Why are they called isothermal coordinates? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:01:33Z http://mathoverflow.net/feeds/question/32169 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32169/why-are-they-called-isothermal-coordinates Why are they called isothermal coordinates? Spiro Karigiannis 2010-07-16T13:44:15Z 2011-04-01T10:56:07Z <p>On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:</p> <p>$$g_{ij} = e^{f} \delta_{ij}$$</p> <p>My question is: Why are such coordinate systems called "isothermal"? It must have something to do with classical thermal physics. I tried looking for a reason online, with no success.</p> <p>It is well known that when the dimension $n=2$, there always exist isothermal coordinates, and this is probably where they were first introduced. So maybe the nomenclature has something to do with heat diffusion in the plane?</p> <p>(The reason I ask is because I am planning to give a seminar talk next week giving a proof that such coordinates exist when $n=2$, and thought it would be nice to explain to the students where the name comes from...)</p> http://mathoverflow.net/questions/32169/why-are-they-called-isothermal-coordinates/32171#32171 Answer by soulphysics for Why are they called isothermal coordinates? soulphysics 2010-07-16T13:57:42Z 2010-07-16T14:03:03Z <p>According to <a href="http://books.google.com/books?id=TGw98Z6Cv-EC&amp;lpg=PA518&amp;dq=isothermal%2520coordinates&amp;pg=PA518#v=onepage&amp;q=%2522isothermal%2522%2520is%2520due%2520to%2520Lame&amp;f=false" rel="nofollow">Gray, Abbena and Salamon</a>, that's the name given to such coordinate systems by Gabriel Lamé in his 1833 study of heat transfer. The reason is, if you've got a thermally isolated surface of constant heat conduction, the constant coordinate lines are isotherms iff the coordinates are isothermal.</p> http://mathoverflow.net/questions/32169/why-are-they-called-isothermal-coordinates/32172#32172 Answer by Willie Wong for Why are they called isothermal coordinates? Willie Wong 2010-07-16T13:58:52Z 2010-07-16T13:58:52Z <p>Isothermal coordinates are harmonic. In other words, it solves $\triangle_g u = 0$. So locally it is a stationary solution of the heat equation. In physics, for a steady state distribution of temperatures, each level set is called an isotherm. </p> http://mathoverflow.net/questions/32169/why-are-they-called-isothermal-coordinates/34930#34930 Answer by Some Math Guy for Why are they called isothermal coordinates? Some Math Guy 2010-08-08T15:25:39Z 2010-08-08T15:25:39Z <p>Isothermal coordinates on a two-dimensional manifold are not harmonic unless the manifold has zero mean curvature (as an immersed submanifold of Euclidean space, for example.) The plane is a minimal surface (well, you COULD screw around with the parametrization :D ) and so isothermal coordinates would be conformal or anti-conformal there.</p> <p>In higher dimensions, isothermal coordinates may not exist but harmonic coordinates do; Einstein introduced them in (about) 1916.</p>