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Re: Re: Gray's Differential Geometry error?


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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