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Re: Re: Gray's Differential Geometry error?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg63231] Re: [mg63215] Re: Gray's Differential Geometry error?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Tue, 20 Dec 2005 04:19:28 -0500 (EST)
*References*: <dnv48c$och$1@smc.vnet.net> <do3lkh$o22$1@smc.vnet.net> <200512191201.HAA10976@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 19 Dec 2005, at 21:01, Steven T. Hatton wrote:
>
> 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?
I do not have Gray's book here (I do own it but tend to keep it on a
different continent than the one I am on now ;-)) but it is trivial
to show that the above can't possibly be correct. Just put b = -a.
Then obviously s is 0 so the expression for the derivative is 0.
However, the function z[t] has a non zero derivative, unless a = b = 0.
It is also quite easy to guess what Gray's expression should have
been. Let's do it with the help of Mathematica (although I first
worked it out by hand):
FullSimplify[D[(a - b)/(2*Exp[I*t]) + ((a + b)*Exp[I*t])/2, t]]
I*b*Cos[t] - a*Sin[t]
FullSimplify[(I/2)*(s*e + c/e)*(s*e - c/e) /. {s -> Sqrt[a + b],c ->
Sqrt[a - b], e -> Exp[(t/2)*I]}]
I*b*Cos[t] - a*Sin[t]
Andrzej Kozlowski
Tokyo, Japan
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