Thursday, September 04, 2008

the less0n p1an that i can't seem to write.

one thing is going to bother me about teaching tomorrow.

we're covering the curvature of vector-valued functions r(t) and their associated curves, and there is a formula for curvature k(t) only in terms of the first and second derivatives r'(t) and r"(t), respectively.

it's a useful formula. you'd otherwise be stuck computing the unit tangent vector and its derivative (w.r.t. the parameter t) which would involve the length of the tangent vector and messy square roots.

the proof -- a clever computation -- isn't terribly intuitive but makes sense, once you make an observation or two. so i'm going to skip it; there's too much to cover, anyway.

however, i hate giving out formulas without explaining them: if you understand why a formula is true, then you're more likely to remember it correctly and less likely to make mistakes with it.

the converse isn't necessarily true, of course.

however, i'd rather not turn my calculus course into a "memorize and use these formulas" sort of course. there is that bias already in mathematics, just as how history is "just names and dates."

the trouble is, i can't find an intuitive way to explain this formula, other than the proof. at some point i should decide something .. and this lesson plan is taking longer than i'd like: there's still some research that i want to do, tonight, and of course ..

.. NSF grant proposals are in a month.
arghhhhhh. i hate myself already.


Leonid said...

The formula says that curvature = normal acceleration divided by square of velocity. In terms of the radius of curvature (R=1/k), it's equivalent to the formula a=v^2/R which they may (or may not) remember from elementary physics.

Damn it, I am exactly two weeks behind you in Calc 3. Were your students introduced to 3D coords/vectors in Calc 2?

janus said...

L: i don't remember that from elementary physics, either. the bit about radius of curvature: i might put it into play in the lesson ..

.. in 20 minutes or so.

as for the 3-dimensional vector geometry, the answer is: "yes, they covered it." i also have a few first years who placed into multivariable and never learned the stuff, so i can't say that "everyone" knows it.

we are also skipping some sections about lines, planes, surfaces .. essentially 3-dimensional geometry, really .. which speeds us up.

say, when did classes start, for you guys?

Leonid said...

Our first class was Aug 25. I think Calc 3 begins too slowly here. As for the a=v^2/R formula, it can be derived geometrically: draw two tangent vectors of length v close to each other. Their difference is approximately v*angle, which is v*(v/R)*dt. Hence the formula.