"FORGET ABOUT EXISTENCE AND UNIQUENESS OF SOLUTIONS" what? most important thing about differential equations.
Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
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Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#22I 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."
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Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#23> 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.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#24Earlier 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…
> And, if regard the material on differential equations as essentially nonsense, then good luck getting NSF grants for research in the subject Luckily, in this world, NSF grant writers are not typically undergraduate students, and there's plenty of money flowing towards ODE research (here are some examples: http://www.nsf.gov/awardsearch/showAward?AWD_ID=1600381&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID…
Nice list.
I saw emphasis on control, optimization, numerical methods, and applications. Nice.
And, sure, PDE's stand to get more.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#25"We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations"
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 systems of differential equations rather than second order differential equations. There is nothing like trying to do Laplace transforms of matrices of functions to demonstrate how important it is to avoid careless errors...
(On one memorable occasion I tried to solve the same problem 12 times and came up with 11 different answers - none of which were correct!)
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#26"We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations"
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…
I'm sorry, but why are all the "you"s missing from your comment?
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#27My 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 go deal with it.
It was totally pointless and life-sucking. It was not until I got to the 3rd year and learned about specific applications in physics (like heat dissipation, strings, and springs), and later in finance (stochastic calculus) and biology (e.g. Lotka-Volterra) when I realized how many wonderful and extremely useful applications they have.
Have this course started with that, things would be completely different.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#28I 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 frequently they were what made him better able to solve a problem than an engineer who wasn't comfortable using them.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#29I 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."
My undergrad was at a Big Ten university. The analog signal processing course in EE teaches how to use differential equations to solve circuits. Right after they taught it, we learned laplace transform and never looked at a differential equation again.
Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
#30"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.