In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the...

basis Spinor spherical harmonics Spin-weighted spherical harmonics Sturm–Liouville theory Table of spherical harmonics Vector spherical harmonics Zernike polynomials...

The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for...

expanded into radiating spherical vector spherical harmonics. The internal field is expanded into regular vector spherical harmonics. By enforcing the boundary...

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree ℓ = 10 {\displaystyle \ell =10} . Some of these...

zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions...

derivatives of a vector-valued function List of canonical coordinate transformations Sphere – Set of points equidistant from a center Spherical harmonic – Special...

spherical harmonics. The vector Laplace operator, also denoted by ∇ 2 {\displaystyle \nabla ^{2}} , is a differential operator defined over a vector field...

standard lighting equations with spherical functions that have been projected into frequency space using the spherical harmonics as a basis. To take a simple...

harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically...

dependence of radiation is recovered. Multipole expansion Spherical harmonics Vector spherical harmonics Near and far field Quadrupole formula Hartle, James...

expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component...

where i is the imaginary unit, k is a wave vector of length k, r is a position vector of length r, jℓ are spherical Bessel functions, Pℓ are Legendre polynomials...

vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical...

transformation of complex spherical harmonics to real form is by a unitary transformation, we can simply substitute real irregular solid harmonics and real multipole...

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

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's...

exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. However, a spherical harmonics series expansion...

mechanics and spherical harmonic functions. While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using...

polynomials or wavelets for instance, and in higher dimensions into spherical harmonics. For instance, if en are any orthonormal basis functions of L2[0...

which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector r ′ {\displaystyle \mathbf {r} '} has coordinates...

V(r)} depends only on the vector magnitude of the position vector, that is, the radial distance from the origin (hence the spherical symmetry of the problem)...

often partially replaced by cubic harmonics for a number of reasons. These harmonics are usually named tesseral harmonics in the field of condensed matter...

alternatively use the left-most vector position. Comparison of vector algebra and geometric algebra Del in cylindrical and spherical coordinates – Mathematical...

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional...

symmetry group O(3) as a certain multipole (or the corresponding vector spherical harmonic), but does not radiate to the far field. In photonics, anapoles...

needs. Furthermore, there was no widely accepted formulation of spherical harmonics for acoustics, so one was borrowed from chemistry, quantum mechanics...

the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmonic" was...

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold...

conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments. The wave equation in one spatial...