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 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 The labels on the components reflect that is the radial component of the vector field, while and are transverse components.
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 : They are also orthogonal in Hilbert space: An additional result at a single point is, for all,
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 separates as follows: where and Also note that this action becomes symmetric, i.e. the off-diagonal coefficients are equal to, for properly normalized VSH.
Examples
First vector spherical harmonics
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Expressions for negative values of m are obtained by applying the symmetry relations.
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 . here - associated Legendre polynomials, and - any of spherical Bessel functions. Vector spherical harmonics are defined as: Here we use harmonics real-valued angular part, where, but complex functions can be introduced in the same way. Let us introduce the notation. 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 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 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.
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.
