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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
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