Math 402/502: Advanced Calculus II - Spring 2024

Instructor: Cristina Pereyra

Homework Problems

MATH 402/502 - HOMEWORK PROBLEMS - Spring 2024

Homework 10 (due on Tuesday April 23, 2024) This is the last homework. Choose 3 of the 5 problems to submit. You can do the other two problem for bonus points.

  1. Exercise 6.6.5 (some examples of contractions and strict contractions).
  2. Exercise 6.6.8 (nearby contractions have nearby fixed points).
  3. (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].
  4. Exercise 6.7.3 (application of the Inverse Function Theorem).
  5. (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 9 (due Friday April 12, 2024)) Do 3 problems out of the 5 proposed. The other 2 problems you can do for bonus points.

  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. Exercise 6.4.5 (application of the chain rule).
  5. Exercise 6.5.1 (an example of a continuous differentiable function whose second order mixed partial derivatives do not coincide).
Reading assignment: Sections 6.1-6.5

Homework 8 (due on Friday April 5, 2024) Do 4 problems out of the 6 proposed. The other 2 problems you can do for bonus points.

  1. Exercise 3.4.3 (The space of continuous functions in B(X->Y) is complete when Y is a complete metric space).
  2. Exercise 4.1.2 (examples of power series).
  3. Exercises 4.2.5 and 4.2.6 (every polynomial is real analytic in R).
  4. Exercise 4.5.1 (basic properties of the exponential).
  5. Exercise 4.5.5 (definition and basic properties of the logarithm. Most references in the Hints are to Tao's Analysis I, except for Theorems 4.5.2 and 4.1.6.).
  6. Exercise 4.7.1 (Trigonometric identities).
Reading Assignment: Chapter 4.

Homework 7 (due on Friday March 29, 2024 at 11:59pm)

  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 3.3.4 (uniformly convergent sequence of continuous functions then f_n(x_n) converges to f(x) if x_n converges to x)
  3. Exercise 3.3.6 and 3.3.7 (uniform limits preserve boundedness, compare to Exercise 3.2.4. Pointwise convergence does not preserve boundedness).
  4. Exercise 3.6.1 (if a series of integrable functions converges uniformly on [a,b] then can interchange integration and series).
  5. Exercise 3.7.3 (interchange of derivative and series).
  6. (Bonus) Exercise 3.4.2 (convergence in the metric space of bounded functions is equivalent to uniform convergence)..
Reading Assignment: Sections 3.1-3.7.

Homework 6 (due on Friday March 1st, 2024 at 11:59pm)

  1. Exercise 2.3.2 (Maximum principle).
  2. Exercise 2.3.4 (Composition preserves uniform continuity).
  3. Exercise 2.3.5 (Pairing of uniformly continuous functions --called direct sum in early editions-- is uniformly continuous).
  4. Exercise 2.3.6 (Uniform continuity is preserved by some arithmetic functions but not all. Focus on addition and multiplications only).
  5. Exercise 2.4.1 (Discrete metric and (dis)connectedness).
Reading Assignment Sections 2.3 and 2.4.

Homework 5 (due on Friday Feb 23, 2024 at 11:59pm)

  1. Exercise 1.5.2 (Compact metric spaces are both bounded and closed).
  2. Exercise 2.1.3 (Composition preserves continuity.)
  3. Exercise 2.2.3 (Absolute value preserves continuity).
  4. Exercise 2.2.11 (Continuous in each variable does not imply jointly continuous).
  5. Exercise 2.4.4 (Continuity preserves connectedness, see definitions in Section 2.4).
Reading Assignment: Sections 2.1, 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 Friday Feb 16, 2024 at 11:59pm) 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 (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. (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, 1.2, 1.4, 1.5 in Tao's Analysis II book. I will not lecture on relative topology Section 1.3 but you are always welcome to read more than what is assigned.

Homework 3 (due on Thursday Feb 8, 2024 at 11:59pm) Exercises from Tao's Analysis I, Chapter 11 on Riemann Integration.
Choose two of the three problems, if you choose to do all you get bonus points:

  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 11.9 and 11.10. I will not discuss the Stieltjes integral (Section 11.8), but you can read it.

Homework 2 (due on Thursday Feb 1st, 2024 at 11:59pm) Exercises from Tao's Analysis I, Chapter 11 on Riemann Integration:

  1. Exercise 11.3.4 and 1.3.5 (deal only with upper Riemann sums and upper Riemann integrals).
  2. Exercise 11.4.1 (b)(d) (Laws of Riemann integration. See the hint in the book for part (b). It will also be useful to remember that sup(-A)=-inf(A) and inf(-A)=-sup(A)).
  3. Exercise 11.5.2 (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 Thursday 1/25/2024 at 11:59pm): 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). Hint: use Theorem 11.1.13 (length is finitely additive) as needed.
  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). Hint: Read hints in the book!
  5. Exercise 11.2.4(c)(d)(e) (Laws of p.c. integration). Hint: It is fair to use prior properties to prove a given property.
Reading Assignment Sections 11.1, 11.2 and 11.3

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Last updated: Jan 17, 2024