7th, 2019)

  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).
  4. Bonus Exercise 11.9.1 (About a function F defined by integration of a R.I. function f, so that F is not differentiable on the rationals --you will need f from Exercise 9.8.5. which you can do for more bonus points--)
    Exercise 9.8.5: In this exercise we give an example of a function which has a discontinuity at every rational point, but is continuous at every irrational. Since the rationals are countable, we can write them as Q={q(0), q(1), q(2),...}, where q:N-->Q is a bijection from N to Q. Now define a function g:Q-->R by setting g(q(n)):= 2^{-n} for each natural number n; thus g maps q(0) to 1, q(1) to 2^{-1}, etc. Since sum_{n=0}^{\infty} g(r) is absolutely convergent, we see that sum_{r in Q} g(r) is also absolutely convergent. Now define the function f: R-->R by f(x):= sum_{r in Q: r< x} g(r) .
    Since sum_{r in Q} g(r) is absolutely convergent, we know that f(x) is well-defined for every real number x.
    1. (a) Show that f is strictly monotone increasing. (Hint: you will need to use that rationals are dense in R.)
    2. (b) Show that for every rational number r, f is discontinuous at r. (Hint: since r is rational, r=q(n) for some natural number n. Show that f(x) >= f(r) + 2^{-n} for all x>r.)
    3. (c) Show that for every irrational number x, f is continuous at x. (Hint: first demonstrate that the functions f_n(x):=sum_{r in Q:r=2^{-n}} g(r) are continuous at x, and that |f(x)-f_n(x)|<= 2^{-n}.)
Reading assignment: Sections 11.9 and 11.10.

Homework 2 (due on Thursday Jan 31st, 2019)

  1. Exercise 11.3.5 (comparing only upper Riemann sums with upper Riemann integrals).
  2. Exercise 11.4.1 (b)(d) (Riemann integration law: show the case scalar a>0).
  3. 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 1/24/2019): Exercises from Tao's Analysis I, 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 11.2.1 (f piecewise constant w.r.t. P then it is piecewise constant w.r.t. a refinement P')
  3. 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),
  4. Exercise 11.2.3 (Piecewise constant integral is independent of the partition)
Reading Assignment Sections 11.1, 11.2 and 11.3

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Last updated: February 22, 2019