Talk:Isothermal coordinates

started new article

This is very basic, just to get the page started. Please improve it if you can. Artie P.S. 14:02, 26 March 2007 (UTC)

I cleaned up the definition by making it slightly less technical, and started a properties list. There's still work to be done, however. Haseldon 18:06, 2 June 2007 (UTC)

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Yours truly AlexNewArtBot 00:46, 27 March 2007 (UTC)

correctness of the formulas

I think the formulas for lambda and mu are incorrect. At least, I get different answers when I calculate. I started from

ds^2 = E dx^2 + 2F dx dy + G dy^2,   

substituted dx = (1/2) (dz + d \bar z) and dy = (1/2i) (dz - d\bar z), expanded to three terms in dz^2, d \bar z ^2, and dz d \bar z then start with ds^2 = lambda | dz + mu d \bar z |^2 and expand that and compare the coefficients. The result is not the stated formulas. Mu should have a factor of 4 in the denominator and lambda should have a 4 where it has a 2. This is important because for existence and uniqueness we need mu < 1.

why isothermal?

Why are these coordinates called "isothermal"? Does this have anything to do with temperature? Tkuvho (talk) 15:54, 6 June 2012 (UTC)

I do not know the origin of the name, but I guess the invariance of the Laplace operator under conformal maps tells you that solving the Laplace equation or the heat equation in isothermal coordinates is essentially the same as solving it in Euclidean space. At least locally. I'll see if I can find something on the history of the name. —Kusma (t·c) 18:01, 6 June 2012 (UTC)

how does one find u harmonic without a critical point in a neighborhood of a given point?

The assertion "u can be chosen to be harmonic near a given point, i.e. Δ u = 0, with du non-vanishing" non-trivial. The author should provide a reference.Lost-n-translation (talk) 15:17, 24 October 2016 (UTC)