Andy Latto <andy.latto@pobox.com> wrote:
Keith F. Lynch <kfl@keithlynch.net> wrote:
One of the most astonishing (to me) calculus theorems is that any smooth curve (in the sense that all of its derivatives are continuous) can only be extended in one way. Any non-zero-length segment of such a curve, no matter how short, uniquely determines the rest of the curve. Also, if you know all the derivatives at any one point, then you know the value at every point.
What's most astonishing to me is the fact that calculus classes, while usually not actually making any false statements, seem designed to mislead people into thinking that this statement is true. It's completely false.
Thanks for the correction. I don't know where I learned it. I own several calculus books. Most of them are older than I am (and I am not young), but presumably this has been known to be false for centuries. What I do know is I didn't learn it in any calculus class, as I've never taken a calculus class. I picked up my math on the street. That's what comes from hanging out with the wrong crowd, I guess.
and for x < 1 with the function f(x) = -47 e(- (1/x^2))
Why 47? Will other numbers work?
Why do so many people think the false "theorem" is true? One reason is that when we consider functions from the complex numbers to the complex numbers, rather than from the reals to the reals, the theorem is true.
At least that rescues the Riemann zeta function. Nobody ever seems to suggest that it has more than one possible analytic continuation. What about from the reals to the complex numbers or vice versa?
I think the real reason the false theorem is believed is that calculus texts place great emphasis on the notion of the Taylor series of a function, and want to emphasize its importance. Its importance would seem diminished if the texts let out the dirty secret that even if the Taylor series converges, there is no reason to expect that it converges to the original function, rather than to some completely different function, even on a neighborhood of 0.
That's disturbing. I'll admit to worshiping at the altar of the power series. Mainly because what else is there? At least I didn't (directly) use them when I had to raise e to the power of a matrix. Instead I divided the matrix by 2^n, then added the identity matrix to it, then squared it n times. I tried several values for n, and used the one that had results closest to its neighbors. (Is this a known algorithm, or is it original with me?)
Since it seems plausible that my own motion is smooth, i.e. I never experience infinite anything (distance, velocity, acceleration, jerk, snap, crackle, or pop (yes, those are the official terms for successive derivatives of position)), that implies that all my past and future travels are predetermined.
Only if you think it plausible that your motion is well-defined and smooth for complex values of t, and that there is reason that the resulting function is differentiable in the strong form of being differentiable as a function from C to C, rather than as a function from R^2 to R^2.
Does it help that my position, velocity, etc., are all three-vectors? Probably not. For one thing, if my position is defined as my center of gravity, whenever I trim a fingernail there's an instant when the clipping ceases to be part of me. At that instant my center of gravity instantaneously shifts. It's arbitrary exactly when in the process of trimming a nail that happens, but I don't think there's anything arbitrary about it happening, or about it being instantaneous.