to be isothermal.) Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around
the mathematical field of differential geometry, the existence of isothermal coordinates for a (pseudo-)Riemannian metric is often of interest. In the case
dimensions, certain harmonic coordinates known as isothermal coordinates have been studied since the early 1800s. Harmonic coordinates in higher dimensions were
coordinate system given by f is isothermal. Conversely, consider a diffeomorphism f that does give us isothermal coordinates. We then have μ ( z ) = ( u x
spacetime)#Proper coordinates or Fermi coordinates Geodesic normal coordinates Fermi–Walker transport Christoffel symbols Isothermal coordinates Manasse, F.
(\log r),} where r denotes the geodesic distance from the point. In isothermal coordinates, first considered by Gauss, the metric is required to be of the
(Euclidean) coordinates while preserving angles. Liouville's equation appear in many areas of physics, including Emden–Chandrasekhar equation for isothermal stars
linearly independent, so that u and v give local isothermal coordinates. The existence of isothermal coordinates can be proved by other methods, for example
existence of a canonical form is equivalent to the existence of isothermal coordinates for the associated Riemannian metric A ( x , y ) d x 2 + 2 B ( x
manifold is conformally flat, a consequence of the existence of isothermal coordinates. Indeed, the existence of a conformally flat scale amounts to solving