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.




Contents






  • 1 Definition


  • 2 Main Properties


    • 2.1 Symmetry


    • 2.2 Orthogonality


    • 2.3 Vector multipole moments


    • 2.4 The gradient of a scalar field


    • 2.5 Divergence


    • 2.6 Curl


    • 2.7 Laplacian




  • 3 Examples


    • 3.1 First vector spherical harmonics




  • 4 Applications


    • 4.1 Electrodynamics


    • 4.2 Fluid dynamics




  • 5 See also


  • 6 References


  • 7 External links





Definition


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:



  • Ylm=Ylmr^,{displaystyle mathbf {Y} _{lm}=Y_{lm}{hat {mathbf {r} }},}{displaystyle mathbf {Y} _{lm}=Y_{lm}{hat {mathbf {r} }},}

  • Ψlm=r∇Ylm,{displaystyle mathbf {Psi } _{lm}=rnabla Y_{lm},}{displaystyle mathbf {Psi } _{lm}=rnabla Y_{lm},}

  • Φlm=r×∇Ylm,{displaystyle mathbf {Phi } _{lm}=mathbf {r} times nabla Y_{lm},}{displaystyle mathbf {Phi } _{lm}=mathbf {r} times nabla Y_{lm},}


with r^{displaystyle {hat {mathbf {r} }}}{hat {mathbf {r} }} being the unit vector along the radial direction in spherical coordinates and r{displaystyle mathbf {r} }mathbf {r} the vector along the radial direction with the same norm as the radius, i.e., r=rr^{displaystyle mathbf {r} =r{hat {mathbf {r} }}}{displaystyle mathbf {r} =r{hat {mathbf {r} }}}. 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


E=∑l=0∞∑m=−ll(Elmr(r)Ylm+Elm(1)(r)Ψlm+Elm(2)(r)Φlm).{displaystyle mathbf {E} =sum _{l=0}^{infty }sum _{m=-l}^{l}left(E_{lm}^{r}(r)mathbf {Y} _{lm}+E_{lm}^{(1)}(r)mathbf {Psi } _{lm}+E_{lm}^{(2)}(r)mathbf {Phi } _{lm}right).}{displaystyle mathbf {E} =sum _{l=0}^{infty }sum _{m=-l}^{l}left(E_{lm}^{r}(r)mathbf {Y} _{lm}+E_{lm}^{(1)}(r)mathbf {Psi } _{lm}+E_{lm}^{(2)}(r)mathbf {Phi } _{lm}right).}

The labels on the components reflect that Elmr{displaystyle E_{lm}^{r}}E_{{lm}}^{r} is the radial component of the vector field, while Elm(1){displaystyle E_{lm}^{(1)}}E_{{lm}}^{{(1)}} and Elm(2){displaystyle E_{lm}^{(2)}}E_{{lm}}^{{(2)}} are transverse components (w.r.to the radius vector r{displaystyle mathbf {r} }mathbf {r} ).



Main Properties



Symmetry


Like the scalar spherical harmonics, the VSH satisfy


Yl,−m=(−1)mYlm∗,Ψl,−m=(−1)mΨlm∗,Φl,−m=(−1)mΦlm∗,{displaystyle {begin{aligned}mathbf {Y} _{l,-m}&=(-1)^{m}mathbf {Y} _{lm}^{*},\mathbf {Psi } _{l,-m}&=(-1)^{m}mathbf {Psi } _{lm}^{*},\mathbf {Phi } _{l,-m}&=(-1)^{m}mathbf {Phi } _{lm}^{*},end{aligned}}}{displaystyle {begin{aligned}mathbf {Y} _{l,-m}&=(-1)^{m}mathbf {Y} _{lm}^{*},\mathbf {Psi } _{l,-m}&=(-1)^{m}mathbf {Psi } _{lm}^{*},\mathbf {Phi } _{l,-m}&=(-1)^{m}mathbf {Phi } _{lm}^{*},end{aligned}}}

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 r{displaystyle mathbf {r} }mathbf {r} :


Ylm(r)⋅Ψlm(r)=0,Ylm(r)⋅Φlm(r)=0,Ψlm(r)⋅Φlm(r)=0.{displaystyle {begin{aligned}mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Psi } _{lm}(mathbf {r} )&=0,\mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Phi } _{lm}(mathbf {r} )&=0,\mathbf {Psi } _{lm}(mathbf {r} )cdot mathbf {Phi } _{lm}(mathbf {r} )&=0.end{aligned}}}{displaystyle {begin{aligned}mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Psi } _{lm}(mathbf {r} )&=0,\mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Phi } _{lm}(mathbf {r} )&=0,\mathbf {Psi } _{lm}(mathbf {r} )cdot mathbf {Phi } _{lm}(mathbf {r} )&=0.end{aligned}}}

They are also orthogonal in Hilbert space:


∫Ylm⋅Yl′m′∗dΩ=δll′δmm′,∫Ψlm⋅Ψl′m′∗dΩ=l(l+1)δll′δmm′,∫Φlm⋅Φl′m′∗dΩ=l(l+1)δll′δmm′,∫Ylm⋅Ψl′m′∗dΩ=0,∫Ylm⋅Φl′m′∗dΩ=0,∫Ψlm⋅Φl′m′∗dΩ=0.{displaystyle {begin{aligned}int mathbf {Y} _{lm}cdot mathbf {Y} _{l'm'}^{*},dOmega &=delta _{ll'}delta _{mm'},\int mathbf {Psi } _{lm}cdot mathbf {Psi } _{l'm'}^{*},dOmega &=l(l+1)delta _{ll'}delta _{mm'},\int mathbf {Phi } _{lm}cdot mathbf {Phi } _{l'm'}^{*},dOmega &=l(l+1)delta _{ll'}delta _{mm'},\int mathbf {Y} _{lm}cdot mathbf {Psi } _{l'm'}^{*},dOmega &=0,\int mathbf {Y} _{lm}cdot mathbf {Phi } _{l'm'}^{*},dOmega &=0,\int mathbf {Psi } _{lm}cdot mathbf {Phi } _{l'm'}^{*},dOmega &=0.end{aligned}}}{displaystyle {begin{aligned}int mathbf {Y} _{lm}cdot mathbf {Y} _{l'm'}^{*},dOmega &=delta _{ll'}delta _{mm'},\int mathbf {Psi } _{lm}cdot mathbf {Psi } _{l'm'}^{*},dOmega &=l(l+1)delta _{ll'}delta _{mm'},\int mathbf {Phi } _{lm}cdot mathbf {Phi } _{l'm'}^{*},dOmega &=l(l+1)delta _{ll'}delta _{mm'},\int mathbf {Y} _{lm}cdot mathbf {Psi } _{l'm'}^{*},dOmega &=0,\int mathbf {Y} _{lm}cdot mathbf {Phi } _{l'm'}^{*},dOmega &=0,\int mathbf {Psi } _{lm}cdot mathbf {Phi } _{l'm'}^{*},dOmega &=0.end{aligned}}}

An additional result at a single point r{displaystyle mathbf {r} }mathbf {r} (not reported in Barrera et al, 1985) is, for all l,m,l′,m′{displaystyle l,m,l',m'}l,m,l',m',


Ylm(r)⋅Ψl′m′(r)=0,Ylm(r)⋅Φl′m′(r)=0.{displaystyle {begin{aligned}mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Psi } _{l'm'}(mathbf {r} )&=0,\mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Phi } _{l'm'}(mathbf {r} )&=0.end{aligned}}}{displaystyle {begin{aligned}mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Psi } _{l'm'}(mathbf {r} )&=0,\mathbf {Y} _{lm}(mathbf {r} )cdot mathbf {Phi } _{l'm'}(mathbf {r} )&=0.end{aligned}}}


Vector multipole moments


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


Elmr=∫E⋅Ylm∗dΩ,Elm(1)=1l(l+1)∫E⋅Ψlm∗dΩ,Elm(2)=1l(l+1)∫E⋅Φlm∗dΩ.{displaystyle {begin{aligned}E_{lm}^{r}&=int mathbf {E} cdot mathbf {Y} _{lm}^{*},dOmega ,\E_{lm}^{(1)}&={frac {1}{l(l+1)}}int mathbf {E} cdot mathbf {Psi } _{lm}^{*},dOmega ,\E_{lm}^{(2)}&={frac {1}{l(l+1)}}int mathbf {E} cdot mathbf {Phi } _{lm}^{*},dOmega .end{aligned}}}{displaystyle {begin{aligned}E_{lm}^{r}&=int mathbf {E} cdot mathbf {Y} _{lm}^{*},dOmega ,\E_{lm}^{(1)}&={frac {1}{l(l+1)}}int mathbf {E} cdot mathbf {Psi } _{lm}^{*},dOmega ,\E_{lm}^{(2)}&={frac {1}{l(l+1)}}int mathbf {E} cdot mathbf {Phi } _{lm}^{*},dOmega .end{aligned}}}


The gradient of a scalar field


Given the multipole expansion of a scalar field


ϕ=∑l=0∞∑m=−llϕlm(r)Ylm(θ,ϕ),{displaystyle phi =sum _{l=0}^{infty }sum _{m=-l}^{l}phi _{lm}(r)Y_{lm}(theta ,phi ),}{displaystyle phi =sum _{l=0}^{infty }sum _{m=-l}^{l}phi _{lm}(r)Y_{lm}(theta ,phi ),}

we can express its gradient in terms of the VSH as


∇ϕ=∑l=0∞∑m=−ll(dϕlmdrYlm+ϕlmrΨlm).{displaystyle nabla phi =sum _{l=0}^{infty }sum _{m=-l}^{l}left({frac {dphi _{lm}}{dr}}mathbf {Y} _{lm}+{frac {phi _{lm}}{r}}mathbf {Psi } _{lm}right).}{displaystyle nabla phi =sum _{l=0}^{infty }sum _{m=-l}^{l}left({frac {dphi _{lm}}{dr}}mathbf {Y} _{lm}+{frac {phi _{lm}}{r}}mathbf {Psi } _{lm}right).}


Divergence


For any multipole field we have


∇⋅(f(r)Ylm)=(dfdr+2rf)Ylm,∇⋅(f(r)Ψlm)=−l(l+1)rfYlm,∇⋅(f(r)Φlm)=0.{displaystyle {begin{aligned}nabla cdot left(f(r)mathbf {Y} _{lm}right)&=left({frac {df}{dr}}+{frac {2}{r}}fright)Y_{lm},\nabla cdot left(f(r)mathbf {Psi } _{lm}right)&=-{frac {l(l+1)}{r}}fY_{lm},\nabla cdot left(f(r)mathbf {Phi } _{lm}right)&=0.end{aligned}}}{displaystyle {begin{aligned}nabla cdot left(f(r)mathbf {Y} _{lm}right)&=left({frac {df}{dr}}+{frac {2}{r}}fright)Y_{lm},\nabla cdot left(f(r)mathbf {Psi } _{lm}right)&=-{frac {l(l+1)}{r}}fY_{lm},\nabla cdot left(f(r)mathbf {Phi } _{lm}right)&=0.end{aligned}}}

By superposition we obtain the divergence of any vector field:


∇⋅E=∑l=0∞∑m=−ll(dElmrdr+2rElmr−l(l+1)rElm(1))Ylm.{displaystyle nabla cdot mathbf {E} =sum _{l=0}^{infty }sum _{m=-l}^{l}left({frac {dE_{lm}^{r}}{dr}}+{frac {2}{r}}E_{lm}^{r}-{frac {l(l+1)}{r}}E_{lm}^{(1)}right)Y_{lm}.}{displaystyle nabla cdot mathbf {E} =sum _{l=0}^{infty }sum _{m=-l}^{l}left({frac {dE_{lm}^{r}}{dr}}+{frac {2}{r}}E_{lm}^{r}-{frac {l(l+1)}{r}}E_{lm}^{(1)}right)Y_{lm}.}

We see that the component on Φℓm is always solenoidal.



Curl


For any multipole field we have


∇×(f(r)Ylm)=−1rfΦlm,∇×(f(r)Ψlm)=(dfdr+1rf)Φlm,∇×(f(r)Φlm)=−l(l+1)rfYlm−(dfdr+1rf)Ψlm.{displaystyle {begin{aligned}nabla times left(f(r)mathbf {Y} _{lm}right)&=-{frac {1}{r}}fmathbf {Phi } _{lm},\nabla times left(f(r)mathbf {Psi } _{lm}right)&=left({frac {df}{dr}}+{frac {1}{r}}fright)mathbf {Phi } _{lm},\nabla times left(f(r)mathbf {Phi } _{lm}right)&=-{frac {l(l+1)}{r}}fmathbf {Y} _{lm}-left({frac {df}{dr}}+{frac {1}{r}}fright)mathbf {Psi } _{lm}.end{aligned}}}{displaystyle {begin{aligned}nabla times left(f(r)mathbf {Y} _{lm}right)&=-{frac {1}{r}}fmathbf {Phi } _{lm},\nabla times left(f(r)mathbf {Psi } _{lm}right)&=left({frac {df}{dr}}+{frac {1}{r}}fright)mathbf {Phi } _{lm},\nabla times left(f(r)mathbf {Phi } _{lm}right)&=-{frac {l(l+1)}{r}}fmathbf {Y} _{lm}-left({frac {df}{dr}}+{frac {1}{r}}fright)mathbf {Psi } _{lm}.end{aligned}}}

By superposition we obtain the curl of any vector field:


∇×E=∑l=0∞∑m=−ll(−l(l+1)rElm(2)Ylm−(dElm(2)dr+1rElm(2))Ψlm+(−1rElmr+dElm(1)dr+1rElm(1))Φlm).{displaystyle nabla times mathbf {E} =sum _{l=0}^{infty }sum _{m=-l}^{l}left(-{frac {l(l+1)}{r}}E_{lm}^{(2)}mathbf {Y} _{lm}-left({frac {dE_{lm}^{(2)}}{dr}}+{frac {1}{r}}E_{lm}^{(2)}right)mathbf {Psi } _{lm}+left(-{frac {1}{r}}E_{lm}^{r}+{frac {dE_{lm}^{(1)}}{dr}}+{frac {1}{r}}E_{lm}^{(1)}right)mathbf {Phi } _{lm}right).}{displaystyle nabla times mathbf {E} =sum _{l=0}^{infty }sum _{m=-l}^{l}left(-{frac {l(l+1)}{r}}E_{lm}^{(2)}mathbf {Y} _{lm}-left({frac {dE_{lm}^{(2)}}{dr}}+{frac {1}{r}}E_{lm}^{(2)}right)mathbf {Psi } _{lm}+left(-{frac {1}{r}}E_{lm}^{r}+{frac {dE_{lm}^{(1)}}{dr}}+{frac {1}{r}}E_{lm}^{(1)}right)mathbf {Phi } _{lm}right).}


Laplacian


The action of the Laplace operator Δ=∇⋅∇{displaystyle Delta =nabla cdot nabla }{displaystyle Delta =nabla cdot nabla } separates as follows:


Δ(f(r)Zlm)=(1r2∂∂rr2∂f∂r)Zlm+f(r)ΔZlm,{displaystyle Delta left(f(r)mathbf {Z} _{lm}right)=left({frac {1}{r^{2}}}{frac {partial }{partial r}}r^{2}{frac {partial f}{partial r}}right)mathbf {Z} _{lm}+f(r)Delta mathbf {Z} _{lm},}{displaystyle Delta left(f(r)mathbf {Z} _{lm}right)=left({frac {1}{r^{2}}}{frac {partial }{partial r}}r^{2}{frac {partial f}{partial r}}right)mathbf {Z} _{lm}+f(r)Delta mathbf {Z} _{lm},}

where Zlm=Ylm,Ψlm,Φlm{displaystyle mathbf {Z} _{lm}=mathbf {Y} _{lm},mathbf {Psi } _{lm},mathbf {Phi } _{lm}}{displaystyle mathbf {Z} _{lm}=mathbf {Y} _{lm},mathbf {Psi } _{lm},mathbf {Phi } _{lm}} and


ΔYlm=−1r2(2+l(l+1))Ylm+2r2Ψlm,ΔΨlm=2r2l(l+1)Ylm−1r2l(l+1)Ψlm,ΔΦlm=−1r2l(l+1)Φlm.{displaystyle {begin{aligned}Delta mathbf {Y} _{lm}&=-{frac {1}{r^{2}}}(2+l(l+1))mathbf {Y} _{lm}+{frac {2}{r^{2}}}mathbf {Psi } _{lm},\Delta mathbf {Psi } _{lm}&={frac {2}{r^{2}}}l(l+1)mathbf {Y} _{lm}-{frac {1}{r^{2}}}l(l+1)mathbf {Psi } _{lm},\Delta mathbf {Phi } _{lm}&=-{frac {1}{r^{2}}}l(l+1)mathbf {Phi } _{lm}.end{aligned}}}{displaystyle {begin{aligned}Delta mathbf {Y} _{lm}&=-{frac {1}{r^{2}}}(2+l(l+1))mathbf {Y} _{lm}+{frac {2}{r^{2}}}mathbf {Psi } _{lm},\Delta mathbf {Psi } _{lm}&={frac {2}{r^{2}}}l(l+1)mathbf {Y} _{lm}-{frac {1}{r^{2}}}l(l+1)mathbf {Psi } _{lm},\Delta mathbf {Phi } _{lm}&=-{frac {1}{r^{2}}}l(l+1)mathbf {Phi } _{lm}.end{aligned}}}

Also note that this action becomes symmetric, i.e. the off-diagonal coefficients are equal to 2r2l(l+1){displaystyle {frac {2}{r^{2}}}{sqrt {l(l+1)}}}{displaystyle {frac {2}{r^{2}}}{sqrt {l(l+1)}}}, for properly normalized VSH.



Examples



First vector spherical harmonics



  • l=0{displaystyle l=0}l=0.

Y00=14πr^,Ψ00=0,Φ00=0.{displaystyle {begin{aligned}mathbf {Y} _{00}&={sqrt {frac {1}{4pi }}}{hat {mathbf {r} }},\mathbf {Psi } _{00}&=mathbf {0} ,\mathbf {Phi } _{00}&=mathbf {0} .end{aligned}}}{displaystyle {begin{aligned}mathbf {Y} _{00}&={sqrt {frac {1}{4pi }}}{hat {mathbf {r} }},\mathbf {Psi } _{00}&=mathbf {0} ,\mathbf {Phi } _{00}&=mathbf {0} .end{aligned}}}


  • l=1{displaystyle l=1}l=1.

Y10=34πcos⁡θr^,Y11=−38πeiφsin⁡θr^,{displaystyle {begin{aligned}mathbf {Y} _{10}&={sqrt {frac {3}{4pi }}}cos theta ,{hat {mathbf {r} }},\mathbf {Y} _{11}&=-{sqrt {frac {3}{8pi }}}e^{ivarphi }sin theta ,{hat {mathbf {r} }},end{aligned}}}{displaystyle {begin{aligned}mathbf {Y} _{10}&={sqrt {frac {3}{4pi }}}cos theta ,{hat {mathbf {r} }},\mathbf {Y} _{11}&=-{sqrt {frac {3}{8pi }}}e^{ivarphi }sin theta ,{hat {mathbf {r} }},end{aligned}}}

Ψ10=−34πsin⁡θθ^,Ψ11=−38πeiφ(cos⁡θθ^+iφ^),{displaystyle {begin{aligned}mathbf {Psi } _{10}&=-{sqrt {frac {3}{4pi }}}sin theta ,{hat {mathbf {theta } }},\mathbf {Psi } _{11}&=-{sqrt {frac {3}{8pi }}}e^{ivarphi }left(cos theta ,{hat {mathbf {theta } }}+i,{hat {mathbf {varphi } }}right),end{aligned}}}{displaystyle {begin{aligned}mathbf {Psi } _{10}&=-{sqrt {frac {3}{4pi }}}sin theta ,{hat {mathbf {theta } }},\mathbf {Psi } _{11}&=-{sqrt {frac {3}{8pi }}}e^{ivarphi }left(cos theta ,{hat {mathbf {theta } }}+i,{hat {mathbf {varphi } }}right),end{aligned}}}

Φ10=−34πsin⁡θφ^,Φ11=38πeiφ(iθ^−cos⁡θφ^).{displaystyle {begin{aligned}mathbf {Phi } _{10}&=-{sqrt {frac {3}{4pi }}}sin theta ,{hat {mathbf {varphi } }},\mathbf {Phi } _{11}&={sqrt {frac {3}{8pi }}}e^{ivarphi }left(i,{hat {mathbf {theta } }}-cos theta ,{hat {mathbf {varphi } }}right).end{aligned}}}{displaystyle {begin{aligned}mathbf {Phi } _{10}&=-{sqrt {frac {3}{4pi }}}sin theta ,{hat {mathbf {varphi } }},\mathbf {Phi } _{11}&={sqrt {frac {3}{8pi }}}e^{ivarphi }left(i,{hat {mathbf {theta } }}-cos theta ,{hat {mathbf {varphi } }}right).end{aligned}}}


  • l=2{displaystyle l=2}l=2.

Y20=145π(3cos2⁡θ−1)r^,Y21=−158πsin⁡θcos⁡θeiφr^,Y22=14152πsin2⁡θe2iφr^.{displaystyle {begin{aligned}mathbf {Y} _{20}&={frac {1}{4}}{sqrt {frac {5}{pi }}},(3cos ^{2}theta -1),{hat {mathbf {r} }},\mathbf {Y} _{21}&=-{sqrt {frac {15}{8pi }}},sin theta ,cos theta ,e^{ivarphi },{hat {mathbf {r} }},\mathbf {Y} _{22}&={frac {1}{4}}{sqrt {frac {15}{2pi }}},sin ^{2}theta ,e^{2ivarphi },{hat {mathbf {r} }}.end{aligned}}}{displaystyle {begin{aligned}mathbf {Y} _{20}&={frac {1}{4}}{sqrt {frac {5}{pi }}},(3cos ^{2}theta -1),{hat {mathbf {r} }},\mathbf {Y} _{21}&=-{sqrt {frac {15}{8pi }}},sin theta ,cos theta ,e^{ivarphi },{hat {mathbf {r} }},\mathbf {Y} _{22}&={frac {1}{4}}{sqrt {frac {15}{2pi }}},sin ^{2}theta ,e^{2ivarphi },{hat {mathbf {r} }}.end{aligned}}}

Ψ20=−325πsin⁡θcos⁡θθ^,Ψ21=−158πeiφ(cos⁡2θθ^+icos⁡θφ^),Ψ22=158πsin⁡θe2iφ(cos⁡θθ^+iφ^).{displaystyle {begin{aligned}mathbf {Psi } _{20}&=-{frac {3}{2}}{sqrt {frac {5}{pi }}},sin theta ,cos theta ,{hat {mathbf {theta } }},\mathbf {Psi } _{21}&=-{sqrt {frac {15}{8pi }}},e^{ivarphi },left(cos 2theta ,{hat {mathbf {theta } }}+icos theta ,{hat {mathbf {varphi } }}right),\mathbf {Psi } _{22}&={sqrt {frac {15}{8pi }}},sin theta ,e^{2ivarphi },left(cos theta ,{hat {mathbf {theta } }}+i,{hat {mathbf {varphi } }}right).end{aligned}}}{displaystyle {begin{aligned}mathbf {Psi } _{20}&=-{frac {3}{2}}{sqrt {frac {5}{pi }}},sin theta ,cos theta ,{hat {mathbf {theta } }},\mathbf {Psi } _{21}&=-{sqrt {frac {15}{8pi }}},e^{ivarphi },left(cos 2theta ,{hat {mathbf {theta } }}+icos theta ,{hat {mathbf {varphi } }}right),\mathbf {Psi } _{22}&={sqrt {frac {15}{8pi }}},sin theta ,e^{2ivarphi },left(cos theta ,{hat {mathbf {theta } }}+i,{hat {mathbf {varphi } }}right).end{aligned}}}

Φ20=−325πsin⁡θcos⁡θφ^,Φ21=158πeiφ(icos⁡θθ^−cos⁡2θφ^),Φ22=158πsin⁡θe2iφ(−iθ^+cos⁡θφ^).{displaystyle {begin{aligned}mathbf {Phi } _{20}&=-{frac {3}{2}}{sqrt {frac {5}{pi }}}sin theta ,cos theta ,{hat {mathbf {varphi } }},\mathbf {Phi } _{21}&={sqrt {frac {15}{8pi }}},e^{ivarphi },left(icos theta ,{hat {mathbf {theta } }}-cos 2theta ,{hat {mathbf {varphi } }}right),\mathbf {Phi } _{22}&={sqrt {frac {15}{8pi }}},sin theta ,e^{2ivarphi },left(-i,{hat {mathbf {theta } }}+cos theta ,{hat {mathbf {varphi } }}right).end{aligned}}}{displaystyle {begin{aligned}mathbf {Phi } _{20}&=-{frac {3}{2}}{sqrt {frac {5}{pi }}}sin theta ,cos theta ,{hat {mathbf {varphi } }},\mathbf {Phi } _{21}&={sqrt {frac {15}{8pi }}},e^{ivarphi },left(icos theta ,{hat {mathbf {theta } }}-cos 2theta ,{hat {mathbf {varphi } }}right),\mathbf {Phi } _{22}&={sqrt {frac {15}{8pi }}},sin theta ,e^{2ivarphi },left(-i,{hat {mathbf {theta } }}+cos theta ,{hat {mathbf {varphi } }}right).end{aligned}}}

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 ω{displaystyle omega }omega and complex amplitude


J^=J(r)Φlm,{displaystyle {hat {mathbf {J} }}=J(r)mathbf {Phi } _{lm},}{displaystyle {hat {mathbf {J} }}=J(r)mathbf {Phi } _{lm},}

and the corresponding electric and magnetic fields, can be written as


E^=E(r)Φlm,B^=Br(r)Ylm+B(1)(r)Ψlm.{displaystyle {begin{aligned}{hat {mathbf {E} }}&=E(r)mathbf {Phi } _{lm},\{hat {mathbf {B} }}&=B^{r}(r)mathbf {Y} _{lm}+B^{(1)}(r)mathbf {Psi } _{lm}.end{aligned}}}{displaystyle {begin{aligned}{hat {mathbf {E} }}&=E(r)mathbf {Phi } _{lm},\{hat {mathbf {B} }}&=B^{r}(r)mathbf {Y} _{lm}+B^{(1)}(r)mathbf {Psi } _{lm}.end{aligned}}}

Substituting into Maxwell equations, Gauss' law is automatically satisfied


∇⋅E^=0,{displaystyle nabla cdot {hat {mathbf {E} }}=0,}{displaystyle nabla cdot {hat {mathbf {E} }}=0,}

while Faraday's law decouples as


∇×E^=−iωB^⇒{l(l+1)rE=iωBr,dEdr+Er=iωB(1).{displaystyle nabla times {hat {mathbf {E} }}=-iomega {hat {mathbf {B} }}quad Rightarrow quad left{{begin{array}{l}displaystyle {frac {l(l+1)}{r}}E=iomega B^{r},\displaystyle {frac {dE}{dr}}+{frac {E}{r}}=iomega B^{(1)}.end{array}}right.}{displaystyle nabla times {hat {mathbf {E} }}=-iomega {hat {mathbf {B} }}quad Rightarrow quad left{{begin{array}{l}displaystyle {frac {l(l+1)}{r}}E=iomega B^{r},\displaystyle {frac {dE}{dr}}+{frac {E}{r}}=iomega B^{(1)}.end{array}}right.}

Gauss' law for the magnetic field implies


∇⋅B^=0⇒dBrdr+2rBr−l(l+1)rB(1)=0,{displaystyle nabla cdot {hat {mathbf {B} }}=0quad Rightarrow quad {frac {dB^{r}}{dr}}+{frac {2}{r}}B^{r}-{frac {l(l+1)}{r}}B^{(1)}=0,}{displaystyle nabla cdot {hat {mathbf {B} }}=0quad Rightarrow quad {frac {dB^{r}}{dr}}+{frac {2}{r}}B^{r}-{frac {l(l+1)}{r}}B^{(1)}=0,}

and Ampère-Maxwell's equation gives


∇×B^=μ0J^+iμ0ε0ωE^⇒−Brr+dB(1)dr+B(1)r=μ0J+iωμ0ε0E.{displaystyle nabla times {hat {mathbf {B} }}=mu _{0}{hat {mathbf {J} }}+imu _{0}varepsilon _{0}omega {hat {mathbf {E} }}quad Rightarrow quad -{frac {B^{r}}{r}}+{frac {dB^{(1)}}{dr}}+{frac {B^{(1)}}{r}}=mu _{0}J+iomega mu _{0}varepsilon _{0}E.}{displaystyle nabla times {hat {mathbf {B} }}=mu _{0}{hat {mathbf {J} }}+imu _{0}varepsilon _{0}omega {hat {mathbf {E} }}quad Rightarrow quad -{frac {B^{r}}{r}}+{frac {dB^{(1)}}{dr}}+{frac {B^{(1)}}{r}}=mu _{0}J+iomega mu _{0}varepsilon _{0}E.}

In this way, the partial differential equations have been transformed into 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.,


∇⋅v=0,0=−∇p+η∇2v,{displaystyle {begin{aligned}nabla cdot mathbf {v} &=0,\mathbf {0} &=-nabla p+eta nabla ^{2}mathbf {v} ,end{aligned}}}{displaystyle {begin{aligned}nabla cdot mathbf {v} &=0,\mathbf {0} &=-nabla p+eta nabla ^{2}mathbf {v} ,end{aligned}}}

with the boundary conditions


v=0(r=a),v=−U0(r→∞).{displaystyle {begin{aligned}mathbf {v} &=mathbf {0} quad (r=a),\mathbf {v} &=-mathbf {U} _{0}quad (rto infty ).end{aligned}}}{displaystyle {begin{aligned}mathbf {v} &=mathbf {0} quad (r=a),\mathbf {v} &=-mathbf {U} _{0}quad (rto infty ).end{aligned}}}

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


U0=U0(cos⁡θr^−sin⁡θθ^)=U0(Y10+Ψ10).{displaystyle 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).}{displaystyle 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).}

The last expression suggests an expansion in spherical harmonics for the liquid velocity and the pressure


p=p(r)Y10,v=vr(r)Y10+v(1)(r)Ψ10.{displaystyle {begin{aligned}p&=p(r)Y_{10},\mathbf {v} &=v^{r}(r)mathbf {Y} _{10}+v^{(1)}(r)mathbf {Psi } _{10}.end{aligned}}}{displaystyle {begin{aligned}p&=p(r)Y_{10},\mathbf {v} &=v^{r}(r)mathbf {Y} _{10}+v^{(1)}(r)mathbf {Psi } _{10}.end{aligned}}}

Substitution in the Navier–Stokes equations produces a set of ordinary differential equations for the coefficients.



See also



  • Spherical harmonics

  • Spin spherical harmonics

  • Spin-weighted spherical harmonics

  • Electromagnetic radiation

  • Spherical basis




References




  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



  • Vector Spherical Harmonics at Eric Weisstein's Mathworld







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