Vector spherical harmonics

The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

Given a scalar spherical harmonic Yℓm(θ, φ), we define three VSH: with

being the unit vector along the radial direction in spherical coordinates and

the vector along the radial direction with the same norm as the radius, i.e.,

The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.

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

are transverse components (with respect to the radius vector

Like the scalar spherical harmonics, the VSH satisfy

which cuts the number of independent functions roughly in half.

The orthogonality relations allow one to compute the spherical multipole moments of a vector field as

By superposition we obtain the divergence of any vector field:

By superposition we obtain the curl of any vector field:

Also note that this action becomes symmetric, i.e. the off-diagonal coefficients are equal to

Expressions for negative values of m are obtained by applying the symmetry relations.

For instance, a magnetic multipole is due to an oscillating current with angular frequency

Substituting into Maxwell equations, Gauss's law is automatically satisfied

Gauss' law for the magnetic field implies

In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates.

In the component form vector spherical harmonics are written as:

For electric harmonics, the radial part decreases faster than angular, and for big

We can also see that for electric and magnetic harmonics angular parts are the same up to permutation of the polar and azimuthal unit vectors, so for big

electric and magnetic harmonics vectors are equal in value and perpendicular to each other.

The solutions of the Helmholtz vector equation obey the following orthogonality relations:[7]

Under rotation, vector spherical harmonics are transformed through each other in the same way as the corresponding scalar spherical functions, which are generating for a specific type of vector harmonics.

The behavior under rotations is the same for electrical, magnetic and longitudinal harmonics.

In the calculation of the Stokes' law for the drag that a viscous fluid exerts on a small spherical particle, the velocity distribution obeys Navier–Stokes equations neglecting inertia, i.e.,

In spherical coordinates this velocity at infinity can be written as

The last expression suggests an expansion in spherical harmonics for the liquid velocity and the pressure

are spherical Bessel functions, with help of plane wave expansion one can obtain the following integral relations:[11]

[12][11] For vector spherical harmonics the following relations are obtained: