The FMM has also been applied in accelerating the iterative solver in the Rokhlin and is based on the multipole expansion of the vector Helmholtz equation.
22.
A related approach is to divide the charges into those nearby the point of observation, and those far enough away to allow a multipole expansion.
23.
Formally, the field can be expressed as a multipole expansion : A dipole field, plus a quadrupole field, plus an octupole field, etc.
24.
While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank.
25.
As in the multipole expansion, we can replicate the highly non-trivial high-energy by a generic set of effective point-like interactions.
26.
The field of a point dipole has a particularly simple form, and the order-1 term in the multipole expansion is precisely the point dipole field.
27.
The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay.
28.
However, in the multipole expansion of a " magnetic " field, the " monopole " term is always exactly zero ( for ordinary matter ).
29.
Theoretically, an electric dipole is defined by the first-order term of the multipole expansion, and consists of two equal and opposite charges infinitely close together.
30.
The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion
