You might be interested in the theory of generalised functions. A generalised function (sometimes called a 'functional') has an unlimited number of derivatives unlike an ordinary function. Even step functions or impulse functions have derivatives at all points in the theory generalised functions. This was not considered possible until the first half of the 20century. But I am not a mathematician so please don't ask me to explain.
Harry Grimer at [EMAIL PROTECTED] wrote: > In previous posts the idea of a series of increasing > orders of derivatives of length with respect to time > has been chewed over. > > It will be helpful to recapitulate the discussion by > using excepts from the Vortex archives. > > On the subject of imagining what these derivative mean > physically, we have. > > > ======================================================== > ...Funny you should say that, Richard, because I've been > pondering how one could physically visualize high order > derivatives of distance with respect to time. > > dL/dT ......VELOCITY .......moving scenery > - no problem > > d2L/dT2 ....ACCELERATION ...being pushed back in ones > seat as the plane takes off > - no problem > > d3L/dt3 ....JERK............Mmm..more difficult - being > hit over the head with a > bottle perhaps? > > d4L/dT4 ....JOUNCE..........I have no feeling whatsoever > for this or high derivatives. > > But the failure to visualise these higher order derivative > is because I am thinking in terms of straight line motion. > If I think instead in terms of circular motion, or better > still, helical motions, then things become very much easier. > > If I allow myself to be pinned to the wall of a fairground > centrifuge then I can experience being "pushed back in my > seat on a continuous basis. By imposing a circular motion > on this circular motion to form an open vortex helix I can > visualize the next derivative, though I am well past the > age where I would want to experience it - and so on - and > so forth. > > ======================================================== > > On the topic of visualisation Keith Nagel wrote, > > ====================================================== > Also, you mentioned Jerk and Jounce ( sounds like > a b-list rap group ). I've also puzzled over the > physical meaning of these terms. It's rather like > trying to imagine higher dimensional shapes. One > dimension up is about all I can muster, which in > this case is Jerk. Standing on a carousel, with > the speed increasing and decreasing sinusoidally, > ought to do it. Perhaps a better term would be > "projectile vomiting" rather than jerk, huh??? (grin). > ====================================================== <snip>
