# The Unapologetic Mathematician

## A Short Rant about Electromagnetism Texts

I’d like to step aside from the main line to make one complaint. In refreshing my background in classical electromagnetism for this series I’ve run into something that bugs the hell out of me as a mathematician. I remember it from my own first course, but I’m shocked to see that it survives into every upper-level treatment I’ve seen.

It’s about the existence of potentials, and the argument usually goes like this: as Faraday’s law tells us, for a static electric field we have $\nabla\times E=0$; therefore $E=\nabla\phi$ for some potential function $\phi$ because the curl of a gradient is zero.

What?

Let’s break this down to simple formal logic that any physics undergrad can follow. Let $P$ be the statement that there exists a $\phi$ such that $E=\nabla\phi$. Let $Q$ be the statement that $\nabla\times E=0$. The curl of a gradient being zero is the implication $P\implies Q$. So here’s the logic:

\displaystyle\begin{aligned}&Q\\&P\implies Q\\&\therefore P\end{aligned}

and that doesn’t make sense at all. It’s a textbook case of “affirming the consequent”.

Saying that $E$ has a potential function is a nice, convenient way of satisfying the condition that its curl should vanish, but this argument gives no rationale for believing it’s the only option.

If we flip over to the language of differential forms, we know that the curl operator on a vector field corresponds to the operator $\alpha\mapsto*d\alpha$ on $1$-forms, while the gradient operator corresponds to $f\mapsto df$. We indeed know that $*ddf=0$ automatically — the curl of a gradient vanishes — but knowing that $d\alpha=0$ is not enough to conclude that $\alpha=df$ for some $f$. In fact, this question is exactly what de Rham cohomology is all about!

So what’s missing? Full formality demands that we justify that the first de Rham cohomology of our space vanish. Now, I’m not suggesting that we make physics undergrads learn about homology — it might not be a terrible idea, though — but we can satisfy this in the context of a course just by admitting that we are (a) being a little sloppy here, and (b) the justification is that (for our purposes) the electric field $E$ is defined in some simply-connected region of space which has no “holes” one could wrap a path around. In fact, if the students have had a decent course in multivariable calculus they’ve probably seen the explicit construction of a potential function for a vector field whose curl vanishes subject to the restriction that we’re working over a simply-connected space.

The problem arises again in justifying the existence of a vector potential: as Gauss’ law for magnetism tells us, for a magnetic field we have $\nabla\cdot B=0$; therefore $B=\nabla\times A$ for some vector potential $A$ because the divergence of a curl is zero.

Again we see the same problem of affirming the consequent. And again the real problem hinges on the unspoken assumption that the second de Rham cohomology of our space vanishes. Yes, this is true for contractible spaces, but we must make mention of the fact that our space is contractible! In fact, I did exactly that when I needed to get ahold of the magnetic potential once.

Again: we don’t need to stop simplifying and sweeping some of these messier details of our arguments under the rug when dealing with undergraduate students, but we do need to be honest that those details were there to be swept in the first place. The alternative most texts and notes choose now is to include statements which are blatantly false, and to rely on our authority to make students accept them unquestioningly.

February 18, 2012 -

1. Exactly what’s wrong with almost everything these days.

Comment by Avery Andrews | February 18, 2012 | Reply

2. Thank you!! In this particular case it’s completely inexcusable because for many students, this material is already so intimidating, any who are alert enough to catch the mistake will probably assume they’re misunderstanding something elsewhere.

(There’s some sort of weird opposite Occam’s Razor at work here: if you’re neck deep in computations and something doesn’t add up, the error (one is tempted to reason) must be in the complicated computations, not in the basic logic!)

Comment by Sam Alexander | February 18, 2012 | Reply

3. This is your _one_ problem with E&M texts? How about all the integrating through the singularity at the origin with nary a mention, or, for that matter, the fact that the entire subject is a fiction. I once read an article that pointed out that if one follows through the arguments of classical E&M, a point particle will accelerate on its own to infinity (something about the interaction of its electric and magnetic fields).

Comment by Greg Friedman | February 18, 2012 | Reply

4. It’s not the only problem, Greg, but it’s among the most universal. Many, if not most, texts introduce delta functions and are explicit about the fact that they’re not going into the whole theory of singular distributions. They also do tend to bring up divergent integrals involved in self-interactions, and point out that resolving these problems is beyond their scope.

My problem here is not just that the standard physics pedagogy plays fast and loose with the math; it’s that they don’t admit they’re playing fast and loose with the math the way they do elsewhere.

Incidentally, I have a similar problem with multivariable calculus courses fudging the difference between points and position vectors, so mathematicians aren’t perfect either.

Comment by John Armstrong | February 18, 2012 | Reply

5. This is an incredible post, to be reblogged immediately!

I think that contemporary science education is veritably plagued by poor presentations. Just think of the illustration of gravity as a bowling ball on a mattress. It’s gravity that suppresses it in the first place!

Thanks for this,
NS

Comment by notedscholar | February 20, 2012 | Reply

6. Hi, John, I wonder if we could have a compiled version of all your notes on electromagnetism from the past few months’ blog entries.

Comment by Shuhao Cao | February 20, 2012 | Reply

7. Admittedly I don’t remember reading the textbook for the course 20 years ago, but I distinctly remember the words “Helmholtz decomposition” from at least one of the lectures. No mention of de Rham cohomology, of course.

Given that the Helmholtz theorem is “the fundamental theorem of vector calculus”, I think it’s reasonable to state it in a physics class without proof, as long as you actually state it.

Comment by Pseudonym | February 21, 2012 | Reply

8. Helmholtz is one possibility, and it works fine in real, three-dimensional space. It doesn’t generalize so well, though.

Comment by John Armstrong | February 21, 2012 | Reply

9. […] said should be all that new to a former physics major, although at some points we’ve infused more mathematical rigor than is typical. But now I want to go in a different […]

Pingback by Maxwell’s Equations in Differential Forms « The Unapologetic Mathematician | February 22, 2012 | Reply

10. Here’s the fundamental reason for your discomfort: as a mathematician, you don’t realize that scalar and vector potentials have *no physical significance* (or for that matter, do you understand the distinction between objects of physical significance and things that are merely convenient mathematical devices?).

It really doesn’t matter how scalar and vector potentials are defined, found, or justified, so long as they make it convenient for you to work with electric and magnetic fields, which *are* physical (after all, if potentials were physical, gauge freedom would make no sense).

On rare occasions (e.g. Aharonov-Bohm effect), there’s the illusion that (vector) potential has actual physical significance, but when you realize it’s only the *differences* in the potential, it ought to become obvious that, once again, potentials are just mathematically convenient devices to do what you can do with fields alone.

P.S. We physicists are very happy with merely achieving self-consistency, thankyouverymuch. Experiments will provide the remaining justification.

Comment by Peter Erge | March 8, 2012 | Reply

• Eh, here’s *my* discomfort – what in the world does *physical significance* mean? Talk about vacuous handwaving….

Thankyouverymuch.

Comment by Occam's Strop | June 17, 2012 | Reply

• I’m sorry. Here I was, thinking I was among scientists, albeit those with more theoretical bent than most are comfortable with. Among scientists, something with “physical significance” means something that can be measured in a lab. (And among scientists, this simple fact needs no elaboration, unless I wanted to insult your intelligence, or, more likely, a lack thereof.)

Comment by Peter Erge | June 17, 2012 | Reply

11. […] comment just came in on my short rant about electromagnetism texts. Dripping with condescension, it states: Here’s the fundamental reason for your discomfort: as a […]

Pingback by A Continued Rant on Electromagnetism Texts and the Pedagogy of Science « The Unapologetic Mathematician | March 8, 2012 | Reply

12. I thought my courses in Electro-Magnetism were the most confusing I had as an under-grad.

If memory serves me correctly we worked our way through about one problem per 90 minute class
with the TA going up to the Professor before, during break and after class and correcting his mistakes
which were duly noted at the start of the next class period.

It seems we need to do two things:

a) Explain Vector Calculus very, very well – double the amount of time in class and work out the physics
behind the calculus step by step and let the students struggle with problems in class instead of just
going through one proof after another.

b) Have very good and very explicit practical problems in Electro-Magnetism so that it is as clear as
can be what is going on and why the calculus reveals the physics and vice versa.

The amount of time students in the sciences spend in the classroom and lab should not be modeled
after the amount of time that Liberal Arts students spend in class. 15 credit hours a terms may translate
into 15 hours of lecture for them, but 15 credit hours of science should translate into 30 hours of class
and lab time.

Instead of 4 years to earn B.S. why not 5 years – there is so much more to learn and it is so much
more important to learn it well. For the typical English Major – who is not going on to Grad School
what real difference in their future work will it make if they confuse Dryden with Donne ?

Comment by George Watson | July 8, 2012 | Reply

13. “I have a similar problem with multivariable calculus courses fudging the difference between points and position vectors”

Of course the scalar product of two points in space has intrinsic meaning. Doesn’t it? And naturally you can multiply a point by a number and get another point. Right?

This really bothered me in first semester physics, not because I knew anything about vector spaces, but simply because I could not make sense of the various things they were doing with those arrows.

Naturally I was too proud to memorize formulas or do calculations by rote.

40+ years later, I still don’t know whether I was being pig-headed or sensible.

Comment by Ralph Dratman | July 19, 2012 | Reply

14. How about when computing the electric field INSIDE a volume charge distribution using the integral form of Coulomb’s Law? The denominator of the integrand contains the term (r-r’), where r and r’ are the “field point” and “source point”, respectively. The field point is treated as a constant during this integration, as integration is with respect to the source point. If the field point r is allowed to be within the domain of integration, then the integrand is undefined at said when r’ coincides with r! Division by zero! What the hell! This would never fly in my vector analysis text.

I took 2 quarters undergrad engineering E&M this past year. I just realized this problem approx. 1 week ago and it is driving me crazy.

Comment by Matthew Kvalheim | August 12, 2012 | Reply

15. Peter Erge’s comment, if I understand it correctly, suggests that a physicist’s use of mathematics does not require that the physics and the math correspond precisely at every possible point — only that the mathematics can be used to make some predictions which have been found to match experimental observations over a certain range of configurations and measurement.

Although Erge’s approach leaves familiar logical gaps, I think it does describe how physicists have to use mathematics.

Comment by Ralph Dratman | August 12, 2012 | Reply

• That’s exactly my point—and mathematicians take something away from this exchange, too: what use would distributions have been except that some of physicists’ favorite “functions” (such as the Dirac delta function, which, BTW, is the result of doing the 1/r integral for a point charge (going from the differential form to integral form of Maxwell’s equations) in the complaint by Mr. Kvalheim above) are poorly defined as traditional functions?

When one happens to know the answer (or has a way to check the answer), one can afford to be a bit sloppy—for the sake of faster progress, not of sloppiness itself.

Comment by Peter Erge | August 12, 2012 | Reply

16. Ralph and Peter, you’re still ignoring what I’ve said over and over and over and over again here: I’m talking about how physics is taught, not how it’s used. And I’m not even asking for rigor in how it’s taught; I’m asking for a explicit mention of the fact that a given point is not rigorous.

Seriously, you’re not even arguing the same point. Just stop before you embarrass yourselves further.

Comment by John Armstrong | August 12, 2012 | Reply

• Nice dodge, but that can’t be it.

Physics textbooks seldom claim the kind of mathematical rigor you are insisting on. For example, most electrodynamics textbooks (the classic textbooks Griffiths (undergrad level) and Jackson (grad level) both do this) will introduce electric potential in electrostatics—i.e. many chapters before Faraday’s law has even been introduced (after magnetostatics) and hence there were no reasons to worry about changing electric fields in the first place.

You have inserted the rigor/logic that physicists don’t even bother to imagine and demanded that physicists hold up to mathematicians’ standards. That is unreasonable, not because physicists aren’t capable of that, but because we like to teach something substantial—and have the class end within a semester at the same time.

Comment by Peter Erge | August 12, 2012 | Reply

• I also strongly agree with John Armstrong. The texts absolutely should point out, in some way, that the physics and the math differ. In fact, I remember reading the strange assertion you mention (that some potential function must exist because the curl of the gradient is zero). It immediately made no sense to me, even though I had no idea how the statement needed to be qualified. In fact, I am pretty surprised to read that all you need is a contractible space to make it true. I don’t usually have a longing to see a proof, but in that case I did want some kind of motivation for believing it — if only because it is so much easier to remember things that make sense.

Comment by Ralph Dratman | August 12, 2012 | Reply

17. Here is the exact rewrite I am asking for, Peter. Before:

… therefore $E=\nabla\phi$ for some potential function $\phi$ because the curl of a gradient is zero.

After:

… therefore choosing $E$ of the form $\nabla\phi$ for some potential function $\phi$ will always satisfy this condition because the curl of a gradient is zero. We will assume that $E$ is of this form from now on.

Is that really so objectionable?

Comment by John Armstrong | August 12, 2012 | Reply

• Yes. Because it detracts from the subject of the class: physics, not mathematical nitpicking.

Comment by Peter Erge | August 12, 2012 | Reply

• I seriously disagree. In fact, I would personally prefer a little bit more explanation than is to be found in the edit John provides — even though it is adequate mathematically. I fail to see how a footnote (of any length) could disrupt flow of an argument or the comprehension of same by students.

If you really worry about that, Peter, just tell them something like, “You are not responsible for any of the material in footnotes.” Then the ones who would be distracted won’t even look at them.

The lack of consistency in logic and notation bothered me very much as an undergraduate physics student, and certainly not because I was a nitpicker. I just sometimes had trouble following the flow of ideas in certain areas, and I think, speaking very generally, that more mathematical explanation, when done properly, can translate into more sense.

Comment by Ralph Dratman | August 12, 2012 | Reply

18. While I find that the majority of my engineering classmates don’t seem to care about the lack of rigor in our electromagnetics classes (or any class for that matter), it has always bothered me.

I find myself agreeing with the point of view of Mr. Armstrong. In sweeping details under the rug, teachers frequently leave me feeling as though I am thinking about the subject matter completely wrong- because, as far as I know, nobody else had ever run into the same problem as myself.

I have seen Griffiths’ Intro to Electrodynamics mentioned by a few others on this site. This was not the text used in my classes, but I am reading it now (about 150 pages in). I find this text EXTREMELY helpful, because although it does not seem rigorous, it is filled with footnotes where Dr. Griffiths at least acknowledges when he is not mentoning all the details. This has already provided me with a lot of insight.

Another example- using a delta distribution for the divergence of 1/r^2 fields seemed a little suspect to me. Griffiths just sort of throws it in as though “oh, the divergence theorem doesn’t work here for mystery reasons. But it does if we put this thing called a delta function in.” I never truly understood Gauss’ Law until, while reading Vector Calculus by Marsden & Tromba, I saw a proof of Gauss’ Law using the divergence theorem for unions of simple elementary regions (without any delta distributions).

(On a side note, can anyone explain why the integration I mentioned in my previous post is valid? Or at least point me to a resource that does? I apologize if this is too off topic)

Comment by Matthew Kvalheim | August 12, 2012 | Reply

19. I am a physicist and I tend to agree with John – the presentation of relevant mathematics in physics books is not always sufficient. That gap should be filled mainly by dedicated class lead by teacher who understands the subtleties of mathematics for physicists, and also there should be at least footnotes setting things right like Mr. Griffiths makes.

However, Peter has also a good point in that physicist should know and usually knows what is potential much sooner than he understands curl and div operators, so there is no real harm done to physics, as we always deal with simply-connected space R^3.

then the integrand is undefined at said when r’ coincides with r! Division by zero! What the hell!

Matthew, the integral you worry is free of problem if the charge density is finite function in some neighbourhood of the field point in question(where you seek the field). The singularity is indeed in the expression, but is often integrable and the result is finite and non-problematic.

Comment by Jan Lalinsky | December 4, 2012 | Reply

• Jan, thanks for responding to my question. Do you mean that the integral is then defined in an “improper” sense? I.e., would you simply integrate over the charge distribution, but excluding a ball which encloses the field point, and let the radius of this ball approach zero? Or am I thinking about this incorrectly?

I guess I’m just curious how one actually proves what you said.

Comment by Matthew Kvalheim | December 5, 2012 | Reply