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More generally, if g is C k, α (with k larger than one) and Ric(g) is C l, α relative to some coordinate charts, then the transition function to a harmonic coordinate chart will be C k + 1, α, and so Ric(g) will be C min(l, k), α in harmonic coordinate charts. So, by the previous result, g will be C min(l, k) + 2, α in harmonic coordinate ...
A coordinate condition selects one (or a smaller set of) such coordinate system(s). The Cartesian coordinates used in special relativity satisfy d'Alembert's equation, so a harmonic coordinate system is the closest approximation available in general relativity to an inertial frame of reference in special relativity.
Unlike the harmonic and synchronous coordinate conditions, some commonly used coordinate conditions may be either under-determinative or over-determinative. An example of an under-determinative condition is the algebraic statement that the determinant of the metric tensor is −1, which still leaves considerable gauge freedom. [ 4 ]
The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase. The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines. Visual Array of Complex Spherical Harmonics Represented as 2D Theta/Phi Maps
Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties: Regularity. Any harmonic map heat flow is smooth as a map (a, b) × M → N given by (t, p) ↦ f t (p). Now suppose that M is a closed manifold and (N, h) is geodesically complete. Existence.
This page was last edited on 24 October 2021, at 00:09 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Q k. Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to ...
This page was last edited on 7 December 2010, at 12:32 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.