What Is R3 Math

[Nicodemo Aidara]

Inone of David Lodge's comic novels about academia, the English-professor characters play a game called "Humiliation," where they take turns admitting classic works of literature that they haven't read, scoring one point for each of the other players who has read it. The winner is an American who confesses he's never read Hamlet.

It's a very Internet-age sort of thing to do, though these days the prize seems to be a published essay arguing that the canonical work they haven't read actually shouldn't be read at all, by anyone. It pops up now and again as a basis for book clubs, and you occasionally find it adapted into other fields.

If I were to take a stab at a physics version of "Humiliation," my play would likely be this: To the best of my knowledge, I have never used Noether's theorem to calculate anything. This, despite it regularly being hailed in terms like "the backbone on which all of modern physics is built," and "as important a theorem in our understanding of the world as the Pythagorean theorem," and "possibly the most profound idea in science." I know what it is, and have used it rhetorically, but I've never really worked through a proof of it (if I were going to, I'd probably try this one), and I'm fairly certain I've never used it to do a calculation where I identified a symmetry in something and determined the associated conservation law, or anything like that.

How did I manage to get a Ph.D. without ever doing something that's supposedly so fundamental? Mostly because I'm an experimentalist in low-energy physics. I took the required classes in graduate school, and a few courses beyond that (some subject-specific electives and some stuff that I expected I might someday need to teach), but once I passed the qualifying exam, I moved into the lab, and was concerned more with technical details of vacuum pumps and lasers and electronic circuits and computer data acquisition and analysis.

While you probably could start from first principles and describe our experiments in terms of a Lagrangian with identifiable translation symmetries and the like, it's really not remotely necessary. The conserved quantities we worry about are garden-variety energy, momentum and angular momentum, and don't require all that much justification. There's rarely any need for calculus of variations in analyzing atomic physics data, and on those occasions when some bit of advanced math proves necessary, we were generally happy to pass that off to professional theorists.

I was thinking about this because I had dinner last week at a conference where I sat with a colleague and some students from my undergrad alma mater. One of the students was fretting that he hadn't been able to take enough math to be fully prepared for graduate school--I think the course he was regretting not being able to fit into his schedule was Complex Analysis. My colleague and I both tried to reassure him that he would be just fine, as neither of us could recall ever using that material outside of a "Mathematical Methods for Physics" course.

But then, my colleague is also an experimentalist, working in a similar low-energy regime, so he had a similar graduate school experience. Had we been sitting with a high-energy theorist, things might've been different.

I get asked sometimes "What math do I need to take to study physics?," and the true answer is "It depends on what sort of physics you want to do." Which, unfortunately, often comes off as unhelpful. But it is true, as the above illustrates--if your goal is to work in a lab with lasers and atoms, you don't need nearly as much math as if you plan to discover a Theory of Everything.

1) Vector Calculus: Even experimentalists need to know the basics of integration and differentiation in multiple dimensions. You need to understand gradient and curl and related operations on vector fields, and have a solid conceptual understanding of what it means to integrate along a path, over a surface, or through a full volume. If nothing else, if you hold out hope of an academic job, you'l need to teach this stuff someday.

2) Basic Differential Equations: My social media feeds yesterday saw a lot of re-shares of a Sidney Coleman quote about theoretical physicists solving the harmonic oscillator over and over again. There's a lot of truth to that--a huge range of problems can be made to look like small variations on the harmonic oscillator, so we spend a lot of time on that. The harmonic oscillator is one of the handful of differential equations with nice, friendly, easy-to-work-with solutions, and anybody working in physics needs to know how to work with all of those. And also the general technique for working with differential equations outside that handful, which boil down to "find a way to make it look like a perturbation on one of the equations we do know how to solve."

3) Basic Linear Algebra: The most compact and elegant expression of quantum mechanics is written in the language of linear algebra: vectors, matrices, eigenvalue problems, etc. Language from linear algebra even permeates the wave-mechanics versions of quantum mechanics, which can be a little confusing for students who haven't seen the math yet. It's absolutely essential to get this stuff down, because there's no getting away from it.

4) Basic Statistics: Stats are obviously essential for experimentalists who need to quantify the uncertainty in their measurements, but even theory has uncertainty, thanks to the need to put in experimental parameters. Anybody working in physics will need to have some understanding of standard deviations, error propagation, averaging techniques, etc. This material is also incredibly useful for understanding lots of public policy debates, so it's a win-win: it makes you a better physicist, and also a better citizen.

Beyond that core, though, what you need to know to work in physics varies enormously depending on what field you're in. My field of atomic, molecular and optical physics has tons of linear algebra, because we're basically doing applied quantum mechanics. If you're doing more classical optics--light primarily as a wave, not a particle--you'll need a ton more experience with special functions and solutions to differential equations. Particle and nuclear theory push on into a lot more calculus of variations, and so on-- thus, the central role they see for Noether's theorem-- and if you go into gravity and relativity, you need to learn stuff about differential geometry and the like that doesn't really show up at all in the list above. And, of course, the experiment-vs.-theory divide is enormous--if you're going to be an experimentalist, you need a solid conceptual foundation, but not much calculational technique, but if you're going to do theory, you need a whole lot more.

Of course, that suggests that maybe we also need a lab-skills version of "Humiliation," for experimentalists to torment our theory colleagues. Players could go around and score points for things like "I've never changed the oil in a diffusion pump," or "I've never used a grating spectrometer," or the near-certain winner "I've never soldered two wires together." Maybe we'll give that a shot the next time I'm at a physics conference...

I am Sara Van Der Werf, a 24-year mathematics teacher in Minneapolis Public Schools. I have taught math in grades 7-12 as well as spent several years leading mathematics at the district office. I currently teach Advanced Algebra at South High School and I'm also the current President of the Minnesota Council of Teachers of Mathematics (MCTM). I am passionate about encouraging and connecting with mathematics teachers.I'd love to connect via twitter. Join the community. Tweet me @saravdwerf.

Before you can plan your math courses, you need to know where to start! At PCC your math placement is based on your high school math courses and GPA. This will determine which math class you can start with. Check LancerPoint for your placement!

Additional ways to place into a math class is to use your scores from an AP exam you took in high school or to transfer with math courses from other colleges. Visit the prerequisite office to find out more.

Once you know which math level to start with, you need to understand where you are headed. We offer different courses and sequences depending on which level of math you need to reach for your major and academics goals.

If you are planning to major in Science, Technology, Engineering, or Mathematics (STEM), you will need to take Calculus and beyond. As you plan your courses, follow the STEM math sequence to get you on the track to fulfill math requirements and transfer to a four-year degree program in a STEM major.

Transferring to a four-year university, regardless of your major, requires that you fulfill a general math requirement. But, if you are not pursuing a STEM major, you most likely will NOT need to take calculus. If this is your goal, you should follow the SLAM sequence which will you lead you to statistics and liberal arts mathematics and fulfill your general math requirement for transfer.

Similar to transfer students, earning an Associate Degree or Certificate at PCC requires that you fulfill a general math requirement. If the degree or certificate program is not a STEM major, you will not need to take Calculus. If this is your goal, you should follow the SLAM sequence which will you lead you to statistics and liberal arts mathematics and fulfill your general math requirement for a degree or certificate.

Now that you know where to start and where you're headed, you can select the math sequence to follow. Your academic goals determine the math sequence you select and your placement determines where you start in the sequence.

The Hechinger Report is a national nonprofit newsroom that reports on one topic: education. Sign up for our weekly newsletters to get stories like this delivered directly to your inbox. Consider supporting our stories and becoming a member today.

3a8082e126