Advanced Calculus

ASSIGNMENTS

10:30 - 11:45 am M,W,F MB 124

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Assignments must be handed in at the beginning of class on the due date.

This is a graduate level class: You are expected to continue developing as an independent, intellectually honest scholar. Collected assignments is one of the instruments to assess your progress in this direction. Not every item covered in class is featured in the hand-in assignments, and not every item featured on the hand-in assignments is covered in class. 

One of the learning outcomes of the course is to write clearly for specific purposes and audiences. This specific outcome will be fostered through the standards I will apply to collected work (homework assignments, examinations, projects). A problem solution is an expository work which explains in full detail, step-by-step, to anyone with access to the textbook for the course  how to obtain a solution to the problem. Note that if you were to find that a problem is not well-defined, or requires proving a statement that is false, you are expected to point this out in your work on the problem, providing the details that substantiate your findings.

You may discuss the homework problems with fellow enrollees in this course only, and you may consult any published literature on the subject. However, you must present your solutions in your own words, and you must properly attribute ideas and information gained from other sources. For the purposes of the homework assignments it is important to produce a scholarly document that demonstrates your grasp of the material and your ability to properly and
thoroughly explain the solutions to these problems. Answers without accompanying exposition that shows that you can coherently and logically explain the mathematical reasons justifying your answers will not receive any credit.

Content requirements for homework assignments:
a. The exposition of a solution must contain appropriate prose.
b. The exposition of a solution must, where appropriate, identify the mathematical steps taken, in logical order.
c. The exposition of a solution must properly reference the mathematical facts.  

Format requirements for homework assignments:
a. Your hand-in should be an e-mailed .tex document plus an in-class hard-copy of a typewritten document with at least 11 point font and margins of at least 1 inch on each side. Prepare your final documents in time for the due time and date. It is required that you use the LaTeX typesetting program. Documents not typeset in LaTeX will not be  accepted for grading, nor will partially handwritten work be graded.

b. The upper right-hand corner of the cover page of your document must contain in this order:
     Your name
     The homework set number
     The due date of the homework
c. A multiple page document must be stapled in the top left corner.
d. The problems must be presented in the order they are assigned.
e. Each problem solution must begin with the full statement of the original problem.
f. There must be a visible separation between problems.

Specific requirements:
The nature of some of the assignments may induce additional content and format requirements specific to that assignment. For these assignments the additional specific requirements will be stated with the assignment.

Policy:
1) No late hand ins will be accepted. Hand in on time, or take a zero for the assignment.
2) Hand ins not meeting ALL the format and specific requirements above will not be graded. Instead a zero will be recorded for the assignment.

Assignment 1. Due Sept. 08:

  1. (514 only) Rudin p. 43, Number 9
  2. (414/514) Rudin p. 44, Numbers 10, 11

Assignment 2. Due Sept. 18:

  1. (514 only) Prove: If a metric space is compact, then it has a countable dense subset.
  2. (414/514)
    • Let X be the set of rational numbers between square root of 7 and square root of 8. Show that, as subset of the set of rational numbers, X is closed and bounded, but not compact.
    • Rudin, p. 45, number 23.

Assignment 3 - Group Assignment 01. Due Sept. 29:

C[0,1] denotes the set of functions continuous on [0,1]. For f and g in C[0,1] define d(f,g) = sup{|f(x)-g(x)|x in [0,1]}.

  1. Compute:
    • d(x^2, x^3)
    • d(e^x, e^(2x))
  2. Prove that d is a metric on C[0,1].
  3. Let 0 denote the function that is constantly equal to 0. Define U = {f in C[0,1]: d(f,0) <= 1}.
    • Is U closed?
    • What is the interior of U?
    • Is U compact?
  4. Define B = {f in C[0,1]: d(f,0) = 1}.
    • Is B closed?
    • Is B compact?
    • Is B uncountable?

Assignment 4 - Group Assignment 02. Due Oct. 06:

  1. A specific sequence (f_n: n = 1, 2, 3, ...) in C[0,1] was defined in class.
    • Determine if the terms of this sequence are elements of B.
    • Compute d(f_m,f_n) for arbitrary values of m and n.
    • Is (f_n: n = 1, 2, 3, ...) a Cauchy sequence?
  2. C[0,1] under function pointwise addition, and multiplication by real numbers, is a vector space. Consider D = {f in C[0,1]: f is differentiable on (0, 1)}.
    • Is D[0,1] under these same operations a vector subspace of C[0,1]?

Assignment 5 - Group Assignment 03. Due Oct. 13:
For f and g in C[0,1] define d_1(f,g) to be the integral over [0,1] of the function|f-g|

    • Compute d_1(x^2,x^3).
    • Compute d_1(e^x,e^(2x))
  1. Determine of d_1 is a metric on C[0,1].
  2. Put U_1 = {f in C[0,1]: d_1(f,0)<=1}.
    • Is U_1 = U?
    • Is U_1 closed in the metric d_1?

Assignment 6 - Group Assignment 04. Due Oct. 20:
This assignment continues comparison of the metrics d and d_1 on C[0,1].

  1. For each positive integer n define the function f_n on [0,1] so that for x< 1/2 - 1/(n+4), f_n(x) = 0, for x > 1/2 + 1/(n+4), f_n(x) = 1, and otherwise f_n(x) = (n+4)*x/2 - (n+2)/4. Determine if the sequence (f_n:n=1,2,...) is a Cauchy sequence in the metric space (C[0,1],d_1).
  2. Define D = {f in C[0,1]: f is differentiable on (0,1)}.
    • Is D a dense subset of the metric space (C[0,1],d)?
    • Is D a dense subset of the metric space (C[0,1],d_1)?
  3. Show that U is a subset of U_1. Is U a closed (in the sense of metric d_1) subset of U_1?

Assignment 7 - Group Assignment 05. Due Nov. 03:
For f in C[0,1] define G(f) to be the antiderivative of f with constant term 0.

  1. Compare the values of d(f,g) and d_1(G(f),G(g)) to determine which is larger, where
    • f(x) = x^2 and g(x)=x^3.
    • f(x) = x^k and g(x)=x^m where k and m are positive integers.
    • f(x) = e^x and g(x)=e^(2x).
  2. Determine if the set {G(f):f in U} is bounded in the metric d_1.

Assignment 8 Due Nov. 06:

  1. (514 only) Rudin p. 99, Number 6
  2. (414/514) Rudin p. 99, Number 7

Assignment 9 - Group Assignment 06. Due Nov. 10:
For f in D^2[0,1] define H(f) to be the derivative of f.

  1. Compare the values of d(f,g) and d_1(f,g):
    • Is there a c>0 such that for all f and g in C[0,1], d(f,g) <= c*d_1(f,g)?
    • Is there a c>0 such that for all f and g in C[0,1], d_1(f,g) <= c*d(f,g)?
  2. For each combination of the metrics d and d_1 on the domain D^2[0,1] and range D[0,1], determine if the function H is continuous.

Assignment 10 - Due Nov. 17:

  1. (514 only) Rudin p.100, Number 21
  2. (414/514) Rudin p. 99, Number 18

Assignment 11 - Group Assignment 07. Due Nov. 17:

  1. Which of the following is (1) continuous, (2) uniformly continuous? On R^k, use the metric d_k((x_1,...,x_k), (y_1,...,y_k)) = sqrt((x_1-y_1)^2 + ... + (x_k-y_k)^2):

    • A: R^4 --> R^2 defined by A(x_1,x_2,x_3,x_4) = (x_1+x_3, x_2+x_4)
    • D: R^6 --> R defined by A(x_1,x_2,x_3,x_4,x_5,x_6) = x_1x_4 + x_2x_5 + x_3x_6
    • C: R^6 --> R^3 defined by A(x_1,x_2,x_3,x_4,x_5,x_6) =( x_2x_6 - x_3x_5, x_3x_4 - x_1x_6, x_1x_5 - x_2x_4)
    • I: C[0,1]-->R: f |--> I(f) = definite integral of f on the interval [0,1], (a) using the metric d on C[0,1] (b) using the metric d_1 on C[0,1].