Our topic is “arc lengths for curves given parametrically”. Good news right away: the first display is (or would be were I not graphically challenged unto uselessness… probably oughta get some pencilpad gizmo or something) the drawing—a curve in the plane with line segments connecting a few points—that accompanies the derivation of the main “formula” (in the sense of “memorize this first of course”) for the section at hand (1):
The first thing I’d like to’ve said about that is that it’s easier to make sense of this formula than of something like . This may be a matter of taste but I doubt it. It appears to me that by thinking of the x and y co-ordinates of a point of the curve as functions of a parameter, we are removing an artificial asymmetry that exists in the “y is a function of x” point of view. By not even having to ask which co-ordinate is to be considered “independent” and which “dependent”, we get a simpler proof and a more symmetric, principle-revealing, easily generalized thing-of-beauty and joy-forever, formula (1).
The next thing is probably along the lines of “this derivation would be well worth looking at even if it were entirely review”. Touching various parts of the drawing with chalk or pencil (or in our case, alas, whiteboard marker) while talking about the symbols leads in the best-case scenario to thinking like a physicist… one becomes able to perform certain calculations by manipulating some visual-seeming “object” in the time-and-space of one’s imagination. This is something I’m not very good at; quite often like the Algebra major I am I’m reduced to pure symbolism and have to think of certain combinations of letters (and suchlike symbols) undergoing certain transformations according to rules.
What’s always been clear to me—to my considerable frustration—is that, once one “sees” the meaning of, say (rather than thinking of it as a piece of “code” to be “plugged in” as part of some calculation), a proof like that for the Area of a Surface of Revolution becomes simply an exercise in writing down the obvious.
The “geometric” material in this course was mostly invented for the needs of physical science (the study of electromagnetic forces in particular). If you get good at it now, probably you could go on to do well in ODE and grab hold of the half of the Program that almost completely eluded me.
(Instant footnote: “ODE” means “Ordinary Differential Equations” in mathmajorese [everybody else says “diff, E, Q”; ugh]—the subject that follows our course in a typical STEM major (Science, Tech, Engineering, Math) and which is typically [but not for me] followed by PDE—Partial Differential Equations—also known, in math departments, as “Applied Mathematics”. Math for physics, in a word…)
Any new “formula” should be taken for a test drive. In the last lecture, I looked at two problems that somehow eluded me last time… the line segment and the arc of a circle. These are the simplest interesting examples; one habit of mind I’ve found very useful is to look for these. I blogged about this idea last quarter in VME. Specifically, I did x(t) = t/3, y(t) = t/4, for (the segment connecting the origin to (4, 3)… like a beginner I pulled this parametrization out of thin air in my eagerness to apply formula (1)… today I’ll probably say some more about how to find it…); then I said “here’s a problem from today’s quiz; I’m afraid the class as a whole isn’t ready; somebody do it at the board and I’ll give the whole class full credit”… and put up x =cos(t), y=sin(t), and somebody stepped up and did it. Yay, this class.
As it turns out, there are only a few “types” of arc length problem where it’s practical to work out exact answers with paper-and-pencil methods so suchlike basic-example motivation might be even more crucial than necessary. As simple a curve as the parabola is already brutally tough for Calc III according to my admittedly very limited understanding: the curve —a textbook problem—requires a tricky substitution in the integral (supplied by a very capable student last week) even in order to transform it into another tricky integral, one I blogged about last year.
Then there’s the cycloid (and related curves). And some carefully constructed quadratics, designed so that “middle terms” will “cancel” in a convenient way. And that’s all I know. Probably we’ll do one from this last category today. The first exam is Monday…