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In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian
Harmonic_coordinates
Geographic coordinate specifying north-south position
ellipsoidal-harmonic coordinates or simply ellipsoidal coordinates (although that term is also used to refer to geodetic coordinate). These coordinates are the
Latitude
Geographic coordinate system
triad is also known as Earth ellipsoidal coordinates (not to be confused with ellipsoidal-harmonic coordinates). Longitude measures the rotational angle
Geodetic_coordinates
Special mathematical functions defined on the surface of a sphere
called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials
Spherical_harmonics
Functions in mathematics
generally, minimal submanifolds are harmonic immersions of one manifold in another. Harmonic coordinates are a harmonic diffeomorphism from a manifold to
Harmonic_function
Method to choose coordinate systems
form of equations for the evolution of the four coordinates, and indeed in some cases (e.g. the harmonic coordinate condition) they can be put in that form
Coordinate_conditions
Second-order partial differential equation
analyzing the behavior of harmonic functions at infinity. Laplace's equation in two independent variables in rectangular coordinates has the form ∂ 2 ψ ∂ x
Laplace's_equation
Quantum mechanical model
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually
Quantum_harmonic_oscillator
Coordinates comprising two distances and an angle
line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpendicular distance ρ from the main axis; the point
Cylindrical_coordinate_system
Coordinates comprising a distance and two angles
three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point
Spherical_coordinate_system
Solution to the Einstein field equations
Schwarzschild coordinates Kruskal–Szekeres coordinates Eddington–Finkelstein coordinates Gullstrand–Painlevé coordinates Lemaître coordinates (Schwarzschild
Schwarzschild_metric
Type of geometry in mathematics
various authors use this name in slightly different ways. Relative to harmonic coordinates, the condition of Ricci-flatness for a Riemannian metric can be interpreted
Ricci-flat_manifold
Periodic motion of the atoms of a molecule
eigenvalues can be found in. In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave
Molecular_vibration
Geometric figure which approximates the Earth's shape
triad is also known as Earth ellipsoidal coordinates (not to be confused with ellipsoidal-harmonic coordinates). In 1687 Isaac Newton published the Principia
Earth_ellipsoid
Coordinate system in two dimensions
In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates. Log-polar coordinates in the plane consist
Log-polar_coordinates
Solutions of the Laplace equation in spherical polar coordinates
physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions R
Solid_harmonics
Concept in mathematics
differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential
Harmonic_map
Differential operator in mathematics
density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions
Laplace_operator
Nonlinear optical process
Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems
Second-harmonic_generation
French mathematical physicist (1923–2025)
equations. She expressed the Einstein field equations in vacuum using harmonic coordinates, previously introduced by Théophile De Donder and Cornelius Lanczos
Yvonne_Choquet-Bruhat
Wilson's normal mode analysis
of arbitrary linear internal coordinates, while F represents the (harmonic) potential energy in terms of these coordinates. The GF method gives the linear
GF_method
Three-dimensional orthogonal coordinate system
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about
Toroidal_coordinates
Defunct theory of electromagnetism
limit, which gives the Einstein equations of general relativity in harmonic coordinates. Already in his philosophical writing on time measurements (1898)
Lorentz_ether_theory
Hungarian-American mathematician (1893–1974)
distant parallelism. Another innovation of Lanczos in this context was harmonic coordinates, which he introduced independently of Théophile De Donder. These
Cornelius_Lanczos
Energy related to Earth's gravity
ellipsoidal-harmonic coordinates (not to be confused with geodetic coordinates). It can also be expressed as a series expansion in terms of spherical coordinates;
Geopotential
Coordinate condition in general relativity
coordinate system(s). The Cartesian coordinates used in special relativity satisfy d'Alembert's equation, so a harmonic coordinate system is the closest
Harmonic_coordinate_condition
Formulation of classical mechanics
each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration
Lagrangian_mechanics
Sets of coordinates on phase space which can be used to describe a physical system
coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are
Canonical_coordinates
Mathematical objects more general than vectors
The spherical coordinates are not involved here. The rules for using harmonic symmetric tensors are demonstrated that directly follow from their properties
Harmonic_tensors
Point found separated from another, given a point pair
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following
Projective_harmonic_conjugate
Solutions to Laplace's equation
In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, ∇ 2 V = 0
Cylindrical_harmonics
Solutions of the Helmholtz equation
spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical harmonics. They
Spheroidal_wave_function
Sub-area of scientific computing for solving General Relativity equations
of light for the propagation of coordinate effects (e.g., using harmonic coordinates coordinate conditions). The second problem is that as the black holes
Numerical_relativity
mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular
Zonal_spherical_harmonics
conformal. In local coordinates, x {\displaystyle x} on M {\displaystyle M} and y {\displaystyle y} on N {\displaystyle N} , the harmonicity of ϕ {\displaystyle
Harmonic_morphism
David Finkelstein (Eddington–Finkelstein coordinates), Vladimir Aleksandrovich Fock (textbook, harmonic coordinates), Robert L. Forward (gravitational-wave
List of contributors to general relativity
List_of_contributors_to_general_relativity
System configuration relative to another
In analytical mechanics, generalized coordinates are a set of parameters used to represent the configuration of a system in a configuration space. These
Generalized_coordinates
Vector operator in vector calculus
transported by the flow for any period of time. In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x
Divergence
nowhere-vanishing derivative. Isothermal coordinates are constructed from such a function in the following way. Harmonicity of u is identical to the closedness
Isothermal_coordinates
Formulation of classical mechanics using momenta
{\boldsymbol {q}})} is called phase space coordinates. (Also canonical coordinates). In phase space coordinates ( p , q ) {\displaystyle ({\boldsymbol
Hamiltonian_mechanics
Geometric figure
hyperbola in Elements of Dynamic (1878) by W. K. Clifford. He describes quasi-harmonic motion in a hyperbola as follows: The motion ρ = α cosh ( n t + ϵ ) +
Unit_hyperbola
Mathematical series
the Cartesian coordinates x, y, and z, while the second is in terms of spherical harmonics which depend on spherical polar coordinates. The Cartesian
Multipole_expansion
div-curl systems, Maxwell's equations, Einstein's equations (with four harmonic coordinates) and Yang-Mills equations (with gauge conditions) are well-determined
First-order partial differential equation
First-order_partial_differential_equation
Hamiltonian operator for molecules
these conditions arise in a natural way from a harmonic analysis in mass-weighted Cartesian coordinates. In order to simplify the expression for the kinetic
Molecular_Hamiltonian
Solutions of Lamé's equation
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It
Lamé_function
Fourth-order PDE in continuum mechanics
biharmonic function. Any harmonic function is biharmonic, but the converse is not always true. In two-dimensional polar coordinates, the biharmonic equation
Biharmonic_equation
Pattern of oscillating motion in a system
spheroidal and toroidal modes. Antiresonance Critical speed Harmonic oscillator Harmonic series (music) Infrared spectroscopy Leaky mode Mechanical resonance
Normal_mode
Quantum chemistry rule regarding vibronic transitions
In the semiclassical picture of vibrations (oscillations) of a simple harmonic oscillator, the necessary conditions can occur at the turning points, where
Franck–Condon_principle
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Method of solution for certain mechanical problems
In classical mechanics, action-angle variables are a set of canonical coordinates that are useful in characterizing the nature of commuting flows in integrable
Action-angle_coordinates
Algorithm on pulse-width modulation
requirements. One active area of development is in the reduction of total harmonic distortion (THD) created by the rapid switching inherent to these algorithms
Space_vector_modulation
Foundational principle in quantum physics
quantified by the uncertainty principle. Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators
Uncertainty_principle
Three-dimensional orthogonal coordinate system
independent constants for each harmonic. An alternative and geometrically intuitive set of oblate spheroidal coordinates (σ, τ, φ) are sometimes used,
Oblate_spheroidal_coordinates
Mathematical concept
certain Möbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane.
Poisson_kernel
Partial differential equations
second kind, which is a toroidal harmonic. Here the expansion has been written in terms of cylindrical coordinates ( R , φ , z ) {\displaystyle (R,\varphi
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
Curve from a cone intersecting a plane
plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables which can be written in
Conic_section
Extension of the scalar spherical harmonics for use with vector fields
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH
Vector_spherical_harmonics
Atomic model
often partially replaced by cubic harmonics for a number of reasons. These harmonics are usually named tesseral harmonics in the field of condensed matter
Cubic_harmonic
Laws describing planetary orbits
{1-\varepsilon ^{2}}}}.\end{aligned}}} The semi-latus rectum p is the harmonic mean between rmin and rmax: p = ( r max − 1 + r min − 1 2 ) − 1 , p a =
Kepler's laws of planetary motion
Kepler's_laws_of_planetary_motion
Quasiparticle of mechanical vibrations
atoms are effectively screened. Secondly, the potentials V are treated as harmonic potentials. This is permissible as long as the atoms remain close to their
Phonon
Force directed to the center of rotation
directly with this issue, local coordinates are preferable, as discussed next. Local coordinates mean a set of coordinates that travel with the particle
Centripetal_force
Invariant in projective geometry
point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio
Cross-ratio
Generalized sphere of dimension n (mathematics)
j = n − 1 {\displaystyle j=n-1} in concordance with the spherical harmonics. The standard spherical coordinate system arises from writing R n {\displaystyle
N-sphere
Isomorphism of projective spaces in geometry
version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies
Homography
Algorithm in molecular physics
a single harmonic oscillator. In simple terms, it means that the algorithm fails to generate a canonical distribution for a single harmonic oscillator
Nosé–Hoover_thermostat
Decomposition of periodic functions
first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums converge to the square wave. Function s 6
Fourier_series
Multivariate derivative (mathematics)
coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions). We consider general coordinates, which we write as x1, …, xi,
Gradient
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
Laplace–Beltrami operator. By standard elliptic theory, u can be chosen to be harmonic near a given point, i.e. Δ u = 0, with du non-vanishing. By the Poincaré
Uniformization_theorem
Property of certain dynamical systems
particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either
Integrable_system
South Korean physicist, academic, author and researcher
(1973). Covariant Harmonic Oscillators and the Quark Model. Phys. Rev. D 8, 3521-3526 Kim, Y.S. & Noz, M.E. (1977). Covariant Harmonic Oscillators and the
Young_Suh_Kim
Spectroscopic technique
_{zx}^{2})\right]}}} In the double-harmonic approximations, the potential energy is expanded to the second order near equilibrium (harmonic force fields), while polarizability
Raman_spectroscopy
American mathematician (born 1950)
known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps. Schoen was born in Celina, Ohio, on October 23, 1950. In 1968, he
Richard_Schoen
Formulation of the principle of stationary action
that the true evolution q(t) of a system described by N generalized coordinates q = (q1, q2, ..., qN) between two specified states q1 = q(t1) and q2
Hamilton's_principle
Mathematical description of quantum state
functions in this case are the spherical harmonics. The Legendre polynomials are ingredients in the spherical harmonics. Most problems with rotational symmetry
Wave_function
Function describing an electron in an atom
combinations of mℓ and −mℓ orbitals, and are often labeled using associated harmonic polynomials (e.g., xy, x2 − y2) which describe their angular structure
Atomic_orbital
Special functions
spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere
Spin-weighted spherical harmonics
Spin-weighted_spherical_harmonics
Abstract coordinate system
sufficient to fully define a reference frame. Using rectangular Cartesian coordinates, a reference frame may be defined with a reference point at the origin
Frame_of_reference
Theoretical description of Earth's gravimetric shape
are the spherical coordinates which satisfy the partial differential equation (6) (the Laplace equation) are called spherical harmonic functions. They take
Geopotential spherical harmonic model
Geopotential_spherical_harmonic_model
Property of a mass in motion
generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example
Momentum
Speed and direction of a motion
v_{n}}}}} If s1 = s2 = s3 = ... = s, then average speed is given by the harmonic mean of the speeds v ¯ = n ( 1 v 1 + 1 v 2 + 1 v 3 + ⋯ + 1 v n ) − 1 =
Velocity
Archimedean solid with 62 faces
pentagonal faces, with 60 vertices, and 120 edges. Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short
Rhombicosidodecahedron
Integral of the Gaussian function, equal to sqrt(π)
ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and
Gaussian_integral
Signal processing algorithm
short-time Fourier transform) by mapping the data to time-frequency coordinates that are nearer to the true region of support of the analyzed signal
Reassignment_method
Type of polynomial
∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} , so f {\displaystyle f} is a harmonic function. This implies f {\displaystyle f} has maxima and minima only on
Multilinear_polynomial
Frenet–Serret formulas Gauss's law Gradient Green's theorem Green's identities Harmonic function Helmholtz decomposition Hessian matrix Hodge star operator Inverse
List of multivariable calculus topics
List_of_multivariable_calculus_topics
Type of vector space in math
For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit
Hilbert_space
Generalization of definite integrals to functions of multiple variables
transformation to polar coordinates (see the example in the picture) which means that the generic points P(x, y) in Cartesian coordinates switch to their respective
Multiple_integral
Key result in Hamiltonian mechanics and statistical mechanics
systems do. The general setting for conjugate position and momentum coordinates is available in the mathematical setting of symplectic geometry. Liouville's
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Change in the position of an object
are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient
Motion
Q^{i\alpha }} , and the corresponding fermionic coordinates are θ i α {\displaystyle \theta ^{i\alpha }} . Harmonic superspace is given by the product of ordinary
Harmonic_superspace
Basis used to express spherical tensors
angular momentum in quantum mechanics and spherical harmonic functions. While spherical polar coordinates are one orthogonal coordinate system for expressing
Spherical_basis
Theorem of dynamical systems
independent harmonic oscillators in the example of the springs). More precisely, there exists a canonical transformation to action-angle coordinates in which
Liouville–Arnold_theorem
Mathematics of smooth surfaces
involving a theorem of Sacks & Uhlenbeck (1981) on removable singularities of harmonic maps of finite energy. Gauss 1902. Struik 1987, pp. 50–53 Wells 2017, pp
Differential geometry of surfaces
Differential_geometry_of_surfaces
Mathematical identities
the point. When the Laplacian is equal to 0, the function is called a harmonic function. That is, Δ f = 0. {\displaystyle \Delta f=0.} For a tensor field
Vector_calculus_identities
Analog graphical calculator
coordinate system invented by d'Ocagne rather than standard Cartesian coordinates. A nomogram consists of a set of n scales, one for each variable in an
Nomogram
Framework of distances and directions
One can freely move in space but not in time. Thus, time and space coordinates are treated differently both in special relativity (where time is sometimes
Space
Branch of physics describing the motion of objects without considering forces
These systems may be rectangular like Cartesian, Curvilinear coordinates like polar coordinates or other systems. The object trajectories may be specified
Kinematics
Mechanical oscillations about an equilibrium point
transient input, or a random input. The periodic input can be a harmonic or a non-harmonic disturbance. Examples of these types of vibration include a washing
Vibration
Matrix of partial derivatives of a vector-valued function
function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However, a function does not need to be differentiable
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Topological space that locally resembles Euclidean space
harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics,
Manifold
HARMONIC COORDINATES
HARMONIC COORDINATES
Boy/Male
Muslim
Harmony
Boy/Male
American, Australian, British, Chinese, Christian, English, French, German, Greek, Hebrew
Man of the Army; Army Man; Noble; Name of a Place During Biblical Period; Hardy Man; Variant of Herman
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Boy/Male
French American Hebrew
Girl/Female
Greek Latin
Daughter of Ares.
Girl/Female
Latin
Harmony.
Female
English
Variant spelling of English Harmony, HARMONIE means "concord, harmony."
Boy/Male
Welsh
Harmony.
Girl/Female
American, Australian, British, Christian, English, French, Greek, Latin
A State of Order or Agreement; Unity; Concord; Harmony; Agreement
Girl/Female
Latin American
Concord.
Male
English
English surname transferred to forename use, from the German personal name Harman, HARMON means "bold/hardy man."
Boy/Male
Indian
Harmony
Girl/Female
Christian & English(British/American/Australian)
Harmony
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Girl/Female
Latin
Harmony.
Surname or Lastname
Irish (mainly County Louth)
Irish (mainly County Louth) : generally of English origin (see 1); but sometimes also used as a variant of Harman or Hardiman, i.e. an Anglicized form of Gaelic Ó hArgadáin (see Hargadon).English : variant spelling of Harman 1.
Female
English
English name derived from the vocabulary word harmony, from Greek Harmonia, HARMONY means "concord, harmony."
Girl/Female
American, British, English, Greek, Latin
A State of Order or Agreement; Unity; Concord; Musically in Tune; A Tuneful Sound
Female
Greek
(ΑÏμονία) Greek name HARMONIA means "concord, harmony." In mythology, this is the name of the daughter of Ares and Aphrodite. Her Latin name is Concordia.
Girl/Female
American, Australian, British, Chinese, Christian, English, French, Greek, Latin
A State of Order or Agreement; A Beautiful Blending; Agreement; Concord; Musical Combination of Chords; Harmony; Joining
HARMONIC COORDINATES
HARMONIC COORDINATES
Boy/Male
Tamil
Intelligent, Innovative
Girl/Female
Muslim
Resembling the full moon.
Surname or Lastname
English
English : habitational name from a minor place called Studding’s Farm in Herstmonceaux, Sussex, or possibly from an unidentified place in Devon.
Boy/Male
Hindu
The most valuable stone, Whichiis in the possession of Lord Vishnu
Girl/Female
Indian
Blue Eyes
Boy/Male
Arabic, Muslim
Sheltered; Well Protected
Boy/Male
Australian, Hebrew
Joyful
Girl/Female
Indian, Sanskrit
With the Scent of a Palm Tree
Boy/Male
Arabic, Indian, Muslim, Sindhi
Opening
Boy/Male
Tamil
Hemanga | ஹேமாஂகாÂ
Golden bodied
HARMONIC COORDINATES
HARMONIC COORDINATES
HARMONIC COORDINATES
HARMONIC COORDINATES
HARMONIC COORDINATES
n.
One who understands the principles of harmony or is skillful in applying them in composition; a musical composer.
v. i.
To agree in vocal or musical effect; to form a concord; as, the tones harmonize perfectly.
a.
Relating to harmony, -- as melodic relates to melody; harmonious; esp., relating to the accessory sounds or overtones which accompany the predominant and apparent single tone of any string or sonorous body.
a.
Not harmonic; inharmonious; discordant; dissonant.
n.
Concord or agreement in facts, opinions, manners, interests, etc.; good correspondence; peace and friendship; as, good citizens live in harmony.
n.
A literary work which brings together or arranges systematically parallel passages of historians respecting the same events, and shows their agreement or consistency; as, a harmony of the Gospels.
a.
Not harmonic.
n.
Alt. of Harmonite
a.
Alt. of Harmonical
n.
A musical note produced by a number of vibrations which is a multiple of the number producing some other; an overtone. See Harmonics.
n.
The just adaptation of parts to each other, in any system or combination of things, or in things, or things intended to form a connected whole; such an agreement between the different parts of a design or composition as to produce unity of effect; as, the harmony of the universe.
n.
See Harmonic suture, under Harmonic.
v. i.
To agree in action, adaptation, or effect on the mind; to agree in sense or purport; as, the parts of a mechanism harmonize.
pl.
of Harmony
v. t.
To accompany with harmony; to provide with parts, as an air, or melody.
n.
One who shows the agreement or harmony of corresponding passages of different authors, as of the four evangelists.
n.
One of a religious sect, founded in Wurtemburg in the last century, composed of followers of George Rapp, a weaver. They had all their property in common. In 1803, a portion of this sect settled in Pennsylvania and called the village thus established, Harmony.
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.
a.
Concordant; musical; consonant; as, harmonic sounds.
a.
Of, pertaining to, or obtained from, carbon; as, carbonic oxide.