# Gbur’s Mathematical Methods

REVIEW: Gregory J. Gbur (2011), Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. [Post also available in PDF.]

As it was, I had to wait until this past January, at the ScienceOnline 2011 conference. These annual meetings in Durham, North Carolina feature scientists, journalists, teachers and students, all blurring the lines between one specialization and another, trying to figure out how the Internet can help us do and talk science. Lots of the attendees had books recently published or soon forthcoming, and the organizers arranged a drawing. We could each pick a book from the table, with all the books anonymized in brown paper wrapping. Greg “Dr. Skyskull” Gbur had brought fresh review copies of his textbook. Talking it over, we realized that if somebody who wasn’t a physics person got a mathematical methods textbook, they’d probably be sad. So, we went to the table and hefted the offerings until we found one which weighed enough to be full of equations, and everyone walked away happy.

MMfOPE is, as the kids say, exactly what it says on the tin. It begins with vector calculus and concludes with asymptotic analysis, passing through matrices, infinite series, complex analysis, Fourierology and ordinary and partial differential equations along the way. Each subject is treated in a way which physicists will appreciate: mathematical rigour mortis is not stressed, but when more careful or Philadelphia-lawyerly treatments are possible, they are indicated, and the ways in which their subtleties can become relevant are pointed out. In addition, issues like the running time and convergence of numerical algorithms are, where appropriate, addressed.

# “More Decimal Digits”

On occasion, somebody voices the idea that in year $N$, physicists thought they had everything basically figured out, and that all they had to do was compute more decimal digits. I won’t pretend to know whether this is actually true for any values of $N$ — when did one old man’s grumpiness become the definitive statement about a scientific age? — but it’s interesting that not every physicist with an interest in history has supported the claim.

One classic illustration of how the old guys with the beards knew their understanding of physics was incomplete involves the specific heats of gases. How much does a gas warm up when a given amount of energy is poured into it? The physics of the 1890s was unable to resolve this problem. The solution, achieved in the next century, required quantum mechanics, but the problem was far from unknown in the years before 1900. Quoting Richard Feynman’s Lectures on Physics (1964), volume 1, chapter 40, with hyperlinks added by me:

In network science, one can study the dynamics of a network — nodes being added or removed, edges being rewired — or the dynamics on the network — spins flipping from up to down in an Ising model, traffic flow along subway routes, an infection spreading through a susceptible population, etc. These have often been studied separately, on the rationale that they occur at different timescales. For example, the traffic load on the different lines of the Boston subway network changes on an hourly basis, but the plans to extend the Green Line into Medford have been deliberated since World War II.

In the past few years, increasing attention has been focused on adaptive networks, in which the dynamics of and the dynamics on can occur at comparable timescales and feed back on one another. Useful references:

# Of Predators and Pomerons

Consider the Lagrangian density

$\mathcal{L} (\tilde{\phi},\phi) = \tilde{\phi}\left((\partial_t + D_A(r_A – \nabla^2)\right)\phi – u\tilde{\phi}(\tilde{\phi} – \phi)\phi + \tau \tilde{\phi}^2\phi^2.$

Particle physicists of the 1970s would recognize this as the Lagrangian for a Reggeon field theory with triple- and quadruple-Pomeron interaction vertices. In the modern literature on theoretical ecology, it encodes the behaviour of a spatially distributed predator-prey system near the predator extinction threshold.

Such is the perplexing unity of mathematical science: formula X appears in widely separated fields A and Z. Sometimes, this is a sign that a common effect is at work in the phenomena of A and those of Z; or, it could just mean that scientists couldn’t think of anything new and kept doing whatever worked the first time. Wisdom lies in knowing which is the case on any particular day.

[Reposted from the archives, in the light of John Baez’s recent writings.]

# Vertices on the Blogohedron

Meanwhile, our long cybernetic nightmare continues. Thoughtlessness abounds. Things fall apart; the ice sheets cannot hold. Bloggers demand their due from a system which is ill-prepared to provide it. Satirists strive mightily to invent new jokes about Twilight.

Some things I’ve enjoyed reading recently:
Continue reading Vertices on the Blogohedron

# Textbook Cardboard and Physicist’s History

By the way, what I have just outlined is what I call a “physicist’s history of physics,” which is never correct. What I am telling you is a sort of conventionalized myth-story that the physicists tell to their students, and those students tell to their students, and is not necessarily related to the actual historical development, which I do not really know!

Richard Feynman

Back when Brian Switek was a college student, he took on the unenviable task of pointing out when his professors were indulging in “scientist’s history of science”: attributing discoveries to the wrong person, oversimplifying the development of an idea, retelling anecdotes which are more amusing than true, and generally chewing on the textbook cardboard. The typical response? “That’s interesting, but I’m still right.”

Now, he’s a palaeontology person, and I’m a physics boffin, so you’d think I could get away with pretending that we don’t have that problem in this Department, but I started this note by quoting Feynman’s QED: The Strange Theory of Light and Matter (1986), so that’s not really a pretence worth keeping up. When it comes to formal education, I only have systematic experience with one field; oh, I took classes in pure mathematics and neuroscience and environmental politics and literature and film studies, but I won’t presume to speak in depth about how those subjects are taught.

So, with all those caveats stated, I can at least sketch what I suspect to be a contributing factor (which other sciences might encounter to a lesser extent or in a different way).

Suppose I want to teach a classful of college sophomores the fundamentals of quantum mechanics. There’s a standard “physicist’s history” which goes along with this, which touches on a familiar litany of famous names: Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, Werner Heisenberg, Ernst Schrödinger. We like to go back to the early days and follow the development forward, because the science was simpler when it got started, right?

The problem is that all of these men were highly trained, professional physicists who were thoroughly conversant with the knowledge of their time — well, naturally! But this means that any one of them knew more classical physics than a modern college sophomore. They would have known Hamiltonian and Lagrangian mechanics, for example, in addition to techniques of statistical physics (calculating entropy and such). Unless you know what they knew, you can’t really follow their thought processes, and we don’t teach big chunks of what they knew until after we’ve tried to teach what they figured out! For example, if you don’t know thermodynamics and statistical mechanics pretty well, you won’t be able to follow why Max Planck proposed the blackbody radiation law he did, which was a key step in the development of quantum theory.

Consequently, any “historical” treatment at the introductory level will probably end up “conventionalized.” One has to step extremely carefully! Strip the history down to the point that students just starting to learn the science can follow it, and you might not be portraying the way the people actually did their work. That’s not so bad, as far as learning the facts and formulæ is concerned, but you open yourself up to all sorts of troubles when you get to talking about the process of science. Are we doing physics differently than folks did N or 2N years ago? If we are, or if we aren’t, is that a problem? Well, we sure aren’t doing it like they did in chapter 1 of this textbook here. . . .