For tedious reasons, I find myself faced with giving what will basically be a pure math lecture next Friday. I need to introduce a bunch of mathematical apparatus that we will need in the coming weeks, and I know that the Math department doesn’t cover these topics in any of the classes that these students have taken. If I want them to be able to use this stuff, I need to teach it myself.
Formal mathematics is probably my least favorite part of teaching physics. I’m very much inclined toward the “swashbuckling physicist” approach to math, in which we cavalierly assume that all sorts of picky details will resolve themselves when we eventually have to compare our calculations to experiment. In the fields I deal with, it’s usually obvious when a potential solution is un-physical, and we just discard those. As a result, I’ve always had a very casual approach to dealing with math, which occasionally puts me in a bad place when I have to introduce math to students.
In this particular case, the main idea that they need to get is the Euler relationship between complex exponentials and the trigonometric functions, usually expressed as:
ei x = cos(x) + i sin(x)
It says that when you take the exponential of an imaginary number, the result you get is the sum of a couple of trig functions. It’s a wonderfully elegant result, that has all sorts of nice properties when you start dealing with the physics of waves and oscillations. And it’s absolutely indispensible when you talk about quantum mechanics, as wavefunctions are necessarily complex quantities.
the problem is, I have absolutely no idea how to introduce it to these particular students…
Part of the problem is that I have a deep loathing of just stating anything as a fact to be memorized. That’s the sort of behavior that gives physics a bad reputation among students of other disciplines, and on top of that, it’s not terribly effective. Without some context, it becomes just a bit of trivia taken on faith, and students are prone to forget it before the end of the class period, let alone the final exam.
When I introduce physics formulae, I always make a point to provide some justification. Either I derive the results from first principles, or I appeal to physical intuition and common sense. I think that works better than just stating results, both in terms of student retention of the important facts, and also for giving a more complete picture of how the field works. There are a few places where I’m forced to punt– the Schrödinger equation being the obvious example– but I try to at least make plausibility arguments for everything.
I’m sort of stumped on the Euler relationship, though, because the only way I know to justify it relies on mathematics the students haven’t had yet. You can make a swashbuckling-physicist argument that it obviously has to be true based on the series expansions of ex, cos(x) and sin(x), but they haven’t seen series expansions yet, either, and I don’t have time to introduce that as well.
I’m open to suggestions, here. Does anybody know a good way to introduce the Euler relationship without reference to series expansions? Or am I just stuck with Proof by Blatant Assertion?