MATH 621 Notes

Supplementary Notes

I will post additional notes or information on this page as the semester progresses. Class exercise handouts will be kept at the top of the page. Web links could also be posted here, so let me know if you find something that you think would be useful or interesting to your classmates. If I agree, I will post it.


Class Exercise Handouts

(12/05) The Cantor Function and Integration

(11/16) Derivation of Derivative Formulas

(11/14) Continuity and Product Spaces

(10/17) Sequences and Series

(10/10) Series

(10/05) Sequences

(09/21) Metric and Semi-metric Spaces

(09/19) Countability and Cardinality

(09/05) The real number system


Additional Notes

12/14
  1. Appendix H (page 791) of these notes on Calculus in 3D by Zbigniew Nitecki of Tufts University presents an example to show that surface area cannot be defined, analogously to the length of a curve, as a supremum of areas of piecewise-linear approximating surfaces. (The crucial pictures are on page 794).
12/05
  1. There is a nice discussion of the result that a function convex on an open interval is continuous there on Mathematics Stack Exchange. (The discussion includes a simple example which shows that the corresponding result on a closed interval is false in general. You might want to try and think of such an example yourself before reading the page.)
11/18
  1. Here are solutions to the midterm exam. Please let me know of any typos, or more serious errors or omissions you find.
09/31
  1. Kurt Bryan, at Rose-Hulman University, has a nice write-up of the construction of the completion of a metric space (Exercises 23 and 24 of Chapter 3 in Rudin).
09/26
  1. Here is a solution to Exercise 6 of Chapter 1 in Rudin (defining exponentials).
09/17
  1. Since we are talking about building the real number system, you might be interested in a recent discovery of the first recorded use of a symbol for zero.