CAG@ICH#46
I'm beginning to think everything comes together somewhere.
- William Wharton, Birdy
the Eulerian derivative
A back issue of the Bulletin arranged thematically around contemporary assessments of some work of Leonard Euler (in celebration of the tercentennial of his birth) today introduced me to his definition of the derivative (in the aptly-titled "Euler's Definition of the Derivative"). Though not stated explicitly in the article, the concept tastes strongly of rational maps and birational equivalence of varieties in algebraic geometry, both discussed in the most depth yet today in Dr. Green's class and presented in an earlier article of the volume. The basic ideas are quite natural: Define things implicitly and forget about removable discontinuities. For example, the rational function y = (x + 1)(x + 2)/(x + 1) is equal to the simpler y = x + 2 except when x = -1, when it is undefined, even though we know that y should be 1 when x is -1. If, however, we think of the curve of the function instead as the variety (zero set) of the implicit expression y(x + 1) - (x + 1)(x + 2), then the missing point (-1,1) explodes (or, more technically, "blows up") into the collection of points (-1,y), with y anything at all. What we want, though, is to "cancel" the factor (x + 1) from both sides of the equality, or factor it out of both terms in the expression, leaving us, in the latter case, with the expression y - x - 2, whose variety is a perfectly pleasant 1-dimensional curve (in fact a line). This can immediately be done in the world of algebraic varieties (under rational maps, not morphisms of varieties) via the inclusion V(y - (x + 1)(x + 2)/(x + 1)) --> V(y - x &ndash 2), where "V" denotes "variety", which has an inverse going from "almost all" of the latter variety to the former, making the varieties birationally equivalent. All this says is that the former variety behaves nicely everywhere except on a (Zariski-)closed subset, so basically everywhere but a few points (in this case one).
Euler's derivative puts the ratio (y - y0)/(x - x0) in this context, essentially giving it meaning even when x0 = y0, provided the relationship between x and y behaves nicely enough to allow the "cancellation" of birational equivalence. For example, if the variables are related implicitly by the equation x = y2, then a change in y of k occurs in tandem with a change in x of (y + k)2 - y2 = 2k + k2, so the ratio of these changes is k/(2k + k2), an expression morally equivalent to 1/(2 + k). Projectively we might even express this derivative as a value (point) in P1, written [1,2+k], which is well-defined even when k = -2 as [1,0], the "point at infinity".
I realize that my exposition is utterly abstruse except where it's borrowed shamelessly from the hyperlinked article. But this might be shaped into an acceptable way for students to be presented with calculus, being that their loathing for ε-δ proofs is matched only by their irrelevance to practical calculus, provided related courses adopted compatible expositions in tandem to avoid course-to-course inconsistencies. Or perhaps a variety of definitions for the same concept is a healthy experience even for mathematically disinclined applied scientists?
uneasy chair
I should feel like the worst of leeches, regularly planting myself amid hundreds of books in desperate need of sale at the cost of an espresso and an (absolutely delicious) cookie. The Easy Chair may well deserve better. My guilt is tempered by (a) my rule of browsing at least a shelf every visit in search of a title that screams to my tastes and (b) my shrinking dip into the red, which i hope to eliminate for good by a simple change in direct deposit percentages. In particular, last time here i happened to sit right beside Doug Hoekstra's Bothering the Coffee Drinkers, which put (a) and (b) in stark contrast. This songwriter supreme (i'm led to assess by the several positive reviews and comparisons available) is sufficiently obscure, i would suppose by choice, to have no presence on Pandora and no illegal downloads easily accessible (at least by someone outside the vein of the free-music subculture). However, the bit of playback available at hisSpace places his albums, or at least a sample album, on my wish list, and his book will be my next substantial cash purchase, provided no one else discovers it while i reallocate my own funds. (You can read a sample essay here.)
derivative humor
I'm far from unique in declaring Audrey Hepburn to be my favorite female actor of all time, nor in acknowledging her limited range (dramatically and vocally) or basing my admiration more on her socioeconomic status-straddling roles than on her abilities as an actor, her humanitarian work, or her generally amiability as a person. And apparently i'm also in good company when it comes to making obscure references to her roles, as coaxed by leisurely walks almost anywhere a camera crew will fit; if one can stroll down it, Audrey has done so, gracefully as a breeze. It serves as an intimate reminder among admirers to have the sources of her appeal to ourselves so recalled: For me a significant piece is the confident and collected (sometimes exaggeratedly so) demeanor with which she does the most mundane things. Watch her confront an enraged child, or watch a puppet show, or eat a danish. As though the world is provoking her simply by surrounding her, and she'd better keep up appearances. She'll break, of course; we spend most of the movie waiting for it. But she endures, without needing to have some sort of personal epiphany that completely changes her style. The world is still provoking her, but she knows, toward the end, that the world is just as much a player in her game as is she in its. Paradoxically, it is the delusion of balance, or harmony, or duality, between our inner selves and the "outside" that keeps us sane.