I do integration of arbitrary trigonometric functions for that. If the meetings are extremely long and boring, I start calculating satellite orbits by hand and compare to ephemerides.
Stranger
Probably with enough practice, you could do the AC method in your head pretty quickly. But then, the quadratic formula isn’t so tough with integer roots. And while solving quadratics has proven useful to me many times, what isn’t useful is doing endless lists of them as rapidly as possible.
I took a doctorate level class on computational fluid mechanics, where I learned about tensors:
I mean I kind of learned about them, enough to pass the class. That was a good 25 years ago, and at this point I’m pretty sure I couldn’t even explain to anyone what a tensor is.
I took a doctorate-level class on general relativity from the math department (physics profs wouldn’t touch it). Course numbers 665 and 666. Tensors were the beginning; by the end, I’m not even sure what I was doing. I could mechanically do the Einstein notation, but I never grokked it like other physics. Electro-dynamics was easy in comparison. (And quantum mechanics is beginner stuff in physics grad school.)
Totally worth it, just to have “Math 666” on my transcript. Good times, good times.
With a quadratic equation, you can divide all coefficients by the first coefficient and then if b^2/4-c is a positive square number (1, 4, 9, 16, 25, …) then you have real integer roots; otherwise you don’t and need to use the quadratic formula. There is no reason to use “George” or the “AC method” or any weird algorithms that only work on a tiny subset of quadratics. In fact, with the quadratic formula you can calculate the regular roots in your head about as fast as you can write them down with a little practice. It is also really easy to demonstrate by proof, so it isn’t as if it is some kind of esoteric trick. There are a lot of numerical tricks like casting out nines and using logrithms to estimate complicated multiplication and division problems but finding roots of quadratics is already such a trivial operation that there isn’t reason to use any kind of ‘tricks’ for some small subset of solutions.
Stranger
“It’s a trap!”
Stranger
The thing with tensors is, a lot of different fields use them. But so far as I’ve ever seen, only relativists deal with them using the Einstein notation, which is simple, compact, and easy to both use and understand. Everyone else who uses them uses this long, elaborate, and hard-to-understand notation. I don’t know why the Einstein notation hasn’t caught on more widely.
@Senegoid, thanks to working as a substitute for years, I’ve seen the inside of math classes in seven different high schools in the past decade, as well as the state standards and standardized tests. Proofs are definitely still emphasized in high school geometry. Maybe not quite as much as when I was in high school in the early 90s, but then, mine was an honors class.
What?! That’s totally insane. Like doing stock prices in fractions instead of decimals.
What confused me was people talking about “dyads” and what not without using the word “tensor”; finally, after a linear algebra class (not the very first semester one) in which the instructor explained multi-linear maps quite clearly I understood it, and agree that it is impossible to get away from (e.g., a matrix is a type of tensor, so is a linear map, etc.) In particular you cannot do much differential or algebraic geometry (or relativity theory and so forth) without the notion.
Einstein notation is useful; the general principle, though, is not to introduce a system of coordinates or non-canonical equivalences at all where they are not necessary, and your life will be much easier.
I used Einstein summation notation when doing work on creating a closed-form simulation for 3D mechanisms because it was way too clunky to use long form, and in general it widely (albeit probably not exclusively) used in continuum mechanics because it would be painful to do analytical work otherwise, and it reflects how you actually handle tensor operations algorithmically.
Stranger
Same here, Differential Equations. I have a Physics and EE degrees, and you would think that as a required prerequisite that this would be at least somehow useful somewhere. The whole rest of my education never once needed it, and it only ever occasional showed up in a problem set as an “haha, remember this obscure trick?” challenge. Professionally I’ve never had to use it, as real life never produced the very specific problem cases that it might be useful for.
One of my math professors called it his most hated class to teach, as it is merely rote cookbook lookups and steps to work specifically tailored problems, without any flexibility, elegance, or thought.
This math is beyond my ability but some of you may enjoy it.
I’ve been lucky enough to have actually have been paid to solve a differential equation or two in my career, but I know that’s unusual.
On the other hand, even when dealing with systems that could be solved that way (because of non-linearities, etc.) I think the background in DiffEQ was helpful because (I find) theres often some insight into the system’s actual behavior can be gained by looking at the linear differential equations that are “close” to the real sysyrm momentarily.
I loved math in school (hey, numbers behave better than people). But out here in Real Life, I’m surprised at how much of the advanced math I learned but I’ve never used. But some of the basics are indispensable.
And thanks to Pythagoras, I know how much distance I’m saving by cutting diagonally across a park (or a busy intersection, or a minefield).
And how do ordinary mortals figure out percentages without remembering “Part over the whole times 100”? I use that all the time, often surprised by the result when my guess would’ve been way off.
But THE MOST used math, for me, are the basics: + – x ÷
Doing those fast in my head is quite the superpower.
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Anecdote: But there was a time that my math-y brain was a detriment: I was captivated in front of the eggs at the market, doing 3D elliptical math to figure out the volume of each size to determine the best deal. Girlfriend sidled up and said “If it’s less than 10¢ more than the smaller size, it’s more economical.”
(Damn, social science major won out over the real science major.)
A lot of math teaching is about the history of it, when you look at what they teach and when. But the only math textbook which went very deeply into the history was university calculus apart from a couple interested teachers.
When I got a computer science degree I had to take two semesters of calculus, differential equations, and linear algebra. I don’t think I ever used any of it on the job. That’s not to say that these things are useless–they are essential for many professions, just not mine.
Your experience may have been different, but I did not find them teaching much history at all, much less “a lot” of it, calculus courses no exception. Sure, many theorems had names like “Darboux’s Theorem”, but I do not recall much discussion of the history of the theorem nor indeed whether it was Darboux who proved it at all, and who was that guy anyway.
Not that history is ignored, but it remains a sideline or something mentioned in chapter endnotes.
Sorry, I meant that the order in which things are taught to some extent parallels their historical development. Very, very little math from the last fifty or hundred or more years is taught to most undergraduate or secondary school students at all. Many spent a lot of time on Euclid, though.
A specific, widely used calculus text by Stewart comes in different versions. The full one incorporates a lot of mathematicians and famous problems and applications. Most do not really cover the sideline.
That’s intriguing; what modern stuff is in there?
The thing about Euclid is, he is “elementary”. What do you replace him with? Grothendieck? Then you’ll really have people complaining that what they learned was “useless”, not to mention the kids will not like it any more the way they currently sometimes perk up at classical geometry puzzles. Though ISTM a class like “Computational Geometry” would be suitable for let’s say undergraduates: they would get to play with computers (possibly picking up some programming skills), solve many real-world problems and applications instead of learning only pure theory, etc.
Math purists insist that by today’s standards Euclid wasn’t logically rigorous enough.