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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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