MATH 334

This is my course review for MATH 334, Advanced Calculus. I took the course in Autumn 2015 with James Morrow and teaching-assistant Will Dana.

Content

Just follows Folland’s Advanced Calculus. Morrow also posts various supplementary material, mostly as PDFs, on the course website.

I’m actually starting to appreciate Folland’s text a lot more after having “warmed up” to it. I think the Amazon reviews and some remarks elsewhere online put me off the text without having really examined it fully myself. I still like Tao’s “redundant” writing style, Spivak’s charm, etc., but I think Folland is a solid text as well. I wouldn’t recommend it for self-studying analysis, however; I think I benefit significantly from referencing Tao (Analysis II), Pugh (Real Mathematical Analysis), and Morrow’s lectures/supplementary notes.

Homework

So far problem sets seem to take even longer than they did in MATH 13X and also seem to require greater sophistication to complete.

One thing I find odd is the great difference in difficulty between the lecture and homework. In previous math classes, the homework problems were perhaps just one step beyond what was covered in lecture. Yet in 334, it seems like the homework problems are both more numerous (and therefore fast-paced, since there is a fixed due-date) and difficult than what is covered in lecture.

In fact another strange thing: in math 13X, the homework was due on Monday, so the lecture covered essentially what we would need for the homework that we would work on during the weekend (since everyone seemed to put it off until then anyway). However in math 334, the homework is due on Friday, so how do we use the weekend? Since there are so many homework problems in comparison to 13X, it’s prudent to use the weekend as a “head-start”; yet if we do this, the lecture becomes just a boring reiteration of what the book already covered canonically. Indeed, the lecture content seems way too easy (and yet many still seem to be asking questions), and in no way of use in solving the homework problems. Yet another point here: the TA in-class session is on Friday, after the homework is due, and we cannot ask questions about the next homework…just…what?—All this to say that the homework routine I had built up last year is of no use this year; I’m actually still not certain how I should approach the structure imposed by 334.

These concerns are echoed in a reddit comment:

One of my personal gripes with the class is that often, we would have a homework due Thursday on material we’d just covered in class Wednesday, or worse yet not covered at all. The textbook is awful, which makes studying on your own for those homeworks a pain.

Some interesting things Morrow said today when discussing the multivariable chain rule:

After writing it down as

\[ \frac{\partial z}{\partial x_1} = \frac{\partial z}{\partial y_1} \frac{\partial y_1}{\partial x_1} + \cdots + \frac{\partial z}{\partial y_n} \frac{\partial y_n}{\partial x_1} \]

he said (paraphrased):

Writing it down this way is okay, but you have to understand what the symbols mean […] It’s an efficient way to write it down […] I think you’ll get used to it […] The exercises [in the book] are meant to show you that it’s easy to mess it up […] Chemists don’t tell you which variable they’re keeping constant […] There’s a lot that’s implicit and it’s hard to tell what’s going on.

I think it has been repeated many times that physicists are somewhat more sloppy with notation than mathematicians, but even then I’ve always felt that they people around me who do math don’t care enough about notation. I thought it was nice that Morrow seemed to sympathize with this.


The “pep talk” after the first midterm was by Matt Junge, a sixth(?) year PhD student here who did poorly in 33X (3.1, 3.4, 3.4 GPA in the sequence, with his midterm scores in 335–6 consistently \(\mu - \sigma\))—but who now has five published papers and is good at math. His main points were to not give up, continue on doing math (take at least one course in the 400s), lose one’s ego (don’t be afraid to make mistakes), LaTeX your homework, read the textbook before lecture, and so on. The content of the pep talk was not very novel, but I enjoyed hearing from Matt (he seemed like a cool guy).


I’d like to say that this course is not just mathematical (make no mistake that mathematics is at the center of the course) but also very cultural/philosophical/pedagogical. I think Morrow does a good job (if you attend lecture!) of explaining the historical developments of concepts, and often likes to quote various mathematicians. (I think my younger self would have greatly appreciated that.) The course also emphasizes alternative explanations. Morrow often presents proofs and definitions sightly differently than in Folland, which makes the course pedagogically interesting. (These are also later posted to the course website so you don’t have to absorb or write down anything in lecture, really.) I will still maintain my general attitude that lectures aren’t very useful for learning, but I think Morrow is much better at this than any other lecturer I’ve seen so far at UW.


Some more reflection; somewhat stream-of-consciousness and needs a complete rewrite later.

I’ve been “philosophizing” more lately about math. It’s true that I used to think mathematics is what I wanted to do all my life, and a means by which I would make a living. It’s also true that at some point in the last two years, I decided this wasn’t something I wanted to do, after all, and that I would probably work as a programmer (kind of a resignation, really, because I still don’t think it’s too exciting to be a programmer).

I’ve mentioned this elsewhere, but to reiterate: with prodding from a few people, I’ve become much more “real-world oriented” in the past two years, to the point where I question the worth of a lot of what I do in school (besides classifying it as “signalling”). Even when I learn something I find somewhat interesting or useful, I am conscious that e.g. the problems I solve on homework assignments are just “toy problems” that are designed to be solvable with a few hours’ thought.

I think MATH 334 is a really interesting class because it induces in me a philosophical state of mind, much more so than in any humanities class I have taken (though I haven’t taken very many at UW) that deliberately tries to inculcate a “liberal arts” state of mind. It sounds a bit silly to say, but besides a few exceptions, the people in math 13X/334 are the people I have most closely interacted with at UW—which isn’t saying much—and it’s where I get a more extended opportunity to observe people1, unlike the shallow observations from elsewhere at UW.—So that’s often what I think about in class, for example. I’ve found it increasingly difficult to focus on lectures, so these thoughts about interpersonal interactions, the progression of an individual’s life, etc. enter my mind.


Another thing that just occurred to me: I rarely actually interacted with Morrow. Most of my interactions were with my classmates or the TA. In fact, the only time I remember talking to Morrow was when I went to pick up my two midterms (I missed both of the days when he handed them back). He showed another student and me how there are ladybugs that would come into his office(?) in the fall/winter, and also seemed surprised that I hadn’t even picked up my first midterm before the second midterm (his office hours were at 8:30 so I never managed; this was right after lecture so more manageable).

I guess it’s not too surprising; I hardly interacted with my math 13X instructor either (the only interactions I remember: visiting during office hours because I wanted to enroll in the course, picking up exams, asking a few question during lecture). I guess the difference is that so far this year I haven’t had to ask any questions in lecture…


  1. This is something I used to be able to do a lot in high school, but the quarter system at UW makes is difficult to keep in contact with the same people, especially for someone as passive as me.