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- Multipole_expansion abstract "A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.The multipole expansion is expressed as a sum of terms with progressively finer angular features. For example, the initial term — called the zero-th, or monopole, moment — is a constant, independent of angle. The following term — the first, or dipole, moment — varies once from positive to negative around the sphere. Higher-order terms (like the quadrupole and octupole) vary more quickly with angles. A multipole moment usually involves powers (or inverse powers) of the distance to the origin, as well as some angular dependence.In principle, a multipole expansion provides an exact description of the potential and generally converges under two conditions: (1) if the sources (e.g., charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources (e.g., charges) are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called exterior multipole moments or simply multipole moments whereas, in the second case, they are called interior multipole moments. The first (the zeroth-order) term in the expansion is called the monopole moment, the second (the first-order) term is denoted as the dipole moment, and the third (the second-order), fourth (the third-order), etc. terms are denoted as quadrupole, octupole, etc. moments.".
- Multipole_expansion wikiPageID "1425534".
- Multipole_expansion wikiPageLength "26784".
- Multipole_expansion wikiPageOutDegree "92".
- Multipole_expansion wikiPageRevisionID "703485553".
- Multipole_expansion wikiPageWikiLink Adrien-Marie_Legendre.
- Multipole_expansion wikiPageWikiLink Atomic_number.
- Multipole_expansion wikiPageWikiLink Axial_multipole_moments.
- Multipole_expansion wikiPageWikiLink Barnes–Hut_simulation.
- Multipole_expansion wikiPageWikiLink Category:Potential_theory.
- Multipole_expansion wikiPageWikiLink Category:Vector_calculus.
- Multipole_expansion wikiPageWikiLink Clebsch–Gordan_coefficients.
- Multipole_expansion wikiPageWikiLink Complex_number.
- Multipole_expansion wikiPageWikiLink Coordinate_system.
- Multipole_expansion wikiPageWikiLink Coulombs_law.
- Multipole_expansion wikiPageWikiLink Cylindrical_multipole_moments.
- Multipole_expansion wikiPageWikiLink Dipole.
- Multipole_expansion wikiPageWikiLink Directional_derivative.
- Multipole_expansion wikiPageWikiLink Distribution_(mathematics).
- Multipole_expansion wikiPageWikiLink Einstein_notation.
- Multipole_expansion wikiPageWikiLink Electric_potential.
- Multipole_expansion wikiPageWikiLink Electromagnetic_field.
- Multipole_expansion wikiPageWikiLink Electromagnetic_radiation.
- Multipole_expansion wikiPageWikiLink Ewald_summation.
- Multipole_expansion wikiPageWikiLink Expectation_value_(quantum_mechanics).
- Multipole_expansion wikiPageWikiLink Fast_multipole_method.
- Multipole_expansion wikiPageWikiLink Field_(physics).
- Multipole_expansion wikiPageWikiLink Function_(mathematics).
- Multipole_expansion wikiPageWikiLink Gravitational_field.
- Multipole_expansion wikiPageWikiLink Gravitational_potential.
- Multipole_expansion wikiPageWikiLink Gravitational_wave.
- Multipole_expansion wikiPageWikiLink Group_representation.
- Multipole_expansion wikiPageWikiLink Kronecker_delta.
- Multipole_expansion wikiPageWikiLink Laplace_expansion_(potential).
- Multipole_expansion wikiPageWikiLink Laplaces_equation.
- Multipole_expansion wikiPageWikiLink Laurent_series.
- Multipole_expansion wikiPageWikiLink Legendre_polynomials.
- Multipole_expansion wikiPageWikiLink Leslie_Greengard.
- Multipole_expansion wikiPageWikiLink Linear_combination.
- Multipole_expansion wikiPageWikiLink Magnetic_field.
- Multipole_expansion wikiPageWikiLink Magnetic_potential.
- Multipole_expansion wikiPageWikiLink Mass.
- Multipole_expansion wikiPageWikiLink Mathematical_physics.
- Multipole_expansion wikiPageWikiLink Mathematics.
- Multipole_expansion wikiPageWikiLink Molecular_symmetry.
- Multipole_expansion wikiPageWikiLink Molecule.
- Multipole_expansion wikiPageWikiLink Moment_(mathematics).
- Multipole_expansion wikiPageWikiLink Monopole_(mathematics).
- Multipole_expansion wikiPageWikiLink Orthogonal_basis.
- Multipole_expansion wikiPageWikiLink Orthogonality.
- Multipole_expansion wikiPageWikiLink Particle_accelerator.
- Multipole_expansion wikiPageWikiLink Potential.
- Multipole_expansion wikiPageWikiLink Quadrupole.
- Multipole_expansion wikiPageWikiLink Quadrupole_magnet.
- Multipole_expansion wikiPageWikiLink Radius.
- Multipole_expansion wikiPageWikiLink Scalar_(mathematics).
- Multipole_expansion wikiPageWikiLink Selection_rule.
- Multipole_expansion wikiPageWikiLink Separation_of_variables.
- Multipole_expansion wikiPageWikiLink Series_(mathematics).
- Multipole_expansion wikiPageWikiLink Series_expansion.
- Multipole_expansion wikiPageWikiLink Solid_harmonics.
- Multipole_expansion wikiPageWikiLink Spherical_coordinate_system.
- Multipole_expansion wikiPageWikiLink Spherical_harmonics.
- Multipole_expansion wikiPageWikiLink Spherical_multipole_moments.
- Multipole_expansion wikiPageWikiLink Symmetry.
- Multipole_expansion wikiPageWikiLink Symmetry_group.
- Multipole_expansion wikiPageWikiLink Taylor_series.
- Multipole_expansion wikiPageWikiLink Tensor.
- Multipole_expansion wikiPageWikiLink Toroidal_moment.
- Multipole_expansion wikiPageWikiLink Vector_potential.
- Multipole_expansion wikiPageWikiLink Vladimir_Rokhlin,_Jr..
- Multipole_expansion wikiPageWikiLinkText "Multipole expansion".
- Multipole_expansion wikiPageWikiLinkText "monopole".
- Multipole_expansion wikiPageWikiLinkText "more complicated geometries".
- Multipole_expansion wikiPageWikiLinkText "multipole expansion".
- Multipole_expansion wikiPageWikiLinkText "multipole moment".
- Multipole_expansion wikiPageWikiLinkText "multipole series".
- Multipole_expansion wikiPageWikiLinkText "multipole".
- Multipole_expansion wikiPageWikiLinkText "non-dipolar".
- Multipole_expansion wikiPageWikiLinkText "octupole".
- Multipole_expansion wikiPageWikiLinkText "polarization".
- Multipole_expansion wikiPageWikiLinkText "this article".
- Multipole_expansion subject Category:Potential_theory.
- Multipole_expansion subject Category:Vector_calculus.
- Multipole_expansion hypernym Series.
- Multipole_expansion type TelevisionShow.
- Multipole_expansion type Function.
- Multipole_expansion type Redirect.
- Multipole_expansion comment "A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general.".
- Multipole_expansion label "Multipole expansion".
- Multipole_expansion sameAs Q1027847.
- Multipole_expansion sameAs Multipolentwicklung.
- Multipole_expansion sameAs Desarrollo_multipolar.
- Multipole_expansion sameAs Sviluppo_in_multipoli.
- Multipole_expansion sameAs 다중극_전개.
- Multipole_expansion sameAs Rozwinięcie_multipolowe.
- Multipole_expansion sameAs Expansão_multipolar.
- Multipole_expansion sameAs m.050jl5.
- Multipole_expansion sameAs Мультиполь.
- Multipole_expansion sameAs Мультипольний_момент.