Multipole expansion

A series representation of a field or potential in terms of source moments of increasing spatial order. Its leading terms describe large-scale behavior, while higher terms capture finer details of the source distribution.

Monopole: The scalar, zeroth-order term in a multipole expansion, usually determined by total source strength. It is the leading term and often dominates far from a localized source.
Taylor series: A power series that represents a function using its derivatives at a point. Multipole expansions similarly organize a field into successively higher-order contributions.
Electrostatics: The study of electric charges and fields that are stationary or change slowly. Charge distributions generate potentials commonly approximated by multipole terms.
Cartesian tensor: A tensor represented by components that transform under rotations of Cartesian coordinates. Cartesian moments encode the same source structure without spherical-harmonic indexing.
Dipole: A first-order source moment describing separation of positive and negative strength or charge. It supplies the next spatial detail when the monopole contribution vanishes or is insufficient.
Legendre polynomials: A sequence of orthogonal polynomials that solves Legendre's differential equation. They appear in the axisymmetric expansion of inverse-distance potentials.
Gravitational potential: The scalar potential whose spatial gradient determines gravitational acceleration. Mass distributions produce exterior potentials with monopole and higher gravitational moments.
Finite element method: A numerical method that approximates solutions to differential equations over discretized regions. It resolves local fields directly, unlike a far-field moment expansion.
Quadrupole: A second-order source moment describing anisotropic structure beyond the dipole. It captures directional source structure that lower moments cannot represent.
Addition theorem for spherical harmonics: An identity expressing products or sums of spherical harmonics at different directions. It enables the expansion of a point-source potential into angular multipole terms.

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