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This course ended on December 9, 2020 |
Math 331: Foundations of Analysis
Department of Mathematics and Computer Science
Hobart and William Smith Colleges
Fall 2020.
Instructor: David J. Eck (eck@hws.edu)
Syllabus: http://math.hws.edu/eck/courses/math331_f20.html
Monday, Wednesday, Friday, 1:20–2:20 PM
Room Eaton 111.
| Homework | |||
| Homework 1, Due September 2, |
Homework 2, Due September 11, |
Homework 3, Due September 18, |
Homework 4, Due September 26, |
| Homework 5, Due October 3, |
Homework 6, Due October 15, |
Midterm Exam, Due October 26, |
Homework 7, Due November 3, |
| Homework 8, Due November 10, |
Homework 9, Due November 17, |
Final Exam, Due 1:30 PM, December 10, |
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Fifteenth Week: November 30 and December 2
The final two classes of the semester will be online, using the usual Zoom meeting link for Math 331, which can be found on the Canvas page for this course. In these classes, we will return to metric spaces to look at some properties of metric spaces that generalizes properties of the real numbers: compactness, completeness, and (if there is time) connectedness. This material is covered in these online readings:
However, we will not have time to cover everything from these readings.
Fourteenth Week: November 23
We have our final in-person class of the semester on Monday, November 23, before breaking for Thanksgiving. After Thanksgiving, we will reconvene for a few online classes to end the semester.
We started Section 4.6 last week, covering series of function and starting power series. This week, we will finish the material on power series. That will bring us to the end of Chapter 4 and to the end of the part of the textbook covered by this course. For the end of the course after break, we will return to metric spaces.
Here is my short summary about series of functions and power series:
Thirteenth Week: November 16, 18, and 20
A test is scheduled for this Friday, November 20. A study guide for the test is available.
Aside from the test and an opportunity for some questions and review on Wednesday, we will continue with Sections 4.5 and 4.6. We ended last week with the definitions of pointwise and uniform converges for a sequence of functions. We will look at consequences of uniform convergence and then, as time permits, move on to series of functions and power series. This material is not on the test.
Twelfth Week: November 9, 11, and 13
We have hardly begun work on infinite series, and we will spend much of the week on Sections 4.3 and 4.4. However, we should get to Section 4.5 by Friday at the latest. That second covers sequences of functions. It defines two types of convergence for sequences of functions: pointwise convergence and uniform convergence. The tree main theorems show how continuity, integrability, and differentiability work with uniform convergence. A summary of the definitions and theorems from Section 4.5 is here:
The web page of animations of examples and exercises from Section 4.5 can be found here:
Foundations of Analysis: Sequences of Functions
Remember that the second in-class test is coming up next week, on Friday, November 13.
Eleventh Week: November 2, 4, and 6
We need to finish up with Taylor polynomials by proving Taylor's Theorem with Remainder. After that, we will move on to Chapter 4. We have already covered Section 4.1. We will cover Sections 4.2 and 4.3 and part of 4.4 this week. The main topics in Section 4.2 are monotone sequences and Cauchy sequences. Section 4.3 introduces infinite series and convergence tests for infinite series. Section 4.4 covers absolute and conditional convergence. I plan to cover absolute convergence earlier than the textbook so that I can give stronger versions of the Ratio and Root Tests for convergence. My summary of this material is here:
Tenth Week: October 26, 28, and 30
We will spend part of Monday's class looking at some of the problems from the midterm. (That is, unless there is some reason why not all exams could be returned on time; in that case, the discussion will be postponed.)
Since we did not get to the Fundamental Theorems of Calculus last week, we will cover Section 3.6 this week. We will then cover Section 3.7, Taylor polynomials and Taylor's theorem with remainder. If time permits, we might move on to Chapter 4 by the end of the week. Here is a short summary of Taylor polynomials:
Ninth Week: October 19, 21, and 23
A take-home exam will be distributed mid-week and will be due by class time next Monday.
We will spend the week working on theorems about integration, including the two Fundamental Theorems of Calculus. The reading is Sections 3.5 and 3.6. Here is a reading guide, which is really just a list of the major theorems that we will cover:
Eighth Week: October 12, 14, and 16
This week, we will finish our look at the derivative and move on to the Riemann integral. The new reading is Sections 3.4 and 3.5. Note that this puts us a little ahead on the schedule from the syllabus. I have cut back on the time devoted to the derivative in the hope of being able to devote more time to sequences and series of functions at the end of the course. Here is a short guide to the definition of the Riemann integral:
Seventh Week: October 5, 7, and 9
After the test on Monday, we will return briefly to metric spaces to finish our discussion of Limits and Continuity in metric spaces. After that, we return to the textbook to begin Chapter 3, which covers differentiation and integration. We will get a start on differentiation, Sections 3.1 to 3.3. and we will finish it next week. Two very short reading guides on this material can be found here:
You can expect the test to be returned on Friday.
Sixth Week: September 28 and 30; October 2
There is a test coming up next week on Monday, October 5. An information sheet about the test is available.
We will finish the material on open and closed sets in metric spaces. That will be the last of the material that will be covered on next Monday's test.
We will spend some time reviewing for the test. In the remaining time this week, we will move on to new material that will not be on the test.
Our next topic will be a very short look at "subspaces" of metric spaces. The reading is the first half of the second section from the metric space web site, Subspaces and Product Spaces. (We will not discuss product spaces in this course.) After that, the next metric space topic is Limits and Continuity, but before starting that section, we will do a short review of infinite sequences in R, which are covered in Section 4.1 of the textbook.
Fifth Week: September 21, 23, and 25
We are running a little behind the original schedule from the syllabus. We still need to finish the Extreme Value Theorem and uniform continuity, from Section 2.6, which were scheduled for last week. That will finish Chapter 2. After that, we will be taking a short excursion from the textbook before moving on to Chapter 3, by looking at metric spaces. We should be able to start metric spaces before the end of class on Wednesday. You should read the short introduction and the section on open and closed sets. The reading guide for the metric space material is included in
Fourth Week: September 14, 16, and 18
We have not yet finished with limits. We still have some things from Section 2.3 to do, and all of Section 2.4 (although we will not spend much time on 2.4). But we should be able to turn to our next topic, continuity, on Wednesday. The new reading is Chapter 2, Sections 5 and 6. Those sections cover continuity and prove two important theorems about continuity on a closed interval: the Intermediate Value Theorem and the Extreme Value Theorem (called the Max-Min Theorem in the textbook). Also covered is a stronger form of continuity on an interval called uniform continuity, and it is proved that any continuous function on a closed, bounded interval is in fact uniformly continuous. The reading guide for the material on continuity is
Third Week: September 7, 9 and 11
After finishing up the Heine-Borel and Bolzano-Weirstrass theorems, we will move on to Chapter 2, which covers limits and continuity. The reading for the week is Sections 2.0 through 2.4, but we will not spend any class time on 2.0 or 2.1, and we might not get to 2.4 until next week. For much of the rest of the semester, we will be looking at topics that you have already seen in Calculus I and II, but we will be taking a theoretical, proof-oriented approach that uses the rigorous view of the real number system that we have been pursuing. Section 2.2 and 2.3 cover the formal definition and theory of the limit of a function at a point, with proofs that use the epsilon-delta definition of limit. The reading guides for this material, including guide 6 continued from last week, are
Homework 2 is due on Friday. Homework should be turned in on time. However, if circumstances make that difficult for you, you should consult with me about getting an extension.
Second Week: August 31; September 2 and 4
We still have to finish up some material from Section 1.2 on the Least Upper Bound Property and the Archimedian Property of the real numbers. The new reading for the week is Sections 1.3 and 1.4. Sections 1.3 gives a set of axioms for the real numbers as a "complete, ordered field." Section 1.4 covers two important theorems about the real numbers: the Heine-Borel theorem and the Bolzano-Weirstrass theorem. That will probably take us into the beginning of next week. The "reading guides" for this material are
My office hours on Zoom this week will be Tuesday and Thursday, 12:30 to 2:00 and 6:30 to 8:00. You can use the Canvas Calendar feature to make an appointment during those times. Note that I will usually be in my office between classes on Monday, Wednesday, and Friday, from about 11:15 to 1:00. If you want to meet either on Zoom or in person during those times, you should arrange an appointment by email or by talking to me after class, or you should call my office (315-781-3398) to see whether I am available.
First Week: August 24, 26, and 28
Welcome to the course!
The reading for the week is Chapter 1, Sections 1.0 through 1.2. Please read them! We might start covering Section 1.3 in class on Friday. From Section 1.1, you should be familiar with irrational numbers and the Fundamental Theorem of Arithmetic, but we will not spend a great deal of time on that. Section 1.2 gives a "construction" of the set of real numbers; that is, it defines a specific mathematical object that has the properties that we expect the real numbers to have.
I plan to write short guides to the readings for each lecture, emphasizing the most important parts in the reading and sometimes adding information or perspective that is not in the textbook. The guides for the first week are here:
The first homework assignment is already available. You are encouraged to write up your solutions in LaTeX, using a free account at overleaf.com, but I will also accept handwritten work scanned to a PDF file, at least at first. For more information about submitting homework, see
http://math.hws.edu/eck/math331/f20/submitting-homework.html