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Re: Gray's Differential Geometry error?
In article <do684q$b0a$1 at smc.vnet.net>, "Steven T. Hatton" <hattons at globalsymmetry.com> wrote: > > 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. i rather suspect that it was an unintentional error. i have learned that people who know the answer don't always get the derivation right. > > 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. > good. it isn't. within each ( ) the coefficient of E^(-I t/2) should be sqrt (a-b) instead of sqrt (a+b), like eq 2.5 on p. 39. > > 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? the correct expression comes from factoring (U^2 - V^2) as (U+V)* (U-V). i think it's a cute proof, but i have reservations. how does he know the foci are at ± srqt(a^2 - b^2) ? it's true, of course, but i derive that from the "sum of the two distances is constant" rather than the other way around. i also suspect that mathematica should be used to simply compute angles and distances rather than do algebra. finally, we could take the differential geometry questions to email if you like; my eddress is at the bottom of the post. hope this helps. thanks for the questions. vale, rip -- NB eddress is r i p 1 AT c o m c a s t DOT n e t