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Lessons I wish I had learned before teaching differential equations [pdf] (1997)

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Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#221

Post by JadeNB »

wiredfool wrote: post #71
EsportToys wrote: post #10

Speaking of 2nd-order linear ODEs w/ const factor (a.k.a. mass-spring-damper systems), I wrote a blog post[0] deriving all possible general solutions in a concise matrix that makes it easily implementable in code. The following is the complete solution in Lua: function sprung_response(t,pos,vel,k,c,m) local decay = c/2/m local omega = math.sqrt(k/m) local resid = decay*decay-omega*omega local scale = math.sqrt(math.a…

My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis. Ultimately it all comes down to choosing the most convenient basis functions for the questions you're answering.

> My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis.

If I'm understanding correctly, you want another = there for a standard damped spring: F = ma = -Cv - Kx.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#222

Post by yuppiemephisto »

monista wrote: post #28
ianai wrote: post #13

Earlier quoted context omitted.

They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through https://mathworld.wolfram.com/NonstandardAnalysis.html I have long thought such would be easier to work with on modern computers.

Note though, that nonstandard analysis isn't compatible with more "intuitionistic" https://en.wikipedia.org/wiki/Axiom_of_determinacy (in place of axiom on choice), which free you from Banach–Tarski paradox and have some other appealing properties.

Can do it in pure ZF:

https://www.youtube.com/watch?v=YE68yptMMeQ

https://u.cs.biu.ac.il/~katzmik/spot.html

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#223

Post by jojobas »

This sounds misguided. What do you mean, obsolete? It's largely 17th century maths, 19th century understanding was already pretty good. Have there been a pedagogical breakthrough allowing for igniting a spark in a brain much easier? I don't think so.

What do you mean, reduce to the minimum first order equations? I went to the best technical uni in my country (a country known for quite some orthodoxy in teaching), most people would have had some grasp of differential equations before they left high school. Having the most basic equations revisited with actual rigor was eye-opening to whoever I've spoken to about it.

What do you mean, concepts not tricks? Is looking for a solution in a suitable Fourier representation a trick? Certainly is if you ask me. A mother of all tricks hinting at the spectral fabric of the universe.

Last but not least, he sounds like he has some sort of metric on how well his students apply his enlightened teaching later in life vs orthodox inefficient professors'. I have some doubts on this.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#224

Post by denotational »

JadeNB wrote: post #221
wiredfool wrote: post #71

Earlier quoted context omitted.

My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis. Ultimately it all comes down to choosing the most convenient basis functions for the questions you're answering.

> My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis. If I'm understanding correctly, you want another = there for a standard damped spring: F = ma = -Cv - Kx.

F is a forcing function, not the resultant force. It’s often arranged this way with all the derivatives on one side (as opposed to having the resultant force, ma, on one side of the equality by itself) so that it matches the general form of a non-homogeneous second-order linear differential equation.

(At least I assume this is what the original commenter meant!).

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#225

Post by grog454 »

DiffEQ was probably my favorite math class in college. Eventually I created https://play.google.com/store/apps/details?id=simplicial.sof..., originally for a course about computer graphics. I think it lays the foundation for an intuitive understanding of the basic concepts in a way that a text book cannot.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#226

Post by jampekka »

nuancebydefault wrote: post #210
sampo wrote: post #205

Earlier quoted context omitted.

Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.

Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.

Number theory was a big deal already in ancient greece and probably got there from Babylonia, so it's probably safe to say that they didn't live to see the public key crypto usage. :)

But (integer) numbers and their behavior had huge significance in the ancient worldview, and still do even in our days if you look deep enough. And the Pythagoreans et al applied number theory in e.g. music.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#227

Post by nonameiguess »

ajkjk wrote: post #178
anon291 wrote: post #170

Earlier quoted context omitted.

> I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe. You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no f…

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, a…

I think you're generalizing far too much from limited experience. Not all mathematics, even applied mathematics, corresponds neatly to intuitive physics models you build naturally by being an animal that interacts with the world. Quantum mechanics was already given as an example, but I had the opposite experience with cryptography. So much of the training as a CS student was along the lines of "accept that this works" and then practice by implementing something, and sure, it worked, but at no point did it ever become obvious why any of it should work. Until I finally went back and spent a solid year and a half learning group theory and algebraic number theory in a more rigorous way, figuring out all of the proofs we had skipped over, at least trying it myself, but giving up and reading someone else's proof if I couldn't figure it out within a day or two. And in this case, the abstraction definitely came before the application. This is math that was thought to be compeletely useless for centuries, but people worked on it anyway just because they loved math. Lo and behold, it made the Internet possible, so I'm glad someone did it and loved the tedium.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#228

Post by ajkjk »

anon291 wrote: post #203
ajkjk wrote: post #178

Earlier quoted context omitted.

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, a…

> Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In part…

I'm very much not arguing that nobody should do pure math. I'm arguing that teaching pure math to undergrads who are trying to learn applied math is a tragedy and ruins math for a lot of people.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#229

Post by analog31 »

jampekka wrote: post #87

There seems to be something similar in other branches of mathematics too (and lots of other fields). A clear example I recall was studying Fourier transforms in math and I couldn't make any sense of it. To me it was just some by-rote algebra with integrals of exponentials of complex numbers. But then I happened to do some audio signal analysis, and when I saw a magnitude spectrum of a waveform, it was instantly obvio…

It's hard if you don't know any practical examples. Anecdote: I taught a college math course for a semester while I was between jobs. I shared an office with some other teachers including a bright grad student who was TA'ing differential equations.

My degree was in physics, and I had worked in industry. I told him that I wished the math courses had included some engineering applications of differential equations. He looked at me with a straight face and said: "There are no engineering applications of differential equations."

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#230

Post by anon291 »

gbjw wrote: post #211
anon291 wrote: post #202

Earlier quoted context omitted.

> Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can. Once again, I'm not 100% sure what you're saying. If you have something that's of length 1, then you can easily construct the line with a ratio sqrt(2):1. Draw another line of length 1 (use compass and straightedge) at a 90 degree angle. Repeat 4 times until you have a square. Now draw the diagonal. You h…

Yes, apologies -- using the term 'construct' muddied the point as sqrt(2) is a 'constructible' number as you point out. The term 'real' is what's at issue here and I am arguing for a distinction between a 'map-like' real and a 'territory-like' real, the latter of which has some sort of spatiotemporal grounding. > There are some numbers that are not, and perhaps these can truly be said to not exist. So then we have a…

Sure, but no one uses the reals anyway for 'constructible' things. As I pointed out, the computable reals themselves form a closed field and are the things you would find when describing real life.

As for the 'problem'... I personally don't view it that way. In my opinion (and it's just that, since there's no mathematical 'truth' here), I don't believe non-computable reals exist in any meaningful way. I believe this is similar to how we talk about a 'program that can check if another one halts'. Anyone can make that statement and claim that such a thing exists, but it's not at all clear that such a thing exists. But that's a lot different than saying there are X particles in the universe, thus the number X + 1 does not exist. Because x + 1 does exist and you can write a turing machine that can compute it to any precision (or a lambda calculus function that'll give you the next church encoded representation of it, etc).

My point is two fold. Firstly that there are certainly numbers that are greater than the total number of 'stuff' in the universe. Secondly, that there are some numbers that cannot be described in any meaningful way. These can be said to not exist (my belief), but others disagree.

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