Multipole moments in mathematics and mathematical physics are an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of distribution theory, as directional derivatives. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... Mathematical physics is a scientific discipline aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In physics, a field is an assignment of a quantity to every point in space. ... A point source may be a source of light that is treated as having no physical extension, or a source in fluid flow, or in electrostatics, or a source of pollution coming from a single cause, such as a municipal plant. ... This page deals with mathematical distributions. ... In mathematics, the directional derivative of a multivariate differentiable function along a given unit vector intuitively represents the rate of change of the function in the direction of that vector. ...

They are useful because in practice, many fields can be easily approximated with a finite number of multipole moments (though an infinite number is required to reconstruct an arbitrary field). Problems solved once for a given order of multipole moment may be linearly combined to create a final approximate solution for a given source. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

See Multipole expansion for an example in classical electromagnetism. Multipole expansion for electric potentials The (scalar) potential at the point x for an arbitrary charge distribution ρ(x) is given by This can be expanded in (negative) powers of , obtaining (after some work) the multipole expansion where this integral, like the previous one, is over all of space, Pn... Electromagnetism is the physics of electromagnetic fields: a field, encompassing all of space, comprised of electrical and magnetic fields. ...

See also: Dipole A dipole (Greek: dyo = two and polos = pivot) is a pair of electric charges or magnetic poles of equal magnitude but opposite polarity (opposite electronic charges), separated by some (usually small) distance. ...