In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form
where is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)
Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold.
By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally flat if and only if its Weyl tensor vanishes.
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to be isothermal.) Isothermalcoordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermalcoordinates exist around...
coordinate system given by f is isothermal. Conversely, consider a diffeomorphism f that does give us isothermalcoordinates. We then have μ ( z ) = ( u x...
dimensions, certain harmonic coordinates known as isothermalcoordinates have been studied since the early 1800s. Harmonic coordinates in higher dimensions were...
spacetime)#Proper coordinates or Fermi coordinates Geodesic normal coordinates Fermi–Walker transport Christoffel symbols Isothermalcoordinates Manasse, F....
linearly independent, so that u and v give local isothermalcoordinates. The existence of isothermalcoordinates can be proved by other methods, for example...
(\log r),} where r denotes the geodesic distance from the point. In isothermalcoordinates, first considered by Gauss, the metric is required to be of the...
curvature is Liouville's equation in terms of the Laplacian in isothermalcoordinates. The surface integral of the Gaussian curvature over some region...
used the later called Beltrami equation to prove the existence of isothermalcoordinates on analytic surfaces. The essay concludes with examples of conformal...
surface. This can be seen as a consequence of the existence of isothermalcoordinates. In complex analytic terms, the Poincaré–Koebe uniformization theorem...
manifold is conformally flat, a consequence of the existence of isothermalcoordinates. Indeed, the existence of a conformally flat scale amounts to solving...
applications in medical image analysis, computer vision and graphics. Isothermalcoordinates Quasiregular map Pseudoanalytic function Teichmüller space Tissot's...
conformal, i.e. scale the metric by a smooth function. The existence of isothermalcoordinates—which can be proved using either local existence theorems for the...
third depicted the locus of the isothermal chromaticities on the CIE 1931 x,y chromaticity diagram. Since the isothermal points formed normals on his UCS...
1869 by Massieu for the isothermal process (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process. More...
}}_{u}=S_{u}^{2}} (as shown in ref., complex stereographic rather than real isothermalcoordinates are used just for the convenience of completely solving NP equations)...
horocycle in the appropriate Q quadrant. For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted...
case of two-dimensional surfaces, making use of the existence of isothermalcoordinates. Slightly weaker estimates were obtained by Schoen and Simon, although...
Isothermal Community College (ICC) is a public community college in Spindale, North Carolina. Named after its location in the thermal belt, an area in...
values are (X, Y, Z) = (109.85, 100.00, 35.58), and the chromaticity coordinates using the standard observer are (x, y) = (0.44758, 0.40745). Illuminants...
space; that is, a color is specified by a set of three numbers (the CIE coordinates X, Y, and Z, for example, or other values such as hue, colorfulness,...
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