Wednesday, July 1, 2009

Chicken or Egg or ...: Which comes first? CalcIII, LinAlg, or DiffEq?

If I can point to a set of questions that are asked most often of the Math Department, one definitely on the list is the following:
Is it better to take Calculus III or Linear Algebra first?
Throw Differential Equations into the mix, and you get a branching of one's math career into three distinct paths. All three of these courses, at least here at Hopkins, have a full year of single variable calculus as a prerequisite; necessary for technique as well as theory in the case of multivariable calculus (Calculus III) and Differential Equations, and necessary for a sufficient level of "mathematical maturity" in the case of Linear Algebra.

But for many majors, and interests, one must take courses in and well understand two, if not all three, of these topics. So what order makes the most sense?

Its a good question. It turns out, it is not really important....

I am starting a new series about these and other courses under the tag and title "Beyond Calculus II". In this series, I will explain better the idea and focus of these three (and other) courses taken after a full year of calculus is achieved. Here at Hopkins, a large population of our students start their tenure here at this level.

For now, though, let's stick with the topic above. To start:
There is neither multivariable calculus nor differential equations in linear algebra, yet there is a bit of linear algebra in both of the others. In contrast, linear algebra is a more mature course, sometimes requiring more in the way of expanding one's frame of reference mathematically than the other two.
That said, we actually took a look at performance among students who took the two courses 110.202 Calculus III and 110.201 Linear Algebra back to back over a two year period (there are quite a few of them). I will pass on the details of this study, but we found that there was no real preferred order to these two, at least as far as ultimate grades go.

Couple this with the fact that any linear algebra found in either calculus III or differential equations is essentially covered within the courses, and any of the three may be taken in any real order. Hence preference for time slots, professors, and/or friends in the course may be of higher priority in your choice than content.

And one last note, our course 110.302 Differential Equations, fairly standard in content with most sophomore-level courses at American universities, is a course in ordinary differential equations (involving functions of one independent variable, in contrast with partial differential equations). One can easily describe this course as Calculus II.5 (weird notation, hih?). It can be viewed as the proper successor to Calculus II, rather than Calculus III. Just sayin....


Chris said...

From a purely mathematical standpoint it may not make much of a difference which order these classes are taken in, but the question is compounded in particular if one is not a math major.

Biology, natural sciences, and engineering majors should take a close look at their expected schedules to better prepare themselves mathematically for their fields of study; it is here that the order these math classes are taken in may make a more significant difference.

For example, an electrical engineer or budding physicist who intends on taking electromagnetic theory, should consider taking Calculus III (multivariable calculus) early on as professors in this subject will spend most of their lecture time teaching the requisite mathematics to actually understand the physics and work the problems. I recall that I somehow managed to take Calc III after E/M theory and discovered that I already knew most of the math and was simply adding on formalism to what I had already learned.

Students in circuits, control theory, or moving towards quantum mechanics will find themselves better prepared if they've taken linear algebra first.

Students interested in signal processing and communications may be better prepared with analysis, fourier theory, complex analysis, probability, random variables, and stocastic processes.

Mechanical engineering (statics, dynamics, thermodynamics) and biology students may find that they're better prepared by having taken differential equations before the others.

Computer scientists may be best served by skipping all of these three temporarily to take courses in number theory, combinatorics, and abstract algebra first.

Many students who aspire to areas of theoretical physics often find themselves poorly prepared mathematically, if at all, because they stopped at the level of Calculus III/ Linear Algebra/ Differential Equations. They should have continued on to areas of Analysis, Topology, Abstract Algebra, Lie Group Theory, and Differential Geometry.

My personal experience would suggest that students who can plot out the courses they'd like to take in the next two years do so and then ask their advisor(s) and the professors of those courses which mathematics courses they ought to take to best prepare themselves. My personal advice to engineers and scientists would be to adopt an unofficial advisor in the math department as well as your own official advisor.

One thing is definitely certain: regardless of the field, one should never quit learning more mathematics! Just because no one has yet tied an advanced area of mathematics to your field of specialization doesn't mean it won't ever happen; in fact, your undying fame may result because you were the one to do so.

The one thing the history of mathematics has continually shown us is that every branch is interwoven with all of the others. For who would have thought algebra and topology could have been so beautifully united until Poincare showed us the way?

Chris said...

If you'll indulge me to continue...

I've not read any specific studies on the subject, but by observing the top schools like MIT, Caltech, and even Hopkins who churn out most of the doctoral candidates who continue on to become professors and top notch researchers, the one thing they certainly all have in common is that they require a more rigorous mathematical background than most of their ("lower ranked") peers. It is this math preparation which better equips their graduates to do more advanced work with greater understanding and alacrity. Most students joke that Dr. Zucker is a bit crazy when he gives his CalcI rants, but in all honesty, he's very serious and knows that of which he speaks. Alas too many students don't realize this until long after their academic careers are at a close.

To go back for a moment to the case of physics preparation, I have recently been reading portions of Lee Smolin's text "The Trouble with Physics." Though I'm not aware that he mentions it as a specific cause, my personal best guess as to why theoretical physics has been stalled for the past quarter of a century as Smolin posits is that the level of mathematical sophistication to do physics has risen to the level of the state of the art of mathematics. Most major breakthroughs in physics for the last 2000 years were concurrent with or subsequent to major breakthroughs in mathematical theory. (Think about Newton/Calculus, Einstein w/ Relativity, Quantum Mechanics/Partial differential equations and group theory, etc.) Most physicists don't have the level of mathematical training they need to operate at a high enough or sophisticated enough level and rely on mathematicians who don't have the training to do the physics. As a result, we're generally stalled. I'd be curious to know the number of physics Ph.D.'s in the country who have actually had formal mathematical training in advanced real and complex analysis, topology, abstract algebra, advanced linear algebra, algebraic topology, Lie Groups and Representations, differential forms, manifold theory, or differential geometry? My guess is that other than some scant theory they pick up in their physics classes, it's not many. One can also note that although there have definitely been small advances within mathematics as a broad field, there have been no massive advances in theory or completely new branches of mathematics since the death of Poincare. Is this just a coincidence or is it causation? Is it also not telling that Hilbert and Poincare are the last two mathematicians about who it can generally be said that they knew and understood ALL of the highest branches of mathematics within their lifetimes? (For the uninitiated, they died in 1943 and 1912 respectively.)

Chris said...


I know it may not be popular amongst students to say this, and I say it as an engineer, but first and foremost be a mathematician and only then can you be a physicist, an electrical engineer, a mechanical engineer, a chemist, a biologist, or any other -logist you choose. It may be radical, but take an entire year after you've had some intro Physics, Chemsitry, Calc I & II and JUST take math. It shouldn't be too difficult in terms of prerequisites to take CalcIII, LinAlg, DiffEq, Topology, Analysis, Probability and Statistics, number theory, and combinatorics all simultaneously. If you intend to go to graduate school, it will be the best present you could give to yourself.

I'll also say something which I'm sure is not popular amongst the faculty either -- especially the engineering faculty. And that is: TREAT THE MATHEMATICS DEPARTMENT LIKE THE CROWN JEWEL OF THE UNIVERSITY! When you think of the math department do so with the utmost respect -- it's the house that J.J. Sylvester built! (And I seriously doubt 112 years after your passing that you will be as well remembered as he.) Don't think of the math department as the second-class group of faculty who service your whims and who you order to better prepare your students mathematically instead of spending their time pushing the boundaries of their own practice and advancing their own doctoral students. Instead, pick up the mantle yourselves and wear it proudly to lead your students where they need to go; offer to teach an introductory mathematics class yourself in addition to your engineering classes. Lightening the math department's load in this way will allow them to teach more students the more advanced topics you may not have the background for. As a result you, your students, the University, and the world will be all the better because of it.

Richard Brown said...

Chris, I absolutely cannot add a thing to your posts to increase their value (notwithstanding the fact that I am overseas at the monent, and the internet is not easy to find). When I return, I would like to showcase these posts in a diary of their own. Thank you very much for this great contribution.