An inescapable obstacle in learning mathematics is its trade off between allure and difficulty. I do not know, nor can I recall, how public school math teachers sustain a veneer of enthusiasm for the piffle which appears thoughout the agoge. What's that? Why is the quadratic formula important? Well, children, you use it when you're enclosing a fenced area in a square! And, um, maybe to balance your checkbook? Have I mentioned enclosing a fenced area?
It is the very definition of a no-win situation. The truth may set you free, but here the truth would cause students to riot and may well get you fired. You really want to know why the quadratic formula is useful? All right, here's the truth: It's not. There. I said it. All the happy cartoons in the world showing everyone from astronauts to zoo keepers using the quadratic formula in their daily life won't change that. It's a stepping stone. A way-station. If you understood contour integration we might be able to converse about something interesting. But noooo, NCLB says that's too "advanced" for third grade.
Here our harsh mistress smiles in a manner indistinguishable from a sneer. Suffer the little children.
It is only when a topic is mastered that its beauty can be appreciated. A kind of runner's high that eclipses the pain, albeit existing in the life of the mind. In that, the learning of mathematics is no different from any other human endeavour. If you practice your stupid fretting, one day you'll be able to play Eruption. If you do stupid push-ups, one day you will be able to rock climb in Europe. If you learn your stupid Latin, one day you will be able to glory in the Aeneid as Virgil intended. Similarly, If you master the necessary prerequisites, one day you will be able to appreciate mathematics for its beauty, the sublime manna washing over you like a soothing balm.
Or so I'm told. To be honest I've yet to experience sublime manna washing over me like a soothing balm, in math or in any intellectual realm. The closest I've come to-date is Cormac McCarthy's timeless cowboy drama Blood Meridian, which you can open to any page and bask in the exquisite prose on display, assuming you're not distracted by the corpses and the raping and the corpse raping. Louis L'amour, he ain't.
In mathematics, however, the manna has always been less a soothing balm and more a beard of bees. But if there is any one area in which I have some faculty ("mastery" would be far too optimistic a term) it would be ordinary differential equations. Not because there are no topics I find easier (I wield a mean quadratic formula) nor because there do not exist topics that are more universally useful (cf. partial differential equations) but because ODE achieve a balance of allure and difficulty, the spectre tightrope we have been walking today in this exotic circus.
So imagine my joy upon discovering Paul Waltman's A Second Course in Elementary Differential Equations. It is a thin volume, Waltman is, and -- paradoxically -- I don't recommend it to learn the material de novo (that task always falls to Greenberg). No, Waltman is my differential equations novel, the ODE equivalent of an art appreciation class. The book you take with you to Fenway park and read while the Cubs are losing, an $18 chilidog in one hand and your Chuck Taylor's propped up on the empty seat in front of you. The math book you inhale and swirl and taste like a fine cooking sherry. The equivalent of the beat poet works the barista at your local independent coffee shop is always going on about, until the inevitable hostile takeover by Starbucks suffocates her free spirit.
I don't imagine the title rings a bell. Evidently not a best seller, although the book is listed in the usually-excellent Dover catalog. I found it by accident at my campus bookstore, buried at the bottom of a sale bin under a mountain of surplus Olaf plushies and great works of zombie fiction and hastily-produced Mockingbird slash fic. There it lay whimpering in the corner, like Hope after the furies had been savaging it for millennia until Pandora turned them on us.
The cover told a sad tale of ever-increasing enticements to purchase, a stack of yellow price tags one on top of the other that were we to carefully exhume the strata we would discover the original modest asking price down in the bedrock. The top three stickers had gone from $2.98 to $0.99 in short order, and then to $0.98. Presumably, the next price reduction would have been written in sharpie. Would one of you please just steal this so we don't have to pay to have it burned? At least Waltman was spared the humiliation of having a notch band-sawed into his life's work to get it off the books, no pun intended.
But now this slight volume sits on LabKitty's shelf like a rescued stray (♫ in the arms of the angels ♫). And slight it is. Less than 250 pages and presented in three parts (four, if you count the chapter on autonomous systems as being distinct from systems and why would you).
Waltman begins with systems of differential equations, a topic that is simultaneously more complex than scalar equations (because system) and less complex (because attention is restricted to constant coefficient beasties else we can't really do much with them). It is here we encounter that intoxicating mixture of eigenvalues and differential equations -- is there any combination more sumptuous?
The second movement comprises a brief tour of existence theory that for once doesn't feel like the Tom Bombadil chapters in LOTR. Finally come boundary value problems -- a topic often given short shrift in first courses due to their initial value fetish. This is capped by a wonderful half-chapter on Green's functions -- the Yeti of differential equations -- and applied to the kind of ODEs that don't immediately induce a seizure (cf. physics literature). Waltman builds a Green's function petting zoo where we can study and poke a Xanax-laced form of the beast in reasonable safety, taking from the experience experience that might serve us should we one day encounter the brutal specimens which roam the wasteland like beings provoked out of the absolute rock and set nameless and at no remove from their own loomings to wander ravenous and doomed and mute as gorgons.
Any title which begins "A Second Course in..." is often a recipe for disaster, the author assuming a tone of you wouldn't be reading this if you were a beginner, ergo I'm allowed to explain it badly. Waltman does not succumb to this temptation. To be sure, there is reason to heed his titular caution. There are theorems here and proofs, and a few exotic climes -- metric spaces and contraction theorems, bifurcations and Bendixson-Dulac. But these never grind the narrative to a halt. Rather, they add depth to this fictional world, like Tolkien's occasional splash of Dwarven or Lovecraft writing in the tongue of the Ancient Ones. We're all richer for having seen them, Gordon Downie once said of the Rheostatics. So, too, Waltman, who will fill in the gnawing ODE shaped spaces your First Course left behind, your incomplete agoge.
Let Waltman fill your holes at Amazon.
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