Vector spherical harmonics

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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[edit]

Several conventions have been used to define the VSH.[1][2][3][4][5] We follow that of Barrera et al.. 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

The labels on the components reflect that is the radial component of the vector field, while and are transverse components (w.r.to the radius vector ).

Main Properties[edit]

Symmetry[edit]

Like the scalar spherical harmonics, the VSH satisfy

which cuts the number of independent functions roughly in half. The star indicates complex conjugation.

Orthogonality[edit]

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 (not reported in Barrera et al, 1985) is, for all ,

Vector multipole moments[edit]

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

The gradient of a scalar field[edit]

Given the multipole expansion of a scalar field

we can express its gradient in terms of the VSH as

Divergence[edit]

For any multipole field we have

By superposition we obtain the divergence of any vector field:

We see that the component on Φℓm is always solenoidal.

Curl[edit]

For any multipole field we have

By superposition we obtain the curl of any vector field:

Laplacian[edit]

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[edit]

First vector spherical harmonics[edit]

  • .
  • .
  • .

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

Applications[edit]

Electrodynamics[edit]

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 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.

Fluid dynamics[edit]

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.

See also[edit]

References[edit]

  1. ^ R.G. Barrera, G.A. Estévez and J. Giraldo, Vector spherical harmonics and their application to magnetostatics, Eur. J. Phys. 6 287-294 (1985)
  2. ^ B. Carrascal, G.A. Estevez, P. Lee and V. Lorenzo Vector spherical harmonics and their application to classical electrodynamics, Eur. J. Phys., 12, 184-191 (1991)
  3. ^ E. L. Hill, The theory of Vector Spherical Harmonics, Am. J. Phys. 22, 211-214 (1954)
  4. ^ E. J. Weinberg, Monopole vector spherical harmonics, Phys. Rev. D. 49, 1086-1092 (1994)
  5. ^ P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)

External links[edit]