Math 331:
Foundations of Analysis
Department of Mathematics and Computer Science Hobart and William Smith Colleges Fall 2022. Instructor: David J. Eck (eck@hws.edu) Course web site: https://math.hws.edu/eck/math331/ Monday, Wednesday, Friday, 9:50–10:50 AM Room: Coxe 1.
About The Course and the Textbook
Math 331 is a first course in real analysis. For much of the course, we will be talking about topics that you covered in Calculus I and Calculus II: limits, continuity, differentiation, integration, and infinite series and sequences. It is, however, decidedly not just another calculus course. In your calculus courses, you mostly learned some techniques and used them to solve some problems. For example, you memorized a list of rules for differentiation, and you used them to find the derivatives of a lot of functions (often given by formulas that are unlikely ever to turn up in a real-world application). You might have done some simple proofs, and you probably spent some time developing intuition about why certain things are true, but for the most part, "calculus" meant "calculating" based on some memorized formulas.
In this course, you will do very little calculating. Instead, you will be doing mathematics. That is, we will start from first principles and build up a theory from there, based on rigorous logical reasoning. The first principles are the definition and basic properties of the real numbers. The theory that we will develop is the real Calculus, the theory of the infinite and the infinitesimal that has taken some 2500 years to develop.
The textbook for the course is Foundations of Analysis, Second Edition, by David Belding and Kevin Mitchell. The two authors were professors at Hobart and William Smith Colleges until their retirement in 2017. This book was written specifically as a textbook for a pair of courses that they developed, Foundations of Analysis I and II. We will cover the material of the first of those courses, Chapters 1 through 4 in the book.
The textbook was written before Math 135 was invented, and the courses based on it were often the first experience that students had with rigorous mathematics. You should be more prepared for the rigor than those earlier students were. I plan to take advantage of that fact by adding some material on metric spaces, which are often covered in similar analysis courses. We will see how certain fundamental properties of the real numbers can be generalized to a more abstract setting. The readings for this part of the course can be found online at https://math.hws.edu/eck/metric-spaces/.
About the Pandemic
The pandemic is not really over, and it might still disrupt the class. Remember that you should not come to class if you are sick. If you need to miss class because of illness, please consult with me. Similarly, if I am sick, I plan to either try to teach the course remotely or find another professor to substitute for me.
Unless the Colleges or my department mandates otherwise, masks are optional in the classroom. I do not plan to wear a mask while lecturing, but I will probably wear one in lab where I will be working closely with students.
Homework Assignments
Homework will be assigned and collected weekly. I hope that most people in the course will be willing to write their solutions using LaTeX, most likely at overleaf.com. I might require LaTeX after the first couple of assignments. If your work is done at overleaf.com, and if you give me access to your project there, I will be able to view and comment on your work online. For more information about submitting homework, see
https://math.hws.edu/eck/math331/f22/submitting-homework.html
Note that a majority of the homework assignments will involve proving things. In most cases, I will give you feedback on your proofs and a chance to revise them to raise your grade.
You are encouraged to discuss homework assignments with me both before turning them in and after receiving feedback. It is possible that I will ask to meet with you about your revisions.
You are allowed and even encouraged to discuss homework problems with other students in the class. However, you should always write up your own solutions, with your own reasoning and in your own words.
Project and Presentation
Instead of an exam, the final evaluation for the course will be a project and presentation. For the project, you will write a short paper on some mathematical topic related to the course. Group projects with two or three students are possible, depending on the topic. All project topics must be different. You will give a presentation about your project during the scheduled final exam period or possibly, in a few cases, the last two class periods. Here are a few possible topics, to give you some idea of what I am looking for:
- A non-Archimedean ordered field: the rational functions
- Lebesgue measure and the Lebesgue integral
- Fourier series: infinite series of trigonometric functions
- Introduction to complex analysis
- Defining the real numbers using Cauchy sequences of rational numbers
- Surprising mathematical objects from Counterexamples in Analysis
- More properties of metric spaces
More possibilities will undoubtedly come up during the semester. More details about the projects will be available after Fall break.
Tests and Final Evaluation
I plan to give two in-class tests that will cover mostly definitions and basic concepts, including some straightforward proofs. The two in-class tests are planned for Monday, October 3, and Friday, November 18.
There will also be two take-home tests, around the same time as the two exams. We will discuss the exact timing. The take home tests differ from homework in that you will not be allowed to collaborate in any way on the tests, and you should use only your textbook, your notes, and the course web site as resources. I might require that you present some of your work for the test at an in-person meeting with me.
The final evaluation for the course will be in the form of final project presentations.
Grading
Here is the weighting for the various components of the course.
First in-class test: 13% First take-home test: 13% Second in-class test: 13% Second take-home test: 13% Project and presentation: 13% Homework: 35%
About Office Hours
I expect to hold regular, walk-in hours in my office, Lansing 313. Times will be announced and posted on the course web site as soon as my schedule for the semester is finalized. I can also do some meetings by appointment, and appointments for Zoom meetings are also a possibility.
Of course, email is always a good way to contact me. My email address is eck@hws.edu. I welcome comments and questions by email, and I will usually respond to them fairly quickly.
Attendance and Late Homework Policies
I assume that you understand the importance of attending class, and you should always plan to be in class, if possible. However, I do not take attendance into account as part of your grade. And of course, if you are sick, you should not be in class. If you do need to miss class, it is a good idea to talk to me about the material that you missed.
If you need to miss an in-class test, you should make arrangements with me in advance if possible. If that is not possible, you should contact me as soon as you can. If you have a sufficient reason for missing the test, we can arrange a make-up test.
I do not in general accept late homework, but I do try to be reasonable about exceptional cases. If you miss a homework deadline for some reason, you should always discuss it with me.
No Technology During Lecture
I ask that you refrain from using any technology (beyond pen/pencil and paper) in lecture, unless you have a verified need to take notes on computer. This includes laptops, tablets, and cell phones.
There is substantial research showing that taking notes on paper can improve retention of the material, compared to note-taking on computer. My real advice is to take notes in outline form, noting down important ideas and examples, and to make a more formal copy of the notes after class, filling in any missing details. There is also research showing that the multitasking that you are likely to engage in if you have a computer open in front of you is detrimental to learning.
Statement from the Center for Teaching and Learning
Disability Accommodations: If you are a student with a disability for which you may need accommodations and are new to our office, you should self-identify for services by logging into the Accommodate Portal (https://hws-accommodate.symplicity.com/public_accommodation/) and completing the Accommodation Intake Form. Disability related accommodations and services will be provided when the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: https://www.hws.edu/centers/ctl/disability-services.aspx.
Returning students may request accomodations by logging into the Accommodate Portal and requesting a semester accommodation letter. Should you need to meet to add or discuss accommodations, please schedule an appointment in the Accommodate Portal (directions are on the CTL website, should you need them).
Please direct questions about this process or Disability Services at HWS to ctl@hws.edu or x 3351. Jamie Slusser, Disability Services Administrator & Accommodation Specialist, and Christen Davis, Associate Director of CTL for Disability Services, are the main contact staff for Disability Services.
Tentative Schedule
Here is a tentative schedule for the course. Readings are from the textbook, except for those about metric spaces. We should cover all of the topics listed here, but will probably not follow this schedule exactly. See the course web site for accurate scheduling information, posted weekly. In addition to the readings listed here, I will post short "reading guides" on the web that will also be part of the assigned reading; again, see the course web site for details.
Dates | Topics and Readings |
---|---|
Aug. 22, 24, 26 |
The Real Numbers; Dedekind cuts; The least upper bound axiom. Sections 1.0, 1.1, 1.2 (start). |
Aug. 29, 31; Sept. 2 |
Axioms for the real numbers; The Heine-Borel Theorem. Sections 1.2 (finish), 1.3, 1.4 (start). |
Sept. 5, 7, 9 |
The Bolzano-Weirstrass Theorem; Limit definitions. Sections 1.4 (finish), 2.0, 2.1, 2.2, 2.3 (start). (Note: We will not spend much class time on 2.0 and 2.1.) |
Sept. 12, 14, 16 |
More on limits; Continuity; The Intermediate Value Theorem. Sections 2.3 (finish), 2.4, 2.5. |
Sept. 19, 21, 23 |
Uniform Continuity; the Extreme Value Theorem; Metric spaces. Section 2.6. Introduction to metric spaces and Open and Closed Sets. |
Sept. 26, 28, 30 |
Infinite Sequences and their limits. Sections 4.0, 4.1 (review of real-number sequences). Metric spaces: Limits and Continuity |
Oct. 3, 5, 7 |
The derivative, Sections 3.0, 3.1, 3.2. First In-class Test, Monday, October 3. |
Oct. 12, 14 |
Fall break, Oct. 10 and 11. Mean Value Theorem; The Riemann integral. Sections 3.4, 3.5 (start). |
Oct. 17, 19, 21 |
More on integration; The Fundamental Theorem of Calculus. Sections 3.5 (finish), 3.6 (start). |
Oct. 24, 26, 28 |
Finish integration; Taylor Polynomials. Sections 3.6 (finish), 3.7. |
Oct. 31; Nov. 2, 4 |
Monotone and Cauchy sequences; Infinite series. Sections 4.2, 4.3 (start). |
Nov. 7, 9, 11 |
Convergence tests; Absolute and conditional convergence. Sections 4.3 (finish), 4.4, 4.5 (start). |
Nov. 14, 16, 18 |
Sequences and series of functions. Sections 4.5 (finish), 4.6 (start). Second In-class Test, Friday, November 18. |
Nov. 21 |
Thanksgiving break, Nov. 22–27. More on series of functions. Section 4.6 (finish). |
Nov. 28, 30; Dec. 2 |
Properties of Metric Spaces. Compactness, Completeness, and Connectedness. |
Dec. 7 |
Presentations during scheduled final exam period, Wednesday, December 7, 7:00–10:00 PM. |