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

**References**:**Re: Gray's Differential Geometry error?***From:*"Steven T. Hatton" <hattons@globalsymmetry.com>