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The essential point is that their geometric assumptions, via some of the results discussed below on harmonic radius, give good control over harmonic coordinates on regions near infinity. By the use of a partition of unity, these harmonic coordinates can be patched together to form a single coordinate chart, which is the main objective. [19]
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions x α (regarded as scalar fields) satisfies d'Alembert's equation .
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.
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 ...
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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.