CAAM 401 Homework Assignments: Fall 2003

Section numbers refer to the text (Strichartz).
  1. Assigned 26 Aug 03: Read Chapter 1 in its entirety. Then:

    1. 1.1.3, problems 2 and 3 (do you believe the assertion about a.?)

    2. 1.2.3, problems 1, 3, and 4.

    3. Prove that for every two natural numbers p and q, there is an n so that np is greater than q. Use this result to prove the Archimedean Property of the rational numbers: if p/q is a positive rational, then there exists a natural number n so that p/q is greater than 1/n.

      In your proof of the first part, use only the commutative and associative laws of addition and multiplication, and the distributive property of multiplication over addition, for natural numbers. For natural numbers p and q, define "p is greater than q" to mean "there exists a natural number r so that p = q+r". You can use this structural fact also: if p is a natural number, then either p=1 or p is greater than 1.

      [obviously I need to start typesetting these assignments...]

    Solutions (pdf)

  2. Assigned 2 Sept 03: Read Chapter 2, sections 2.1 and 2.2 through p. 47. Then do

    1. 2.1.3 (p. 37): problems 2, 3, 4, 6, 8

    2. 2.2.4 (p. 48): problems 1 (just do commutative), 7, 8, and 9 [Note: problems 7, 8, and 9 do NOT use the construction of the real numbers, only the field axioms. That is, ASSUME that the reals form an ordered field, as defined in the text, and that the triangle inequality is true for real numbers, and go from there. You don't have to understand the proof that the reals form an ordered field to do these problems - you just have to understand the ordered field properties.]

    Solutions (pdf)

  3. Assigned 9 Sept 03: Read Chapter 2, rest of section 2.2 and section 2.3. Read section 3.1. Then do

    1. 2.2.4 (p. 48): problems 5, 11

    2. 2.3.3 (p. 54): problems 2, 3, 8, 9, 10

    Solutions (pdf)

  4. Assigned 16 Sept 03: Read the remainder of section 3.1 and section 3.2. Then do these problems (pdf).

    5. Solutions (pdf)

  5. Assigned 23 Sept 03: Read section 3.3. Then do

    1. 3.2.3 (p. 98): 4 [20 pts - state as a theorem, write a careful proof]; 7 [10 pts], 13 [15 pts], 14 [10 pts]

    2. 3.3.1 (p. 106): 1 [15 pts - see note below], 3 [5 pts], 4 [10 pts], 6 [15 pts]

    Note re 3.3.1.1: What he means is: "Suppose that F is a set of compact sets. Show that the intersection of (all of) the sets in F is compact, and that if F is finite so is the union."

    6. Solutions (pdf)

  6. Assigned 30 Sept 03: Read sections 4.1 and 4.2. Then do these problems (pdf). THIS ASSIGNMENT IS PLEDGED.

    7. Solutions (pdf)

  7. Assigned 7 Oct 03 - due 21 Oct O3:

    1. 4.1.5 (p. 125): 1, 2, 3, 7, 10, 13, 14

    2. 4.2.4 (p. 138): 3, 4, 8, 10, 11, 12

    These problems are based on Ch. 4. Each problem in this set is worth 8 points, except for the last which is worth 12.

    8. Solutions (pdf)

  8. Assigned 21 Oct 03 - due 30 Oct 03:

    1. 5.1.3 (p. 152): 1, 2, 3, 4, 5

    2. 5.2.4 (p. 163): 1, 2, 6, 11, 12, 13

    Each problem is worth nine points, except for the first which is worth 10.

    9. Solutions (pdf)

  9. Assigned 21 Oct 03 - due 6 Nov 03:

    1. 5.3.4 (p. 176): 1, 2, 4, 5, 7, 8, 9

    2. 6.1.5 (p. 217): 1, 2, 4, 9, 11

    Each of these is worth 8, except for 6.1.5.4 which is worth 12. In addition, you can get...

    Extra Credit (10 pts): 5.3.4.3 (p. 176). Hint: use the ideas on p. 129, which depend on the "structure theorem for open sets", p. 88.

    10. Solutions (pdf)

  10. Assigned 6 Nov 03 - due 13 Nov 03. THIS ASSIGNMENT IS PLEDGED. Each problem is worth 20 points.

    1. 5.2.4 (p. 165): 8, 9

    2. 5.3.4 (p. 177): 10

    3. 6.1.5 (p. 218): 13 ["points where f changes sign" = points where f is positive on one side, negative on the other side (for at least a little ways), and zero at the point itself.]

    4. 6.1.5 (p. 219): 14, part a. [Extra credit (10 pts): do part b as well; the trapezoidal rule is defined at the bottom of p. 216.]

    11. Solutions (pdf)

  11. Assigned 13 Nov 03 - due 18 Nov O3

    These problems are based on the class lectures on linear algebra.

  12. Assigned 18 Nov 03 - due 25 Nov 03

    These problems are based on the class lectures on linear algebra, and on Ch. 9, sections 1 and 2, in the text.

  13. Assigned 25 Nov 03 - due 04 Dec 03. You may substitute "normed vector space" for "metric space" whereever this substitution makes sense, if you like.

    1. 9.1.4 (p. 366): 1 (the "dot" product is the usual Euclidean inner product, introduced in class), 2, 3, 15 (each 15 pts)

    2. 9.2.5 (p. 384): 4 (25 pts), 9 (15 pts)

  14. Assigned 04 Dec 03 - due last day of finals. THIS ASSIGNMENT IS PLEDGED. Each problem worth 25 pts.

    1. 9.1.4 (p. 366): 8

    2. Prove that if a Cauchy sequence in a normed vector space (not assumed complete) has a convergent subsequence, then the sequence itself is convergent. [This is a restatement of 9.2.5.5.]

    3. 9.2.5 (p. 384): 10, 11.