[FRIAM] Diff & Contin. to Nick!

[Owen Densmore]

[I sent this earlier but it apparently failed to be sent to the list.]

On Jul 25, 2007, at 10:53 PM, Peter Lissaman wrote:

>> 2. DIFFERENTIABILITY AND CONTINUITY (Nicholas Thompson)
> Nick: Let me be your math consultant! Taught that stuff at Caltech  
> many years!! The mathematicians are horn swogglin' you with mis- 
> understood function theory! A'course the f'n roof is continuous. If  
> it weren't the rain would come through! It is trivial to write a  
> continuous function, f(x) defined for 0<x <=c and g(x) defined for  
> c<x<1 with f(c) = g(c), with the peak at x=c  and a different slope  
> for x=c, than for x>c.  But the function is continuous. Just like a  
> roof ridge.  A geometric function has, at each point, some degree  
> of continuity, denoted by C N, where N is the order of the first  
> discontinuous derivative. The triangular roof frame rafter is C1,  
> meaning continuous in ordinate, discontinuous in slope. Smoother  
> shapes have continuity of higher derivatives. Analytic functions  
> have infinite continuity (thanks to M. Cauchy!). Airfoils have to  
> be very smooth, but they can't be infinity smooth, since we need to  
> tailor the pressure distribution to control separation, and the  
> trailing edge must usually be sharp.   Some of my airfoils of the  
> olden days, when we did this by hand, were C16
....

This is abs-fab! .. I hadn't realized that continuity had been  
categorized in quite this way.  The Mother Of Truth sez:
   http://en.wikipedia.org/wiki/Parametric_continuity
.. which alas is still a stub.  I bet you'd be popular if you filled  
it out!

     -- Owen