Atrophy

Upon entering college, I was fairly convinced I wanted to study economics, and not just the fuzzy track of basic theory, but the kind that at least tried to be mathematically rigorous. So, in my freshman year, I took Math 21a. Multivariable Calculus.

Here are four True/False questions (out of a total of 20) on my first hourly exam, in October 2006:

A line intersects a hyperbolic paraboloid always in 2 distinct points.

1. There is a quadric surface, each of whose intersections with the coordinate planes is either an ellipse or a parabola.
2. The level curves f(x, y) = 1 and f(x, y) = 2 of a smooth function f never intersect.
3. The equation x^2 + y^2 − z^2 = −1 defines a one-sheeted hyperboloid.

Then a series of short answer questions, such as:

Imagine the planet Earth as the unit sphere in 3D space centered at the origin. An asteroid is approaching from the point P = (0, 4, 3) along the path ~r(t) = ((4 − t) sin(t), (4 − t) cos(t), 3 − t).

a) When and where will it first hit the Earth?
b) What velocity will it have at the impact?


I went to office hours all the time and developed nervous habits; after each exam, I exited the auditorium with both a diminished sense of self and ragged nails. Nonetheless, I'm pretty sure that I could answer at least half the T/F questions on sight back then. When I look at the questions now, my mind can't even glean meaning from the scattered symbols. One-sheeted hyperboloid? Quadric surface? Re-enacting Armageddon?

This was three years ago. I don't think I will ever need this knowledge again, though my ego would love it if I could still juggle x, y, and z axes and recognize an elliptic paraboloid just from its function. What bugs me is that I so easily lost not just a formula but an entire body of knowledge. And I notice that after each summer, I read less quickly, communicate less smoothly, process ideas with less acuity. Every skill atrophies with disuse, but I am unaccustomed to thinking of thinking as a skill. So much of who we are seems to reside in our consciousness, which includes our ability to comprehend calculus but also encompasses the whole range of beliefs, responses, and impulses that accompany us through the day and makes us us (to invoke Descartes in a totally irresponsible way).

It's midterm season, so I see copies of old Math 21a hourlies -- now posted as practice exams -- floating around the computer lab. And I am reminded that what I take for granted is not a constant. After all, I've read Flowers for Algernon.