Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
math.toronto.eduI couldn't agree with the last point more.
> A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives...
I hated the DE cleass I took in college and it was largely because I felt like it was nothing but a bag of tricks. I very distinctly remember one problem that seemed unsolvable until the teacher showed that you had to substitute a "2" with "1/2 + 3/2". And then, to make matters worse, he put the exact same problem on the test. So we were being rewarded, not for really understanding the core basic concepts, but for memorizing the tricks needed to solve specific problems.
I was working as a programmer at the NIH for a senior scientist who was an MD & math PhD. In the course of the work he was doing, we came across a differential equation that needed solving. My undergraduate math degree was not enough to crack it in a day. I tried Mathematica, which also choked on it (probably should have tried Mathematica first.) My boss spent a couple of hours and came up with no explicit solution. I was preparing to move forward with numerical approach, but my boss made a phone call to a friend who came into his office and copied the problem off the whiteboard.
Two days later he had an explicit solution based on two really non-obvious (bizarre) substitutions. I came away very, very impressed. The guy earned a 'we are indebted to ...' footnote in the paper.
I guess the point is someone has to come up with the tricks.
Nothing so quickly emphasizes that a bag of tricks can turn into actual math more than a linear algebra course. In a math degree you can literally take the same course twice. At a first year level you will learn to perform all these tricks with matrices and get a hint of inner product spaces. In 2nd or 3rd year you'll do the reverse and justify why matrix algebra works in the first place.
But the harder a subject, the longer it feels like learning a bag of tricks. The first partial differential equations course feels like you are working on only 3 problems for 4 months.
edit: I had a prof whose first DE course was at the graduate level. At the oral exam he was asked to give an example of a differential equation and all he could do was point to phi on the blackboard.
I'm thankful that my calculus courses were extremely heavy in showing how the tricks actually worked. It wasn't on the level of grad school analysis, but the professor pulled no punches and we went extremely in depth.
Years later when I was learning multi-variable calculus, I found most of it easy because, even having forgotten most of the tricks within those years, the method behind the madness was still there.
Illustrated: http://www.wastedtalent.ca/node/716
I don't know how correct this is http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ but it makes the whole pre-college math education looks like a bag of trick. Here principles are both concrete and abstracted yet it all blends as one.
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 think the problem is that not enough time is spent explaining these very common techniques - what the rationale behind them is, hot to intuit that it could be applied to a specific problem, and how to get from that intuition to the actual answer / application.
Often, there's a very large skill gap between the student and the teacher (especially at college level, where many of the career mathematicians live and breathe maths), and these things are hand-waved away as obvious. Even worse is when the teacher doesn't actually know, and is just presenting the material straight from a guide. The way most course material seems to be set up is to skip over the middle part, and as a result the best short term strategy is to learn it as a bag of tricks.
I love maths, but I think its universal applicability and beauty get lost due to the way it's generally taught.
You know that from study and experience. If that underlying concept wasn't the lesson, like the parent indicated, it would I deep end up a trick in the bag.
The toughest part of teaching, I think, must be really knowing that your students are internalizing the core rules/principles/concepts behind the examples you teach with.
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.
It would be great if that were how it was taught, but when I took the class it was taught as a trick. No theory behind it, just the prof on the board saying, "But look! :swap: And now you can integrate it."
It becomes a technique once you realise it is a specific case of change of bases. Even just getting to basic theorem of algebra does not provide this insight.
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.
I don't remember it being presented that way at all. If it really is a core technique, I would expect it to have been emphasized strongly and the test to have a problem with a different expansion instead of the exact same 2 -> 1/2 + 3/2 problem that he had done in class.
>I felt like it was nothing but a bag of tricks... had to substitute a "2" with "1/2 + 3/2".
Many techniques in math are "tricks" like this. Think of solving a quadratic by completing the square, or integrating by substitution or integrating by partial fractions, etc. You could arrive at these techniques on your own, but that is a lot of trial-and-error, deep understanding of theory, and applying it, which all takes a huge amount of time. Meanwhile, previous mathematicians figured this out and we get to benefit from their work. ;)
Maybe your instructor didn't present it well - plopping out the answer without a good enough explanation of the technique, why it works, etc.
The implication seems to be that students can't do the trial-and-error, understand then apply the theory. I agree with parent comment that this is awfully defeatist.
But even if it is futile to teach true understanding, why are we knowingly teaching computation in its place? What if we left the computation to the computers from the get-go? Could we then have enough time to teach true understanding?
I can't tell you how much time I spent trying to memorize my multiplication table – a 12x12 grid of numbers that for most students became arbitrary 3-number sequences. "3, 3, 9" is different from "3 sets of 3 is equal to 9." Most students learn the former in place of the latter.
>The implication seems to be that students can't do the trial-and-error, understand then apply the theory. I agree with parent comment that this is awfully defeatist.
All teaching is a balance between various factors, and perhaps the instructor didn't get right in this specific instance.
Ideally the instructor would teach the theory, allow some trial-and-error as students grapple with new information, and then step in with prodding towards how to do it - but fundamentally, there is a limit on how the instructor can let the class wander without needing to move on to cover the rest of the material (this wasn't a special topics class covering how to solve this one specific problem).
Hence, go through the information but provide the technique involved after suitable time passes. That's just the nature of teaching/learning when the students don't have infinite time to essentially re-invent the material they are trying to learn.
Yes, trial and error isn't viable at all.
But Google is a better teacher of that kind of stuff than any professor can claim to be. If this is the kind of knowledge you pass on during education, it's basically worthless.
I had an almost identical experience with my ODE class. This is exactly when I started hating math, and moved out of computer science (which was part of the math department at the university I was at).
It took me a long time to return to learning higher maths, this time on my own according to the needs of my job, which is far less an ideal environment than University.
If ODE contains "a few basic concepts that they will remember for the rest of their lives" I never learned it, and I wonder what those concepts would be.
Indeed, Rota's approach to teaching seems to mirror the application of mathematics in general...
He acknowledges that concepts have prerequisites, and that certain topics are misleading or dead ends. You might even call the act of building a pedagogical scope and sequence for a topic graph traversal!
I wish I could have taken one of his classes. I wonder if there are other mathemetician/philosophers out there who have taken the problem of teaching as seriously as the math itself.
you just made me angry at all my college math classes again. grr.
did anyone have an undergrad math education that was NOT like this?
on the upside it made some people feel very smart
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 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.
I took, we called it Diff-e-q, my freshman year in college and the professor died a couple weeks into the course and the new guy was not pleased about having to teach it. We did the applied aspects with the springs and the little ants running away from a candle heating the corner of a plate.
What I didn't like about it was just all the memorization. I had no desire to memorize a bunch of formulas that I knew full well in the real world I'd look up in a table or type into a computer. What I wanted to learn how to do was solve problems using math, not memorize patterns of formulas to apply to problems.
So I didn't memorize them and instead went to work and earned money to pay for college. Still passed the class but it was one of my lowest grades. It's a hard class even without the memorization.
It's a little spooky that the author, Gian-Carlo Rota, passed away the night before teaching a class.
That is spooky. Both from heart problems too. Maybe the stress of teaching the class takes a toll. Heart attacks are pretty common among men of that age, though.
Coincidence, of course. Spooky. For sure.
Strongly agree. The way ODEs are taught at the sophomore level violates the beauty of math by teaching a plug-and-pray method of solution. For this form of equation, try this form of solution, if it doesn't work, try this one, then this. If none work, oops. At least once the Laplace method is taught things get a little better.
While I didn't exactly enjoy my course taught that way, as a human plugging in those methods manually, it was sort of interesting from the perspective of later being a fairly heavy user of computer algebra systems (CASs). The bag-of-tricks approach they teach in school is really how such systems work in practice, and tons of practical problems will be solved that way, either with you doing it by hand, or using software that does it for you. Software like Maxima, Maple, Mathematica, Sage, etc. consists of a huge pile of case analysis techniques and methods that pattern-match on specific equation forms that can be solved with the method in question. A CAS does feel a bit more satisfying because it feels like the pile of techniques is at least being given some kind of formalization and order, versus me just trying to remember them. Although the amount of order is not quite as much as one might like; even using a CAS there's still a lot poring over documentation to find the function that works in your case, which will go a bit faster if you remember enough of the textbook methods to recognize what you're looking for.
That's not to say this is anything like what mathematicians do, especially PhD mathematics researchers. But a lot of applied mathematics in engineering is not that far off from what's taught in a university ODE class.
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...)
I wonder if there are people that really click and appreciate purely abstract definitions of mathematical concepts. Many books and courses (especially in Europe it seems, watching U.S.A classes, Strang's for instance, feel a lot more pragmatic and applied). Sure after a few years of confusion one might finally click and realize all that was hidden behind a lemma, an identity or a symbol, and then that definition goes from painful to beautiful.
I hated ODEs until I took MIT's robotics course and had to solve non-linear ones using non-linear optimization techniques. Since then I love them ;-)
This has to be Russell Tedrake's Underactuated Robotics class right? Love that course, there's an online version now if I remember correctly.
In the OP, the author Gian-Carlo Rota started out with:
> One of many mistakes of my youth was writing a textbook in ordinary differential equations. It set me back several years in my career in mathematics. However, it had a redeeming feature: it led me to realize that I had no idea what a differential equation is.
Wow! Good to see that he wrote this. Looking at his book,
Garrett Birkhoff and Gian-Carlo Rota, Ordinary Differential Equations, Ginn and Company, Boston, 1962.
I got the same impression! I couldn't see what the heck they were driving at. Instead, they seemed to flit around with a lot of tiny topics of little or no interest for little or no reason.
Want to understand ordinary differential equations, read Coddington:
Earl A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1961.
Then for more, to make such equations much more important, read some deterministic optimal control theory, e.g., Athans and Falb
Michael Athans and Peter L. Falb, Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill Book Company, New York,
that, BTW, also has some good introductory, but very useful, material on ordinary differential equations.
More generally, want to know what to study in a subject that will be useful? Okay, one approach is to go to more advanced material that is an application of that subject and see what that material emphasizes for prerequisites, e.g., sometimes quite clear in an appendix.
E.g., Athans and Falb say quite clearly what is important in ordinary differential equations for their work.
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 engineers, they will fund the time it takes to write the book. I would spare couple of dollars even if you said that it will take 5 years. I believe, there will be even companies that will give you funds upfront. If you reach out for help there will be people who will help you collect example problems from different fields to replace couple of "salt tank problems".
I am not a mathematician, so I don't know how mathematics textbooks are written and how much effort goes into them, so feel free to point out that this idea is stupid.
Rota died almost 17 years ago, I'm sorry.
> 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.
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 (a) linear equations with constant coefficients, and (b) Laplace transform methods.
They also offered (at the time) a 3rd, optional course called Boundary Value Problems that was focused on several physics-motivated BVPs like with Laplace's equation, heat equation, wave equation, Young's modulus, and others, and that course heavily used Fourier and Laplace methods.
We did have word problems, but they were almost exclusively "salt tank" problems. Literally, every word problem described a tank of water or pre-mixed brine solution, with some description of either more salt or more water being added or removed, either gradually or in discrete injections.
The fact that every problem was an infamous "salt-tank problem" essentially made its status as a word problem irrelevant. This seems like it wouldn't be that helpful but actually it was really nice. You got so used to the different pieces that comprised the modeling problem that when you went off and did something in other courses, like circuit systems or conservation systems in mechanical engineering, you knew how to translate the problem to 'salt tank' form, which really covered a huge range of practical problems.
As a math major, one fault I noticed of this method was that it did not make the connections to linear algebra very clear. It took me another few semesters afterward to catch up on that part, but I can understand how engineering majors cared less about that.
I don't know what Rose-Hulman does for this curriculum now, but it would be cool to somehow take a "snapshot" of their methods for it and compare it with other experiences like this OP.
> 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.
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."
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.
Same with my EE undergrad. Then I got to grad school and decided to take a course in Linear Systems, which is when I realized my ODEs course taught me nothing.
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.
I took the differential equation course over a summer, which means an entire semester was condensed into four weeks. I passed the class but I couldn't tell you a single application for differential equations at the time.
Don't know if the professor could've done anything better or if they had no choice but to plow through it because of the time constraints.
"He looked at me with a straight face and said: "There are no engineering applications of differential equations."
This made me laugh; I think he wanted you to laugh too.
Some DE courses could be better described as _histories_ of the applications of mathematics. Consequently they appear to be little more than a patchwork of tricks and hints. My DE professor, as adept with applications as theory, weaved this patchwork together quite skillfully and held our interest.
BTW his DE skills paid his bills extremely well (oil companies have lots of DEs to solve), while the university position allowed time for theory and was frosting on the cake.
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.
> 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 not get converted into heat. The usual way of presenting this is to show that if one Fourier transforms in time, you get the wrong answer. The fact that the system begins at a certain state, and is thus an initial value problem, needs to be respected.
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 to Fourier series.
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<0 => f(t) = 0) where as the fourier transform is less picky.
"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 equations.
> 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=1408295&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1318480&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1505215&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1418042&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1346876&Hist...) and of course there is much more cash going into PDEs.
Nice list.
I saw emphasis on control, optimization, numerical methods, and applications. Nice.
And, sure, PDE's stand to get more.
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!)
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.
I'm sorry, but why are all the "you"s missing from your comment?
In my opinion, the first course on ODEs shouldn't be concerned with the transmission of information.
That is where the wave equation comes in, in the first course on PDEs.
> 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 fine. After all, even if the solution is explicit, like sin(x), or especially a special function, then we still need to use numerical techniques to actually evaluate that explicit solution.
The article is not questioning the "theory", the identified problem is the teaching at undergraduate level. As it was/is commonly taught, it is neither pure nor applied. Rather it is categorisations and tricks, little of which has any practical value.
As a body of work Differential equations are so messy that theorists landed on so many disparate results. As such Differential equations courses are commonly taught as a "survey of the land" type of courses, so they tend to be incoherent. On the other hand if the teaching focused on practicality there is a lot of commonality among the practical cases.
I don't get it, either. Bessel functions certainly do have engineering applications.
I recently saw them in a grad eng class, but I agree with the article - from what I saw there is no need to give them the math professor treatment. You can use them as a piece of trivia - ie pde of type x has this set of basis functions - now apply the principles of basis functions to solve your your problem.
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.
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 should teach academics an SEO course.
Even worse: instead of having a discussion on college mathematics pedagogy and differential equations, you decided this was the most relevant thing to discuss, made even more totally irrelevant by the fact that 1) it's a futile suggestion, 2) the original author is dead, and 3) you haven't heard of full-text search or file renaming.
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.
What's wrong with any of the pieces of reference management software out there? I use Endnote, others may have different preferences. Reference management and document retrieval is a problem that popped up together with the first library, and library classification has evolved alongside. I wouldn't say it's a solved problem, but it's a known problem with known attempts at solution. Attempting to roll your own is so reminiscent of OpenSSL, and they keep getting into trouble with their habit of inventing the wheel.
Everybody has a different system, and why should they standardize on yours?
What kind of a system is surname.pdf, New-Pub_New2new.pdf, USENIX_.PDF, and Paper.pdf.
I like to say: Show me how you organize your files, and I'll tell you how good of a computer user you are.
How good of a computer user do you have to be to sort by modified date? :P
Lots of things can touch modification date.
It's ironic, because professors and instructors often ask their students to name their research papers, essays, and projects with a well-defined, searchable naming scheme.
surname.pdf is almost certainly a half-hearted (because that is all that is possible) way to fit into someone else's naming scheme. Otherwise, it would be cv.pdf, application.pdf, etc. because those are the file names that make sense on my computer.
What you don't see is the directory structure that that file lives in, which gives almost all the necessary information.
~/work/projects/reinventwheel/paper/manuscript.pdf should and does contain more information than "manuscript.pdf". If you're not renaming files when you get them and putting them in the correct context on your machine, I would say it is you that is doing it wrong.
I also got tired of organizing papers and websites (various articles I print to pdf), I just throw it all into Mendeley Desktop now and apply my own tags and so some minor modifications on author/title. I go with the "last name - year - title" naming and this works well enough for me.
I have an order of magnitude fewer docs that you; I can't imagine hand organizing that many!
you're welcome :)
http://pgbovine.net/publications.htm
this is something i try to do for all my papers, although of course people will still need to manually re-name to fit their own conventions. but at least it's better than Guo.pdf
The academic equivalent of the dreaded "resume.pdf".
When you download you have the option to pick a name (and not use the data provided by content-disposition)
The last point may well be the best, "TEACH CONCEPTS, NOT TRICKS"
I am taking a refresher class in calculus II from a community college and am pained by the tricks being taught and feel that the students are being deprived of learning core concepts. I am taking the trouble to read up and not too focused on the solution techniques that the professor emphasizes. However this doesn't do too much for grades. I wish he focused on concepts and application.
But the bag of tricks is a feature of US undergrad education seemingly everywhere, except perhaps in pockets at the very highest level. The tricks permit the student to pass the test so they can go on to do something else. Whoever sits in a chemistry course isn't there for the chemistry, they are mostly there to go up to medical school or allied health science. Of course there are entrance exams.
We have stopped putting things into context, i.e. we do not provide an education any longer. The sideswipe remark in the original paper about Prof. Neanderthaler is also very real.
He was the author of Indiscrete Thoughts[1], a great book on Mathematics.
[1] Review here: http://www.maa.org/publications/maa-reviews/indiscrete-thoug...
Could someone give some examples of applications of numerical methods for solving differential equations that are relevant to a HN crowd? Also, where might I find some introductory material that teaches it well, according to the the suggestions in the OP?
I'm not sure what the HN crowd finds useful as a group, but I personally use a couple of different techniques in my work (geophysics). One of them is finite difference and finite element methods. I have a book called 'Introduction to Numerical Geodynamic Modeling' by Teras Gerya that teaches finite difference modeling of plate tectonic phenomena, particularly of 2nd order differential equations (Poisson equation and variants) that are useful in modeling heat flow, diffusion and so forth through space and time. It's a great book but written for a specialized audience. I've used finite element models a lot, and the occasional boundary element method, but never written any.
I also use Green's functions, which are equations that describe the response of a medium to an impulse (think the propagation of sound waves from a source, though I do different stuff), by using convolution.
But I think jofer is the only other geophysicist on HN so we're probably not representative. Nonetheless, a lot of HNers have a physics, classical engineering or chemistry background and use similar tools... just not to find out what happened tens of millions of years ago.
http://longnow.org/essays/richard-feynman-connection-machine...
"By the end of that summer of 1983, Richard had completed his analysis of the behavior of the router, and much to our surprise and amusement, he presented his answer in the form of a set of partial differential equations. To a physicist this may seem natural, but to a computer designer, treating a set of boolean circuits as a continuous, differentiable system is a bit strange. Feynman's router equations were in terms of variables representing continuous quantities such as "the average number of 1 bits in a message address." I was much more accustomed to seeing analysis in terms of inductive proof and case analysis than taking the derivative of "the number of 1's" with respect to time. Our discrete analysis said we needed seven buffers per chip; Feynman's equations suggested that we only needed five. We decided to play it safe and ignore Feynman.
The decision to ignore Feynman's analysis was made in September, but by next spring we were up against a wall. The chips that we had designed were slightly too big to manufacture and the only way to solve the problem was to cut the number of buffers per chip back to five. Since Feynman's equations claimed we could do this safely, his unconventional methods of analysis started looking better and better to us. We decided to go ahead and make the chips with the smaller number of buffers.
Fortunately, he was right. When we put together the chips the machine worked. The first program run on the machine in April of 1985 was Conway's game of Life."
https://www.udacity.com/course/differential-equations-in-act...
I'm sure there are others, but this is a pretty good introduction. It focuses quite a bit on numerical solutions (using python programs that are automatically graded.)
Does anyone else get the urge to spend the next few hours (or days) trying to work out (at least the basics of) everything Rota is talking about here?
Mathematicians have so much fun..
Although I agree that word problems can be somewhat distracting in a DE course, I do not think they should be totally avoided. One of the main reasons it was difficult to grasp the concepts we studied in our first DE courses is that they were too abstract. When later we studied other subjects, such as electric circuits or fluid dynamics, everything started making much more sense.
In my opinion, the ideal way of learning would be to first have very basic (only conceptual) introductory courses of applied fields, where we find some basic equations that we do not know how to solve. And then, we study DE to learn the techniques to solve these problems, avoiding direct references but keeping in mind where we are going with all this.
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!
I got an A- in my differential equations class in college. I still wasn't sure what a differential equations was at the end. My pattern matching skills got a good workout, though.
I had a similar experience with similar grade and outcome. I finally started understanding things when I took PDEs and so on.
Can anybody comment on that? (page 8 paragraph 1 ):
Professional mathematicians have avoided facing up to density functions by a variety of escapes, such as Stieltjes integrals, measures, etc. But the fact is that the current notation for density functions in physics and engineering is provably superior, and we had better face up to it squarely
In physics and engineering, you traditionally talk about things like "delta functions", write expressions involving them as if they are actual functions, etc. This is notationally very convenient but may be misleading because these things are not really functions.
So, what are they really? Well, the key things you can do with them are (1) "boring" linear algebra operations (you can add and subtract them, and multiply them by scalars) and (2) multiplying by some function and taking the integral. E.g., what delta(x) -- the Dirac delta function -- really is, is a thing such that when you compute integral f(x) delta(x) dx, you get f(0).
And so pure mathematicians have ways of dealing with them that make this property more explicit. The theory of distributions says: no, these aren't functions, they're linear functionals on the space of functions (e.g., the delta function is the thing that maps f to f(0)). So now you're no longer allowed to write them as integrals, which means that the very close analogy between "distributions" and ordinary functions is obscured, and e.g. if you need to do a change of variables you can no longer just do it the same way you already know about from doing integrals.
Alternatively, the theory of signed measures says: no, these aren't functions, they're kinda like probability distributions except that the total "weight" doesn't need to be 1 and the density can be negative in places. They are naturally applied not to points but to sets of points. (E.g., the delta function is the signed measure that gives a measure of 1 to any set including 0 and a measure of 0 to any other set.) Now you are allowed to write those integrals, but instead of writing integral f(x) delta(x) dx you need to write integral f(x) dH(x) where H(x) is the "Heaviside step function", so instead of delta(x) appearing there you have (morally) its integral, and again if you want to change variables or something you need to know a new set of rules for what you do to the measure.
Note: I have skated over some technicalities. They are quite important technicalities. Sorry about that.
The sloppy non-rigorous physicists' and engineers' notation, where you just pretend the damn thing is a function and manipulate it as you would any other function, is more convenient. (Right up to the point where you do some manipulation that is safe for actual functions but gives nonsense when applied to singular things like delta functions, and get the wrong answer.)
It's a little like calculus notation. The "Leibniz" notation we all use these days writes derivatives as dy/dx as if dx and dy were just small numbers (compare: we write integrals against distributions as integral f(x) delta(x) dx as if delta were just a function), which is kinda nonsensical if you take it too seriously but very convenient because it makes things like dz/dy dy/dx = dz/dx "obvious", which is not just coincidence but has something to do with the fact that derivatives really are kinda like quotients (in fact, they are limits of quotients). Similarly, using "function" notation for distributions lets you write things like "integral f(x) delta(x-3) dx" and see that "of course" that's f(3), and this convenience isn't mere coincidence but has something to do with the fact that distributions really are kinda like functions (and in fact every distribution "is" a limit of functions).
Newton had a different notation for derivatives. It didn't have a conceptual error baked into it (pretending that derivatives just are quotients), but it turns out that that's a useful conceptual error and that's part of why everyone uses Leibniz's notation these days.
+1 to teaching concepts - but I don't agree with the author's opposition to word problems and learning to apply a "bag of tricks". I'd argue that "deciphering vaguely-phrased word problems and figuring out which of a selection of tricks to apply" is the MOST transferrable skill somebody can take away from a math class, because it's a major component of working in lots of other fields. For instance, the biggest difficulties I've observed in novice developers are in breaking apart a big challenge ("write a program that solves this Sudoku board") into digestible / implementable pieces and in understanding which piece of information they already know can get the result they want.
I agree, but I think the "word problems" the author was referring to are much lower quality than the ones you're thinking of. I imagined some highly-contrived exercises where all of the relevant information is already pre-processed for you, removing any need for problem decomposition. For example, "if the angle between the ground and a tree's shadow is 45 degrees and a 50 ft tall telephone pole that's 10 ft away from the tree casts a shadow..." (substitute a similar differential equation problem). As you point out, half the fun is defining a problem and breaking it down, and these kinds of word problems don't give you a chance to do that.
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.
Read Arnold, read Arnold, he is the master of us all.
Here's an attempted expansion.
If you have an equation of the form dy/dx = f(x) and you want "solve" it, what you are typically looking for is to write y = g(x), right? In other words, the solution to this differential equation is some curve in the x-y plane. This applies more generally, e.g. to situations where you end up with an "implicit" solution like h(y) = g(x): you still get an equation relating x and y which can then be represented as some set of points in the x-y plane (the ones that satisfy that equation).
Now say f(x) happens to have the form a(x,y)/b(x,y). You can consider the system of two differential equations: dy/dt = a(x,y), dx/dt = b(x,y). Solving this system gives x and y as functions of t. Picking any particular value of t gives values of x and y, which gives you a point in the x-y plane. The first key point is that the set of points produced by this procedure as you plug in all possible values of t is exactly the set of points for which the h(y) = g(x) equation above holds. In other words, the solution to the two-equation system encapsulates all the information about the solution to the original equation.
The second key point is that the solution to the two-equation system has _more_ information than the solution to the original equation. In particular, it has the actual values of dx/dt and dy/dt for every given value of t, which don't correspond to anything in our original problem. Their _ratio_ does correspond to something in our original problem: the slope of the tangent line to the solution curve (dy/dx). But the exact values themselves are somewhat arbitrary, as long as their ratio is correct. Put another way, our original problem's solution is a curve in the x-y plane, while the solution of our two-equation system is a curve together with a description for how fast to move along it as t changes. That's the "velocity" bit in Rota's article.
OK, but if how fast we move along the curve doesn't really matter, maybe we can choose to move along it in a nice way that makes it particularly simple to figure out what the shape of the curve is. Our only constraint is that at any given point along the curve the ratio of dx/dt and dy/dt is fixed, because in our original problem we have a fixed dy/dx if we're given values of x and y. So if, at every point (x,y) we multiply dx/dt and dy/dt by the same number (which can depend on x and y) then we get a system of two equations that has different solutions for x and y as functions of t, but the graph of the resulting thing in the x-y plane still looks the same. That's the integrating factor bit; we just formalize it by saying that we multiply both dx/dt and dy/dt by the same function q(x,t), which is exactly what it means to multiply them both at every point by some number that might depend on that point.
The hard part, of course, is choosing a q(x,y) that makes things work out nicely and makes it easy to solve our two equations to get x(t) and y(t).
Here's a concrete example that might help:
Say dy/dx = x/y. We rewrite this in the form dy/dt = x, dx/dt = y. This isn't terribly convenient to solve, so we multiply by q(x,y) = 1/(2xy) to get a new system: dy/dt = 1/(2y), dx/dt = 1/(2x). At this point, maybe you just look at it and go, ah, y = sqrt(t + C1), x = sqrt(t + C2), or maybe you figure out some other way to get there. In any case, now you see that t + C1 = y^2, t + C2 = x^2, so x^2 - y^2 = C for some constant (C2-C1, but both are arbitrary, so this is just some single arbitrary constant). And that's your (implicit) solution for the original differential equation: a hyperbola, or more precisely a family of hyperbolas each of which satisfies the equation.
To illustrate the point about velocities, let's just consider C = 1, so x^2 - y^2 = 1. The point (sqrt(2), 1) lies on this curve. At this point, dy/dx = x/y = sqrt(2). On our original formulation of the parametric system, dy/dt = sqrt(2), dx/dt = 1 at this point. In our reformulation with the integrating factor, dy/dt = 1/2 and dx/dt = 1/(2*sqrt(2)). So the two formulations have us moving along the hyperbola at different speeds at this point as t changes, but they're moving along the same hyperbola.
Does that help at all?
It helped, thanks!
Wow. I managed to skip Diff-e-q in my C.S. curriculum, and I always wondered if was the worse for it. Perhaps I was prescient.
I somehow managed to get a bachelor's in mathematics without ever taking a DE class (snuck through on a year when program requirements were being rewritten). Everyone I've ever told this to has been aghast, and yet when I ask what I missed out on, no one really has a response other than "I thought everyone had to take it".
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?
I believe he's referring to Strogatz's introductory textbook "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering".
Amazon link: http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-...
"FORGET ABOUT EXISTENCE AND UNIQUENESS OF SOLUTIONS" what? most important thing about differential equations.
As an engineer, I promise you, I give not one single solitary conscious fuck about the uniqueness of solutions to differential equations. Mostly I just hit things with Fourier or Laplace transforms as appropriate until they stop moving.
As an engineer working in some optimization problems where is utterly important if a solution is a local or a global minimum, I assure you I give many fucks about the uniqueness of solutions.
I totally agree with your way of working for most applications, it is what I do most of the time too, and I agree with the article in that this point is given more importance in DE courses than it really is for engineers, but there are perfectly valid use cases for these theorems in engineering.
I'm glad someone got something out of that section of the course besides a bad grade and the lingering suspicion that if it had been taught out of the engineering school it would have made more sense.
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.
There is definitely pedagogical value. This issue is covered by the author, but when studying ODEs for the first time, at some point the student will come across the fact that exp(x) is a solution to y' = y, which is easy enough, but needs to be convinced that exp(x) is the solution up to linear combinations. To most students this is not at all obvious! Lack of explanation here is doing the student a disservice.
Very late to the conversation, but I emailed this to my uncle and he wrote back:
At ANSTO I worked with solution of simultaneous first order differential equations, as arose from Newtons law of cooling for a 4-body calorimeter. That was fun.
I 100% believe DE 1 and 2 are courses used to weed out computer science students who don't meet a certain criteria. Whether this is good or bad is highly debatable imo.
a = F/m
Want motivation? Just saying.
It seems there are lots of posts complaining about how ODE's are taught here. I am planning to study them in the next months. I can only learn maths through applied mathematics and always need to know the why before I can get the how. Can anyone please point me in the right direction for online materials which will help me self-learn, taking into account my learning preferences?
I would say skip studying "pure ODEs" then examine what you need specifically depending on your area
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.
> As a matter of fact, the need for proving existence theorems was not felt until the end of the nineteenth century, and I refuse to believe that someone like Cauchy or Riemann did not think of them. More probably, they thought about the possibility of proving existence theorems, but they rejected it as inferior mathematics.
...
> Most often, some student will retort with the dreaded question: “So what?”
Insecure snob.