Vector spherical harmonics

Several conventions have been used to define the VSH.{{cite journal | last1=Barrera | first1=R G | last2=Estevez | first2=G A | last3=Giraldo | first3=J | title=Vector spherical harmonics and their application to magnetostatics | journal=European Journal of Physics | publisher=IOP Publishing | volume=6 | issue=4 | date=1985-10-01 | issn=0143-0807 | doi=10.1088/0143-0807/6/4/014 | pages=287–294| bibcode=1985EJPh....6..287B | citeseerx=10.1.1.718.2001 | s2cid=250894245 }}{{cite journal | last1=Carrascal | first1=B | last2=Estevez | first2=G A | last3=Lee | first3=Peilian | last4=Lorenzo | first4=V | title=Vector spherical harmonics and their application to classical electrodynamics | journal=European Journal of Physics | publisher=IOP Publishing | volume=12 | issue=4 | date=1991-07-01 | issn=0143-0807 | doi=10.1088/0143-0807/12/4/007 | pages=184–191| bibcode=1991EJPh...12..184C | s2cid=250886412 }}{{cite journal | last=Hill | first=E. L. | title=The Theory of Vector Spherical Harmonics | journal=American Journal of Physics | publisher=American Association of Physics Teachers (AAPT) | volume=22 | issue=4 | year=1954 | issn=0002-9505 | doi=10.1119/1.1933682 | pages=211–214| bibcode=1954AmJPh..22..211H | s2cid=124182424 | url=http://pdfs.semanticscholar.org/e950/1438ddec10ff6baef4614cf3b66f4e27c44b.pdf | archive-url=https://web.archive.org/web/20200412143916/http://pdfs.semanticscholar.org/e950/1438ddec10ff6baef4614cf3b66f4e27c44b.pdf | url-status=dead | archive-date=2020-04-12 }}{{cite journal | last=Weinberg | first=Erick J. | title=Monopole vector spherical harmonics | journal=Physical Review D | publisher=American Physical Society (APS) | volume=49 | issue=2 | date=1994-01-15 | issn=0556-2821 | doi=10.1103/physrevd.49.1086 | pages=1086–1092| pmid=10017069 |arxiv=hep-th/9308054| bibcode=1994PhRvD..49.1086W | s2cid=6429605 }}P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)

We follow that of Barrera et al.. Given a scalar spherical harmonic {{math|Y_ℓm(θ, φ)}}, we define three VSH:

• $\backslash mathbf\{Y\}\_\{\backslash ell\; m\}\; =\; Y\_\{\backslash ell\; m\}\backslash hat\{\backslash mathbf\{r\}\},$
• $\backslash mathbf\{\backslash Psi\}\_\{\backslash ell\; m\}\; =\; r\backslash nabla\; Y\_\{\backslash ell\; m\},$
• $\backslash mathbf\{\backslash Phi\}\_\{\backslash ell\; m\}\; =\; \backslash mathbf\{r\}\backslash times\backslash nabla\; Y\_\{\backslash ell\; m\},$

with $\backslash hat\{\backslash mathbf\{r\}\}$ being the unit vector along the radial direction in spherical coordinates and $\backslash mathbf\{r\}$ the vector along the radial direction with the same norm as the radius, i.e., $\backslash mathbf\{r\}\; =\; r\backslash hat\{\backslash mathbf\{r\}\}$. 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

$$\backslash mathbf\{E\}\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash sum\_\{m=-\backslash ell\}^\backslash ell\; \backslash left(E^r\_\{\backslash ell\; m\}(r)\; \backslash mathbf\{Y\}\_\{\backslash ell\; m\}\; +\; E^\{(1)\}\_\{\backslash ell\; m\}(r)\; \backslash mathbf\{\backslash Psi\}\_\{\backslash ell\; m\}\; +\; E^\{(2)\}\_\{\backslash ell\; m\}(r)\; \backslash mathbf\{\backslash Phi\}\_\{\backslash ell\; m\}\backslash right).$$

The labels on the components reflect that $E^r\_\{\backslash ell\; m\}$ is the radial component of the vector field, while $E^\{(1)\}\_\{\backslash ell\; m\}$ and $E^\{(2)\}\_\{\backslash ell\; m\}$ are transverse components (with respect to the radius vector $\backslash mathbf\{r\}$).

Like the scalar spherical harmonics, the VSH satisfy

$$\backslash begin\{align\}$$

\mathbf{Y}_{\ell,-m} &= (-1)^m \mathbf{Y}^*_{\ell m}, \\

\mathbf{\Psi}_{\ell,-m} &= (-1)^m \mathbf{\Psi}^*_{\ell m}, \\

\mathbf{\Phi}_{\ell,-m} &= (-1)^m \mathbf{\Phi}^*_{\ell m},

\end{align}

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

The VSH are orthogonal in the usual three-dimensional way at each point $\backslash mathbf\{r\}$:

$$\backslash begin\{align\}$$

\mathbf{Y}_{\ell m}(\mathbf{r}) \cdot \mathbf{\Psi}_{\ell m}(\mathbf{r}) &= 0, \\

\mathbf{Y}_{\ell m}(\mathbf{r}) \cdot \mathbf{\Phi}_{\ell m}(\mathbf{r}) &= 0, \\

\mathbf{\Psi}_{\ell m}(\mathbf{r}) \cdot \mathbf{\Phi}_{\ell m}(\mathbf{r}) &= 0.

\end{align}

They are also orthogonal in Hilbert space:

$$\backslash begin\{align\}$$

\int\mathbf{Y}_{\ell m}\cdot \mathbf{Y}^*_{\ell'm'}\,d\Omega &= \delta_{\ell\ell'}\delta_{mm'}, \\

\int\mathbf{\Psi}_{\ell m}\cdot \mathbf{\Psi}^*_{\ell'm'}\,d\Omega &= \ell(\ell+1)\delta_{\ell\ell'}\delta_{mm'}, \\

\int\mathbf{\Phi}_{\ell m}\cdot \mathbf{\Phi}^*_{\ell'm'}\,d\Omega &= \ell(\ell+1)\delta_{\ell\ell'}\delta_{mm'}, \\

\int\mathbf{Y}_{\ell m}\cdot \mathbf{\Psi}^*_{\ell'm'}\,d\Omega &= 0, \\

\int\mathbf{Y}_{\ell m}\cdot \mathbf{\Phi}^*_{\ell'm'}\,d\Omega &= 0, \\

\int\mathbf{\Psi}_{\ell m}\cdot \mathbf{\Phi}^*_{\ell'm'}\,d\Omega &= 0.

\end{align}

An additional result at a single point $\backslash mathbf\{r\}$ (not reported in Barrera et al, 1985) is, for all $\backslash ell,m,\backslash ell\text{'},m\text{'}$,

$$\backslash begin\{align\}$$

\mathbf{Y}_{\ell m}(\mathbf{r}) \cdot \mathbf{\Psi}_{\ell'm'}(\mathbf{r}) &= 0, \\

\mathbf{Y}_{\ell m}(\mathbf{r}) \cdot \mathbf{\Phi}_{\ell'm'}(\mathbf{r}) &= 0.

\end{align}

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

$$\backslash begin\{align\}$$

E^r_{\ell m} &= \int \mathbf{E}\cdot \mathbf{Y}^*_{\ell m}\,d\Omega, \\

E^{(1)}_{\ell m} &= \frac{1}{\ell(\ell+1)}\int \mathbf{E}\cdot \mathbf{\Psi}^*_{\ell m}\,d\Omega, \\

E^{(2)}_{\ell m} &= \frac{1}{\ell(\ell+1)}\int \mathbf{E}\cdot \mathbf{\Phi}^*_{\ell m}\,d\Omega.

\end{align}

Given the multipole expansion of a scalar field

$$\backslash phi\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash sum\_\{m=-\backslash ell\}^\backslash ell\; \backslash phi\_\{\backslash ell\; m\}(r)\; Y\_\{\backslash ell\; m\}(\backslash theta,\backslash phi),$$

we can express its gradient in terms of the VSH as

$$\backslash nabla\backslash phi\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash sum\_\{m=-\backslash ell\}^\backslash ell\backslash left(\backslash frac\{d\backslash phi\_\{\backslash ell\; m\}\}\{dr\}\; \backslash mathbf\{Y\}\_\{\backslash ell\; m\}+$$

\frac{\phi_{\ell m}}{r}\mathbf{\Psi}_{\ell m}\right).

For any multipole field we have

$$\backslash begin\{align\}$$

\nabla\cdot\left(f(r)\mathbf{Y}_{\ell m}\right) &= \left(\frac{df}{dr} + \frac{2}{r}f\right) Y_{\ell m}, \\

\nabla\cdot\left(f(r)\mathbf{\Psi}_{\ell m}\right) &= -\frac{\ell(\ell+1)}{r} f Y_{\ell m}, \\

\nabla\cdot\left(f(r)\mathbf{\Phi}_{\ell m}\right) &= 0.

\end{align}

By superposition we obtain the divergence of any vector field:

$$\backslash nabla\backslash cdot\backslash mathbf\{E\}\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash sum\_\{m=-\backslash ell\}^\backslash ell\; \backslash left(\backslash frac\{dE^r\_\{\backslash ell\; m\}\}\{dr\}+\backslash frac\{2\}\{r\}E^r\_\{\backslash ell\; m\}-\backslash frac\{\backslash ell(\backslash ell+1)\}\{r\}E^\{(1)\}\_\{\backslash ell\; m\}\backslash right)Y\_\{\backslash ell\; m\}.$$

We see that the component on {{math|Φ_ℓm}} is always solenoidal.

For any multipole field we have

$$\backslash begin\{align\}$$

\nabla\times\left(f(r)\mathbf{Y}_{\ell m}\right) &= -\frac{1}{r}f\mathbf{\Phi}_{\ell m}, \\

\nabla\times\left(f(r)\mathbf{\Psi}_{\ell m}\right) &= \left(\frac{df}{dr}+\frac{1}{r}f\right)\mathbf{\Phi}_{\ell m}, \\

\nabla\times\left(f(r)\mathbf{\Phi}_{\ell m}\right) &= -\frac{\ell(\ell+1)}{r}f\mathbf{Y}_{\ell m}-\left(\frac{df}{dr} + \frac{1}{r} f\right)\mathbf{\Psi}_{\ell m}.

\end{align}

By superposition we obtain the curl of any vector field:

$$\backslash nabla\backslash times\backslash mathbf\{E\}\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash sum\_\{m=-\backslash ell\}^\backslash ell$$

\left(-\frac{\ell(\ell+1)}{r}E^{(2)}_{\ell m}\mathbf{Y}_{\ell m}-\left(\frac{dE^{(2)}_{\ell m}}{dr}+

\frac{1}{r}E^{(2)}_{\ell m}\right)\mathbf{\Psi}_{\ell m}+

\left(-\frac{1}{r}E^r_{\ell m}+\frac{dE^{(1)}_{\ell m}}{dr}+\frac{1}{r}E^{(1)}_{\ell m}\right)\mathbf{\Phi}_{\ell m}\right).

The action of the Laplace operator $\backslash Delta\; =\; \backslash nabla\backslash cdot\backslash nabla$ separates as follows:

$$\backslash Delta\backslash left(f(r)\backslash mathbf\{Z\}\_\{\backslash ell\; m\}\backslash right)\; =\; \backslash left(\backslash frac\{1\}\{r^2\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; r^2\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; r\}\; \backslash right)\backslash mathbf\{Z\}\_\{\backslash ell\; m\}$$

+ f(r)\Delta \mathbf{Z}_{\ell m},

where $\backslash mathbf\{Z\}\_\{\backslash ell\; m\}\; =\; \backslash mathbf\{Y\}\_\{\backslash ell\; m\},\; \backslash mathbf\{\backslash Psi\}\_\{\backslash ell\; m\},\; \backslash mathbf\{\backslash Phi\}\_\{\backslash ell\; m\}$ and

$$\backslash begin\{align\}$$

\Delta\mathbf{Y}_{\ell m} &= -\frac{1}{r^2}(2+\ell(\ell+1))\mathbf{Y}_{\ell m} +\frac{2}{r^2}\mathbf{\Psi}_{\ell m}, \\

\Delta\mathbf{\Psi}_{\ell m} &= \frac{2}{r^2}\ell(\ell+1)\mathbf{Y}_{\ell m} -\frac{1}{r^2}\ell(\ell+1)\mathbf{\Psi}_{\ell m}, \\

\Delta\mathbf{\Phi}_{\ell m} &= -\frac{1}{r^2}\ell(\ell+1)\mathbf{\Phi}_{\ell m}.

\end{align}

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

{{multiple image|perrow=3

| image1 = YE1grid.png

| caption1 = $\backslash mathbf\{\backslash Psi\}\_\{1m\}$

| image2 = YE2grid.png

| caption2 = $\backslash mathbf\{\backslash Psi\}\_\{2m\}$

| image3 = YE3grid.png

| caption3 = $\backslash mathbf\{\backslash Psi\}\_\{3m\}$

| image4 = YB1grid.png

| caption4 = $\backslash mathbf\{\backslash Phi\}\_\{1m\}$

| image5 = YB2grid.png

| caption5 = $\backslash mathbf\{\backslash Phi\}\_\{2m\}$

| image6 = YB3grid.png

| caption6 = $\backslash mathbf\{\backslash Phi\}\_\{3m\}$

| footer = Visualizations of the real parts of $\backslash ell=1,2,3$ VSHs. Click to expand.

}}

{{unordered list

| $\backslash ell=0$.

$$\backslash begin\{align\}$$

\mathbf{Y}_{00} &= \sqrt{\frac{1}{4\pi}}\hat{\mathbf{r}}, \\

\mathbf{\Psi}_{00} &= \mathbf{0}, \\

\mathbf{\Phi}_{00} &= \mathbf{0}.

\end{align}

| $\backslash ell=1$.

$$\backslash begin\{align\}$$

\mathbf{Y}_{10} &= \sqrt{\frac{3}{4\pi}}\cos\theta\,\hat{\mathbf{r}}, \\

\mathbf{Y}_{11} &= -\sqrt{\frac{3}{8\pi}}e^{i\varphi}\sin\theta\,\hat{\mathbf{r}},

\end{align}

$$\backslash begin\{align\}$$

\mathbf{\Psi}_{10} &= -\sqrt{\frac{3}{4\pi}}\sin\theta\,\hat{\mathbf{\theta}}, \\

\mathbf{\Psi}_{11} &= -\sqrt{\frac{3}{8\pi}}e^{i\varphi}\left(\cos\theta\,\hat{\mathbf{\theta}}+i\,\hat{\mathbf{\varphi}}\right),

\end{align}

$$\backslash begin\{align\}$$

\mathbf{\Phi}_{10} &= -\sqrt{\frac{3}{4\pi}}\sin\theta\,\hat{\mathbf{\varphi}}, \\

\mathbf{\Phi}_{11} &= \sqrt{\frac{3}{8\pi}}e^{i\varphi}\left(i\,\hat{\mathbf{\theta}}-\cos\theta\,\hat{\mathbf{\varphi}}\right).

\end{align}

| $\backslash ell=2$.

$$\backslash begin\{align\}$$

\mathbf{Y}_{20} &= \frac{1}{4}\sqrt{\frac{5}{\pi}}\,(3\cos^{2}\theta-1)\,\hat{\mathbf{r}}, \\

\mathbf{Y}_{21} &= -\sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,e^{i\varphi}\,\hat{\mathbf{r}}, \\

\mathbf{Y}_{22} &= \frac{1}{4}\sqrt{\frac{15}{2\pi}}\,\sin^{2}\theta\,e^{2i\varphi}\,\hat{\mathbf{r}}.

\end{align}

$$\backslash begin\{align\}$$

\mathbf{\Psi}_{20} &= -\frac{3}{2}\sqrt{\frac{5}{\pi}}\,\sin\theta\,\cos\theta\,\hat{\mathbf{\theta}}, \\

\mathbf{\Psi}_{21} &= -\sqrt{\frac{15}{8\pi}}\,e^{i\varphi}\,\left(\cos 2\theta\,\hat{\mathbf{\theta}}+i\cos\theta\,\hat{\mathbf{\varphi}}\right), \\

\mathbf{\Psi}_{22} &= \sqrt{\frac{15}{8\pi}}\,\sin\theta\,e^{2i\varphi}\,\left(\cos\theta\,\hat{\mathbf{\theta}}+i\,\hat{\mathbf{\varphi}}\right).

\end{align}

$$\backslash begin\{align\}$$

\mathbf{\Phi}_{20} &= -\frac{3}{2}\sqrt{\frac{5}{\pi}}\sin\theta\,\cos\theta\,\hat{\mathbf{\varphi}}, \\

\mathbf{\Phi}_{21} &= \sqrt{\frac{15}{8\pi}}\,e^{i\varphi}\,\left(i\cos\theta\,\hat{\mathbf{\theta}}-\cos 2\theta\,\hat{\mathbf{\varphi}}\right), \\

\mathbf{\Phi}_{22} &= \sqrt{\frac{15}{8\pi}}\,\sin\theta\,e^{2i\varphi}\,\left(-i\,\hat{\mathbf{\theta}}+\cos\theta\,\hat{\mathbf{\varphi}}\right).

\end{align}

}}

Expressions for negative values of {{mvar|m}} are obtained by applying the symmetry relations.

{{clear}}

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 $\backslash omega$ and complex amplitude

$$\backslash hat\{\backslash mathbf\{J\}\}=\; J(r)\backslash mathbf\{\backslash Phi\}\_\{\backslash ell\; m\},$$

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

$$\backslash begin\{align\}$$

\hat{\mathbf{E}} &= E(r)\mathbf{\Phi}_{\ell m}, \\

\hat{\mathbf{B}} &= B^r(r)\mathbf{Y}_{\ell m}+B^{(1)}(r)\mathbf{\Psi}_{\ell m}.

\end{align}

Substituting into Maxwell equations, Gauss's law is automatically satisfied

$$\backslash nabla\backslash cdot\backslash hat\{\backslash mathbf\{E\}\}=0,$$

while Faraday's law decouples as

$$\backslash nabla\backslash times\backslash hat\{\backslash mathbf\{E\}\}=-i\backslash omega\backslash hat\{\backslash mathbf\{B\}\}\backslash quad\backslash Rightarrow\backslash quad$$

\begin{cases} \dfrac{\ell(\ell+1)}{r}E = i\omega B^r, \\

\dfrac{dE}{dr} +\dfrac{E}{r}= i\omega B^{(1)}.\end{cases}

Gauss' law for the magnetic field implies

$$\backslash nabla\backslash cdot\backslash hat\{\backslash mathbf\{B\}\}\; =\; 0\backslash quad\backslash Rightarrow\; \backslash quad\backslash frac\{dB^r\}\{dr\}+\backslash frac\{2\}\{r\}B^r\; -\; \backslash frac\{\backslash ell(\backslash ell+1)\}\{r\}B^\{(1)\}=0,$$

and Ampère–Maxwell's equation gives

$$\backslash nabla\backslash times\backslash hat\{\backslash mathbf\{B\}\}\; =\; \backslash mu\_0\; \backslash hat\{\backslash mathbf\{J\}\}\; +\; i\backslash mu\_0\backslash varepsilon\_0\backslash omega\; \backslash hat\{\backslash mathbf\{E\}\}\; \backslash quad\backslash Rightarrow\backslash quad\; -\backslash frac\{B^r\}\{r\}+\backslash frac\{dB^\{(1)\}\}\{dr\}+\backslash frac\{B^\{(1)\}\}\{r\}\; =\; \backslash mu\_0J+i\backslash omega\backslash mu\_0\backslash varepsilon\_0E.$$

In this way, the partial differential equations have been transformed into a set of ordinary differential equations.

File:VSHwiki.svg

In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates.Bohren, Craig F. and Donald R. Huffman, Absorption and scattering of light by small particles, New York : Wiley, 1998, 530 p., {{ISBN|0-471-29340-7}}, {{ISBN|978-0-471-29340-8}} (second edition){{Cite book|first=J. A. |last=Stratton|title=Electromagnetic Theory|url=https://archive.org/details/electromagnetict0000stra |url-access=registration |location= New York|publisher= McGraw-Hill|year= 1941}}

In this case, vector spherical harmonics are generated by scalar functions, which are solutions of scalar Helmholtz equation with the wavevector $\backslash mathbf\; k$.

$$\backslash begin\{array\}\{l\}$$

{\psi_{e m n} = \cos m \varphi P_{n}^{m}(\cos \vartheta) z_{n}({k} r)} \\

{\psi_{o m n} = \sin m \varphi P_{n}^{m}(\cos \vartheta) z_{n}({k} r)}

\end{array}

here $P\_\{n\}^\{m\}(\backslash cos\; \backslash theta)$ are the associated Legendre polynomials, and $z\_\{n\}(\{k\}\; r)$ are any of the spherical Bessel functions.

Vector spherical harmonics are defined as:

; longitudinal harmonics : $\backslash mathbf\{L\}\_\{^e\_o\; m\; n\}=\backslash mathbf\{\backslash nabla\}\; \backslash psi\_\{^e\_o\; m\; n\}$

; magnetic harmonics : $\backslash mathbf\{M\}\_\{^e\_o\; m\; n\}=\backslash nabla\; \backslash times\backslash left(\backslash mathbf\{r\}\; \backslash psi\_\{^e\_o\; m\; n\}\backslash right)$

; electric harmonics : $\backslash mathbf\{N\}\_\{^e\_o\; m\; n\}=\backslash frac\{\backslash nabla\; \backslash times\; \backslash mathbf\{M\}\_\{^e\_o\; m\; n\}\}\{\{k\}\}$

Here we use harmonics real-valued angular part, where $m\; \backslash geq\; 0$, but complex functions can be introduced in the same way.

Let us introduce the notation $\backslash rho\; =\; kr$. In the component form vector spherical harmonics are written as:

$$\backslash begin\{aligned\}$$

{\mathbf{M}_{e m n}(k, \mathbf{r}) =\qquad {\frac{-m}{\sin (\theta)} \sin (m \varphi) P_{n}^{m}(\cos (\theta))} z_{n}(\rho)\mathbf{e}_{\theta}} \\

{{{}-\cos (m \varphi) \frac{d P_{n}^{m}(\cos (\theta))}{d \theta}} }z_{n}(\rho) \mathbf{e}_{\varphi}

\end{aligned}

$$\backslash begin\{aligned\}$$

{\mathbf{M}_{o m n}(k, \mathbf{r}) =\qquad {\frac{m}{\sin (\theta)} \cos (m \varphi) P_{n}^{m}(\cos (\theta)) }}z_{n}(\rho) \mathbf{e}_{\theta} \\

{{}-\sin (m \varphi) \frac{d P_{n}^{m}(\cos (\theta))}{d \theta} z_{n}(\rho) \mathbf{e}_{\varphi}}

\end{aligned}

$$\backslash begin\{aligned\}$$

{\mathbf{N}_{e m n}(k, \mathbf{r}) =\qquad \frac{z_{n}(\rho)}{\rho} \cos (m \varphi) n(n+1) P_{n}^{m}(\cos (\theta)) \mathbf{e}_{\mathbf{r}}} \\

{{{{}+\cos (m \varphi) \frac{d P_{n}^{m}(\cos (\theta))}{d \theta}}}} \frac{1}{\rho} \frac{d}{d \rho}\left[\rho z_{n}(\rho)\right] \mathbf{e}_{\theta} \\

{{{}-m \sin (m \varphi) \frac{P_{n}^{m}(\cos (\theta))}{\sin (\theta)}}} \frac{1}{\rho} \frac{d}{d \rho}\left[\rho z_{n}(\rho)\right] \mathbf{e}_{\varphi}

\end{aligned}

$$\backslash begin\{aligned\}$$

\mathbf{N}_{o m n} (k, \mathbf{r}) =\qquad \frac{z_{n}(\rho)}{\rho} \sin (m \varphi) n(n+1) P_{n}^{m}(\cos (\theta)) \mathbf{e}_{\mathbf{r}} \\

{}+\sin (m \varphi) \frac{d P_{n}^{m}(\cos (\theta))}{d \theta} \frac{1}{\rho} \frac{d}{d \rho}\left[\rho z_{n}(\rho)\right] \mathbf{e}_{\theta} \\

{}+{m \cos (m \varphi) \frac{P_{n}^{m}(\cos (\theta))}{\sin (\theta)}} \frac{1}{\rho} \frac{d}{d \rho}\left[\rho z_{n}(\rho)\right] \mathbf{e}_{\varphi} \end{aligned}

There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big $\backslash rho$ 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 $\backslash rho$ electric and magnetic harmonics vectors are equal in value and perpendicular to each other.

Longitudinal harmonics:

$$\backslash begin\{aligned\}$$

\mathbf{L}_{^e_o{m n}}(k, \mathbf{r})

{}=\qquad &\frac{\partial}{\partial r} z_{n}(k r) P_{n}^{m}(\cos \theta){^{\cos }_{\sin }} {m \varphi} \mathbf{e}_r \\

{}+{} &\frac{1}{r} z_{n}(k r) \frac{\partial}{\partial \theta} P_{n}^{m}(\cos \theta) {^{\cos }_{\sin}} m \varphi \mathbf{e}_{\theta} \\

{}\mp{} &\frac{m}{r \sin \theta} z_{n}(k r) P_{n}^{m}(\cos \theta) {^{\sin }_{\cos}} m \varphi \mathbf{e}_{\varphi}

\end{aligned}

The solutions of the Helmholtz vector equation obey the following orthogonality relations:

$$\backslash begin\{align\}$$

\int_{0}^{2 \pi} \int_{0}^{\pi} \mathbf{L}_{^e_omn} \cdot \mathbf{L}_{^e_omn} \sin \vartheta d \vartheta d \varphi &= (1 + \delta_{m,0}) \frac{2 \pi}{(2 n+1)^{2}} \frac{(n+m) !}{(n-m) !} k^{2}\left\{n\left[z_{n-1}(k r)\right]^{2}+(n+1)\left[z_{n+1}(k r)\right]^{2}\right\} \\[3pt]

\int_{0}^{2 \pi} \int_{0}^{\pi} \mathbf{M}_{^e_omn} \cdot \mathbf{M}_{^e_omn} \sin \vartheta d \vartheta d \varphi &= (1 + \delta_{m,0}) \frac{2 \pi}{2 n+1} \frac{(n+m) !}{(n-m) !} n(n+1)\left[z_{n}(k r)\right]^2 \\[3pt]

\int_{0}^{2 \pi} \int_{0}^{\pi} \mathbf{N}_{^e_omn} \cdot \mathbf{N}_{^e_omn} \sin \vartheta d \vartheta d \varphi &= (1 + \delta_{m,0}) \frac{2 \pi}{(2 n + 1)^{2}} \frac{(n + m) !}{(n - m) !} n(n + 1)\left\{(n + 1)\left[z_{n-1}(k r)\right]^{2} + n\left[z_{n+1}(k r)\right]^{2}\right\} \\[3pt]

\int_{0}^{\pi} \int_{0}^{2 \pi} \mathbf{L}_{^e_omn} \cdot \mathbf{N}_{^e_omn} \sin \vartheta d \vartheta d \varphi &= (1 + \delta_{m,0}) \frac{2 \pi}{(2 n+1)^{2}} \frac{(n+m) !}{(n-m) !} n(n+1) k\left\{\left[z_{n-1}(k r)\right]^{2}-\left[z_{n+1}(k r)\right]^{2}\right\}

\end{align}

All other integrals over the angles between different functions or functions with different indices are equal to zero.

File:RotationwikiVSH.svg

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-matricesD. A. Varhalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum [in Russian], Nauka, Leningrad (1975){{cite journal|first1=Huayong|last1=Zhang|first2=Yiping|last2=Han|title=Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients|journal=J. Opt. Soc. Am. B|volume=25|number=2|pages=255–260|year=2008|doi=10.1364/JOSAB.25.000255|bibcode=2008JOSAB..25..255Z }}{{cite journal|first1=Seymour|last1=Stein|title=Addition theorems for spherical wave functions|journal=Quarterly of Applied Mathematics|volume=19|number=1|pages=15–24|year=1961|doi=10.1090/qam/120407}}

\hat{D}(\alpha, \beta, \gamma) \mathbf{Y}_{JM}^{(s)}(\theta, \varphi)= \sum_{M' = -J}^J [D^{(J)}_{MM'}(\alpha, \beta, \gamma)]^* \mathbf{Y}_{JM'}^{(s)}(\theta, \varphi),

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.

\hat{I}\mathbf{N}_{JM}(\theta, \varphi)= (-1)^J \mathbf{N}_{JM}(\theta, \varphi),

and magnetic ones have the opposite parity:

\hat{I}\mathbf{M}_{JM}(\theta, \varphi)= (-1)^{J+1} \mathbf{M}_{JM}(\theta, \varphi),

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.,

$$\backslash begin\{align\}$$

0 &= \nabla\cdot \mathbf{v}, \\

\mathbf{0} &= -\nabla p + \eta \nabla^2\mathbf{v},

\end{align}

with the boundary conditions

$$\backslash mathbf\{v\}\; =\; \backslash begin\{cases\}$$

\mathbf{0} & r = a, \\

-\mathbf{U}_0 & r \to \infty.

\end{cases}

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

$$\backslash mathbf\{U\}\_0\; =\; U\_0\backslash left(\backslash cos\backslash theta\backslash ,\; \backslash hat\{\backslash mathbf\{r\}\}\; -\; \backslash sin\backslash theta\; \backslash ,\backslash hat\{\backslash mathbf\{\backslash theta\}\}\backslash right)\; =\; U\_0\; \backslash left(\backslash mathbf\{Y\}\_\{10\}\; +\; \backslash mathbf\{\backslash Psi\}\_\{10\}\backslash right).$$

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

$$\backslash begin\{align\}$$

p &= p(r)Y_{10}, \\

\mathbf{v} &= v^r(r) \mathbf{Y}_{10} + v^{(1)}(r) \mathbf{\Psi}_{10}.

\end{align}

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

Here the following definitions are used:

$$\backslash begin\{align\}$$

Y_{e m n} &= \cos m \varphi P_{n}^{m}(\cos \theta) \\

Y_{o m n} &= \sin m \varphi P_{n}^{m}(\cos \theta)

\end{align}

\mathbf{X}_{^e_o m n}\left(\frac{\mathbf{k}}{k}\right) = \nabla \times\left(\mathbf{k} Y_{^o_e m n}\left(\frac{\mathbf{k}}{k}\right)\right)

\mathbf{Z}_{^o_e m n}\left(\frac{\mathbf{k}}{k}\right) = i \frac{\mathbf{k}}{k} \times \mathbf{X}_{^e_o m n}\left(\frac{\mathbf{k}}{k}\right)

In case, when instead of $z\_n$ are spherical Bessel functions, with help of plane wave expansion one can obtain the following integral relations:{{cite web|url=http://www.fresnel.fr/perso/stout/SHMs.pdf|first1=B.|last1=Stout|title=Spherical harmonic lattice sums for gratings|editor-last=Popov|editor-first=E|at=Gratings: theory and numeric applications|publisher=Institut Fresnel, Universite d'Aix-Marseille 6 |year=2012}}

\mathbf {N}_{pmn}^{(1)}(k, \mathbf r) = \frac{i^{-n}}{4\pi} \int \mathbf Z_{pmn}\left(\frac{\mathbf{k}}{k}\right) e^{i \mathbf k \cdot \mathbf r} d\Omega_k

\mathbf {M}_{pmn}^{(1)}(k, \mathbf r) =\frac{i^{-n}}{4\pi} \int \mathbf X_{pmn}\left(\frac{\mathbf{k}}{k}\right) e^{i \mathbf k \cdot \mathbf r} d\Omega_k

In case, when $z\_n$ are spherical Hankel functions, one should use the different formulae.{{cite journal|first1=R. C.|last1=Wittmann|doi=10.1109/8.7220|title=Spherical wave operators and the translation formulas|journal=IEEE Transactions on Antennas and Propagation|volume=36|number=8|pages=1078–1087|year=1988|bibcode=1988ITAP...36.1078W |url=https://zenodo.org/record/1262852 }} For vector spherical harmonics the following relations are obtained:

\mathbf{M}_{p m n}^{(3)}(k, \mathbf{r}) = \frac{i^{-n}}{2 \pi k} \iint_{-\infty}^{\infty} d k_{ \|} \frac{e^{i\left(k_{x} x+k_{y} y \pm k_{z} z\right)}}{k_{z}} \mathbf{X}_{p m n}\left(\frac{\mathbf{k}}{k}\right)

\mathbf{N}_{p m n}^{(3)}(k, \mathbf{r}) = \frac{i^{-n}}{2 \pi k} \iint_{-\infty}^{\infty} d k_{ \|} \frac{e^{i\left(k_{x} x+k_{y} y \pm k_{z} z\right)}}{k_{z}} \mathbf{Z}_{p m n}\left(\frac{\mathbf{k}}{k}\right)

where $k\_\{z\}\; =\; \backslash sqrt\{k^\{2\}-k\_\{x\}^\{2\}-k\_\{y\}^\{2\}\}$, index $(3)$ means, that spherical Hankel functions are used.

• Spherical harmonics
• Spinor spherical harmonics
• Spin-weighted spherical harmonics
• Electromagnetic radiation
• Spherical basis

• [http://mathworld.wolfram.com/VectorSphericalHarmonic.html Vector Spherical Harmonics at Eric Weisstein's Mathworld]

Category:Vector calculus

Category:Special functions

Category:Differential equations

Category:Applied mathematics

Category:Theoretical physics