The interesting thing about evaluating surface integrals for parametrized functions of three variables is that you don't actually need the function that defines the curvature of the surface--just two directional derivatives of the same. Instead of just integrating for a volume, you instead determine arc length--in two directions at once! This is interesting because you can determine the area of linear surfaces of complex shape using a polar coordinate system without having to parameterize the surface's incidental angle of inclination, without having to derive the κ-curvature seperately, and without having to stuff a complex surface equation into the boundaries of a definite integral. The down side is that this can be a bit counter-intuitive, because you're just taking volume times arc length, and appears to neophytes like myself that the result should just be the volume of a distortion of the initial function. In the end, you just have to trust the equations and hope you don't screw them up too badly.

...some random thoughts on Calculus III. I had some random thoughts on why I hate harpsichords in chamber music, but I forgot to write them down before I stopped caring. That's a greek kappa up there, by the way; if all you see is a funky-looking box, your browser isn't set up for internationalization. Silly Yank. I hadn't realized how many different character sets had been rigged for i18n in the UTF-8 standard, though... it's boring stuff, I suppose, but UTF-8 could eventually enable anyone anyware to use the same computer, regardless of where they're from or what language they speak. This is more useful than is sounds, and much more important to the future of this small world than you might think. It is at the same time another very trivial fact, of little use to most people right now than for odd things, like enabling me to show off a κ to you, properly define the area ℝ³ in a math equation, or throw out a ♧ or ☾ from the "dingbat" encoding sheet. Like I said, trivial. No idea why someone snuck the Lucky Charms shapes into the standard... but there they are. You can check to see how i18n-ready your browser is with a quick proof sheet here (or a slightly longer proof sheet here) or check out an edited mirror of the official UTF-8 encoding charts.

i18n is fun. If I weren't going after an engineering degree, I would probably have been a linguist. Or maybe a poet. ...no, I think I'll leave the poetry to contemporary poet Joan(s). Yeah.

Note to self: it's called a nabla...