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? -- The Mathematica Wiki: http://www.mathematica-users.org/ Math for Comp Sci http://www.ifi.unizh.ch/math/bmwcs/master.html Math for the WWW: http://www.w3.org/Math/