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

math.toronto.edu

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

#51

Post by vlasev »

pkrumins wrote: post #6

One lesson academics should learn: pdf-naming-skills.pdf. I've been collecting interesting scientific papers and publications since early 2000 (I've a collection of 10,000 or so) and I've not yet seen a single academic, not even a computer scientist, who understands how to name your documents right so that when I download them I could quickly find them. I've to rename every single pdf. It's infuriating. Someone shoul…

I remember writing a little script in Python that used a pdf library to try to grab the title from inside the pdf and rename the file. It worked half of the time. I too have too many pdfs and it's annoying to have to rename every single one of them.

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

#53

Post by mattb314 »

vcarl wrote: post #47
batbomb wrote: post #41

Earlier quoted context omitted.

I'm not sure what that problem would have been, but an integral part (no pun intended) of DEs is expansions. It's not part of a bag of tricks, it's a very common technique used to solve a problem.

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...

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

#54

Post by tnecniv »

p4wnc6 wrote: post #11

I'm proud to report that at my undergrad institution, Rose-Hulman Institute of Technology, they very successfully adhered to these rules (and I was taking ODEs there way back in 2005). They had a custom textbook created for their 2-course ODE sequence that several of the faculty collaborated on. Though it did contain content on uniqueness theorems and some proofs, far and away the biggest two items hammered in were (…

> It took me another few semesters afterward to catch up on that part, but I can understand how engineering majors cared less about that.

It's definitely something engineers care about, but not at the undergraduate level.

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

#55

Post by tnecniv »

btilly wrote: post #25
graycat wrote: post #4

Earlier quoted context omitted.

Yes, and once students discover this, then good luck in getting students in the class! And, if regard the material on differential equations as essentially nonsense, then good luck getting NSF grants for research in the subject! Actually, can communicate a lot of good information in a course in differential equations, but to do this apparently need some exposure to some of the leading applications of differential equ…

The first course in differential equations is ordinary differential equations. Facility with that subject is needed before you can tackle more useful topics like control theory and partial differential equations. To a surprising extent, WHAT you learn about ODEs does not matter as much as developing enough familiarity with them that you can layer more complex stuff on top. That said the course I took focused on syste…

The Laplace transform was the bane of my existence as an EE. Not because it was difficult conceptually or mechanically for me, but because the problems I had on exams were such that I tended to make careless mistakes that only manifested after a page of work. I was lucky if I found them in time.

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

#56

Post by danidiaz »

Where can I find an expansion of the intuitive explanation given for integrating factors?

> It is of the utmost importance to explain the relation between the solutions of the differential equation and the solutions of the system. The solutions of the system are trajectories, they are parametric curves endowed with a velocity given by the vector field. The solutions of the corresponding differential equation are integral curves, and their graphs are the graphs of the trajectories deprived of velocity. Often, instead of solving the differential equation, it is more convenient to solve the corresponding autonomous system.

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

#57

Post by lamontcg »

mynegation wrote: post #27

My major is computational math, from 15 years ago, from leading Russian university, so it is just anecdata, and by no means should be generalized. I absolutely love mathematics, for me it is the embodiment of pure beauty. Still, I positively, absolutely hated the sophomore course of ODEs. The way it was taught was extremely abstract: here is the equation, this is integration, this is separation, this is your SLP, now…

Heh, all these years I thought it was just me.

Hated sophomore ODEs.

Much later took a mathematical physics course that actually taught me ODEs. Also took a year of applied mathematics that really solidified PDEs and taught me complex analysis. Took another graduate course in calculus of variations that was useful. Another very senior-level physics course taught me greens functions, method of steepest descent, etc.

The Physics and Applied Math profs were vastly better at teaching Math than the Math profs. Only real problem was in the freshman Physics courses where they taught you sloppy vector calculus before you'd seen that from the freshman calculus courses (and generally if you tried to pick up Math directly from the Physics courses that were teaching concepts it was all sloppy physics math -- it worked but you never quite understood why...)

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

#59

Post by gajomi »

I would be very curious to see if Gian Carlo Rota had anything to say about Stephen Strogatz's view on this. Strogatz's text (which was written three years before this article, right before he left MIT), is much beloved by many scientists and engineers, but most mathematician's will have complaints about it.

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

#60

Post by ataspinar »

gajomi wrote: post #59

I would be very curious to see if Gian Carlo Rota had anything to say about Stephen Strogatz's view on this. Strogatz's text (which was written three years before this article, right before he left MIT), is much beloved by many scientists and engineers, but most mathematician's will have complaints about it.

Do you have a link?
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