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In mathematics, '''vector spherical harmonics (VSH)''' are an extension of the scalar [[spherical harmonics]] for the use with [[vector field]]s.


==Definition==
Several conventions have been used to define the VSH.<ref>
R.G. Barrera, G.A. Estévez and J. Giraldo, ''Vector spherical harmonics and their application to magnetostatics'', Eur. J. Phys. '''6''' 287-294 (1985)</ref><ref>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)</ref><ref>
E. L. Hill, ''The theory of Vector Spherical Harmonics'', Am. J. Phys. '''22''', 211-214 (1954)</ref><ref>
E. J. Weinberg, ''Monopole vector spherical harmonics'', Phys. Rev. D. '''49''', 1086-1092 (1994)</ref><ref>P.M. Morse and H. Feshbach, ''Methods of Theoretical Physics, Part II'', New York: McGraw-Hill, 1898-1901 (1953)</ref>
We follow that of Barrera ''et al.''. Given a scalar spherical harmonic <math>Y_{lm}(\theta,\varphi)</math> we define three VSH:


* <math>\mathbf{Y}_{lm} = Y_{lm}\hat{\mathbf{r}}</math>
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* <math>\mathbf{\Psi}_{lm} = r\nabla Y_{lm}</math>
 
* <math>\mathbf{\Phi}_{lm} = \vec{\mathbf{r}}\times\nabla Y_{lm}</math>
 
being <math>\hat{\mathbf{r}}</math> the unitary vector along the radial direction and <math>\vec{\mathbf{r}}</math> the position vector of the point with spherical coordinates <math>r</math>,<math>\theta</math> and<math>\phi</math>. The radial factors are included to guarantee that the dimensions of the VSH are the same as 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]]
 
: <math>\mathbf{E}=\sum_{l=0}^\infty\sum_{m=-l}^l\left(E^r_{lm}(r)\mathbf{Y}_{lm}+E^{(1)}_{lm}(r)\mathbf{\Psi}_{lm}+E^{(2)}_{lm}(r)\mathbf{\Phi}_{lm}\right)</math>
 
The labels on the components reflect that <math>E^r_{lm}</math> is the radial component of the vector field, while <math>E^{(1)}_{lm}</math> and <math>E^{(2)}_{lm}</math> are transverse components.
 
==Main Properties==
 
===Symmetry===
Like the scalar spherical harmonics, the VSH satisfy
 
:<math>\mathbf{Y}_{l,-m} = (-1)^m \mathbf{Y}^*_{lm}\qquad\mathbf{\Psi}_{l,-m} = (-1)^m \mathbf{\Psi}^*_{lm}\qquad\mathbf{\Phi}_{l,-m} = (-1)^m \mathbf{\Phi}^*_{lm}</math>
 
===Orthogonality===
The VSH are orthogonal in the usual three-dimensional way
 
:<math>\mathbf{Y}_{lm}\cdot\mathbf{\Psi}_{lm}=0\qquad\mathbf{Y}_{lm}\cdot\mathbf{\Phi}_{lm}=0\qquad\mathbf{\Psi}_{lm}\cdot\mathbf{\Phi}_{lm}=0</math>
 
but also in the Hilbert space
 
:<math>\int\mathbf{Y}_{lm}\cdot  \mathbf{Y}^*_{l'm'}\,\mathrm{d}\Omega  = \delta_{ll'}\delta_{mm'}</math>
 
:<math>\int\mathbf{\Psi}_{lm}\cdot  \mathbf{\Psi}^*_{l'm'}\,\mathrm{d}\Omega  = l(l+1)\delta_{ll'}\delta_{mm'}</math>
 
:<math>\int\mathbf{\Phi}_{lm}\cdot  \mathbf{\Phi}^*_{l'm'}\,\mathrm{d}\Omega  = l(l+1)\delta_{ll'}\delta_{mm'}</math>
 
:<math>\int\mathbf{Y}_{lm}\cdot  \mathbf{\Psi}^*_{l'm'}\,\mathrm{d}\Omega  = 0</math>
 
:<math>\int\mathbf{Y}_{lm}\cdot  \mathbf{\Phi}^*_{l'm'}\,\mathrm{d}\Omega  = 0</math>
 
:<math>\int\mathbf{\Psi}_{lm}\cdot  \mathbf{\Phi}^*_{l'm'}\,\mathrm{d}\Omega  = 0</math>
 
===Vector multipole moments===
The orthogonality relations allow to compute the spherical multipole moments of a vector field as
 
:<math>E^r_{lm} = \int \mathbf{E}\cdot \mathbf{Y}^*_{lm}\,\mathrm{d}\Omega</math>
 
:<math>E^{(1)}_{lm} = \frac{1}{l(l+1)}\int \mathbf{E}\cdot \mathbf{\Psi}^*_{lm}\,\mathrm{d}\Omega</math>
 
:<math>E^{(2)}_{lm} = \frac{1}{l(l+1)}\int \mathbf{E}\cdot \mathbf{\Phi}^*_{lm}\,\mathrm{d}\Omega</math>
 
===The gradient of a scalar field===
Given the [[multipole expansion]] of a scalar field
 
: <math>\phi = \sum_{l=0}^\infty \sum_{m=-l}^l \phi_{lm}(r) Y_{lm}(\theta,\phi)</math>
 
we can express its gradient in terms of the VSH as
 
: <math>\nabla\phi = \sum_{l=0}^\infty \sum_{m=-l}^l\left(\frac{\mathrm{d}\phi_{lm}}{\mathrm{d}r} \mathbf{Y}_{lm}+
\frac{\phi_{lm}}{r}\mathbf{\Psi}_{lm}\right)</math>
 
===Divergence===
For any multipole field we have
 
:<math>\nabla\cdot\left(f(r)\mathbf{Y}_{lm}\right) = \left(\frac{\mathrm{d}f}{\mathrm{d}r}+\frac{2}{r}f\right)Y_{lm}</math>
 
:<math>\nabla\cdot\left(f(r)\mathbf{\Psi}_{lm}\right) = -\frac{l(l+1)}{r}fY_{lm}</math>
 
:<math>\nabla\cdot\left(f(r)\mathbf{\Phi}_{lm}\right) = 0</math>
 
By superposition we obtain the [[divergence]] of any vector field
 
:<math>\nabla\cdot\mathbf{E} = \sum_{l=0}^\infty \sum_{m=-l}^l \left(\frac{\mathrm{d}E^r_{lm}}{\mathrm{d}r}+\frac{2}{r}E^r_{lm}-\frac{l(l+1)}{r}E^{(1)}_{lm}\right)Y_{lm}</math>
 
we see that the component on <math>\mathbf{\Phi}_{lm}</math> is always [[solenoidal]].
 
===Curl===
For any multipole field we have
 
:<math>\nabla\times\left(f(r)\mathbf{Y}_{lm}\right) =-\frac{1}{r}f\mathbf{\Phi}_{lm}</math>
 
:<math>\nabla\times\left(f(r)\mathbf{\Psi}_{lm}\right) = \left(\frac{\mathrm{d}f}{\mathrm{d}r}+\frac{1}{r}f\right)\mathbf{\Phi}_{lm}</math>
 
:<math>\nabla\times\left(f(r)\mathbf{\Phi}_{lm}\right) = -\frac{l(l+1)}{r}f\mathbf{Y}_{lm}-\left(\frac{\mathrm{d}f}{\mathrm{d}r}+\frac{1}{r}f\right)\mathbf{\Psi}_{lm}</math>
 
By superposition we obtain the [[Curl (mathematics)|curl]] of any vector field
 
:<math>\nabla\times\mathbf{E} = \sum_{l=0}^\infty \sum_{m=-l}^l
\left(-\frac{l(l+1)}{r}E^{(2)}_{lm}\mathbf{Y}_{lm}-\left(\frac{\mathrm{d}E^{(2)}_{lm}}{\mathrm{d}r}+
\frac{1}{r}E^{(2)}_{lm}\right)\mathbf{\Psi}_{lm}+
\left(-\frac{1}{r}E^r_{lm}+\frac{\mathrm{d}E^{(1)}_{lm}}{\mathrm{d}r}+\frac{1}{r}E^{(1)}_{lm}\right)\mathbf{\Phi}_{lm}\right)</math>
 
==Examples==
 
===First vector spherical harmonics===
 
* <math>l=0\,</math>
 
:* <math>\mathbf{Y}_{00}= \sqrt{\frac{1}{4\pi}}\hat{\mathbf{r}}</math>
 
:* <math>\mathbf{\Psi}_{00}= \mathbf{0}</math>
 
:* <math>\mathbf{\Phi}_{00}= \mathbf{0}</math>
 
* <math>l=1\,</math>
 
:*<math>\mathbf{Y}_{10}= \sqrt{\frac{3}{4\pi}}\cos\theta\,\hat{\mathbf{r}}</math>
:*<math>\mathbf{Y}_{11}= -\sqrt{\frac{3}{8\pi}}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta\,\hat{\mathbf{r}}</math>
 
:*<math>\mathbf{\Psi}_{10}= -\sqrt{\frac{3}{4\pi}}\sin\theta\,\hat{\mathbf{\theta}}</math>
:*<math>\mathbf{\Psi}_{11}= -\sqrt{\frac{3}{8\pi}}\mathrm{e}^{\mathrm{i}\varphi}\left(\cos\theta\,\hat{\mathbf{\theta}}+\mathrm{i}\,\hat{\mathbf{\varphi}}\right)</math>
 
:*<math>\mathbf{\Phi}_{10}= -\sqrt{\frac{3}{4\pi}}\sin\theta\,\hat{\mathbf{\varphi}}</math>
:*<math>\mathbf{\Phi}_{11}= \sqrt{\frac{3}{8\pi}}\mathrm{e}^{\mathrm{i}\varphi}\left(\mathrm{i}\,\hat{\mathbf{\theta}}-\cos\theta\,\hat{\mathbf{\varphi}}\right)</math>
 
The expression for negative values of m are obtained applying the symmetry relations.
 
==Applications==
===Electrodynamics===
The VSH are especially useful in the study of [[Electromagnetic_wave_equation#Multipole_expansion|multipole radiation fields]]. For instance, a magnetic multipole is due to an oscillating current with angular frequency <math>\omega\,</math> and complex amplitude
 
:<math>\hat{\mathbf{J}}= J(r)\mathbf{\Phi}_{lm}</math>
 
and the corresponding electric and magnetic fields can be written as
 
:<math>\hat{\mathbf{E}}= E(r)\mathbf{\Phi}_{lm}</math>
 
:<math>\hat{\mathbf{B}}= B^r(r)\mathbf{Y}_{lm}+B^{(1)}(r)\mathbf{\Psi}_{lm}</math>
 
Substituting into Maxwell equations, Gauss' law is automatically satisfied
 
:<math>\nabla\cdot\hat{\mathbf{E}}=0</math>
 
while Faraday's law decouples in
 
:<math>\nabla\times\hat{\mathbf{E}}=-\mathrm{i}\omega\hat{\mathbf{B}}\qquad\Rightarrow\qquad
\left\{\begin{array}{l}\displaystyle \frac{l(l+1)}{r}E = \mathrm{i}\omega B^r \\ \ \\
\displaystyle\frac{\mathrm{d}E}{\mathrm{d}r} +\frac{E}{r}= \mathrm{i}\omega B^{(1)}\end{array}\right.</math>
 
Gauss' law for the magnetic field implies
 
:<math>\nabla\cdot\hat{\mathbf{B}} = 0\quad\Rightarrow \quad\frac{\mathrm{d}B^r}{\mathrm{d}r}+\frac{2}{r}B^r - \frac{l(l+1)}{r}B^{(1)}=0</math>
 
and Ampère-Maxwell's equation gives
 
:<math>\nabla\times\hat{\mathbf{B}} = \mu_0 \hat{\mathbf{J}} + \mathrm{i}\mu_0\varepsilon_0\omega \hat{\mathbf{E}} \quad\Rightarrow\quad -\frac{B^r}{r}+\frac{\mathrm{d}B^{(1)}}{\mathrm{d}r}+\frac{B^{(1)}}{r} = \mu_0J+\mathrm{i}\omega\mu_0\varepsilon_0E</math>
 
In this way, the partial differential equations have been transformed in a set of ordinary differential equations.
 
==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.
 
:<math>\nabla\cdot \mathbf{v} = 0</math>
 
:<math>\mathbf{0} = -\nabla p + \eta \nabla^2\mathbf{v}</math>
 
with the boundary conditions
 
:<math>\mathbf{v} = \mathbf{0}\quad (r=a)</math>
 
:<math>\mathbf{v} = -\mathbf{U}_0\quad (r\to\infty)</math>
 
being <math>\mathbf{U}\,</math> the relative velocity of the particle to the fluid far from the particle. In spherical coordinates this velocity at infinity can be written as
 
:<math>\mathbf{U}_0 = U_0\left(\cos\theta\, \hat{\mathbf{r}} - \sin\theta \,\hat{\mathbf{\theta}}\right) = U_0 \left(\mathbf{Y}_{10} + \mathbf{\Psi}_{10}\right) </math>
 
The last expression suggest an expansion on spherical harmonics for the liquid velocity and the pressure
 
:<math>p=p(r)Y_{10}\,</math>
 
:<math>\mathbf{v} = v^r(r) \mathbf{Y}_{10} + v^{(1)}(r) \mathbf{\Psi}_{10}</math>
 
Substitution in the Navier-Stokes equations produces a set of ordinary differential equations for the coefficients.
 
==See also==
* [[Harmonic Vector Fields]]
* [[Spherical harmonics]]
* [[Spin spherical harmonics]]
* [[Multipole expansion]]
* [[Electromagnetic radiation]]
* [[Spherical coordinates]]
* [[Spherical basis]]
 
==External links==
* [http://mathworld.wolfram.com/VectorSphericalHarmonic.html ''Vector Spherical Harmonics'' at Eric Weisstein's Mathworld]
 
<div class="references">
 
==References==
<references />
 
[[Category:Vector calculus]]

Latest revision as of 01:23, 19 November 2014


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