Vector spherical harmonics

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.
Definition.
Several conventions have been used to define the VSH. 
We follow that of Barrera "et al.". Given a scalar spherical harmonic , we define three VSH:
with formula_4 being the unit vector along the radial direction in spherical coordinates and formula_5 the vector along the radial direction with the same norm as the radius, i.e., formula_6. 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
The labels on the components reflect that formula_8 is the radial component of the vector field, while formula_9 and formula_10 are transverse components (with respect to the radius vector formula_5).
Main Properties.
Symmetry.
Like the scalar spherical harmonics, the VSH satisfy
which cuts the number of independent functions roughly in half. The star indicates complex conjugation.
Orthogonality.
The VSH are orthogonal in the usual three-dimensional way at each point formula_5:
They are also orthogonal in Hilbert space:
An additional result at a single point formula_5 (not reported in Barrera et al, 1985) is, for all formula_17,
Vector multipole moments.
The orthogonality relations allow one to compute the spherical multipole moments of a vector field as
The gradient of a scalar field.
Given the multipole expansion of a scalar field
we can express its gradient in terms of the VSH as
Divergence.
For any multipole field we have
By superposition we obtain the divergence of any vector field:
We see that the component on is always solenoidal.
Curl.
For any multipole field we have
By superposition we obtain the curl of any vector field:
Laplacian.
The action of the Laplace operator formula_26 separates as follows:
where formula_28 and
Also note that this action becomes symmetric, i.e. the off-diagonal coefficients are equal to formula_30, for properly normalized VSH.
Examples.
First vector spherical harmonics.
Expressions for negative values of "m" are obtained by applying the symmetry relations.
Applications.
Electrodynamics.
The VSH are especially useful in the study of multipole radiation fields. For instance, a magnetic multipole is due to an oscillating current with angular frequency formula_41 and complex amplitude
and the corresponding electric and magnetic fields, can be written as
Substituting into Maxwell equations, Gauss' law is automatically satisfied
while Faraday's law decouples as
Gauss' law for the magnetic field implies
and Ampère-Maxwell's equation gives
In this way, the partial differential equations have been transformed into a set of ordinary differential equations.
Alternative definition.
In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates.
In this case, vector spherical harmonics are generated by scalar functions, which are solutions of scalar Helmholtz equation with the wavevector formula_48. 
here formula_50 - associated Legendre polynomials, and formula_51 - any of spherical Bessel functions.
Vector spherical harmonics are defined as:
Here we use harmonics real-valued angular part, where formula_55, but complex functions can be introduced in the same way .
Let us introduce the notation formula_56. In the component form vector spherical harmonics are written as:
There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big formula_61 can be neglected. 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 formula_61 electric and magnetic harmonics vectors are equal in value and perpendicular to each other.
Longitual harmonics:
Orthogonality.
The solutions of the Helmholtz vector equation obey the following orthogonality relations: 
All other integrals over the angles between different functions or functions with different indices are equal to zero.
Rotation and inversion.
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. For example, if the generating functions are the usual spherical harmonics, then the vector harmonics will also be transformed through the Wigner D-matrixes
The behavior under rotations is the same for electrical, magnetic and longitudinal harmonics.
Under inversion, electric and longitudinal spherical harmonics behave in the same way as scalar spherical functions, i.e.
and magnetic ones have the opposite parity:
Fluid dynamics.
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.,
with the boundary conditions
where U is the relative velocity of the particle to the fluid far from the particle. 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
Substitution in the Navier–Stokes equations produces a set of ordinary differential equations for the coefficients.
Integral relations.
Here the following definitions are used:
In case, when instead of formula_78 are spherical bessel functions, with help of plane wave expansion one can obtain the following integral relations:
In case, when formula_78 are spherical hankel functions, one should use the different formulae. For vector spherical harmonics the following relations are obtained:
where formula_84, index formula_85 means, that spherical hankel functions are used.