Advanced Calculus

FALL 2017

DIARY

[Home] [Diary] [Assignments] [LaTeX links] [Exams]

  1. August 23, 2017:
    • Brief discussion of nature of course
    • Introduction to countability
  2. August 25, 2017:
    • Students work on examples
  3. August 28, 2017:
    • Substitute teaching: Countability and uncountability
  4. August 30, 2017:
    • Substitute teaching: Unions, intersections, metric spaces
  5. September 01, 2017:
    • Substitute teaching: Open sets, closed sets, compactness.
  6. September 04, 2017:
    • Labor Day
  7. September 06, 2017:
    • Review of open, closed, Theorem 2.24 and the notion of a topology, and Theorem 2.27
  8. September 08, 2017:
    • Collect Assignment 01
    • Proof that compact metric spaces are closed: Two alternative proofs.
  9. September 11, 2017:
    • Return MATH 414 Assignment 01
    • Proofs that nested sequences of k-cells have nonempty intersection.
    • Proof that k-cell is compact.
    • Proof that when a family of compact sets has the finite intersection property, the intersection of the family is nonempty.
  10. September 13, 2017:
    • Return MATH 514 Assignment 01
    • Discussion of (a)-(f) of assignment problem.
    • Equivalent forms of Compactness in R^k.
  11. September 15, 2017:
    • Collect Assignment 02
    • Equivalent forms of Compactness in R^k, continued.
  12. September 18, 2017:
    • Equivalent forms of Compactness in R^k, continued.
    • Lebesgue's Covering Lemma
  13. September 20, 2017:
    • Lebesgue's Covering Lemma: Proof
  14. September 22, 2017:
    • The metric space C[0,1] - group work.
  15. September 25, 2017:
    • Connectedness
  16. September 27, 2017:
    • Connectedness and the real line
  17. September 29, 2017:
    • Collected group assignment 01
    • Sequences in metric spaces
    • Group work on C[0,1] and D[0,1]
  18. October 02, 2017:
    • Cauchy Sequences
  19. October 04, 2017:
    • A compact metric space is complete.
  20. October 06, 2017:
    • Returned group assignment 01
    • Collected group assignment 02
    • Remarks about compactness criteria for metric spaces.
    • Remarks about Cauchy sequences.
    • Group work on C[0,1] and a potential alternative metric.
  21. October 09, 2017:
    • Rudin, p. 82, Exercise 20
    • Rudin, p. 82, Exercise 21
    • Nowehere dense sets, first category.
  22. October 11, 2017:
    • The Baire Category Theorem.
  23. October 13, 2017:
    • Returned Groups Assignment 02
    • Collected Group Assignment 03
    • Group work on metric spaces.
  24. October 16, 2017:
    • An equivalence relation on the set of Cauchy sequences of a metric space.
    • The completion of a metric space.
  25. October 18, 2017:
    • A metric on the set of equivalence classes.
    • An isometry from the metric space to the space of equivalence classes.
  26. October 20, 2017:
    • Start of Midterm Examination - Follow the "Exams" link at the top of the page for access.
  27. October 23, 2017:
    • Collected Midterm Examination.
    • Collected Group Assignment 04.
    • Definition of Continuity and Uniform Continuity in metric spaces.
  28. October 25, 2017:
    • Consider the function f defined on (0,1) by f(1/2) = 14, and f(x) = 1/x otherwise. Explored whether lim_{x--->1/2}f(x) = 2.
    • Returned Group Assignment 03.
  29. October 27, 2017:
    • Returned Group Assignment 04.
    • Group work on the map G from (C[0,1],d) to (C[0,1],d_1) that assigns to f the antiderivative of f with constant 0.
  30. October 30, 2017:
    • Returned midterm examination.
    • Properties of continuous functions
  31. November 01, 2017:
    • Continuous functions preserve compactness.
    • A continuous real-valued function on a compact metric space achieves a maximum value and a minimum value.
  32. November 03, 2017:
    Let D^n be the set of continuous functions on [0,1] that are n-times differentiable on (0,1).
    • Group work on functions between metric spaces (X,m) and (Y,n) where X, Y are chosen from {C[0,1], D[0,1], D^2[0,1]} and m and n are metrics chosen from {d, d_1}.
  33. November 06, 2017:
    • Collected Assignment 08.
    • Equivalences of continuity.
    • A continuous surjection of a connected metric space is connected.
    • Definition and discussion of uniform continuity.
  34. November 08, 2017:
    • Proof that a continuous function from a compact domain is uniformly continuous (via Exercise 4.10) - Part I.
  35. November 10, 2017:
    • Returned group Assignment 05 (=Assignment 07).
    • Collected group Assignment 06 (=Assignment 09).
    • Group work on the continuity of familiar functions such as addition, dot products, cross products, and definite integration.
  36. November 13, 2017:
    • Returned Assignment 08.
    • Proof that a continuous function from a compact domain is uniformly continuous (via Exercise 4.10) - Part 2.
    • Introduction to Differentiation.
  37. November 15, 2017:
    • If the function f on interval [a,b] has a derivative at the point x, then it is contiuous at x.