Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

math.toronto.edu

Post reply on HN
Print view View original

81–90 of 122 posts

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

#81

Post by lagadu »

analog31 wrote: post #18

I once told a math teacher at a Big Ten university, that I thought their undergrad math instruction for engineers was weak. As an example, I said that I didn't think students learned any engineering applications of differential equations. He looked at me with a straight face and said: "There are no engineering applications of differential equations."

That answer doesn't surprise me in the slightest. I'm a software engineer so I forgot everything taught in those classes the moment I turned the exam in but I've a friend who's a professor of something that mechanical engineers take related to fluids (don't ask me details, we don't even live in the same country any longer); at one point he told me that anything other than numerical analysis is absolutely worthless in the real world because you can't find analytical solutions for almost anything.

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

#82

Post by qubex »

amelius wrote: post #78
gohrt wrote: post #69

Earlier quoted context omitted.

The common techniques ARE a bag of tricks. Feynman was famous for being really good at integrals, because he had memorized the huge bag of tricks. Today, we have Mathematica for that, you don't need to be Feynman.

This makes me wonder. Is Mathematica also applying a bag of tricks (I suppose in a breadth first search), or does it have a more structural approach?

As I recall, Mathematica embodies a (perhaps incomplete) implementation of the Risch algorithm.

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

#83

Post by octonion »

mattb314 wrote: post #53
vcarl wrote: post #47

Earlier quoted context omitted.

I remember a similar problem in Calc 2. I forget the specifics now, but I think it was an integral of some combination of sin/cos that ended up being circular. You had to recognize an opportunity to swap one of the steps for an equivalent, which would lead you to the final solution.

Probably the second example here [1] for those curious (I think the integral of sin(x)*e^x dx is the only place I've seen this used, would love to know if there are other examples). [1] https://en.m.wikipedia.org/wiki/Integration_by_parts#Tabular...

This becomes much more transparent if you realize you're integrating Im(e^x * e^{ix}). And it's no longer a trick but a technique.

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

#84

Post by curt15 »

evanb wrote: post #30
rafinha wrote: post #21

"FORGET ABOUT EXISTENCE AND UNIQUENESS OF SOLUTIONS" what? most important thing about differential equations.

For ODEs you can simply think of solving the autonomous system numerically. Since I have many numerical algorithms to solve such a system, solutions exist. Since (most of) those algorithms are totally deterministic and offer no choices anywhere along the way (except maybe for some initial conditions) the solutions are unique. For PDE's it's much more interesting, as the author points out.

I think the underlying issue is that the technical conditions guaranteeing for existence and uniqueness for ODEs (the Picard-Lindelof theorem) are so easy to satisfy (which is what guarantees that different numerical algorithms will give the same answer) that they're something most students are unlikely to encounter in practice.

That said, I do think there is some pedagogical value in teaching existence/uniqueness even though the result may not be so interesting because it shows students that it's possible to get information about solutions directly from the equation even without explicit formulas available. It also introduces them to the sort of abstract arguments at the core of modern mathematics.

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

#85

Post by musgravepeter »

I thought the comment on exterior differential forms was interesting. I always wanted to delve into those and better understand what a dx all by itself was when separated from dy/dx by simple manipulation. Loved his comment "We justify this sudden introduction of differentials by saying that this is 'just another way or rewriting the differential equation,' or some equally atrocious lie."

I am now lusting after http://www.amazon.ca/Exterior-Analysis-Using-Applications-Di... but it's a bit pricy for a indulgence purchase!

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

#86

Post by madengr »

I remember my undergrad signals and systems class. Instructor said if I use Laplace transforms on any of the problems, I would get no partial credit. I got an A+ for the course.

Also got A+ in DE, but I still don't think a grokked it.

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

#87

Post by bazzargh »

CamperBob2 wrote: post #23
backprojection wrote: post #9

> Some thirty or so years ago, Bessel functions were included in the syllabus, but in our day they are out of the question. > Teaching a subject of which no honest examples can be given is, in my opinion, demoralizing. I don't get this. Differential equations theory is about proving existence and uniqueness of solutions. If you have to use numerical techniques to actually compute the solution, then that's perfectly f…

I don't get it, either. Bessel functions certainly do have engineering applications.

But are they useful now, other than as nomenclature? Bessel functions are defined as the solutions of Bessel's differential equation. It's all a bit circular. (there's the series expansion, but it doesn't gain you much)

30 years ago, if I wanted to plot the result of solving an equation like this, Bessel functions were useful as I'd just reach for Abramowitz&Stegun and look at the tabulated values. But now I have a computer, tabulated special functions don't matter nearly so much.

It's a long time since I had to use Bessel functions, so I could be very wrong, but this might be one of the reasons Rota said that.

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

#88

Post by nimish »

semi-extrinsic wrote: post #77
drjesusphd wrote: post #74

> I do not know how to properly motivate the Laplace transform I feel like this is impossible without going to the complex plane. Like the author said, taking the inverse Laplace transform is no joke. I feel like I never properly understood the Laplace transform until I learned about Landau damping. This is when waves exist, but are damped in a collisionless plasma. This damping is not disspiation and the energy does…

The Laplace transform is somehow the continuous analog of a Taylor series expansion. You don't need complex analysis to motivate it. I sketched this for my students when TAing once upon a time, heavily inspired by this [1] nice MIT lecture. I'm very surprised this isn't standard material. It makes the parallel between Laplace and Fourier transforms so much more intuitive, because you get Taylor series as a parallel t…

expand an analytic function in its taylor series then find its values on the unit circle. There's a Fourier series.

But the Fourier series uses global data than the taylor series which uses point data so they aren't perfect analogs.

A laplace transform is a fourier transform rotated in the complex plane (more or less), and if you allow the transform to take complex "frequencies" then they are basically unified. The difference is that the laplace transform is all about causal functions of time (t f(t) = 0) where as the fourier transform is less picky.

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

#89

Post by jimhefferon »

tomekowal wrote: post #64

Please, write another textbook. The internet changed drastically the process of writing books. I saw people making profit from books available online for free. I saw books written chapter by chapter with errors found quickly by first readers. I saw systems that allow commenting parts that are not clear enough with comments how to clarify them. If you promise to deliver a textbook that teaches skills relevant to engin…

> I don't know how mathematics textbooks are written and how much effort goes into them

They are written in many ways by many different people.

But some of us have started to write books that are Free, in the sense that software is Free. I have a couple and in addition to making the text and the source available I also sell one of them on Amazon and it does OK (see http://joshua.smcvt.edu/linearalgebra), because lots of people prefer a paper version when they really get down to studying.

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

#90

Post by [deleted] »

agf wrote: post #28
analog31 wrote: post #18

I once told a math teacher at a Big Ten university, that I thought their undergrad math instruction for engineers was weak. As an example, I said that I didn't think students learned any engineering applications of differential equations. He looked at me with a straight face and said: "There are no engineering applications of differential equations."

That is really, really hard to believe. I can't say that I remember any discussion of applications in my differential equations course (also at a Big Ten school), but I'm positive the professor could have provided them. I didn't have a particular need for examples in a course, as differential equations were held up as the holy grail of math by my father, an optical engineer -- he used them at work fairly often, and f…

[deleted]
Post reply on HN