Mary Boas.

[NNadir]

So I went to my kid's Masters graduation ceremony this weekend and I got to thinking it's probably been a long time since he took a formal math class. He took four semesters of Calculus in high school; the last one, vector calculus not for college credit, and repeated the course in his Freshman year of college.

So I'm thinking to myself, "Self, that kid needs some review and reference for his Ph.D program."

I had a nice book on partial differential equations for physics somewhere around here, but I can't find it. It's been many years since I last saw it; maybe someone borrowed it and never brought it back.

So I decided to buy a nice math review/reference book as a gift. A kind of general reference for this sort of thing - the reviews are mixed from "Love it!" to "Hate it" - is Erwin Kreyszig's Advanced Engineering Mathematics, 10th Edition.

Since there were a subset of people who whined about this book - it's apparently utilized in upper division undergraduate courses - I asked myself, "Self, what else is out there?"

Well, there's Mary Boas's Mathematical Methods in the Physical Sciences

It's an old book, but a lot of people still like it, apparently. Dr. Boas died in 2010, after having retired from DePaul in 1987. The book dates from the 1960s.

She was a pioneer as this blurb from DePaul indicates: Women’s History at DePaul

An excerpt:

Women’s achievements in American history can be celebrated while simultaneously highlighting the established gender roles of a particular era. A newly-acquired book held in DePaul Special Collections and Archives showcases some of women’s groundbreaking successes. After the War: Women in Physics in the United States shares the stories of women who continued to work in the male-dominated physics field even after men returned stateside as veterans and reentered the workplace following World War II. The new book by Ruth H. Howes and Caroline Herzenberg celebrates important women in physics during this time and the various “strategies they used to survive as physicists.”

Former DePaul faculty member Dr. Mary Boas, one of the featured physicists in the book, was born in Washington state and received both her bachelor’s and master’s degrees from the University of Washington. Boas’ husband Ralph taught mathematics at numerous universities, including Harvard, MIT, and Northwestern. Mary received her PhD in physics from MIT in 1948. After the family moved to the Chicago area in the 1950s, Mary Boas took a teaching position at DePaul University in 1958...

I'll probably end up buying the Kreyszig book in ebook form, but it was nice to come across a pioneering woman scientist, even if, as the article states, she had to put up with "Puff pieces" about her life.

[xocetaceans]

1. That's an interesting bit of history regarding that particular author.

Kreysig does not look bad.
Boas is not a bad book, though.
Arfken can be avoided.

If you want to look at an older pair of books (for fun), take a look at Morse and Feshbach, Methods of Theoretical Physics - Part I and Part II.

There's also Courant and Hilbert: Methods of Mathematical Physics - Vol. I and Vol. II.

These two sets of books are fun to look through. They might not align with the purpose of a review, though.

I'd be a little concerned with a master's degree program (if it were in physics) that has not already significantly addressed applied math. Anyone taking a physics master's degree would have already completed a mathematical methods course at the level (at least) of Boas. Special topics courses usually align with Ph.D. programs more directly; i.e., Lie groups, representation theory, etc.

Of course, places and degree requirements vary, so my comments might not even remotely apply outside of physics. Regardless, Morse and Feshbach is encyclopedic in nature and would be worth considering as a reference or even for its historical value. (Or just for fun....)

[NNadir]

2. My son's MS is in materials science. He's entering a nuclear engineering Ph.D program.

His Master's thesis was concerned with metallurgy chiefly alloy interfaces. It was a one year program.

He will need to shift gears a bit, but we've been discussing topics such as heat transfer, and multi-group neutron diffusion, perhaps in high entropy alloys. I wasn't formally trained in these topics, which makes life easier on some level for me; he will require more rigor I suspect.

I suspect he has fairly deep mathematic ability, but we don't discuss it that much. I know he's done a fair amount modeling, and at one point was going to go to Sweden to work with the Thermo-calc people but Covid put an end to that idea. He know he's fairly fluid computationally.

I'm neurotic and living somewhat vicariously through him. Over the years, I've found that I often need to refer to references. He probably doesn't need all that much from me, but his old man can't help meddling, or cheering him on; I don't know how he takes it; he tolerates me. He has never expressed anxiety about a challenge except maybe once when they sent him overseas after his Freshman year.

When I'm dead, maybe thinking of me doing these things will give him a good laugh. I hope so.

[xocetaceans]

3. That sounds like very cool stuff, though I don't know those subjects myself.

When I saw Boas mentioned, I had to look at your post.

Knowledge is the best thing that humans can give each other: it's our best hope for a better future.

He (or you - if you're not familiar with it already) might enjoy Morse and Feshbach. It is a classic (reference) text, and, at nearly 2,000 pages total, there's bound to be something slightly new or slightly forgotten to be seen in the two volumes.

It does sound like he has a really good mathematical foundation from what you've written. Hopefully, more kids are able to get calculus in high school now. (The internet will help with that.... However, my old high school - for whatever reason - still only offers a "topics in pre-calculus" course as its highest math elective.)

I don't remember exactly, but I think that Reif has a discussion or example relating to neutron diffusion near the end of his book, Fundamentals of Statistical and Thermal Physics. I don't have a copy on hand and just don't quite remember. That might be fun to check out.

As before, all the best to your son's future endeavors in the nuclear engineering Ph.D. program. It's cool that you care to be involved.

[NNadir]

4. Well, I did buy the Kreyszig e-book, and briefly scanned through it.

I have this habit of taking a look at the first couple of pages of any book before drifting into the good stuff.

The ebook is not a *.pdf; I haven't bought ebooks from Wiley; Elsevier lets one download PDFs. Wiley makes you use a program called "VitalSource." It's OK if you get used to it, but it's another password to remember; not good for old people.

It's pretty pedestrian, the first chapter; I find a "pedestrian" first chapter to be a good idea, as people come to a book at all different levels: It's simple first order first order differential equations. One can do most of the problems in one's head, although I have to confess, being old, and having not actually solved many differential equations I encounter for many years - I generally take the authors of papers in which they appear for their word on the solutions - I had to recall very basic stuff like integration by parts, which caused my old brain to pause for a minute or two.

The book has only one reference to neutrons, a problem:

Find the 95% confidence level for the variance of a normal population from the sample...

...Mean energy (keV) of delayed neutron group (Group 3 , half‐life 6s ) for uranium-235 fission: a sample of values with mean 442.5 and variance 9.3.

For me, perhaps not for real nuclear engineers or, ultimately my son, as I am mostly interested in fuel, the governing equations of nuclear engineering are the Bateman equations. The form I like for them is this one:



Cetnar, J.; Stanisz, P.; Oettingen, M. Linear Chain Method for Numerical Modelling of Burnup Systems. Energies 2021, 14, 1520. https://doi.org/10.3390/en14061520

The author of the paper from which this graphic comes also published a much cited paper on the topic of the Bateman equations some years back, this one: Jerzy Cetnar, General solution of Bateman equations for nuclear transmutations, Annals of Nuclear Energy, Volume 33, Issue 7, 2006, Pages 640-645.

Of course, long before this paper was published, there was lots of software to address these coupled equations numerically; they worked fine since most nuclear reactors work fine. This software has been, of course, greatly refined over the years, even as the nuclear properties of nucleus i or nucleus j have been refined.

Many years back, while drifting somewhat aimlessly through the Firestone Library at Princeton, I came across Serber's famous Los Alamos Primer. What I recall of it was how the people on the Manhattan Project were introduced to neutrons pretty much with statistical mechanics equations. The point is that neutron diffusion and neutron absorption are not entirely out of the realm of closely related mathematical physics.

It occurs to me, now that I think about it, that the "Teller" in Brunauer–Emmett–Teller (BET) theory may have had its origins in thinking about neutrons as models of adsorption.

I think as climate change accelerates, we urgently need to look at rather different nuclear fuels beyond simple enriched uranium.

When I contemplate these things, wonderful thoughts eddy through my mind:

One of the interesting things about the Bateman equation is that the λ, φ, σ in the equation are really not constants, they are functions of temperature, particularly in the case of σ for capture owing to Doppler broadening, and functions time and space, albeit localized space. Moreover φ, the neutron flux, is very much a function of ΣN, the composition, also a function of time. Moreover, there are really two temperatures in a reactor, the "Boltzmann temperatures" of neutrons, and the "Boltzmann temperature" of the fuels itself, fuels that I, at least, if not my son, envision as liquids, cooled by gases. Indeed the "Boltzmann temperatures" of neutrons will be coupled to the Bateman ΣN, and these will be functions of space and time. So we have heat equations, the classic differential equations, compositional Bateman equations, and diffusion equations, all coupled, never mind Breit Wigner distributions for resonances.

I'm quite confident that all of this can be handled numerically, and for now it is, and the very powerful computational infrastructure at Oak Ridge, where my son interned, for example can address these systems; more primitive computers have long been sufficient for practical engineering over the years.

But the thought of the depth of mathematics involved is very beautiful, and whether he needs it or not, I encourage him to keep his mathematical muscles stronger than I have kept mine. Perhaps he'll think me quaint, but he accepts these things are the price of having a father.

My son has made it clear to me, and to the people who interviewed him that his interest is more in experiment than in theory, but I'm sure that he will appreciate how beautiful theory is.

I'd do my life differently if I thought these things when I was a kid, but I was a bit of a jerk as a kid. If I accomplished anything, I raised two sons who escaped many of my mistakes, who are more ready to serve the world than I ever was.

In a "do over," I'd definitely be a nuclear engineer, and my son I guess will need to tolerate his Dad's vicarious interest.

My Father, who I loved very much, and who I still miss decades after his death, was in no position to cheer me on in the same way I cheer my son on. My Dad quit school in the 8th grade.

I know this is all esoteric, but thinking about it set me to musing about how things converge. My sons will live in interesting times, and much of that is surely very scary, but it's nice to know they're equipped to do what can be done.

Thanks for your wonderful advice and comments.

[caraher]

5. I still teach from Boas's text

One reason is that it's not hard to find at a reasonable price, which increases the odds of my students actually buying and keeping a paper copy. I'll also admit that I quite enjoy correct students who embed some version of "what does he mean when he says..." in questions referring to a text passage.

I do remember liking Arfken's book, but I'm not sure it's great for my student population.

Back to Boas for a moment... it's just enough to get a student through a physics major, but there are some topics where it's important to bring in other perspectives. The linear algebra treatment is very different from that of more modern books, and ones that make more use of computers.

Right now I'm enjoying a book that I probably wouldn't pick as a text, the No Bullshit Guide to Linear Algebra. Chapter 1 is a very wide ranging survey of preliminary math and logic topics, and I appreciate the humor. As an example (and I won't be able to fully replicate this here; can DU add support for LaTeX? ), an exercise at the end of the chapter is:

Let M denote the set of people who run Monsanto, and H denote the set of people who deserve to burn in hell for all eternity. Write the math statement (for all p in set M, p is in set H) in plain English.

The No BS Guide closes with three chapters of applications, with one on probability and one on quantum mechanics (the third and longest is a grab bag). Along the way, it also integrates links to high-quality YouTube videos (not the author's work) to help with conceptual understanding (and without any illusions that watching such videos in any way substitutes for actually working problems).

[NNadir]

6. Hey, thanks. I'm going to buy the "No Bullshit Guide to Linear Algebra" for my oldest son, the...

...commie artist, the kid who told me he didn't "need" to take math because he was planning on majoring in Art - which he did (and he has a reputable job) - but who found himself writing code for artists and designers, and spends his time reading Gilbert Strang's texts.

He'll get a kick out of it, and love the Monsanto joke.

(For the record, I've worked with Monsanto people, and believe it or not, many of them care as deeply about the world as anyone else.)

[caraher]

7. I'm sure they do

Still an amusing setup. The previous exercise excoriates bankers.

[NNadir]

8. Oh, I agree.

I think the exercise you describe is very funny.

I suspect some of the people I knew a long time ago at Monsanto would agree.

Many years ago, I visited their labs in Saint Louis to give a talk, and there were lots of jokes over lunch about pitchforks, horns, what have you.

I think in the coming years we're going to see some food crises; I'm personally gratified that we have agricultural science, in particular, genomics. Some of the people I met there were excellent scientists, first rate.

As that guy said when he was a kid, "A hard rain's gonna fall..."

I'm sure my son, who's far to the left of where I am, will enjoy those exercises in any case. I'm sure they'll inspire him to give me lectures, not about math, but about politics.