Thread: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry text

On pages 128,129 in the Principal Curvature section of his text Kreyszig states the series of mathematical steps copied in the graphic attached. Beginning with and beyond the differentiation step I cannot follow his mathematics. For example, where does the minus sign come from between the parentheses? Each step condenses many intermediate steps, particularly because of the indices. His expression incorporating the kronecker delta is particularly puzzling. Anyone who can expand -or deconstruct one might say- the mathematical steps would be greatly appreciated.

Hey cyberbaffled.

Consider that when you differentiate something then everything that isn't relative to the independent variable is treated as a constant.

Lets say you have w = f(x0,x1,x2) and you only want to differentiate against x, then if you only want df/dx^0 you can use tensor notation to mask this out by using d_(m,0)*df/dx^m where m is a dummy variable and d_(m,0) is the kronecker delta function where d_(x,y) = 1 if x=y and 0 otherwise.

Thanks for your quick reply, Chiro. I'm only familiar with the asterisk designating the star operator and the carot designating the wedge product. How are you using them in your expression? Also, I've worked three quarters of the way through Robert Wrede's text Vector and Tensor Analysis without encountering the procedures used by Kreyszig. Let me jump ahead of the curve a bit and ask if you know of a good text that thoroughly exercises the procedures I'm inquiring about and that you've explained in a summary fashion?

Well as a quick example, we can have df/dx^m in Einstein Summation Form to be for m = 1 to 3 to be equal to df/dx^1 + df/dx^2 + df/dx^3 where dx^1 is not dx to the power 1 but instead a particular variable given by that index.

Similarly if we have say d_(m,1)*df/dx^m using Einstein summation, for m = 1 to 3 we get 1*df/dx^1 + 0*df/dx^2 + 0*df/dx^3 = df/dx^1 wher d_(x,y) is the kronecker delta function. If you were summing over two arbitrary indices for kronecker delta (like d_(i,j)) then d_(i,j) would be an identity matrix.

Sorry I went away there for awhile. Got preoccupied with a presidential election. Tomorrow I'll focus on your example. Then reply again.

Please find recent attachment regarding origin of minus sign.

How do I describe the process of making a tessellation using mathematical terms?






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