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MathGroup Archive 2005

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Re: Gray's Differential Geometry error?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg63215] Re: Gray's Differential Geometry error?
  • From: "Steven T. Hatton" <hattons at globalsymmetry.com>
  • Date: Mon, 19 Dec 2005 07:01:08 -0500 (EST)
  • References: <dnv48c$och$1@smc.vnet.net> <do3lkh$o22$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

rip pelletier wrote:

> it appears to me that you are correct: the summations on p. 9 are to n
> not N. this implies that the N in the delta-epsilon bound on p. 10 is
> supposed to be n also, not N. he applies that common bound on p. 10 to
> each of n terms in the sum that defines Bj.

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.

Ironically, this seems to relate to my recent question about local variables
and scoping.  It reveals a close correspondence between the notion of a
bound variable in mathematics, and a local variable in programming.  I
believe it would be wise for mathematicians to adapt the use of the term
"local variable" instead of "dummy variable".  "Bound variable" also seems
preferable to "dummy variable", but connotes something a bit different, and
perhaps something very different.

I will say that in the context of Gray's proof, both N and n are local to
the proof, but global to the individual expressions.  The individual
summation indices are local to the expressions where they appear.  N and n
are globally bound within the proof by their having been given a definite
meaning.  Had they appeared within the proof without definition (explicit
nor implicit), they would have been free variables. 

So binding means giving a definite meaning, whereas locality refers to the
scope of the definition of the variable.

> oh, i'd call it a wonderful little book; the only problem is that it
> isn't little, of course.

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.

The parametric form of the ellipse is given by:

z[t_] = (a - b)/(2Exp[I t]) + ((a + b)Exp[I t])/2

Gray asserts that the derivative of z wrt t can be expressed as:

I/2 (s e + s/e)(s e - s/e) /. {s -> Sqrt[a + b], e -> Exp[t/2I]}

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?

Is there a way to leverage Mathematica to demonstrate this?
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