# Would You Like Some Calculus With Your Physics?

It’s a nice demonstration of the oddity of the blogosphere that a libertarian political blog has become my go-to-source for thoughtful blogging about physics education. Thoreau had two good posts yesterday at Unqualified Offerings, one on the problems created by breaking down incorrect intuition, and another on the lack of calculus in calculus-based physics texts:

The ostensibly calculus-based introductory physics book by Knight is not really a calculus-based book. Sure, integrals and derivatives pop up here and there, but the vast majority of the problems can be solved without them, and calculus is hardly emphasized at all in most of the text and examples. The few problems that do use calculus are generally the hard ones near the end of the problem set, and with very little in the text to prepare them for these problems it’s hard to assign them.

[…]This has been in the back of my mind for a while, but I was able to cope with it because, well, it’s just freshman stuff. But next year I’m supposed to teach the upper division classical mechanics course, and I’m realizing that my students will not have had a truly calculus-based freshman mechanics course, so all of the stuff that I’d like to do must instead be put off until I first redo mechanics (in abbreviated form, of course) with calculus. This does not make me happy.

I’ve noticed much the same thing. In fact, one of my biggest reservations about the Matter and Interactions curriculum is that it has, if anything, even less calculus than the previous intro text we were using.

That may be a little unfair, actually– it has calculus, but in many ways, it’s stealth calculus. The whole text is built around a very computational approach to physics, with lots of time spent on solution methods involving the iterative updating of physical properties over small steps in position or time. These are essentially numerical integrations of the equations of motion, and the book does explicitly say that in several places. But there are essentially no problems involving applied calculus, and all the summary formulae are presented in update form, so I fear that the take-home message is that physics is really algebra-based.

Thoreau’s comment about preparation for upper-level classes is a worry, as well. I’m going to be teaching quantum optics again in the fall (assuming I can cajole enough students into signing up, anyway), and one issue I’ll have to contend with is that a good number of the students will never have seen Maxwell’s equations in differential form. Which makes it a little difficult to get to the wave equation, and set up the necessary background information about the classical model of light as an EM wave.

This is a tough problem, though. There are a lot of problems with trying to make the introductory courses more mathematical, starting with the preparation of our students, many of whom don’t have all that solid a grip on algebra. We list calculus as a co-requisite for intro physics, but the computer system used to handle course registration does not check or enforce prerequisites in any useful way, so we get students in the class who aren’t comfortable with derivatives, let alone differential equations.

On some level, I sort of feel like we should make our students suck it up and deal with the math– after all, the second course in the physics major when I was an undergrad was E&M out of Purcell’s book, and it doesn’t get more mathematical than that. On the other hand, though, I got basically nothing out of that class, and had to re-learn E&M more or less from scratch my junior year. And I was crazy enough to go on to grad school– for the typical wannabe engineer, Purcell would be a slow agonizing death by vector calculus.

Anyway, if anyone knows a foolproof solution for these issues, leave a comment or send me an email, because I’d love to know what to do.

1. #1 Eric Lund
May 8, 2009

[O]ne issue I’ll have to contend with is that a good number of the students will never have seen Maxwell’s equations in differential form. Which makes it a little difficult to get to the wave equation, and set up the necessary background information about the classical model of light as an EM wave.

I know it is possible to get a wave equation out of the integral form of Maxwell’s equations. The professor whose freshman E&M class I TA’ed in grad school wrote up some notes for his class that show the derivation. (I don’t know if I still have my copy of those notes; I haven’t looked for them in a long time.) The problem is that the derivation takes five pages, as opposed to the five lines it takes to derive the wave equation from the differential form.

I also took freshman E&M with Purcell as the textbook, and I second your thought that this is not a viable solution to the problem. In addition to the issues you mentioned, there is the not so small problem that Purcell uses CGS units, so your engineers would be forced to re-learn all of their E&M theory anyway. I still sometimes use Purcell as a reference for translating the CGS blatherings of theorists into SI quantities that correspond to the data (the inside back cover of my edition shows the translations), because even today the translation is not automatic for me.

2. #2 FUG
May 8, 2009

I see the differential form of equations all the time in class, and we use this book: http://www.amazon.com/Scientists-Engineers-Chapters-CengageNOW-2-Semester/dp/0495013129/ref=cm_cr_pr_product_top

I have no perspective on what is too little or too much calculus to have, but I feel like I’m grasping the concepts with an appropriate amount of work, and I have had to use calculus to solve problems, so maybe it’s worthwhile (~ 2 hours per home work assignment, 1 assignment per week, plus a couple extra hours study time for test weeks). I don’t always use calculus, since derived equations are sometimes easier to work with, but usually at least once per assignment.

3. #3 Coriolis
May 8, 2009

My perspective on this is both as a more recent student and a TA going back to it, but I don’t see it as that much of a problem (we use Halliday/resnick/walker). At least at my university, those classes are clearly meant for engineers – and there’s enough complaining with the algebra that is there already. Adding calculus would add more complications then most of the students can handle.

Of course this does mean that the first upper division physics class, in whatever it is comes as quite a shock to physics undergrads, but hey, that’s the beauty of physics education isn’t it hehe.

The one thing we do here that does help somewhat I think is to have a “honors” section for the lower-level physics classes -usually taken by physics majors and interested engineers or other natural science majors. These are much more like the higher level physics classes, in that there is usually only one-two sections, alot more student-teacher interaction, and of course harder problems and everything else. As far as I know though, they still use the same books and perhaps it would be better if they didn’t.

4. #4 Josh
May 8, 2009

As a genetics major with aspirations of becoming a theoretical biologist, I think I feel your pain. For example, almost all biology majors have a requirement of a year of calculus, either in the form of “short calculus” or what appears to be an incredibly condensed “calculus for biology” course, which wham bangs you through calculus and even some probability and diff eq. I myself took the 2 year “math for math/physics/engineering” sequence. Then, they let biologists get away with a touchy-feely version of physics—basically no math involved at all, just concepts. I get the feeling that this kind of dichotomy between “physics for real” and “physics for biologists” is common—and I personally find it insulting.

Moreover, mathematics is heavily de-emphasized in most biology classes, even ones where mathematics is relevant. This leads to the occasional problem when a mathematical concept is critical to understanding a concept. For example, in biochemistry, it is crucial to understand Michaelis-Menten kinetics, which are traditionally derived via a simple ODE. Unfortunately, this ODE scares the hell out of most people. Similar problems arise in ecology, evolution, biophysics, etc. type courses.

And of course, in my domain of theoretical population genetics, the state of affairs makes me pretty sad. I’m currently enrolled in a course (mostly for the fun of it, since I honestly have done so much extra-curricular studying that I know most of it), and the prof skirts over some of the deepest and most elegant derivations, like the probability of fixation and the transit time of a new mutant, as well as some things I think all students should see in the first place, like how to go from the discrete generation version to the continuous time version of some of the master equations. I can’t criticize him, however, because most students simply would not be prepared for it—they probably haven’t evaluated an integral in 3 or 4 years by the time they take this course!

And oh man… don’t get me started on computational biology. The future of biology is by all means going to be heavily computation (bioinformatics, etc.) but those skills are even more de-emphasized than mathematical skills. No one should be able to graduate with any kind of biology major without ONE COURSE in programming at least—preferably in a powerful scripting language like Perl.

5. #5 Cherish
May 8, 2009

I think the smartest thing I ever did was to stop taking any physics until I’d had a full year of calc under my belt. Then every class I took after that was concurrent with my higher level math classes. In fact, I took a lot more math than physics at first.

My personal opinion is that this would be a good requirement for most people who want to be physics majors. It’s a lot easier to learn the math first and then the physics, and I think you get a heckuva lot more out of it. It helps the profs, too, since then they don’t have to go modifying things for the mathematically under-educated.

6. #6 Eric Lund
May 8, 2009

@Josh: It sounds like your curriculum is designed for pre-meds, who in fairness are often a plurality if not a majority of majors in biology and related fields. There is a definite distinction between “pre-med” physics (which is there because med schools insist on it) and “real” physics. I feel your pain on that score; in grad school I TA’ed the pre-med mechanics course once, which I found to be once too often. I also agree that your curriculum does not serve well anybody wishing to go to grad school rather than med school. Talk with your advisor about it; he may confirm my speculation that you are not the target demographic.

@Cherish: I have noticed that in a lot of places most engineering and physical science majors are not expected to take physics until their second term (independent of whether the school is on semesters or quarters). I think the idea is for students to be taking Calc 2 concurrently with Physics 1, allowing quick reinforcement of the math lessons in physics. That made sense for me as a freshman: the math I didn’t understand in calculus I got when it appeared later that week in physics, and sometimes vice versa. Most of the people who do start physics in their first term have placed out of calculus (or at least the first semester thereof).

7. #7 Thoreau
May 8, 2009

First, the one thing I’ll say in favor of the Matter and Interactions curriculum is that they at least try to offer something in place of the calculus that’s getting left out. They offer a more fundamental view of physics, and they try to introduce computation. Whatever the merits or demerits of that curriculum, it’s very different from the approach so many other books are taking where they make it simpler and simpler and simpler.

I don’t have any easy answers here. I’m well aware that even in the curricula that de-emphasize calculus, a lot of students still aren’t getting it. I want to be responsive to that fact, and certainly one plausible response is to do fewer things but try to make sure they at least master those things. OTOH, a lot of them don’t even get that, and I fear that if we just keep responding to the ones who don’t get it then we’re going to spiral downward.

The ostensible divide is that we have physics with calculus for the engineers and physics with algebra for the biologists. In reality, we have physics with a lot of algebra and a bit of calculus here and there for the engineers, and physics where we’re happy if they get the algebra right now and then for the biologists. Maybe that’s all we can expect. But then somebody can look at how many are STILL not getting it and propose to strip out even more math and focus on the remaining basics. So we do that, then some STILL aren’t getting it, so we repeat the process.

The best answer I can come up with, the one that will never fly, is that we should draw a line somewhere (and we can debate where to draw it) and then just start flunking a lot of people. If the goal is a physics class that everyone can master then no level is low enough. But if we have some goal informed by a concern for the subject matter itself, not just the desire to see every student master whatever is offered, then you have to draw a line and start flunking a lot of people.

That solution must obviously wait until I’ve gotten tenure.

8. #8 Thoreau
May 8, 2009

Regarding pre-med physics: There’s even a difference between pre-med physics and biophysics. The MCAT, for whatever reason, includes a lot of “modern physics” and even relativity, as well as lots of standard mechanics. Why? I dunno. I guess they saw it in some standard physics book and decided to write a test on it. You can design a test with medically relevant questions at a low or high level, but either way it will look very different from the current MCAT physics section. It’s not just about the level, it’s about the topics as well.

9. #9 CCPhysicist
May 8, 2009

I agree with both of Thoreau’s criticisms, enough that I should blog about them now that it is summer and I have some free time to get caught up on that sort of thing.

IMHO, there is a lot to the point about an over emphasis on the counter-intuitive parts of physics. Doubly so when students can solve the relevant problem correctly but get the “trick” question wrong. (Example: Whether a slight net force is needed to travel at a constant velocity, a classic.) Since there are a lot of grad students who can get these wrong, perhaps the place to attack them is a bit later when you decide it really matters.

As for the “with calculus” part, there is a lot of calculus in my second semester class. Quite a bit of it is conceptual (what is the charge enclosed, integral of dV is V, integral of dA is A) and thus hard. They really hate to integrate piecewise constant functions for some reason. The rest is chain rule and setting up “word problems” without necessarily having to do the integral.

My first semester has calculus as a corequisite, and I think that is why many of the texts de-emphasize calculus until you get later into the book. We use chain rule when we get to oscillations and waves, and do integrals in thermo, but not much more than that.

10. #10 CCPhysicist
May 8, 2009

Commenting @7 and @8:

Simpler, or less broad? I prefer depth in certain areas to the full span of the mini-PhD that is generally offered in such courses. But I do know of one school where they have dropped all of thermo from the first semester class, and I don’t see any evidence that they have gone deeper into what remains by, say, solving the anharmonic oscillator problem where you keep the cubic term for a pendulum.

I didn’t know the MCAT had relativity on it, but I know that our “trig based” class includes radioactivity and related topics because those are things that doctors actually use. They need optics and circuits a lot more than they need Gauss’ Law.

11. #11 D. C. Sessions
May 8, 2009

OK, some bias to admit here:

1) My lower-division physics classes (and most of the rest) were almost 40 years ago, back when it was “Halliday and Resnick” w. no sign of Walker
2) I’m the parent of two physics majors from a school which requires calculus as a prerequisite for incoming freshmen — if you don’t have it, you take it as a deficiency makeup.

I like that system. Feynman observed forty years ago that he was seeing the majority of his freshmen coming in with basic calculus out of the way already so that Cal Tech could get right into real physics, and it was a Good Thing even then. Thanks to missing the AP exams, I retook freshman calculus, but the main effect was raising my GPA and reinforcing bad habits.

Were it up to me, calculus would be a prerequisite for all serious physics classes [1] and I’d make sure that it was always available in the summer — those who miss the incoming calculus exam could make up the deficit without missing a whole year.

Then again, I’m studying for an advanced degree in Bad Codger Attitude. That, and making plans to go back for the physics PhD I put off in pursuit of a career. I don’t need to be wiping noses while I’m at it.

[1] I’ll except the repeat of high-school physics for the english lit crowd who need a physical science class. I will not except engineers. The lot I have as NCG candidates are crippled enough as it is without watering down basic physics.

12. #12 meichenl
May 8, 2009

I also used Purcell freshman year, and I don’t remember it being particularly mathematical. My impression was that more than any other book I’ve read, Purcell derives results from symmetry principles and physical intuition rather than computation. Those were valuable lessons for me. It’s a book on elementary physics that challenges the student to use a mature physicist’s thought process on simple problems. I don’t think the difficulties students have with it are mathematical. They’re more like “physics growing pains”.

When Purcell does treat mathematical concepts, the intuition he gives is very much “physical”. For example, check out problem 2.16b, in which he shows the identity

div(curl V) = 0

for any vector field V with one diagram and a few lines of text. Or take a look at Figure 2.21. Stokes’ theorem becomes obvious.

He covers about the same ground as Griffiths (conceptually, while omitting some topics), but I don’t remember any “vector integration by parts”, or using Fourier analysis to solve boundary problems with Laplace’s equation. Further, chapter 5 on relativity is about as far removed from a mathematical treatment as is possible. It never mentions the electromagnetic field tensor, and relies mostly on pictures, intuition, and wordy arguments.

It might help to supplement Purcell with Schey’s “Div, Grad, Curl, and All That”.

Of course, none of this applies to students who are not already comfortable with calculus.

13. #13 Thoreau
May 8, 2009

At most schools, 1 quarter or semester of calculus is a prerequisite for the physics sequence for engineers (and generally enrollment in the second calculus course is a corequisite for the physics intro course). Despite all that, we don’t actually use much calculus, and the books reinforce that bad practice.

14. #14 az
May 8, 2009

The way my school handled it (and I thought most schools did this) was to have two-tiered math and physics. One physics track required calculus, the other required trigonometry.
I would think the easiest way to make sure that the students in the calc-based course know calc is to really tighten the requirements on who can take that course: no lower than an A-/B+ in the first 2 semesters of the most rigorous calculus sequence offered (or maybe a 5 on BC calc AP if that’s equivalent). Alternately, you make students take a placement test – if they don’t demonstrate a decent understanding of calculus, they can either go back and take calc again, or take the trig-based class. If you don’t have the power to set your own requirements, or the heart to turn people away, you can give them the results of their test and a sober evaluation of what chances they have to actually succeed in the class. If you’ve done that, then nobody can complain when the class IS rigorously mathematical and they’re doing poorly. They’ve been properly warned.

Of course, this only works if you have a two-tier system when it comes to introductory physics.

I think changes need to be made to the non-calc class too. I’ve taken both sequences (about 15 years apart, the second time I took the algebra/trig-based one because I didn’t think I remembered enough calculus at that point). The algebra-trig version was really stupid, and strangely, much, much harder! Perhaps to some extent this is because I was old + rusty by the time I took it, but I think it was because they left out the necessary math. I guess some people are really good at concepts, but for me (and hopefully I’m not alone on this), I need to see the math to really understand the concepts. I usually need to derive an equation to really understand it, and in the algebra-based class, they’d just give you the equations as if they were pulled out of the air. I only vaguely remembered what I had actually learned in the rigorous physics sequence, but I certainly remembered that material presented was cohesive and often elegant. The algebra version seemed like a mish-mash of random equations pulled out of the instructor’s ass.
I’ve never understood why the hell med schools want their students to have taken physics. I doubt they ever use it, and all it does is create a VERY large group of students who need to take some sort of physics class, but don’t necessarily fit well into either track.

15. #15 Zen
May 8, 2009

Historically, it is only recent that mathematics and physics were taught separately. If you want students to understand mathematics, the best way, is practice and application, which we call physics.

Before college, I had a trigonometry course that I did well in, but it wasn’t until I started doing physics (using sine, cosines and tangents in every problem) that I REALLY comprehended trig.

Perhaps it is time we develop a blended curriculum.

16. #16 CCPhysicist
May 8, 2009

Chad wonders: one issue I’ll have to contend with is that a good number of the students will never have seen Maxwell’s equations in differential form

You can do this in less than a day if they’ve had Calc III. Just put both forms up on the board along with the two theorems from Calc III that relate them to each other, and away you go. The only other thing you need is one identity from the “div grad curl” book and a quick review of ampere and gauss to explain what “div” and “curl” mean physically using fluid dynamics (sinks and paddlewheels) to make it concrete.

Chad says: We list calculus as a co-requisite for intro physics, but the computer system used to handle course registration does not check or enforce prerequisites in any useful way, so we get students in the class who aren’t comfortable with derivatives, let alone differential equations.

I don’t get this at all, Chad. If calculus is a CO-requisite, no pre-req check in the world will guarantee that they’re familiar with a concept like a derivative. They won’t get to that for 3 weeks, at minimum. The first 3 weeks of calculus are spent of the concept of a limit, continuity, epsilon-delta proofs, etc etc. It is then another 2 weeks or more before they get to the applications section where velocity appears as a derivative. I’m fortunate to be in an environment where the people who teach calculus are just down the hall, so we talk about how to approach those first few days. I get them to say that this un-named expression in chapter 2 is actually the “derivative”, so at least we have the language under control from the first day or two rather than 5 weeks later.

DCS says @11: I’m studying for an advanced degree in Bad Codger Attitude.

I am so stealing that one. If you think what you describe is bad, just imagine algebra classes that spend a significant amount of time teaching them how to solve linear or quadratic equations on a TI-83 rather than how to manipulate symbols. I’m sorry, but I just don’t get it and no one can point me to an outcomes-based study that says that approach leads to greater success in calculus or physics.

And I am going to blog about the “conceptual” issues as well as the topic of calculus and physics this weekend.

17. #17 Aki
May 8, 2009

In my undergraduate career, we used Halliday, Resnick and Krane the first year. It seemed to get the job done. The biggest problem we had at the time was a disconnect between our calc. and physics classes. When, as upper-division students, we were invited to comment on ways to bolster retention, we advocated a blended class for freshmen that would force physics and calc. to march in step and allow students to bridge the gap. They didn’t listen.
Now, as a teacher myself, I have my students asking for more calculus and more connections between the physics and the calculus. I’m still trying to find a textbook that serves the need. Suggestions?

18. #18 Chad Orzel
May 8, 2009

I don’t get this at all, Chad. If calculus is a CO-requisite, no pre-req check in the world will guarantee that they’re familiar with a concept like a derivative. They won’t get to that for 3 weeks, at minimum. The first 3 weeks of calculus are spent of the concept of a limit, continuity, epsilon-delta proofs, etc etc. It is then another 2 weeks or more before they get to the applications section where velocity appears as a derivative.

Sorry– Calc II is a co-requisite. The part with integrals. They’re supposed to have taken the first term of calculus before taking physics, but that never gets enforced.

As somebody mentioned above, the physics sequence doesn’t start until the second of our three academic terms– first-year engineering students take physics in the Winter term, after taking math in the Fall term. The idea is to have the students pick up the math they need first, and then take physics.

A couple of people have also suggested a “blended” curriculum, covering both math and physics. We do run one section of an integrated math-physics course. It’s a year-long course, in three parts, covering the first three terms of calculus and the first two terms of physics, team-taught by a math professor and a physics professor.

I’ve only ever taught the first term, which barely gets through Newton’s Laws. It’s an interesting approach, though, and I enjoyed teaching it. It’s been a long, long time since I had any formal math classes, and it was fascinating to see how different the approaches are. It also confirmed my belief that I’m not cut out to be a mathematician. If nothing else, I’m incapable of writing complete sentences on the chalk board in a legible manner.

19. #19 Eofhan
May 8, 2009

I was a Physics major 20 years ago. A bad one, but capable-enough to earn a BA. Even I understand that calculus is fundamental. What the hell happened? . . .

20. #20 Lab Lemming
May 9, 2009

I think much of the problem may lie in the teaching of math, which results in your students not being able to recall or use what they were taught long ago.

As a geologist, I took calculus intensive courses, like geophysics and thermodynamics, interspersed with math-free courses, like paleontology. The result is that as an undergrad, I ended up learning- and forgetting- calculus three times. There has to be a better way to teach math such that scientists can easily pick up the skills that they need after a year or two of not using them. After all, that’s generally how life as a private sector scientist works.

21. #21 Jonathan Vos Post
May 9, 2009

Having gone to a famously good high school and university, I had no idea how badly broken was the Physics/Math connection in American education until I was teaching.

There was no spice of Calculus in the Intro Astronomy (Moons for Goons) that I taught (2 lecture sections of 50 students each), but merely elementary Algebra. That is, it was a “cool stuff on backs of envelopes.” The Astronomy Lab was a Physics lab + computer lab of real Physics, heavily Mathematical, of phenomena central to Astronomy. But, again, no Calculus. Just Intermediate Algebra plus graphing and elementary Statistics.

My wife is a Physics Professor at a private university where I’ve also taught Math. She’s been teaching Physical Science (even less Math) and Physics (with a lab that everybody likes. Again, no Calculus.

A good case can be made that Biology is to Math in the 21st Century as Physics was to math in the 19th and 20th Centuries. But that does not seem to be trickling down yet into the undergrad curriculum.

I have a month to go as Student Teacher at Lincoln High School in L.A. before my credential lets me teach full-time High School Math. Two of my 4 courses, AP Statistics and AP Calculus, have a few Physics problems in the worksheets and textbooks. The Geometry and Algebra classes are nearly Physics-free.

Finally, there are colleges and universities that offer both a B.A. and a B.S. in Economics. The difference? Calculus!

I very strongly disagree with C.P. Snow’s “Two Cultures” hypothesis. But maybe there IS a fuzzy boundary at Calculus between Real Science and Flavor of Science in our schools.

22. #22 Bruce Sherwood
May 9, 2009

I’m one of the authors of the Matter & Interactions curriculum.

Occasionally at Carnegie Mellon a strong student would complain that “this is supposed to be a calculus-based course but we don’t use any calculus” (I’ve not heard this complaint from NCSU students). This always turned out to be a case of the student not realizing that integrals had anything to do with the sum of a large number of small quantities, or that a derivative had anything to do with a ratio of small quantities. What the student perceived as “calculus” was a large number of evaluation formulas for derivatives and integrals. We think our “stealth” version is in fact much closer to the true nature of calculus than is the standard subject (just as the Momentum Principle, dp = Fnet*dt, is what Newton used; F=ma is definitely not Newton’s second law of motion). We’re intrigued that Michael Oehrtmann, a math educator at Arizona State, is developing a calculus course for engineering and science students that emphasizes small, nonzero differences, as do we. His course would be a perfect complement to Matter & Interactions.

I would further argue that iterative calculations which show the time evolution character of the Momentum Principle (in the form dp = Fnet*dt) are very much in the spirit of differential equations, whereas F=ma looks like an algebraic relation (and usually is, since F is usually constant).

The place where calculus per se really kicks in is the Matter & Interactions chapter on finding the electric field of distributed charges, which is fiercely all about calculus. But even there, in a chapter which has been praised by reviewers as being particularly good on how to go from a physical situation to an integrand, we have found that most students are not able to do an example on their own (say, find the electric field along the axis of a uniformly charged rod, some distance beyond one end), because in their “calculus” course the focus was on evaluating integrands, not setting them up (here again, setting up the integrand from a physical situation is one of Oehrtmann’s emphases). In the real world, absent Oehrtmann’s course, we find that it’s not possible to ask average students after two semesters of calculus and a semester of mechanics to set up an integral themselves.

23. #23 dr. dave
May 9, 2009

teaching in a place where there are no physics, engineering, or chemistry majors, this is not a problem i deal with. i teach mostly historically-based physics for students with an interest in the liberal arts.

does anyone know of a good FLUXIONS-based undergrad text? preferably not in Latin? :)

May 9, 2009

At university I was a bit bewildered by simple physics questions, because I didn’t know how to define the problem. For example, we’d get a question such as, “a rock of such a mass falls so far into a bucket. Calculate the changes of energy.” So I’d start adding them up. Of course it lost potential energy and gained kinetic energy–for a while. That part was simple. Then it created a sound and imparted some heat to the bucket. How to calculate the energy lost in producing a sound? How loud a sound? How much heat conversion? I was stumped.

25. #25 Chad Orzel
May 9, 2009

I don’t know why I’m always faintly surprised when the authors of things I post about turn up here. It is available to everyone in the entire world, after all…

Occasionally at Carnegie Mellon a strong student would complain that “this is supposed to be a calculus-based course but we don’t use any calculus” (I’ve not heard this complaint from NCSU students). This always turned out to be a case of the student not realizing that integrals had anything to do with the sum of a large number of small quantities, or that a derivative had anything to do with a ratio of small quantities. What the student perceived as “calculus” was a large number of evaluation formulas for derivatives and integrals. We think our “stealth” version is in fact much closer to the true nature of calculus than is the standard subject (just as the Momentum Principle, dp = Fnet*dt, is what Newton used; F=ma is definitely not Newton’s second law of motion).

I basically agree with this, I think. Certainly, I like the fact that the curriculum starts with momentum, rather than kinematics– that’s one of the strongest points, for me.

I wrote what I did above in large part because I saw some errors on the first mid-term (I’m just over halfway through our intro mechanics course now) that I never saw in previous versions of the class. They seemed to me to be the result of thinking about the problems in an even more algebraic manner than usual. This may be partly due to the somewhat unusual population in the course this term, but it was a striking difference. (I’m hesitant to discuss it in detail on the blog– if you’d like to know more, email me.)

I do think that the finite step method is, in the end, one of the great strengths of the approach, in that it lends itself well to computational solutions, and allows discussion of problems that can’t be solved analytically. I’m not sure how much the students appreciate that, though…

The place where calculus per se really kicks in is the Matter & Interactions chapter on finding the electric field of distributed charges, which is fiercely all about calculus. But even there, in a chapter which has been praised by reviewers as being particularly good on how to go from a physical situation to an integrand, we have found that most students are not able to do an example on their own (say, find the electric field along the axis of a uniformly charged rod, some distance beyond one end), because in their “calculus” course the focus was on evaluating integrands, not setting them up (here again, setting up the integrand from a physical situation is one of Oehrtmann’s emphases). In the real world, absent Oehrtmann’s course, we find that it’s not possible to ask average students after two semesters of calculus and a semester of mechanics to set up an integral themselves.

That’s definitely a problem. I taught that part of the course last spring, and had the same issue.

I got reasonably good results by writing a program to calculate the on-axis field due to a charged rod, and then asking them to modify the code to measure the field for off-axis points. That was an honors class, though, so I’m not sure the results will generalize. I know some of my colleagues were doing the same thing this term– I’ll have to ask them how it went.

26. #26 CCPhysicist
May 9, 2009

This is to Bruce Sherwood: Does Michael Oehrtmann know that he is reinventing the old “infinitesimals” approach to calculus, and that there is a free textbook on the web?

http://www.math.wisc.edu/~keisler/calc.html

It predates calculators, let alone the computer-based text I taught out of back in the 70s, but it might be helpful.

I couldn’t agree more concerning the field due to a rod problem. As I mentioned either here or in my blog or in a comment on Thoreau’s blog, setting up a problem where calculus might be involved is a major weakness in the calculus (or pre-calc or algebra) curriculum. It just doesn’t get done in math classes because the math profs just are not comfortable teaching the non-math subjects (be it physics or biology) that are needed as background to the application.

That is where “linked” classes such as Chad mentions @18 can come into play. The content instructor can cover for the math instructor.

And Chad, you might tell your math colleagues that there was once a textbook for your quarter system where basic calculus (both differential and integral) was taught like arithmetic in the first quarter – to lead into physics – with the abstraction of limits and so on deferred to the second quarter when inverse functions and other complications show up. No epsilons or deltas for the first three weeks. They didn’t leave out the abstraction, they deferred it.

27. #27 Bruce Sherwood
May 10, 2009

Thanks for the tip about the Keisler book. I’ve passed this link on to Oehrtmann, though I’m guessing he knows about it. Many years ago I saw a textbook with this emphasis but lost track of it; maybe it was the Keisler book, though Keisler mentions a mathematician named Robinson who put infinitesimals on a firm footing, and it’s possible I saw a book by Robinson.

28. #28 Bruce Sherwood
May 10, 2009

Having reflected some more on this very interesting discussion, some further thoughts: I’ve taught intro physics for 40 years, starting at Caltech using the Feynman Lectures on Physics for the textbook (which was a fabulous experience). It seems to me that there isn’t a lot of scope for standard calculus in intro mechanics, other than having students calculate positions of the center of mass of odd objects, or moments of inertia of various solids, which would be beyond most students other than in an honors course. What does matter is that prior or concurrent study of calculus is essential for giving the students enough math sophistication to deal with the concepts in mechanics, and to be able to follow some critical derivations. What kind of “calculus” problems did people have in mind for intro mechanics?

The biggy isn’t mechanics but E&M, as has been partially noted in this discussion already. The Great Divide is whether one can use the differential forms of Maxwell’s equations or not. In the intro course (except possibly in an honors course at a very selective university) the answer is a resounding “not”. The Feynman Lectures on Physics and Purcell’s splendid E&M text were the biggest influences on Ruth Chabay and me in writing the Matter & Interactions textbook. We love Purcell. But it is rarely suitable for the intro E&M course, and a bit too low a level for the junior course, hence it doesn’t get used much.

Minor point: Chad says, “…I’ll have to contend with is that a good number of the students will never have seen Maxwell’s equations in differential form. Which makes it a little difficult to get to the wave equation, and set up the necessary background information about the classical model of light as an EM wave.” As another person pointed out, you can get this with the integral equations. In fact, in Matter & Interactions we use only integral forms of Maxwell’s equations to get the full classical model of the interaction of light and matter, though we use a qualitative version of Purcell’s elegant Gauss’s Law argument to say that accelerated charges radiate. But that’s enough, because then the radiative electric field accelerates charges in matter, which re-radiate. Classically, light doesn’t bounce off the wall — you see new light from the accelerated charges in the wall, accelerated by the incoming light.

29. #29 Jonathan Vos Post
May 10, 2009

(1) Chad’s delta experience in teaching is interesting, and he is honest about the problem that both textbook and student characteristics have changed.

(2) Bruce Sherwood is right about Feynman’s Lecture Notes at Caltech. Note that everyone at Caltech (in my 1968-1973 era) came in with strong Calculus experience, which Apostol made rigorous and, with later courses in differential equations and complex analysis raised to a level of mastery. Everyone takes Physics Lab. Note also that the Feynman book has 3 volumes, roughly kinematics of particles under mechanical and electrostatic forces; full-blown Electromagnetics; Quantum Mechanics. I understand that Caltech has tweaked their Physics curriculum, but don’t know details.

(3) Metaphysically and computationally, maybe Wolfram (formerly a Caltech Physics professor) is right that Physics took a wrong turn at Newton, and algorithmics and automata theory is the way to go. Too soon to tell.

(4) Caltech led the world in Astrophysics, and that trickled down to the undergrad curriculum. Now Kip Thorne’s decades of inflouence breaks new ground in computational geometrodynamics. This goes rather deeper that undergrad Calculus.

(5) I am not writing anything about String Theory.

30. #30 CCPhysicist
May 10, 2009

The “real calculus” problem that showed up in my intro (honors) physics class as an undergrad was the anharmonic oscillator (pendulum out to the cubic term in the expansion) and coupled linear pendula, but the calculus I usually have in mind (for my class) are the work integrals in thermo, although motion with velocity dependent drag forces would be nice.

I second the comment about Purcell’s E+M text (I assume you mean “berkeley physics 2”). Fantastic way to learn the subject, and my undergrads still suffer its effects. That is, until they get to the last week of Calc 3 and totally get Stokes Theorem. In our case, it doesn’t hurt that the guy teaching Calc 3 is just down the hall and knows how I teach E+M.

Chad, I’d love hearing about the new problem that crept in.

31. #31 Bruce Sherwood
May 11, 2009

But the work integrals in thermo are typically just used to derive results, they aren’t homework problems. Even the mechanics section of Matter & Interactions makes abundant use of traditional calculus for derivations. Examples:

Factor the momentum vector into the product of magnitude and unit vector and differentiate using the product rule to get a term which represents the rate of change of the magnitude of the momentum and the rate of change of the direction of the momentum; use the Momentum Principle in the form dp/dt = Fnet (as vectors); find the analytical solution for dp/dt = Fnet in the case of the harmonic oscillator (solve the differential equation by the usual scheme of guessing a result and finding what parameters will make that result work in the differential equation); from the Momentum Principle derive the Energy Principle for a point particle starting from dE/dx = dpx/dt; find evaluation formulas for various kinds of potential energy by finding the quantity whose negative gradient is the force (this comes early enough that if calculus is concurrent students haven’t yet seen integrals but can find antiderivatives); etc.

Even the case of motion with velocity dependent drag forces would most likely be a derivation shown to the students, not a homework problem for them to solve using calculus, no?

I’m looking for realistic examples in mechanics where in a regular calculus-based course (not honors, not physics majors) one can imagine assigning a homework problem for which students would independently have to use calculus. I don’t think there are many. And I don’t think there were any when I took calculus-based physics many decades ago, so I don’t think there’s been some major watering down.

A colleague pointed out to me that there has been one change over time. Before most disciplines changed from requiring 3 semesters of physics to 2, it was sometimes the case that the 3rd semester was E&M, and students had already had calculus in 3D, so it was more feasible to consider using differential forms of Maxwell’s equations. Now at most students take calculus in 3D concurrently with E&M.

32. #32 GR
May 12, 2009

It’s hard to think up of anything in intro mechanics that requires calculus — maybe finding the maximum of a trajectory? I think, though, that even if students are just “shown” the derivation, they still should have a firm grasp of calculus. If only there was a way to make sure we all worked through the derivations ourselves…

As a side note about E&M, I first saw the vector calculus version in my intro class, and even though I freely admit I had no idea what the heck was going on it helped later on. One of the things I love about the physics curriculum is how it keeps going back to old topics and adding on new awesomeness.

I just graduated from NCSU (small world I guess…), and having the experience of interacting with both M&I and non-M&I people I feel like I should throw out some observations:

1) M&I and non-M&I students seem to live in different worlds. Students who’ve taken one curriculum seem to have a lot more difficulty with problems from the other one that should be expected. This is kind of disconcerting to me — it’s all the same physics, right? I don’t know if this is because of the type of problems, or the way the problems are asked…

2) One disadvantage of the M&I curriculum that’s been voiced to me by people I’ve tutored is that it introduces the whole machinery of atoms, etc… too quickly. This left them confused as to what actually was going on, and when the various approximations were allowed to be used.

3) I used to think poor math skills were an issue in physics education but now I’m not so sure. I tutored a guy who was amazingly fast at algebraic manipulations, but he just couldn’t figure out what concept was being applied or which formula related to what concept. I’m pretty sure the only way to remedy this is with more “discovery-based” (or whatever the buzzword is) learning. With the precaution that too much exploration often leads to frustration for students because, let’s face it, most of us aren’t Newton. I’m not sure if it’s doing it 100% right (I just can’t quite agree that computer programs are better than “the real world”) but M&I should at least be applauded for bringing that type of learning more into the mainstream.

4) WebAssign and its web-based homework cousins are terrible. Terrible. Terrible. Terrible. Sure, they save time for teachers/TA’s when grading. But they encourage students to not write down problems and just go at it with a calculator, punching in numbers in some combination until they get a right answer*. The x-number-of-tries thing makes students focus solely on “getting the problem right” more than “understanding what’s going on”. And that’s not even touching on the rampant cheating…

5) I still think the best way to “get” physics is to work lots and lots and lots of problems. How many and what weighting they should be given in the grading scheme is something I should probably leave to professionals. My junior-level E&M professor had an interesting strategy. He would have us (on good old paper!) write down the problem, then write out a strategy for solving that problem, and then solve the problem. It was brutal, and I often spent 30+ hours a week on those damn problem sets, but taking the time to stop and really think about the physical situation instead of just jumping into the crank really helped my understanding.

Anyways, just my 2 cents

33. #33 Frank Noschese
May 13, 2009

The College Board has no problem finding ways to put calculus on the AP Physics C exams. Just check out this year’s questions on the Mechanics exam:

http://apcentral.collegeboard.com/apc/public/repository/ap09_frq_physics_c_mech.pdf

Q1. Potential energy function U(x)
Q2. Differential equation for SHM
Q3. Rope with mass slides off the edge of a table

I don’t understand why many colleges don’t give credit for kids who can score a 5 on this exam! Are these questions easier/harder/typical compared to college physics exams?

34. #34 truth is life
May 19, 2009

Frank @33: Those were MUCH more difficult (upon simple inspection) than any of the problems I saw in physics I or II (and I’m a physics major!) Of course, I’m going to the University of Houston, not exactly a big physics school, and they do give you credit for the calculus-based course for getting a 5 on the test.

The most ‘calculus’ I ever did was using integrals and derivatives to save memorizing the different velocity equations (v=at+v_i if a is constant, for example). There were a few problems in E&M where calculus helped some, but for the most part it was just memorizing a ton of algebraic formulas.

Even in the first semester of modern we didn’t really do anything with calculus until the last third of the semester when we started covering the Schrödinger equation. Before that, when we were dealing with Einstein and Bohr/de Brolige/Thompson/etc., it was just algebra.

Bruce Sherwood has it right when he says 1. Calculus is perceived as being just a bunch of memorized formulas (I know I though this way pretty much up until this semester with my analysis course–and even then it’s hard to stop thinking of it this way) and 2. It’s really hard (especially for people thinking like this) to properly formulate the integrals.

35. #35 Fran
May 25, 2009

I’m with Frank (comment 33) on the AP Physics C question. The college board frequently puts free-response questions on the AP exam that require setting up differential equations or integrals. I would LOVE to know how those questions compare to those of “calculus-based” physics courses at Universities. Many of the comments here seem to imply that you would NOT find similar problems on the final exams there.

Thanks for the discussion, everyone! I enjoyed reading it!

36. #36 Mike Heesen
June 19, 2009

I really like calculus and the combination with physics is really awesome for me.

37. #37 Mike Heesen
June 19, 2009

I really like calculus and the combination with physics is really awesome for me.

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