Multidimensional Multivariable Calculus Syllabus
Multivariable calculus is about exploring higher dimensions—so it’s only fitting that our syllabus should be higher-dimensional as well. For a multivariable calculus course I taught in 2021-2022, I made a syllabus on a cube, programatically generated such that each student’s syllabus had a different arrangement of the six faces (or, more precisely, was a random choice from one of the 122,880 possibilities).
I laser-cut a larger-format version into a box:
The laser-cut version collapsed in my hands as I was showing it to the kids. I told V. I was worried that’d be a metaphor for the class as a whole; she pointed out that perhaps instead it’s a metaphor for how we’ll learn to reduce the dimensionality of problems.
Here’s the full text:
The Nueva School
Andrew M.H. Alexander
(third floor pod by the stairwell)
The ultimate goal of our class is to understand the universe better. That’s true of all your classes. In a science class, you might be trying to understand the universe better in its empirical properties. In a literature class, you might be trying to understand the universe better in its human properties. In math classes, we’re trying to understand the universe better in its abstract properties.
In this class: how do we explore higher-dimensional reality? How do we navigate the two-, three-, four-, or even arbitrary-dimensional world? We know how to do calculus in one dimension—how can we do it in multiple dimensions? What things are different? What things are the same? What things are surprising or unexpected? What assumptions were we making that we weren’t even aware of?
In previous math classes, we’ve gotten a good handle on the one-dimensional world. We’ve forged a decent fluency with the two-dimensional world. In MVC, we’ll leave the shallows. We’ll head out into the depths of the higher-dimensional world.
How are we going to learn all of this high-dimensional hootenanny? A couple ways. But mostly, we’ll learn math the way math is supposed to be learned: by doing. In particular: by doing math problems. We’ll do easy problems that build up our technical skills. We’ll do hard problems that push the limits of our intuition. We’ll do these problems together in class. We’ll do them by ourselves at night. Sometimes we’ll get the answer. Sometimes we won’t.
I’ll write notes for you on various topics. You’ll read them to learn various facts. (We’ll minimize the amount of time I spend lecturing—raw information transfer isn’t the best use of our precious time together.) You’ll ask questions when things are unclear.
You’ll have problem sets due every class. You’ll work hard on them, and will come to class ready to discuss. You’ll ask questions. You’ll share your strategies (successful and unsuccessful). We’ll have an active conversation, as a class, in which we puzzle over these problems. We’ll pick through and unpack their nuances, subtleties, and themes.
How are you going to convince me and others that you’ve actually learned all this stuff? I’m fond of hard tests. In zoomland, we couldn’t do that. Even in normalland, we only see each other twice a week and it seems a shame to sacrifice our finite precious moments together to a test. In zoomland I relied on each of you diligently doing your problem sets, with me checking and giving comments (not on every single one, but on many). Perhaps we’ll only do that. I’ve long admired Jana’s oral exams. Maybe we’ll experiment with those.
These are just formal ways—the informal ways are just as important, if not more so. Are you working hard in class? Are you participating? Are you helping your classmates? Are you asking for help yourself? Are you diving into difficult problems with gusto and moxie?
The point is, with all of this stuff, we’ll feel out a way that works for us.
But all this is secondary. We’re here to learn math. First things first. Focus on learning the math as best you can, with all your heart. Everything else will follow.
Most course syllabi are two-dimensional objects. They have to be: written English requires two dimensions. This syllabus is no different. But it’s a two-dimensional syllabus embedded in 3D space, in the form of a cube. (I was going to print it on a Klein bottle, but the i-Lab’s 4D printer was broken.) We’ll study lots of things like that this year: lower-dimensional shapes embedded in higher-dimensional space.
It’s also a syllabus with math problem embedded in it. When you print this, if you linger on the webpage, you’ll notice that every few seconds the six faces of the cube randomly rotate or swap. So, geometrically speaking, each of you have a different syllabus!
Or do you? With the six blocks of text being on any one of the six faces, rotated in any of four orientations, how many different possible syllabi are there? How likely is it that, given how many people are in our class, two of you have printed out the exact same syllabus? (What if you folded it so that the text is on the inside?!?)
If you can figure this out, I’ll give you a prize!