Taylor expansion in " d " / " R " ( see multipole expansion and quadrupole ) allows this potential to be expressed as a series.
32.
It is not possible to construct a field whose multipole expansion contains nonzero odd terms, and that is also consistent with the experimentally-observed behavior of magnetic fields.
33.
When dealing with subsets of the multipole expansion, the base ordering of the components ( according to whatever ordering scheme has been agreed on ) is usually not changed.
34.
This Taylor expansion is known as the multipole expansion of V because the terms in this series can be regarded as energies of two interacting multipoles, one on each monomer.
35.
Even more important : As foundation for the idea of orthonormal function system, which leads to Fourier transformation, multipole expansion, etc and has applications in virtually any kind of physics.
36.
Multipole expansions are also useful in numerical simulations, and form the basis of the Fast Multipole Method of Rokhlin, a general technique for efficient computation of energies and forces in systems of interacting particles.
37.
And in the source itself, or as soon as one enters a region of inhomogeneous materials, the multipole expansion is no longer valid and the full solution of Maxwell's equations is generally required.
38.
If the overlap center is not at an atom, one can move the origin of the multipole expansion to the nearest distributed multipole site, re-expressing the series to account for the change of origin.
39.
This medium can be seen as an idealization growing from the multipole expansion of the potential of an arbitrarily complex charge distribution, truncation of the expansion, and the forcing of the truncated form to apply everywhere.
40.
Formally, the field can be expressed as a multipole expansion : A dipole field, plus a quadrupole field, plus an octopole field, etc . in the Amp�re model, but this can be very cumbersome mathematically.
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