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January 29, 2005

Linear Deficit

I’m teaching advanced mechanics, for juniors and seniors, this semester. And, once again, I’m driven bonkers by one of those educational peculiarities of UT (at least, I think it’s peculiar to UT).

I’m lecturing about damped, coupled harmonic oscillators (surely, the most basic problem in all of advanced mechanics), and I say, “OK, by introducing the velocities as independent dynamical variables, we can write Newton’s 2nd Law as a system of coupled 1st-order linear differential equations.” We can cast those equations in matrix form

(1)ddtV=BV \frac{d}{d t} V = B V


(2)V=(x 1 x n v 1 v n) V = \left(\array{x_1\\ \vdots\\ x_n \\ v_1\\ \vdots\\ v_n}\right)

and BB is a certain 2n×2n2n\times 2n matrix. The solution is

(3)V(t)=e BtV(0) V(t) = e^{B t} V(0)

and I then go on to explain that we can compute the exponential of a matrix if we know how to diagonalize it. Blank stares. And we can diagonalize BB, if we know its eigenvectors and eigenvalues. Uncomfortable rustling. “OK,” I ask this collection of junior and senior physics majors, “who knows how to find the eigenvectors and eigenvalues of a matrix?”

Slightly less than half the class raises their hands.


The problem, you see, is that, unlike most places, where linear algebra is bundled into 2nd-year calculus, the UT Math Department has unbundled it into a separate course. And, nowhere in the curriculum of the UT Physics Department is that course listed as a prerequisite. Thus students can hit my course or, G-d forbid, the Quantum Mechanics course, without so much as a passing acquaintance with linear algebra.

Having taught both courses before, I was prepared for this debacle. A brief review of finding the eigenvectors and eigenvalues of a matrix ensued. And then we exponentiated BtB t and found the general solution to the coupled harmonic oscillator problem.

Finally, we got to case where some of the eigenvalues of BB coincide. In this case (called “critical damping” in the case of the 1D harmonic oscillator), I explained that BB cannot be diagonalized. Instead, the best we can do (but still good enough for exponentiating it) is to put BB in Jordan Canonical Form.

“So, who knows what Jordan Canonical Form is?”

No one raised their hand …

Posted by distler at January 29, 2005 1:49 AM

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12 Comments & 1 Trackback

Re: Linear Deficit


I know the feeling. I am teaching basic mechanics to engineering students at the University of Ottawa and some of them never saw some concepts of linear algebra and calculus. In Ontario, the provincial Government chopped one year of the high school curriculum two years ago to save money… What a bad mistake! Students can get into science and engineering without a single high school physics course and without a full calculus course…



Posted by: Charles Nadeau on January 29, 2005 8:58 AM | Permalink | Reply to this

Re: Linear Deficit

I’m not that surprised that linear algebra is separate course. It is that way here at RIT, though they get a little bit in differential equations. What I find is that even if they have covered the “eigens” in a course you still get blank stares. There are even times when I hear the crickets chirping in a first-year physics course and I talk about calculus. It usually comes back to them as you use it in class, but I can’t count on using certain maths right out the box.

Posted by: Brian on January 29, 2005 9:11 AM | Permalink | Reply to this

Re: Linear Deficit

Frankly,I’ve never heard about linear algebra being bundled with calculus. In most science curicula, linear algebra is always taught as a separate course.

Posted by: didier on January 29, 2005 10:03 AM | Permalink | Reply to this

Re: Linear Deficit

I’ve never heard of linear algebra being taught in calculus either. Actually I was one of those people with a blank look on his face during my first quantum mechanics course as I hadn’t taken any linear algebra yet. I was quite frustrating to be so in over my head at first.

Posted by: allan on January 29, 2005 12:20 PM | Permalink | Reply to this

Re: Linear Deficit

For sure linear algebra is in the calculus sequence at Caltech, but I would bet that it’s separate in most American undergrad programs.

Posted by: CapitalistImperialistPig on January 29, 2005 12:59 PM | Permalink | Reply to this

Re: Linear Deficit

There seems to have been a general decline in educational standards over the years, in parallel with grade inflation.

I went to a public high school where the senior year math courses were linear algebra and calculus, along with a third course which was a hodge podge of topics like probability, statistics, combinatorics, and some linear programming. The linear algebra course was almost identical to the freshman university version, except it didn’t cover the proofs as extensively. The calculus course was almost identical to the freshman university version, except it didn’t cover the more formal things like epsilon-delta limit proofs. These courses were heavily computational more than anything else, where the teachers largely tested us on how fast we could do the calculations without making too many mistakes.

I guess at the high school level that’s about the most the teachers could do considering more than half of the students didn’t care about the subject, but had to take it because they wanted to go to university. Many universities seemed to require these two or three math courses for admissions, even for majors which used very little to no math.

When I took freshman university physics, I came to the realization that the high school physics courses I took were largely watered down versions. It was a bit of a shock for me at the time when I found that many of the freshman classical mechnics and electromagnetism problems required a lot of effort on my part. In comparison, the problems we were assigned in high school physics were relatively trivial and didn’t require a lot of heavy effort.

It could have been a fluke that I just happened to have tough math teachers and “easy” physics teachers, at the public high school I went to. Years later I spoke with several of my old high school teachers, where many of them thought that things were getting “watered down” over the years in the senior level courses. My old physics teachers mentioned that when they first started teaching high school physics in the 60’s or 70’s, the level of difficulty was almost at the same level as freshman university physics courses. Over the years things got watered down when more and more students were flunking the junior and senior level physics courses.

I would suspect that the “watering down” problem seems to have origins even before high school.

Posted by: JC on January 29, 2005 1:08 PM | Permalink | Reply to this

Watered Down?

I’m not sure that this is indicative of a general watering down of the curriculum, but they have drawn things out a bit here.

What is generally (at all of the institutions I’ve been affiliated with before) a 4-semester sequence (two semesters of “Differential and Integral Calculus”, followed by two semesters of “Multivariable Calculus and Linear Algebra”) has been replaced, at UT, by a 3-semester sequence and two à la carte one-semester courses, “Advanced Calculus” and “Linear Algebra”.

The former course is required for upper-division physics courses, but not the latter. Which puts a professor in one of those courses in the awkward position of having to give his students a crash course in linear algebra. Which further means that some unknown fraction of the students will already know the requisite linear algebra, and hence sit bored through your review, while others will be encountering it for the first time, and hence will be more than a bit bewildered.

Wasteful, inefficient, and has driven me nuts for years.

Posted by: Jacques Distler on January 29, 2005 2:21 PM | Permalink | PGP Sig | Reply to this

Re: Watered Down?

When I was an undergrad, the physics majors were required to take three calculus courses, two linear algebra courses, and a differential equations course before we were allowed to take any of the junior or senior level physics courses. (This rule was strictly enforced by the physics department).

Calculus I and II were largely single variable and some ordinary differential equations, while the calculus III was multivariable stuff and vector analysis. (Calculus III required linear algebra I as a prerequisite, which also was strictly enforced by the math department). In freshman year we were required to take calculus I and II, and linear algebra I, which were typically used as “weedout” courses by the math, science, and engineering departments. In sophomore year we were required to take linear algebra II, calculus III, and differential equations I. Linear algebra II was also used as a “weedout” course for some reason.

For some reason the physics department was very strict about folks having the proper prerequisites. They wouldn’t let us waive any of the prerequisites even if we had the knowledge from, say, self-study. This is what drove me nuts when I was an undergrad, since I was already studying and working out many quantum mechanics problems and even some basic quantum field theory on my own even before I finished my freshman undergrad year.

Posted by: JC on January 29, 2005 2:57 PM | Permalink | Reply to this

Re: Linear Deficit

Things are quite different in Canada. We don’t do linear algebra in High School at all. Calculus is available, but only as one of the “locally developed” courses, meaning that the Province and School Board take no responsibility for the content. Its all left to the teachers to develop, test, and deliver.

At university, there are two semisters of calculus in first year. In second year, we start with a semister introduction to multivariate calculus, followed by differential equations. Linear algebra is left to a different class entirely, often taught in parallel. The “advanced calculus” course in third year uses matrix and vector algebra with calculus for the first time.

Having just slogged my way through the third year calculus course, I can attest to the “memory drain” that happens over the summer. It was painful.

Posted by: michael on January 29, 2005 3:57 PM | Permalink | Reply to this

Re: Linear Deficit

Texing up some brief handouts on the math
methods you need to cover as they come up
–and getting straight to the point–might be the best way to get them up to speed, maybe with some problems or worked examples. I know its a pain doing this but usually if you want something done right or explained right it’s best to do it yourself.

Here in the UK there are complaints that each batch of new incoming science students to universities seem less well prepared with each passing year. Lecturers are usually having to devote a lot of time to explaining fundamental material that should already have been covered. It is not just in the sciences either. Some basic literacy skills are often missing. There is also a big problem here in that fewer and fewer students are actually taking courses in the hard sciences, opting instead for soft option ” vocational degrees” in things like “interior design” or “media studies”. (What the press has dubbed “Mickey Mouse” courses.) As a result some 90 physics and engineering departments have actually closed in recent years. A lot of chemistry departments are closing too. It’s a depressing scenario. Science courses should never be dumbed down though. Something is only worth pursuing if it is difficult.

Posted by: Steve M on January 30, 2005 12:33 PM | Permalink | Reply to this
Read the post Mathematical Background
Excerpt: As I was reading Jacques's post about UT's undergraduate math requirements for upper-division physics, I knew he would get at least one comment or trackback about the general watering down of standards, or some such. I was not disappointed.
Tracked: January 30, 2005 1:21 PM

Re: Linear Deficit

Fortunately as a math buff when I was younger I had most of those classes by the time I took the upper tier physics classes as an undergrad. Of course sometimes thats even more difficult, as one is used to different notation and handling of things. Often physics has its own peculariaties that takes getting used too, and quite a few hours of puzzling over things that are by name identical, but mathematically formally nothing to do with one another.

The one exception (and something that drives me nuts with undergrad curricula). I had little to no understanding of probability theory, outside maybe knowing what standard deviations were. When I took stat mech, that was one of the hardest and most confusing moments of my entire career. Hell learning parts of string theory and QFT was easier.

Functional analysis or at least probability and combinatorics should be requisites for any physicist. Its totally ridiculous that they are not prereqs for some of the upper tier physics classes.

Posted by: Frederic on January 31, 2005 3:39 PM | Permalink | Reply to this

Re: Linear Deficit

I think Math has really changed in Canada as they got rid of honor’s classes. When I was in High School in the late 80’s we did a little bit of linear algebgra. Nothing that advanced though. We also did a lot on series. In fact when I went to Dalhousie probably the first half of my calculus class I’d done in grade 12. Down here in the states though I’ve noticed that most people going into the sciences though have graduated with AP calculus and related courses that seemed a bit more advanced than what we did even in honors calculus in Nova Scotia. But my brother tells me they’ve definitely dumbed down all the classes up there the last decade or so.

Regarding pre-requisites, in my physics classes you nearly had completed a math major before you started taking upper division physics classes. I think the first half of mechanics (Lagrangian and perturbation theory) was taking concurrently with linear algebra and ODEs. The advanced mechanics was taught in the physics department and was basically more perturbation theory, PDEs, and doing lots of line integrals. We then had an optional class on tensor analysis which was basically just calculating lots of different change of coordinate systems to get an easy solution. However it definitely came in handy with QM which really isn’t that intense mathematically if you’ve encountered the basic kinds of math before.

I’m sure Cal Tech is a bit more advanced than what we did. But it does seem odd not to require linear algebra for an advanced mechanics class. However to be fair even our first advanced mechanics class had a bit that some people struggled with if they weren’t enrolled in linear algebra alongside it.

I do think that if one can assume more math background that it would enable Junior level physics to be taught a little better. I don’t see the problem with requiring most sophomores to be taking advanced mathematics - especially given the fact that most science students come in a sophomores due to AP classes. If they aren’t quite ready for it then they end up staying an extra year. But so what? I suspect at most universities it is hard for a physics major to graduate in four years.

Posted by: Clark Goble on January 31, 2005 8:52 PM | Permalink | Reply to this

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