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Gray's Differential Geometry error?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg63172] Gray's Differential Geometry error?
*From*: "Steven T. Hatton" <hattons at globalsymmetry.com>
*Date*: Fri, 16 Dec 2005 07:22:30 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
If someone happens to have Alfred Gray's _Differential Geometry of Curves
and Surfaces with Mathematica_ 2nd Edition, I would appreciate a
confirmation regarding my suspicion that he mixed up n and N in the upper
bounds of the summations on pages 8-11, in the proof that the Riemann sum
approximation of the length of a curve in R^n becomes exact in the limit as
the length of the partitions go to 0. Note that when I say "mixed up" I do
not necessarily mean he did something unintentional. Mixed up could be
taken as what happens when you throw different kinds of fruit into a
blender.
It appears to me that, according to his definitions, all the summations on
page 9 should be from 1 to n, and not 1 to N. That doesn't matter so much.
I can be pretty confident of the intent from the context. My real
confusion comes from the epsilon inequality involving N on page 10. I'm
not sure which N he means. Is it the N representing the number of
partitions of the curve, or is it the N representing the number of
dimensions in R^n?
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