TL;DR:
- I’ve uploaded the notes I took for MA 36200, Topics in Vector Calculus, to the course notes page.
- These were, unlike all the other notes I host, not written as I went to class, so they’re not really lecture notes.
- The notes work as an acceptable in-between an actual textbook and actual lecture notes, but I warn that since they weren’t written as I went to class, you should ensure you’re looking at the right content.
- Other resources I think people should look at when doing multivariable calculus:
- Joseph Chen’s lecture notes and lecture videos (chenflix)
- Liam Robinson and Alex Rogers’ website, which gives a way to take practice exams, with full solution videos (boilerexams)
- MIT OCW’s 18.02, Multivariable calculus, taught by Denis Auroux
- 3Blue1Brown’s video visualizing vector functions, divergence and curl
- Andrei Kascha’s vector field visualizer (fieldplay)
The rest of this post is going to talk about my experience taking MA 36200, and how I ended up producing the notes.
So I’m really just relearning MA 26100?
Completing a math major at Purdue when I enrolled requires, among other requirements, the completion of an analysis course and an “advanced calculus selective.” As a second and third year student, I was enrolled in math honors, and thus would have to take the honors versions of these classes, MA 44000 (honors real analysis I) and MA 44200 (honors real analysis II). For a student pursuing a math major, no honors, they would instead take MA 34100 (foundations of analysis) and MA 36200 (topics in vector calculus) instead. Going by names alone, we observe that MA 44000 is the honors version of MA 34100, and by progression, we would infer that MA 44200 is the honors version of MA 36200, so the content covered across the two courses should at least try to be similar.[1]
However, after finishing MA 44000, I decided that I would not pursue honors, and so I would be taking MA 36200 instead. I made the same assumption that the courses would cover more or less the same material, which was proven wrong almost immediately as I took the course. MA 26100 (multivariate calculus) is a prerequisite for linear algebra, which is a prerequisite for analysis which I imagine most students take before MA 36200, yet I found myself learning the same material that was covered in MA 26100, in my first semester. Sure, there were small nuggets of new information, like a treatment of Taylor series in a multivariable context and a single example of epsilon-delta, but there were things in MA 26100 that we did not cover, like curvature and the Frenet-Serret formulas. It felt more like I was retaking the course at another university, where a professor decided to cover different material, rather than an advanced version of the course.
This does not even touch upon the complete miss in material that would otherwise be covered in MA 44200. A friend of mine who took both said there was virtually nothing in common between the two except the discussion of continuity, differentiability and integrability (just the qualifiers) of functions, which we spent a total of 3 lectures on.
However, I held out hope. For those that know me personally, much of my philosophy on how to learn comes from Evan Chen. Chen is in the progress of producing an expository work, the Napkin, which I made use of when I took MA 44000 and made little use of when I took MA 35100[2]. Purdue requires completing MA 26100 before MA 35100, which Chen disagrees with, as you may find in Appendix A.1.i of the Napkin:
I dislike most presentations of linear algebra and multivariable calculus since they miss the two key ideas, namely:
– In linear algebra, we study linear maps between spaces.
– In calculus, we approximate functions at points by linear functions.Evan Chen, An Infinitely Large Napkin
Thus, I believe linear algebra should always be taught before multivariable calculus. In
particular, I do not recommend most linear algebra or multivariable calculus books.
I imagined that since
- Linear algebra was a prerequisite for MA 36200
- We take two classes of linear algebra, the second of which is more rigorous
MA 36200 would be a rigorous reimagination of MA 26100, where linear algebra could finally be used, and I’d finally get to learn what a manifold is.
First homework assignment, we learn how to add vectors and take the inner product. Nothing new, but I’m holding out.
Second homework assignment, and we define lines and planes. Still rather slow.
It never gets better.
I was excited (and scared) when we began learning the basics of topology when we covered continuity and differentiation, but the homework let me know that such ideas were completely irrelevant to the course, and were maybe a single exercise in total.
I was still going to try and do well, but the decision would not be for a pursuit of knowledge.
I avoid repeating history by making the present even worse
When I took MA 26100, I did well on the first exam, did extremely poorly on the second exam, and then did well on the final to get an A- in the course. I had a realization of shock when I found that I would have to get a 100% on the final to get any kind of A in the course without the curve, which was a rather grim tale.
Instead, in MA 36200, I do poorly on the first exam, which is rather disappointing, since the second exam was likely going to be more difficult, if I trace the trend from MA 26100. I did poorly on the first exam partly from being unprepared, but really from just a bad test taking performance. I calculated 
For the second exam, I study a day in advance, redoing all the homework problems, but a similar story happens again. I forget that the equation of a sphere centered at the origin is
so the constant term on the right hand side is the square of the radius, not the radius itself. I don’t recognize an obvious chain rule problem as the chain rule and approach it completely incorrectly, and misread a list of points which made a problem so difficult I did it twice, using up an entire side of paper, and got two different answers.
I want to make it clear that although I had all the knowledge, I did badly on the exams, which happens. If I studied more diligently and earlier, I still would have done better, despite not improving my knowledge. There is an art of preparing for speed and accuracy, once all necessary information is understood, which I mistakenly did not undertake.
I knew I did bad on both exams when I left the classroom. I recognized that if I didn’t fix something, there was a possibility I completely fail the class, which would jeopardize my graduation plans. Once my expectedly disappointing exam grade came out, I calculated my grade to see what possible final grade was still possible. Though the syllabus and professor mentioned that adjustments to grade cutoffs were possible, he indicated that there was also a possibility he would adjust cutoffs negatively, too. Trying to stop myself from overthinking, I just use the standard 73% C cutoff.
Oh.
I already failed.
…
I guess that’s it.
…
Oh, I typed in one of my grades wrong. My homework is out of 25%, and I intermittently calculated that I had earned about 24% of said percent, but I had inputted that I had a 24% average in said category. Ah, that explains things. Just user error. I fixed the calculation:
That’s not exactly a margin I want to play with, but at least it lets me know passing is still possible. But I’ve done things like this before. I’ve won back good grades by doing especially well on final exams in spite of poor performances before, which I’ve done in MA 26100 and MA/STAT 41600 (probability). It’s not a comfortable position in any way, but there’s nothing to do but to sit down and start preparing.
Two weeks of studying for the final
I first decide to redo every single homework problem. This takes maybe three days. I then decide that since the homework problems are the only foothold I have to predict what the final could potentially cover, I really need to understand them. I redo every single homework problem a second time. I also figure that typing them up would be a good idea, and that an index of the problems I’ve done would also be something useful, and decide to completely type up a set of notes based on all the homework problems and the textbook.
I studied 4 days a week-taking the weekends and Wednesday off, and studied for 10-12 hours the other days of the week. It got bad enough that to take breaks on the days I was working, I would just study for another class, one that I didn’t care about, but I did so just so I wouldn’t break from the monotony.
I’d glue my eyes to the screen of MIT OCW’s multivariable calculus[3], forcing myself to listen to Professor Auroux in x2 speed. Normally the retention and comprehension degrades, but I focus and ensure I get every single word.
I produced about 133 handwritten pages of work doing problems, averaging about 11 per day.
I chose to type up the notes about a week in, typing about 20 pages a day the days I actually remembered to keep track.
Across the 2 weeks, I generated 314 pages of academic writing.
The day before the exam, I can’t bring myself to redo the homework problems yet again, so I follow a recommendation of a classmate in a group chat for the course to use Boilerexams, which I then did about 3 final exams for about 60-70 total problems. After this, I’m completely spent, and rest for the big day.
The day of the final
I start the exam, and am pleasantly surprised. The questions are very similar to past MA 26100 exams; not to the point that I have seen the questions before, but are incredibly familiar territory. I manage to do every single question about four times, and I’m glad I did a fourth time, because I caught an error on the last pass, since I relegated a problem to be simple enough to do in my head, only to evaluate 
I submit the exam. I’m still anxious, as I won’t find out my score immediately, but I leave feeling genuinely confident in my performance.
The day after, I check my score.
It worked out.
I end my last final exam of undergrad with a perfect.
On the notes
As I’ve discussed before, I wrote the notes to comprehensively prepare for the final, rather than something I wrote as I took the course and attended lectures. It’s not a worthwhile fight for me to try to convince people that decide not to go to class to go to class, since I don’t really have anything to add that people haven’t already considered, I think.
While my linear algebra notes can feasibly be a replacement for attending lectures, this is only because I took notes with the sole objective of being a replacement for such; that was my philosophy for how to take notes then. Every word my professor said, I probably wrote down, or my take on it, so there’s flavor and personality to them. You won’t find such features in my vector calculus notes, because I wrote them to be useful to me first.
Students that peruse my notes for MA 26100 should be warned that not all content from that course makes it into my notes (as I mentioned before, curvature and the FS formulae), and that I give a rather surface level description of how polar, cylindrical and spherical coordinates work, and just have examples. To that end, I really must caution you and specify that you should only use these when you would like a second opinion of a topic compared to the textbook, or you would like some more commentary than some lecture notes, or need a reference that’s thinner than actual textbook. The other resources listed I recommend greatly.
Footnotes
1. This is hilariously wrong sometimes. When I took MA 44000, we covered sequences, point-set topology, differentiation, integration, series, and function spaces, in incredible depth. A friend of mine taking MA 34100 this most recent semester said they only covered sequences and differentiation, at a very basic level.
2. MA 35100, despite having a fair number of proofs, is fundamentally not a proof based course. A lot of the machinery is hand waved away, which I don’t really consider a problem, since math majors take MA 35301 (linear algebra II, basically), which goes through all of linear algebra incredibly rigorously anyways. Since MA 35100 isn’t that proof based, the Napkin wasn’t a good fit. I (fortunately) didn’t have a reason to use it when I took MA 35301, but I imagine it could have been useful.
3. This doesn’t really belong in the main matter of the post, but I actually cherish MIT OCW’s lectures, especially the section on the last part of vector calculus, on line and surface integrals, as well as the integral theorems of vector calculus. I bring this up only because when I took MA 26100, I didn’t use those lectures, which was a mistake and used chenflix instead. chenflix isn’t bad by any means, but I feel like if you use them, you absolutely have to do the homework without just using a calculus application online or something, which I really don’t think is expressed. In contract, MIT OCW’s are more in-depth, so it might slightly offset the problems of cheating on your homework. Oh, and Professor Leonard‘s videos, which are otherwise great, have a bad treatment on surface integrals. A 3 hour video is just a bit much.
The smartest stupid person 