By DAVID IZRAELEVITZ
Los Alamos
Note to Reader: This Open Book column is part of my occasional “Ex Libris” series about books that have made a difference in my life. Hope you enjoy it.
I was pretty good at Math as an undergrad. I was one or two courses short of a second major in the subject, and even the Math I had to use in graduate school was stuff I had already learned or some uninspired extension of methods I was already very familiar with. I was thus confident of success when I followed my friend Mike’s advice and signed up for 18.305, Advanced Analytic Methods in Science and Engineering, a graduate-level class in the MIT Math Department. After all, he had introduced me to my now-wife, Terry, so his acumen had an impeccable prior reputation.
So, since Mike was so effusive about the mysterious topic of asymptotic expansions, how an approximate solution to a nonlinear differential equation was magically better than an exact one, how it was the hardest class he ever loved, that my toxic mathematical masculinity took over, and I decided that, if Mike can do it and love it, I could too and moreover, I was going to love it as much as he did. The fact that browsing the textbook at the MIT bookstore made it clear that the topics were of absolutely no relevance to my thesis was not going to stop me either. So there!
In fact, the textbook was rather fun to read, interspersed with Sherlock Holmes quotes actually appropriate to the main theme of each chapter. And, unlike most other textbooks I purchased for the sake of decorating another inch of my bookshelves, it was a textbook that actually helped me understand the material and it was very instructive to go through well-documented sample exercises.
The midterm exam came and went, and I was pumped that I performed as well as I did. As one of the few engineers in a class populated mostly by mathematicians and physicists, I thought I was doing a good job of repairing the unjustified reputation held about us by my non-engineer classmates. We are not, I would show them, mathematical Neanderthals who knock simple formulas together until they get some simple machine or model to run. Engineers can do Advanced Analytic Methods as well as anybody.
Well, soon after my successful midterm, I was hit by the analytic equivalent of a Mack truck. The second half of the course expected me to know a variety of incantations worthy of Harry Potter in order to transfigurate some nonlinear differential equation into a form already solved by some famous mathematician (or physicist, argh!) and named after himself. I could follow each example, but if faced with a new problem, I could not reposition my head so as to see a change of variables that uncovered the canonical form of the modified Bernoulli equation (Bernoulli being a mathematician, a physicist, and namesake of many equations). If this sounds like mumbo-jumbo, well, that was me the second half of the 1984 Fall Semester at MIT.
The three-hour open-book final exam time came. From past experience, I knew well that an open-book test just gives the instructor the latitude for unlimited difficulty; after all, you have the book right in front of you! I was doomed.
I was not going to go down without a fight, however. I redid every homework that we had been given, and every solved exercise in the textbook that I had time to investigate. I even took extensive notes with page references to minimize the time I might need to look up something. Six problems, 30 minutes per problem. One cannot get bogged down in any single problem; gotta keep it moving to accumulate at least partial credit.
The reputation for difficulty was well-deserved, and a complete solution to one of the six problems was likely to earn me a B+ or even an A. Partial credit, hit a wall, go to the next one… partial credit, hit a wall, go to the next one… partial credit, hit a wall… wait, didn’t I see this problem before? One of the final exam problems looked like an exercise I had reviewed a few nights before. The problem was identical to my repeated inspection, but I looked again and again for some devilish change in sign, and overlooked exponent that would make the problem worthy of a final exam.
Desperate, I went to the sleepy teaching assistant in front of the class and pointed to the page in the book. He looked back at me with a poker face worthy of the circumstances. I sat down and transcribed the solution provided in the textbook.
The grade of A in 18.305, Advanced Analytic Methods for Scientists and Engineers during the Fall of 1984 did not inspire me to take its Spring term successor, 18.806, Advanced Partial Differential Equations with Applications. More than asymptotic expansions, I now understood that knowing the basics of a subject does not qualify me as an expert who can intuit approaches dependent on creativity, expertise and past experience. For $41.00 plus tax, Bender and Orzag taught me something worth, much, much more.
A little humility.
