Here you can find all 26 lectures of my Real Analysis course at Harvey Mudd College. These lectures were taped in Spring 2010 with the help of Ryan Muller and Neal Pisenti.
The entire course is assembled as a playlist on YouTube. And below are links to individual lectures.
Lecture 1: Constructing the rational numbers
Lecture 2: Properties of Q
Lecture 3: Construction of R
Lecture 4: The Least Upper Bound Property
Lecture 5: Complex Numbers
Lecture 6: The Principle of Induction
Lecture 7: Countable and Uncountable Sets
Lecture 8: Cantor Diagonalization, Metric Spaces
Lecture 9: Limit Points
Lecture 10: Relationship b/t open and closed sets
Lecture 11: Compact Sets
Lecture 12: Relationship b/t compact, closed sets
Lecture 13: Compactness, Heine-Borel Theorem
Lecture 14: Connected Sets, Cantor Sets
Lecture 15: Convergence of Sequences
Lecture 16: Subsequences, Cauchy Sequences
Lecture 17: Complete Spaces
Lecture 18: Series
Lecture 19: Series Convergence Tests
Lecture 20: Functions – Limits and Continuity
Lecture 21: Continuous Functions
Lecture 22: Uniform Continuity
Lecture 23: Discontinuous Functions
Lecture 24: The Derivative, Mean Value Theorem
Lecture 25: Taylor’s Theorem
Lecture 26: Ordinal Numbers, Transfinite Induction
The text for the course was Principles of Mathematical Analysis by Walter Rudin, but you do not need the text to follow these lectures. Also, I realize the board is hard to read, so I’ve supplied some linked lecture notes in the tab above (they may not align perfectly with these lectures, since it’s from a different semester. You can also see my course webpage from the last time I taught the course, with homework schedule.)
Why did you make these videos?
My student Ryan Muller was developing an online learning platform and asked if I’d be interested in recording my Analysis lectures as YouTube videos to embed in it. After some initial hesitancy, I agreed. But after putting them up, I began getting appreciative e-mails from around the world. So I decided to leave them on YouTube.
What textbook do you use?
We use the classic text Principles of Mathematical Analysis by Walter Rudin, and cover the first half of the textbook (Chapter 1 to 5, and part of Chapter 7). You can see what problems I assigned from the last time I taught the course, at my course webpage, linked above.
I wish the resolution were better.
Me too. But that was the technology we had… the recording equipment (mounted in the ceiling of the lecture hall) only records at a lower resolution. But check out my course notes, linked above, if you would like to see some notes that loosely track with the videos.
Is your course a MOOC?
Technically, no. We made these videos before the word MOOC (massively open online classes) became a buzzword. Usually a MOOC has a participation and feedback component and involves a large number of students all working on the same course schedule. My real analysis videos are intended to supplement your own learning.
Have you been using the videos in a ‘flipped classroom’?
No, but you are welcome to. One downside to the videos is that the resolution is not that great, so the blackboard writing is hard to read. Also, the material has not been broken into the bite-size chunks that some flipped classroom models use. But I imagine some instructors have been using it this way, nonetheless — it has been neat to see how popular the videos have been in Sierra Leone. 🙂
Why do you discuss mathematical writing in this course?
Learning to communicate mathematics effectively is one of the goals of my Real Analysis course at Harvey Mudd College. This handout “Guidelines for Good Mathematical Writing“ is my guide to help students think about their writing.
Are you going to do videos for Analysis II?
There are no current plans to do this. We would first need better resources for classroom video capture. But I may do other online projects… stay tuned. 🙂
How do I stay tuned to your other projects?
Can I write you e-mail?
I welcome e-mail, especially stories of how you found the videos useful in your own life situation. If you write, please tell me the country you are writing from, and a little bit about your situation and background. (Are you in college, or have no access to college? Are you a non-traditional student? etc.) Although I may not be able to respond, I do read every e-mail carefully.
How can I thank you?
If you found the videos helpful, you can ‘like’ the videos on YouTube. And for a good read about mathematics, you might enjoy my book.