The real math that describes the behavior of fusion reactors is too complicated to actually carry out, so physicists have to make some simplifying assumptions. They try to choose assumptions so that the results they get are still valid, kind of like when you say, "Keep the change" at the grocery store. You assume that giving up small amounts of money on occasion won't make you go bankrupt, even though you're not actually adding it up all the time to keep track of how much you've given up.
It turns out that one of the simplifying assumptions that physicists have been making over the years does change the result. Giving up the "spare change" does make you "go bankrupt." In this case, ignoring small-scale turbulence makes you lose a significant amount of heat. But figuring that out required some really hairy math and a butt-load of computing power.
EDIT: Ordinary differential equations, yes - we do here in .au, at least in Maths C (advanced maths) in Queensland. Partial differential equations, the kind that are very useful in the kinds of physics simulations discussed - not so much.
And even with that background, our foreign lecturers were regularly disappointed at our lack of mathematics competency in University ("Now I have to waste the next 4 weeks teaching you things we learnt in highschool in my home country before I can move on to what this subject is actually about").
These were Asian/Indian and a Ukrainian lecturer, that I recall.
Ah Russian math lecturers in first year courses. Always making it perfectly clear that as far as they're concerned they might as well be lecturing to a bunch of 8 year olds.
I always wondered if they just said that, as part of bluster. Hard to believe entire classes of graduate math students are at a lower level than all 11th grade Russians. (It wasn't just the first year undergrads I heard of getting scolded by Russian lecturers)
I know Russia values math highly and does push their students, but there's a limit to believability.
Having worked with a few asian/eastern engineers, I do believe that a number of things conspire against western maths education: our culture doesn't value mathematics competency (in some circles people are even afraid to admit to an interest); the way we teach maths here in .au is bloody awful (standardized tests make teachers hate their jobs); and teaching generally is a much less respected/lower-paid profession relative to other occupations compared to better-performing countries (so our quality of teachers is much worse).
I had the same curriculum but was utterly bewildered by derivatives being the first topic. For the simple reason that until that point I had habitually looked at "wholes" in the external world, only then to wonder "what kinds of things progressively make it up?"
I feel like to derive something, you start with the assumption that you know what you're talking about fist. Yet the thing is actually the sum of its parts. Therefore integration seems more natural at first, and only then can things similar to it be derived with confidence.
So the teaching order seemed backwards. I wonder if anyone has felt the same way.
Every school I've ever been at in British Columbia (High School, College, University) - starts with derivatives, and then moves onto integral calculus.
They start with the calculating slope on a curve f(x) being lim h->0 f(x+h)-f(x)/h - spend a few weeks deriving various derivatives, show how you can find tops/bottoms of curves (derivate=0), talk about limit theory a bit, take a bunch more complicated equations derivatives.
Then, once that's figured out - take the inverse, and look at area's under curves, volumes, etc... My brain lost it at integration by parts (so many parts) - too much memorization, and so came an end to my mathematics education in that space.
I'm happy they were taught in that order, mostly on being exasperated with having to memorize all the integration by parts formulas.
Yep! Archimedes was essentially doing integrals 2000 years ago. He got close to infinitesimals, and centuries later we had derivatives and limits. It's strange to teach them in the reverse order (i.e. Most sophisticated theory first, like teaching the reals before the integers).
I have an online series that starts with the historic order which you might like:
I don't know. But I would imagine most high schoolers could at least understand what a differential equation is, if explained. Aside from if they could solve one.
