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Re: Gray's Differential Geometry error?

• To: mathgroup at smc.vnet.net
• Subject: [mg63265] Re: Gray's Differential Geometry error?
• From: rip pelletier <bitbucket at comcast.net>
• Date: Tue, 20 Dec 2005 23:35:52 -0500 (EST)
• References: <dnv48c$och$1@smc.vnet.net> <do3lkh$o22$1@smc.vnet.net> <do684q$b0a$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <do684q$b0a$1 at smc.vnet.net>,
 "Steven T. Hatton" <hattons at globalsymmetry.com> wrote:
> 
> Thanks for the confirmation.  There are few things more frustrating than
> minor notational inconsistencies in the presentation of difficult
> mathematical concepts.  I have to wonder if Gray did that intentionally. 
> Kind of a hidden exercise for the reader.

   i rather suspect that it was an unintentional error. i have learned 
that people who know the answer don't always get the derivation right.

> 
> I was quite disappointed to learn that Dr. Gray has passed away.  I was
> hoping I might be able to meet him.  The book is, indeed, a work of art.
> 
> I do have another question regarding his book.  On page 40, there is part of
> a proof using complex variables.  He shows an equation expressing the the
> derivative of position wrt the curve parameter on an ellipse.  So far I
> have not been able to convince myself that the second form is correct.
> 
   good. it isn't. within each ( ) the coefficient of E^(-I t/2) should 
be sqrt (a-b) instead of sqrt (a+b), like eq 2.5 on p. 39.

> 
> I am inclined to believe this is correct (not a typo), but have not yet show
> it to be.  My suspicion is that it follows from some kind of "completing
> the square" manipulation.  Do you believe the second expression correctly
> expresses dz/dt?

   the correct expression comes from factoring (U^2 - V^2) as (U+V)* 
(U-V).

   i think it's a cute proof, but i have reservations. how does he know 
the foci are at ± srqt(a^2 - b^2) ? it's true, of course, but i derive 
that from the "sum of the two distances is constant" rather than the 
other way around.

   i also suspect that mathematica should be used to simply compute 
angles and distances rather than do algebra.

   finally, we could take the differential geometry questions to email 
if you like; my eddress is at the bottom of the post.

   hope this helps. thanks for the questions.

vale,
   rip

-- 
NB eddress is r i p 1 AT c o m c a s t DOT n e t