Extension of the scalar spherical harmonics for use with vector fields
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 "Yℓm"("θ", "φ"), 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
formula_7
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
formula_12
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:
formula_14
They are also orthogonal in Hilbert space:
formula_15
An additional result at a single point formula_5 (not reported in Barrera et al, 1985) is, for all formula_17,
formula_18
Vector multipole moments.
The orthogonality relations allow one to compute the spherical multipole moments of a vector field as
formula_19
The gradient of a scalar field.
Given the multipole expansion of a scalar field
formula_20
we can express its gradient in terms of the VSH as
formula_21
Divergence.
For any multipole field we have
formula_22
By superposition we obtain the divergence of any vector field:
formula_23
We see that the component on Φ"ℓm" is always solenoidal.
Curl.
For any multipole field we have
formula_24
By superposition we obtain the curl of any vector field:
formula_25
Laplacian.
The action of the Laplace operator formula_26 separates as follows:
formula_27
where formula_28 and
formula_29
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.
\hat{\mathbf{r}}, \\
\mathbf{\Psi}_{00} &amp;= \mathbf{0}, \\
\mathbf{\Phi}_{00} &amp;= \mathbf{0}.
\end{align} ;/math ;
formula_32
formula_33
formula_34
formula_36
formula_37
formula_38
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_39 and complex amplitude
formula_40
and the corresponding electric and magnetic fields, can be written as
formula_41
Substituting into Maxwell equations, Gauss's law is automatically satisfied
formula_42
while Faraday's law decouples as
formula_43
Gauss' law for the magnetic field implies
formula_44
and Ampère–Maxwell's equation gives
formula_45
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_46. 
formula_47
here formula_48 are the associated Legendre polynomials, and formula_49 are any of the spherical Bessel functions.
Vector spherical harmonics are defined as:
Here we use harmonics real-valued angular part, where formula_53, but complex functions can be introduced in the same way.
Let us introduce the notation formula_54. In the component form vector spherical harmonics are written as:
formula_55
formula_56
formula_57
formula_58
There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big formula_59 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_59 electric and magnetic harmonics vectors are equal in value and perpendicular to each other.
Longitudinal harmonics:
formula_61
Orthogonality.
The solutions of the Helmholtz vector equation obey the following orthogonality relations: 
formula_62
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-matrices
formula_63
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.
formula_64
and magnetic ones have the opposite parity:
formula_65
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.,
formula_66
with the boundary conditions
formula_67
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
formula_68
The last expression suggests an expansion in spherical harmonics for the liquid velocity and the pressure
formula_69
Substitution in the Navier–Stokes equations produces a set of ordinary differential equations for the coefficients.
Integral relations.
Here the following definitions are used:
formula_70
formula_71
formula_72
In case, when instead of formula_73 are spherical Bessel functions, with help of plane wave expansion one can obtain the following integral relations:
formula_74
formula_75
In case, when formula_73 are spherical Hankel functions, one should use the different formulae. For vector spherical harmonics the following relations are obtained:
formula_77
formula_78
where formula_79, index formula_80 means, that spherical Hankel functions are used.