**Advanced Calculus
**

__DIARY__

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

- August 23, 2017:
- Brief discussion of nature of course
- Introduction to countability

- August 25, 2017:
- Students work on examples

- August 28, 2017:
- Substitute teaching: Countability and uncountability

- August 30, 2017:
- Substitute teaching: Unions, intersections, metric spaces

- September 01, 2017:
- Substitute teaching: Open sets, closed sets, compactness.

- September 04, 2017:
- Labor Day

- September 06, 2017:
- Review of open, closed, Theorem 2.24 and the notion of a topology, and Theorem 2.27

- September 08, 2017:
- Collect Assignment 01
- Proof that compact metric spaces are closed: Two alternative proofs.

- 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.

- September 13, 2017:
- Return MATH 514 Assignment 01
- Discussion of (a)-(f) of assignment problem.
- Equivalent forms of Compactness in R^k.

- September 15, 2017:
- Collect Assignment 02
- Equivalent forms of Compactness in R^k, continued.

- September 18, 2017:
- Equivalent forms of Compactness in R^k, continued.
- Lebesgue's Covering Lemma

- September 20, 2017:
- Lebesgue's Covering Lemma: Proof

- September 22, 2017:
- The metric space C[0,1] - group work.

- September 25, 2017:
- Connectedness

- September 27, 2017:
- Connectedness and the real line

- September 29, 2017:
- Collected group assignment 01
- Sequences in metric spaces
- Group work on C[0,1] and D[0,1]

- October 02, 2017:
- Cauchy Sequences

- October 04, 2017:
- A compact metric space is complete.

- 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.

- October 09, 2017:
- Rudin, p. 82, Exercise 20
- Rudin, p. 82, Exercise 21
- Nowehere dense sets, first category.

- October 11, 2017:
- The Baire Category Theorem.

- October 13, 2017:
- Returned Groups Assignment 02
- Collected Group Assignment 03
- Group work on metric spaces.

- October 16, 2017:
- An equivalence relation on the set of Cauchy sequences of a metric space.
- The completion of a metric space.

- October 18, 2017:
- A metric on the set of equivalence classes.
- An isometry from the metric space to the space of equivalence classes.

- October 20, 2017:
- Start of Midterm Examination - Follow the "Exams" link at the top of the page for access.

- October 23, 2017:
- Collected Midterm Examination.
- Collected Group Assignment 04.
- Definition of Continuity and Uniform Continuity in metric spaces.

- 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.

- 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.

- October 30, 2017:
- Returned midterm examination.
- Properties of continuous functions

- 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.

- 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}.

- 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.

- November 08, 2017:
- Proof that a continuous function from a compact domain is uniformly continuous (via Exercise 4.10) - Part I.

- 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.

- 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.

- November 15, 2017:
- If the function f on interval [a,b] has a derivative at the point x, then it is contiuous at x.