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 xα (regarded as scalar fields) satisfies d'Alembert's equation. The parallel notion of a harmonic coordinate system in Riemannian geometry is a coordinate system whose coordinate functions satisfy Laplace's equation. Since d'Alembert's equation is the generalization of Laplace's equation to space-time, its solutions are also called "harmonic".
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The harmoniccoordinatecondition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations...
caustics (coordinate singularities). If one combines a coordinatecondition which is Lorentz covariant, such as the harmoniccoordinatecondition mentioned...
general relativity by Albert Einstein and Cornelius Lanczos (see harmoniccoordinatecondition). Following the work of Dennis DeTurck and Jerry Kazdan in 1981...
fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal...
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...
June 2016). "Simultaneous Microgrid Voltage and Current Harmonics Compensation Using Coordinated Control of Dual-Interfacing Converters". IEEE Transactions...
Surface second harmonic generation is a method for probing interfaces in atomic and molecular systems. In second harmonic generation (SHG), the light frequency...
geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential...
calculate the total harmonic distortion (THD). The total harmonic distortion (THD) is the square root of the sum of the squares of the harmonic voltages divided...
harmoniccoordinatecondition. The gravitational field is defined by density, velocity and stress tensor of the Lorentz ether, so that the harmonic conditions...
ellipsoidal-harmonic coordinates or simply ellipsoidal coordinates: §4.2.2 (although that term is also used to refer to geodetic coordinate). These coordinates...
the traditional use of the term "coordinate" to refer to Cartesian coordinates. An example of a generalized coordinate would be to describe the position...
coordinates, we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a...
adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in...
{\displaystyle P.} The constituent frequencies are a discrete set of harmonics at frequencies n P , n ∈ Z , {\displaystyle {\tfrac {n}{P}},n\in \mathbb...
vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which...
spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions...
in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic...
gauge is the so-called "De Donder gauge", also known as the harmoniccondition or harmonic gauge g μ ν Γ σ μ ν = 0 . {\displaystyle g^{\mu \nu }\Gamma...
the velocity component to the corresponding coordinate z2). In each constraint equation, one coordinate is redundant because it is determined from the...
harmonic approximation, which approximates the potential energy surface as a quadratic function of the atomic displacements. This gives the harmonic nuclear...
widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually...
For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit...
2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition where M T {\displaystyle M^{\text{T}}} denotes the transpose of M {\displaystyle...
"Applications of analysis on Lipschitz manifolds". Miniconferences on harmonic analysis and operator algebras (Canberra, 1987). Canberra: Australian National...
June 2016). "Simultaneous Microgrid Voltage and Current Harmonics Compensation Using Coordinated Control of Dual-Interfacing Converters". IEEE Transactions...
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