related sets of orthogonal functions. One uses the technique of separation of variables to extract the corresponding solutions for the radial dependencies. In practice, many fields can be well approximated with a finite number of multipole moments (although an infinite number may be required to reconstruct a field exactly). A typical application is to approximate the field of a localized charge distribution by its monopole and dipole terms. Problems solved once for a given order of multipole moment may be linearly combined to create a final approximate solution for a given source. See also Barnes–Hut simulation Fast multipole method Laplace expansion Legendre polynomials Quadrupole magnets are used in particle accelerators Solid harmonics Toroidal moment References Potential theory Vector calculus Moment (physics)