Syllabus for Math 491 - Real Analysis

Professor: Mark McClure

Course purpose

• To study real analysis: Real analysis is the theory of calculus, which is one of the most important intellectual developments of all time. Together with abstract algebra, it also forms much of the language of modern mathematics.
• To improve your understanding of mathematical proof: The main difference between this class and a calculus class is the point of view. In calculus, the main questions are, "How do I compute derivatives and integrals and what can I do with them?" In real analysis we will be interested in more theoretical questions. For example, "If a function is differentiable, is it necessarily continuous?" Or, "What about the converse statement?".

Materials

• Text: We will use Understanding Analysis by Stephen Abbott. This is an outstanding and readable text from an excellent series of texts.
• Discourse: I'll be setting up an online discussion forum to facilitate questions and answers. You should receive an email soon describing how to log on here.
• LaTeX: Homework and papers will be typed in LaTeX. Mathematical typesetting on Discourse will use LaTeX as well. So, you oughta go ahead and get started with it! The standard way to use it is to download a distribution (it's free):
  • MiKTeX, for PC
  • MacTeX for Mac
  There are online editors for LaTeX these days, like Overleaf. I haven't tried it but I'd be interested to hear thoughts about it.

Evaluation

• Exams: There will be two exams during the semester worth about 100 points apiece. They will both be two day affairs. Likely dates for the exams are:
  • Wednesday-Friday, September 23-25
  • Wednesday-Friday, November 4-6
• Quizzes: There will be two quizzes each two weeks ahead of an exam:
  • Friday, September 1
  • Friday, October 23
• Final exam: There will be a comprehensive, final exam worth around 180 points at 11:30 AM on Wednesday, December 2.
• Discourse: You can earn up to 60 points by participating in our online discussion forum. Note that this is a course requirement; those 60 points are real. There will be a discussion on the site itself indicating exactly how it works.
• Homework: There will be two types of homework:
  • Typed up, turn in assignments.
  • Daily textbook assignments, which will not be collected but offer important practice. These will often be discussed on Discourse.
• Late work: In general, I don't accept late work.
• Cheating: I don't deal with cheating. If I suspect cheating strongly enough, I simply refer you to the provost and fail you for the class.

A bit more on grades

The standard 90-80-70-60 scale will guarantee you an A, B, C, or D. However, it is quite likely that the final scale will be shifted down from this. I cannot give you an exact scale now, since your score will be somewhat influenced by your relative standing in the class; i.e. if Bob has fewer points than Alice, then his grade will certainly not exceed Alice's. I can, however, let you know what skills I believe will merit certain grades.

• To pass this class you must (at a minimum) know the basic definitions and theorems (e.g. continuity, differentiability, IVT, MVT), be able to apply those definitions and theorems to solve some simple problems (e.g. use IVT to approximate polynomial roots), and be able to prove some of the basic results (e.g. that differentiability implies continuity).
• The conditions to earn a C are somewhat similar. However, I would expect you to know {\em most} of the theorems and definitions we encounter and be able to apply them to problem solving and in well written proofs. Some examples of topics I would want the C student to be comfortable with, but not necessarily the D student, include suprema and infima, subsequences, topology, continuity of derivatives, and rearrangements of series. (Note: I don't mean to imply that you don't need to know about these topics to pass. I'm simply saying that I expect the C student to be comfortable with the intricacies of these topics, whereas the D student might not be.)
• To earn a B in this class you must be able to state, apply, and possibly prove virtually all of the basic definitions and theorems, be creative enough to solve new and interesting problems, and consistently write mathematics well.
• In addition to the requirements for a B student, the A student will successfully struggle with some very challenging problems. By "successfully struggle with", I mean that the student will spend some (probably significant) time with the problem, find the crux of the solution, and write a beautifully well written solution or proof to the problem.

Advice

• Learning Mathematics: I believe that mathematics is a wondrous but challenging field. I assume that most people in this class have interest in mathematics and appreciate its unique challenges. There will be times of frustration ahead. Buckle down and work hard.
• Help: You are not in this endeavor alone. You have four major sources of help:
  • Me: I like to talk to people about mathematics. That's why I chose this profession. My full schedule with office hours is shown below. You will almost always find me in my office during my office hours but please feel free to approach me any time you have questions.
  • Your classmates: Most people learn mathematics best by talking through it with others. You will find that you can both learn from and help your fellow classmates. You should get to know one another very well.
  • Discourse: A kinda combination of the previous two that never sleeps!
  • The text: It is very important that you read the text and reread the text and raise questions about the text with me and others until you understand it. We are aiming for a deeper level of understanding than in a lower level course and we are trying to learn to communicate that understanding. You should emulate the text in your mathematical writing.