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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 connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence from the above-mentioned polynomial of degree ; the remaining factor can be regarded as a function of the spherical angular coordinates and only, or equivalently of the orientational unit vector specified by these ...
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 descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion.The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics.
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions .There are two kinds: the regular solid harmonics (), which are well-defined at the origin and the irregular solid harmonics (), which are singular at the origin.
with ^ being the unit vector along the radial direction in spherical coordinates and the vector along the radial direction with the same norm as the radius, i.e., = ^. The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial ...