In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian...

ellipsoidal-harmonic coordinates or simply ellipsoidal coordinates: §4.2.2  (although that term is also used to refer to geodetic coordinate). These coordinates are...

triad is also known as Earth ellipsoidal coordinates (not to be confused with ellipsoidal-harmonic coordinates). Longitude measures the rotational angle...

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...

called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials...

generally, minimal submanifolds are harmonic immersions of one manifold in another. Harmonic coordinates are a harmonic diffeomorphism from a manifold to...

In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates. Log-polar coordinates in the plane consist...

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...

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...

line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpendicular distance ρ from the main axis; the point...

Schwarzschild coordinates Kruskal–Szekeres coordinates Eddington–Finkelstein coordinates Gullstrand–Painlevé coordinates Lemaître coordinates (Schwarzschild...

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...

ellipsoidal-harmonic coordinates (not to be confused with geodetic coordinates). It can also be expressed as a series expansion in terms of spherical coordinates;...

that a non-constant harmonic function cannot assume its maximum value at an interior point. Laplace's equation in spherical coordinates is: ∇ 2 f = 1 r 2...

Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems...

differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential...

eigenvalues can be found in. In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave...

triad is also known as Earth ellipsoidal coordinates (not to be confused with ellipsoidal-harmonic coordinates). In 1687 Isaac Newton published the Principia...

physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions R...

conformal. In local coordinates, x {\displaystyle x} on M {\displaystyle M} and y {\displaystyle y} on N {\displaystyle N} , the harmonicity of ϕ {\displaystyle...

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...

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...

of arbitrary linear internal coordinates, while F represents the (harmonic) potential energy in terms of these coordinates. The GF method gives the linear...

In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters...

coordinate system(s). The Cartesian coordinates used in special relativity satisfy d'Alembert's equation, so a harmonic coordinate system is the closest...

limit, which gives the Einstein equations of general relativity in harmonic coordinates. Already in his philosophical writing on time measurements (1898)...

spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical harmonics. They...

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...

div-curl systems, Maxwell's equations, Einstein's equations (with four harmonic coordinates) and Yang-Mills equations (with gauge conditions) are well-determined...

each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration...