Nabla in verschillende assenstelsels

In de onderstaande tabel staat een overzicht van de vorm die de operator nabla aanneemt in de drie assenstelsels:

Tabel

**Tabel met de operator $\nabla$ in cilinder- en bolcoördinaten**
Operatie	Cartesiaanse coördinaten (x,y,z)	Cilindercoördinaten (ρ,φ,z)	Bolcoördinaten (r,θ,φ)
Relatie	zie:	Cilindercoördinaten	Bolcoördinaten
eenheids- vectoren	$\mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}}$	${\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},{\boldsymbol {\hat {z}}}$	${\boldsymbol {\hat {r}}},{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}$
scalair veld $f\,$	$f(x,y,z)\,$	$f_{c}(\rho ,\phi ,z)=f(\rho \cos(\phi ),\rho \sin(\phi ),z)\,$	$f_{b}(r,\phi ,\theta )=f(r\sin(\theta )\cos(\phi ),r\sin(\theta )\sin(\phi ),r\cos(\theta ))\,$
vectorveld $A\,$	$A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$
$\nabla f$	${\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f_{c} \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f_{c} \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f_{c} \over \partial z}{\boldsymbol {\hat {z}}}$	${\partial f_{b} \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f_{b} \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f_{b} \over \partial \phi }{\boldsymbol {\hat {\phi }}}$
$\nabla \cdot A$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${\frac {1}{\rho }}{\partial (\rho A_{\rho }) \over \partial \rho }+{\frac {1}{\rho }}{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial (r^{2}A_{r}) \over \partial r}+{1 \over r\sin \theta }{\partial A_{\theta }\sin \theta \over \partial \theta }+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$
$\nabla \times A$	${\begin{matrix}({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z})\mathbf {\hat {x}} &+\\({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x})\mathbf {\hat {y}} &+\\({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y})\mathbf {\hat {z}} &\ \end{matrix}}$	${\begin{matrix}({1 \over \rho }{\partial A_{z} \over \partial \phi }-{\partial A_{\phi } \over \partial z}){\boldsymbol {\hat {\rho }}}&+\\({\partial A_{\rho } \over \partial z}-{\partial A_{z} \over \partial \rho }){\boldsymbol {\hat {\phi }}}&+\\{1 \over \rho }({\partial \rho A_{\phi } \over \partial \rho }-{\partial A_{\rho } \over \partial \phi }){\boldsymbol {\hat {z}}}&\ \end{matrix}}$	${\begin{matrix}{1 \over r\sin \theta }({\partial A_{\phi }\sin \theta \over \partial \theta }-{\partial A_{\theta } \over \partial \phi }){\boldsymbol {\hat {r}}}&+\\({1 \over r\sin \theta }{\partial A_{r} \over \partial \phi }-{1 \over r}{\partial rA_{\phi } \over \partial r}){\boldsymbol {\hat {\theta }}}&+\\{1 \over r}({\partial rA_{\theta } \over \partial r}-{\partial A_{r} \over \partial \theta }){\boldsymbol {\hat {\phi }}}&\ \end{matrix}}$
$\Delta f=\nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }(\rho {\partial f_{c} \over \partial \rho })+{1 \over \rho ^{2}}{\partial ^{2}f_{c} \over \partial \phi ^{2}}+{\partial ^{2}f_{c} \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}(r^{2}{\partial f_{b} \over \partial r})+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }(\sin \theta {\partial f_{b} \over \partial \theta })+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f_{b} \over \partial \phi ^{2}}$
$\Delta A=\nabla ^{2}A$	$\mathbf {\hat {x}} \Delta A_{x}+\mathbf {\hat {y}} \Delta A_{y}+\mathbf {\hat {z}} \Delta A_{z}$	${\begin{matrix}{\boldsymbol {\hat {\rho }}}(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\phi } \over \partial \phi })&+\\{\boldsymbol {\hat {\phi }}}(\Delta A_{\phi }-{A_{\phi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \phi })&+\\{\boldsymbol {\hat {z}}}\Delta A_{z}&\ \end{matrix}}$	${\begin{matrix}{\boldsymbol {\hat {r}}}&(\Delta A_{r}-{2A_{r} \over r^{2}}-{2A_{\theta }\cos \theta \over r^{2}\sin \theta }\\\ &-{2 \over r^{2}}{\partial A_{\theta } \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\phi } \over \partial \phi })&+\\{\boldsymbol {\hat {\theta }}}&(\Delta A_{\theta }-{A_{\theta } \over r^{2}\sin ^{2}\theta }\\\ &+{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\phi } \over \partial \phi })&+\\{\boldsymbol {\hat {\phi }}}&(\Delta A_{\phi }-{A_{\phi } \over r^{2}\sin ^{2}\theta }\\\ &+{2 \over r^{2}\sin \theta }{\partial A_{r} \over \partial \phi }+{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \phi })&\ \end{matrix}}$
Niet evidente rekenregels: $\operatorname {div\ grad\ } f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ (Laplaciaan) $\operatorname {rot\ grad\ } f=\nabla \times (\nabla f)=\mathbf {0}$ $\operatorname {div\ rot\ } \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {rot\ rot\ } \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ $\Delta fg=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$

Afleidingen

Cilindercoördinaten

Gradiënt van een scalaire functie

\nabla f=({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}})

De component van grad f in de richting van $\rho$ is:

(\nabla f)_{\rho }={\frac {\partial f}{\partial x}}\cos(\phi )+{\frac {\partial f}{\partial y}}\sin(\phi )={\frac {\partial f_{c}}{\partial \rho }}

De component van grad f in de richting van $\phi$ is:

(\nabla f)_{\phi }=-{\frac {\partial f}{\partial x}}\sin(\phi )+{\frac {\partial f}{\partial y}}\cos(\phi )={\frac {1}{\rho }}{\frac {\partial f_{c}}{\partial \phi }}

Divergentie van een vectorveld

\nabla \cdot A={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}

We drukken de divergentie uit in de componenten van de polaire voorstelling:

A_{x}=A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )

A_{y}=A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )

dus

{\frac {\partial A_{x}}{\partial x}}={\frac {\partial }{\partial x}}\left(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\right)=

{\frac {\partial }{\partial \rho }}\left(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\right){\frac {\partial \rho }{\partial x}}+{\frac {\partial }{\partial \phi }}\left(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\right){\frac {\partial \phi }{\partial x}}=

{\frac {\partial }{\partial \rho }}\left(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\right)\cos(\phi )+{\frac {\partial }{\partial \phi }}\left(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\right)(-{\frac {1}{\rho }}\sin(\phi ))=

{\frac {\partial A_{\rho }}{\partial \rho }}\cos ^{2}(\phi )-\sin(\phi )\cos(\phi ){\frac {\partial A_{\phi }}{\partial \rho }}-{\frac {1}{\rho }}\sin(\phi )\cos(\phi ){\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}A_{\rho }\sin ^{2}(\phi )+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}\sin ^{2}(\phi )+{\frac {1}{\rho }}A_{\phi }\sin(\phi )\cos(\phi )

{\frac {\partial A_{y}}{\partial y}}={\frac {\partial }{\partial y}}\left(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\right)=

{\frac {\partial }{\partial \rho }}\left(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\right){\frac {\partial \rho }{\partial y}}+{\frac {\partial }{\partial \phi }}\left(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\right){\frac {\partial \phi }{\partial y}}=

{\frac {\partial }{\partial \rho }}\left(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\right)\sin(\phi )+{\frac {\partial }{\partial \phi }}\left(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\right)({\frac {1}{\rho }}\cos(\phi ))=

{\frac {\partial A_{\rho }}{\partial \rho }}\sin ^{2}(\phi )+\sin(\phi )\cos(\phi ){\frac {\partial A_{\phi }}{\partial \rho }}+{\frac {1}{\rho }}\sin(\phi )\cos(\phi ){\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}A_{\rho }\cos ^{2}(\phi )+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}\cos ^{2}(\phi )-{\frac {1}{\rho }}A_{\phi }\cos(\phi )\sin(\phi )

Samen leidt dat tot:

\nabla \cdot A={\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {A_{\rho }}{\rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}

Rotatie van een vectorveld

(\nabla \times A)_{x}={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}={\frac {\partial A_{z}}{\partial \rho }}{\frac {\partial \rho }{\partial y}}+{\frac {\partial A_{z}}{\partial \phi }}{\frac {\partial \phi }{\partial y}}-{\frac {\partial A_{y}}{\partial z}}=

{\frac {\partial A_{z}}{\partial \rho }}\sin(\phi )+{\frac {\partial A_{z}}{\partial \phi }}{\frac {1}{\rho }}\cos(\phi )-{\frac {\partial A_{\rho }}{\partial z}}\sin(\phi )-{\frac {\partial A_{\phi }}{\partial z}}\cos(\phi )

(\nabla \times A)_{y}={\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}={\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}{\frac {\partial \rho }{\partial x}}-{\frac {\partial A_{z}}{\partial \phi }}{\frac {\partial \phi }{\partial x}}=

{\frac {\partial A_{\rho }}{\partial z}}\cos(\phi )-{\frac {\partial A_{\phi }}{\partial z}}\sin(\phi )-{\frac {\partial A_{z}}{\partial \rho }}\cos(\phi )+{\frac {\partial A_{z}}{\partial \phi }}{\frac {1}{\rho }}\sin(\phi )

(\nabla \times A)_{z}={\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}={\frac {\partial A_{y}}{\partial \rho }}{\frac {\partial \rho }{\partial x}}+{\frac {\partial A_{y}}{\partial \phi }}{\frac {\partial \phi }{\partial x}}-{\frac {\partial A_{x}}{\partial \rho }}{\frac {\partial \rho }{\partial y}}-{\frac {\partial A_{x}}{\partial \phi }}{\frac {\partial \phi }{\partial y}}=

{\frac {\partial (A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi ))}{\partial \rho }}\cos(\phi )-{\frac {\partial (A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi ))}{\partial \phi }}{\frac {1}{\rho }}\sin(\phi )

-{\frac {\partial (A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi ))}{\partial \rho }}\sin(\phi )-{\frac {\partial (A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi ))}{\partial \phi }}{\frac {1}{\rho }}\cos(\phi )=

{\frac {\partial A_{\rho }}{\partial \rho }}\sin(\phi )\cos(\phi )+{\frac {\partial A_{\phi }}{\partial \rho }}\cos ^{2}(\phi )-{\frac {\partial A_{\rho }\sin(\phi )}{\partial \phi }}{\frac {1}{\rho }}\sin(\phi )-{\frac {\partial A_{\phi }\cos(\phi )}{\partial \phi }}{\frac {1}{\rho }}\sin(\phi )

-{\frac {\partial A_{\rho }}{\partial \rho }}\cos(\phi )\sin(\phi )+{\frac {\partial A_{\phi }}{\partial \rho }}\sin ^{2}(\phi )-{\frac {\partial A_{\rho }\cos(\phi )}{\partial \phi }}{\frac {1}{\rho }}\cos(\phi )+{\frac {\partial A_{\phi }\sin(\phi ))}{\partial \phi }}{\frac {1}{\rho }}\cos(\phi )=

{\frac {\partial A_{\phi }}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}{\frac {1}{\rho }}+A_{\phi }{\frac {1}{\rho }}

Transformeren naar ρ en φ:

(\nabla \times A)_{\rho }=(\nabla \times A)_{x}\cos(\phi )+(\nabla \times A)_{y}\sin(\phi )={\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}

(\nabla \times A)_{\phi }=-(\nabla \times A)_{x}\sin(\phi )+(\nabla \times A)_{y}\cos(\phi )={\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}

Laplaciaan van een scalaire functie

Uit het bovenstaande volgt voor de laplaciaan van een scalaire functie $f$ :

\Delta f=\nabla \cdot \nabla f={\frac {\partial (\nabla f)_{\rho }}{\partial \rho }}+{\frac {(\nabla f)_{\rho }}{\rho }}+{\frac {1}{\rho }}{\frac {\partial (\nabla f)_{\phi }}{\partial \phi }}+{\frac {\partial (\nabla f)_{z}}{\partial z}}=

{\frac {\partial {\frac {\partial f_{c}}{\partial \rho }}}{\partial \rho }}+{\frac {\frac {\partial f_{c}}{\partial \rho }}{\rho }}+{\frac {1}{\rho }}{\frac {\partial {\frac {1}{\rho }}{\frac {\partial f_{c}}{\partial \phi }}}{\partial \phi }}+{\frac {\partial {\frac {\partial f_{c}}{\partial z}}}{\partial z}}={\frac {\partial ^{2}f_{c}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial f_{c}}{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f_{c}}{\partial \phi ^{2}}}+{\frac {\partial ^{2}f_{c}}{\partial z^{2}}}

Laplaciaan van een vectorveld

Uit het bovenstaande volgt voor de laplaciaan van een vectorveld $A$ :

(\Delta A)_{\rho }=(\Delta A)_{x}\cos(\phi )+(\Delta A)_{y}\sin(\phi )=\Delta A_{x}\cos(\phi )+\Delta A_{y}\sin(\phi )=

\left({\frac {\partial ^{2}A_{x}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{x}}{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}A_{x}}{\partial \phi ^{2}}}+{\frac {\partial ^{2}A_{x}}{\partial z^{2}}}\right)\cos(\phi )+

\left({\frac {\partial ^{2}A_{y}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{y}}{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}A_{y}}{\partial \phi ^{2}}}+{\frac {\partial ^{2}A_{y}}{\partial z^{2}}}\right)\sin(\phi )=

{\frac {\partial ^{2}A_{\rho }}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {1}{\rho ^{2}}}\left({\frac {\partial ^{2}A_{x}}{\partial \phi ^{2}}}\cos(\phi )+{\frac {\partial ^{2}A_{y}}{\partial \phi ^{2}}}\sin(\phi )\right)+{\frac {\partial ^{2}A_{\rho }}{\partial z^{2}}}=

{\frac {\partial ^{2}A_{\rho }}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {\partial ^{2}A_{\rho }}{\partial z^{2}}}+

{\frac {1}{\rho ^{2}}}\left({\frac {\partial ^{2}(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi ))}{\partial \phi ^{2}}}\cos(\phi )+{\frac {\partial ^{2}(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi ))}{\partial \phi ^{2}}}\sin(\phi )\right)=

{\frac {\partial ^{2}A_{\rho }}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {\partial ^{2}A_{\rho }}{\partial z^{2}}}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}A_{\rho }}{\partial \phi ^{2}}}-{\frac {1}{\rho ^{2}}}A_{\rho }-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}=

\Delta A_{\rho }-{\frac {1}{\rho ^{2}}}A_{\rho }-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}

En analoog:

(\Delta A)_{\phi }=-(\Delta A)_{x}\sin(\phi )+(\Delta A)_{y}\cos(\phi )=-\Delta A_{x}\sin(\phi )+\Delta A_{y}\cos(\phi )=

-\left({\frac {\partial ^{2}A_{x}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{x}}{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}A_{x}}{\partial \phi ^{2}}}+{\frac {\partial ^{2}A_{x}}{\partial z^{2}}}\right)\sin(\phi )+

\left({\frac {\partial ^{2}A_{y}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{y}}{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}A_{y}}{\partial \phi ^{2}}}+{\frac {\partial ^{2}A_{y}}{\partial z^{2}}}\right)\cos(\phi )=

{\frac {\partial ^{2}A_{\phi }}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \rho }}-{\frac {1}{\rho ^{2}}}\left({\frac {\partial ^{2}A_{x}}{\partial \phi ^{2}}}\sin(\phi )-{\frac {\partial ^{2}A_{y}}{\partial \phi ^{2}}}\cos(\phi )\right)+{\frac {\partial ^{2}A_{\phi }}{\partial z^{2}}}=

{\frac {\partial ^{2}A_{\phi }}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \rho }}+{\frac {\partial ^{2}A_{\phi }}{\partial z^{2}}}-

{\frac {1}{\rho ^{2}}}\left({\frac {\partial ^{2}(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi ))}{\partial \phi ^{2}}}\sin(\phi )-{\frac {\partial ^{2}(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi ))}{\partial \phi ^{2}}}\cos(\phi )\right)=

{\frac {\partial ^{2}A_{\phi }}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \rho }}+{\frac {\partial ^{2}A_{\phi }}{\partial z^{2}}}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}A_{\phi }}{\partial \phi ^{2}}}-{\frac {1}{\rho ^{2}}}A_{\phi }+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}=

\Delta A_{\phi }-{\frac {1}{\rho ^{2}}}A_{\phi }+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}

Bolcoördinaten

Gradiënt van een scalaire functie

\nabla f=({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}})

De component van $\operatorname {grad} f$ in de richting van $r$ is:

(\nabla f)_{r}={\frac {\partial f}{\partial x}}\cos(\phi )\sin(\theta )+{\frac {\partial f}{\partial y}}\sin(\phi )\sin(\theta )+{\frac {\partial f}{\partial z}}\cos(\theta )={\frac {\partial f}{\partial x}}{\frac {\partial x}{\partial r}}+{\frac {\partial f}{\partial y}}{\frac {\partial y}{\partial r}}+{\frac {\partial f}{\partial z}}{\frac {\partial z}{\partial r}}={\frac {\partial f_{b}}{\partial r}}

De component van $\operatorname {grad} f$ in de richting van $\phi$ is:

(\nabla f)_{\phi }=-{\frac {\partial f}{\partial x}}\sin(\phi )+{\frac {\partial f}{\partial y}}\cos(\phi )={\frac {1}{r\sin(\theta )}}\left({\frac {\partial f}{\partial x}}{\frac {\partial x}{\partial \phi }}+{\frac {\partial f}{\partial y}}{\frac {\partial y}{\partial \phi }}\right)={\frac {1}{r\sin(\theta )}}{\frac {\partial f_{b}}{\partial \phi }}

De component van $\operatorname {grad} f$ in de richting van $\theta$ is:

(\nabla f)_{\theta }={\frac {\partial f}{\partial x}}\cos(\phi )\cos(\theta )+{\frac {\partial f}{\partial y}}\sin(\phi )\cos(\theta )-{\frac {\partial f}{\partial z}}\sin(\theta )={\frac {1}{r}}\left({\frac {\partial f}{\partial x}}{\frac {\partial x}{\partial \theta }}+{\frac {\partial f}{\partial y}}{\frac {\partial y}{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {\partial z}{\partial \theta }}\right)={\frac {1}{r}}{\frac {\partial f_{b}}{\partial \theta }}

Divergentie van een vectorveld

\nabla \cdot A={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}

We drukken de divergentie uit in de voorstelling in bolcoördinaten:

A_{x}=A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\,

A_{y}=A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\,

A_{z}=A_{r}\cos(\theta )-A_{\theta }\sin(\theta )

Nu is:

{\frac {\partial A_{x}}{\partial x}}={\frac {\partial A_{x}}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial A_{x}}{\partial \phi }}{\frac {\partial \phi }{\partial x}}+{\frac {\partial A_{x}}{\partial \theta }}{\frac {\partial \theta }{\partial x}}

en analoog voor $y$ en $z$ , zodat:

\nabla \cdot A=

{\frac {\partial }{\partial r}}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right)\cos(\phi )\sin(\theta )+

{\frac {\partial }{\partial \phi }}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right)(-{\frac {1}{r\sin(\theta )}})\sin(\phi )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right){\frac {1}{r}}\cos(\phi )\cos(\theta )+

{\frac {\partial }{\partial r}}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right)\sin(\phi )\sin(\theta )+

{\frac {\partial }{\partial \phi }}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right){\frac {1}{r\sin(\theta )}}\cos(\phi )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right){\frac {1}{r}}\sin(\phi )\cos(\theta )+

{\frac {\partial }{\partial r}}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right)\cos(\theta )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right)(-{\frac {1}{r}})\sin(\theta )

We verzamelen apart:

de termen waarin $A_{r}$ voorkomt:

A_{r}\left({\frac {1}{r}}\sin ^{2}(\phi )+{\frac {1}{r}}\cos ^{2}(\phi )+{\frac {1}{r}}\cos ^{2}(\phi )\cos ^{2}(\theta )+{\frac {1}{r}}\sin ^{2}(\phi )\cos ^{2}(\theta )+{\frac {1}{r}}\sin ^{2}(\theta )\right)={\frac {2}{r}}A_{r}

{\frac {\partial A_{r}}{\partial r}}\left(\cos ^{2}(\phi )\sin ^{2}(\theta )+\sin ^{2}(\phi )\sin ^{2}(\theta )+\cos ^{2}(\theta )\right)={\frac {\partial A_{r}}{\partial r}}

{\frac {\partial A_{r}}{\partial \phi }}\left(-\sin(\phi )\cos(\phi )\sin(\theta )+\cos(\phi )\sin(\phi )\sin(\theta )\right)=0

{\frac {\partial A_{r}}{\partial \theta }}\left(\cos(\phi )\sin(\theta )(-{\frac {1}{r}})\cos(\phi )\cos(\theta )+\sin(\phi )\sin(\theta ){\frac {1}{r}}\sin(\phi )\cos(\theta )+\cos(\theta ){\frac {1}{r}}\sin(\theta )\right)=0

de termen waarin $A_{\phi }$ voorkomt:

A_{\phi }\left({\frac {1}{r\sin(\theta )}}\cos(\phi )\sin(\phi )-{\frac {1}{r\sin(\theta )}}\sin(\phi )\cos(\phi )\right)=0

{\frac {\partial A_{\phi }}{\partial r}}\left(-\sin(\phi )\cos(\phi )\sin(\theta )+\cos(\phi )\sin(\phi )\sin(\theta )\right)=0

{\frac {\partial A_{\phi }}{\partial \phi }}\left(\sin(\phi ){\frac {1}{r\sin(\theta )}}\sin(\phi )+\cos(\phi ){\frac {1}{r\sin(\theta )}}\cos(\phi )\right)={\frac {1}{r\sin(\theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}

{\frac {\partial A_{\phi }}{\partial \theta }}\left(-{\frac {1}{r}}\sin(\phi )\cos(\phi )\cos(\theta )+{\frac {1}{r}}\cos(\phi )\sin(\phi )\cos(\theta )\right)=0

en de termen waarin $A_{\theta }$ voorkomt:

A_{\theta }\left({\frac {1}{r\sin(\theta )}}\sin ^{2}(\phi )\cos(\theta )+{\frac {1}{r\sin(\theta )}}\cos ^{2}(\phi )\cos(\theta )-{\frac {1}{r}}\cos ^{2}(\phi )\sin(\theta )\cos(\theta )-{\frac {1}{r}}\sin ^{2}(\phi )\sin(\theta )\cos(\theta )+{\frac {1}{r}}\sin(\theta )\cos(\theta )\right)

={\frac {1}{r\sin(\theta )}}A_{\theta }\cos(\theta )

{\frac {\partial A_{\theta }}{\partial r}}\left(\cos ^{2}(\phi )\cos(\theta )\sin(\theta )+\sin ^{2}(\phi )\cos(\theta )\sin(\theta )-\sin(\theta )\cos(\theta )\right)=0

{\frac {\partial A_{\theta }}{\partial \phi }}\left(-{\frac {1}{r\sin(\theta )}}\cos(\phi )\sin(\phi )\cos(\theta )+{\frac {1}{r\sin(\theta )}}\sin(\phi )\cos(\phi )\cos(\theta )\right)=0

{\frac {\partial A_{\theta }}{\partial \theta }}\left({\frac {1}{r}}\cos ^{2}(\phi )\cos ^{2}(\theta )+{\frac {1}{r}}\sin ^{2}(\phi )\cos ^{2}(\theta )+{\frac {1}{r}}\sin ^{2}(\theta )\right)={\frac {\partial A_{\theta }}{\partial \theta }}

Samen geeft dat:

\nabla \cdot A={\frac {2}{r}}A_{r}+{\frac {\partial A_{r}}{\partial r}}+{\frac {1}{r\sin(\theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {1}{r\sin(\theta )}}A_{\theta }\cos(\theta )+{\frac {\partial A_{\theta }}{\partial \theta }}=

{\frac {1}{r^{2}}}{\frac {\partial r^{2}A_{r}}{\partial r}}+{\frac {1}{r\sin(\theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {1}{r\sin(\theta )}}{\frac {\partial \sin(\theta )A_{\theta }}{\partial \theta }}

Rotatie van een vectorveld

Voor de divergentie bepaalden we ${\frac {\partial A_{x}}{\partial x}},{\frac {\partial A_{y}}{\partial y}},{\frac {\partial A_{z}}{\partial z}}$ . Nu moeten de andere afgeleiden bepaald worden.

Voor de x-component:

{\frac {\partial A_{z}}{\partial y}}={\frac {\partial A_{z}}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial A_{z}}{\partial \phi }}{\frac {\partial \phi }{\partial y}}+{\frac {\partial A_{z}}{\partial \theta }}{\frac {\partial \theta }{\partial y}}=

{\frac {\partial }{\partial r}}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right)\sin(\phi )\sin(\theta )+

{\frac {\partial }{\partial \phi }}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right){\frac {1}{r\sin(\theta )}}\cos(\phi )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right){\frac {1}{r}}\sin(\phi )\cos(\theta )

{\frac {\partial A_{y}}{\partial z}}={\frac {\partial A_{y}}{\partial r}}{\frac {\partial r}{\partial z}}+{\frac {\partial A_{y}}{\partial \phi }}{\frac {\partial \phi }{\partial z}}+{\frac {\partial A_{y}}{\partial \theta }}{\frac {\partial \theta }{\partial z}}=

{\frac {\partial }{\partial r}}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right)\cos(\theta )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right)(-{\frac {1}{r}})\sin(\theta )

Daaruit volgt:

(\nabla \times A)_{x}={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}=

{\frac {\partial A_{r}}{\partial \phi }}{\frac {1}{r\sin(\theta )}}\cos(\phi )\cos(\theta )+{\frac {\partial A_{r}}{\partial \theta }}{\frac {1}{r}}\sin(\phi )

-{\frac {\partial A_{\phi }}{\partial r}}\cos(\phi )\cos(\theta )+{\frac {\partial A_{\phi }}{\partial \theta }}{\frac {1}{r}}\cos(\phi )\sin(\theta )

-{\frac {\partial A_{\theta }}{\partial r}}\sin(\phi )-{\frac {\partial A_{\theta }}{\partial \phi }}{\frac {1}{r}}\cos(\phi )-A_{\theta }{\frac {1}{r}}\sin(\phi )

Voor de y-component:

{\frac {\partial A_{x}}{\partial z}}={\frac {\partial A_{x}}{\partial r}}{\frac {\partial r}{\partial z}}+{\frac {\partial A_{x}}{\partial \phi }}{\frac {\partial \phi }{\partial z}}+{\frac {\partial A_{x}}{\partial \theta }}{\frac {\partial \theta }{\partial z}}=

{\frac {\partial }{\partial r}}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right)\cos(\theta )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right)(-{\frac {1}{r}})\sin(\theta )

{\frac {\partial A_{z}}{\partial x}}={\frac {\partial A_{z}}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial A_{z}}{\partial \phi }}{\frac {\partial \phi }{\partial x}}+{\frac {\partial A_{z}}{\partial \theta }}{\frac {\partial \theta }{\partial x}}=

{\frac {\partial }{\partial r}}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right)\cos(\phi )\sin(\theta )+

{\frac {\partial }{\partial \phi }}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right)(-{\frac {1}{r\sin(\theta )}})\sin(\phi )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\cos(\theta )-A_{\theta }\sin(\theta )\right){\frac {1}{r}}\cos(\phi )\cos(\theta )

Daaruit volgt:

(\nabla \times A)_{y}={\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}=

{\frac {\partial A_{r}}{\partial \phi }}{\frac {1}{r\sin(\theta )}}\sin(\phi )\cos(\theta )-{\frac {\partial A_{r}}{\partial \theta }}{\frac {1}{r}}\cos(\phi )

-{\frac {\partial A_{\phi }}{\partial r}}\sin(\phi )\cos(\theta )+{\frac {\partial A_{\phi }}{\partial \theta }}{\frac {1}{r}}\sin(\phi )\sin(\theta )

+A_{\theta }{\frac {1}{r}}\cos(\phi )+{\frac {\partial A_{\theta }}{\partial r}}\cos(\phi )-{\frac {\partial A_{\theta }}{\partial \phi }}{\frac {1}{r}}\sin(\phi )

Voor de z-component:

{\frac {\partial A_{x}}{\partial y}}={\frac {\partial A_{x}}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial A_{x}}{\partial \phi }}{\frac {\partial \phi }{\partial y}}+{\frac {\partial A_{x}}{\partial \theta }}{\frac {\partial \theta }{\partial y}}=

{\frac {\partial }{\partial r}}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right)\sin(\phi )\sin(\theta )+

{\frac {\partial }{\partial \phi }}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta ))\right){\frac {1}{r\sin(\theta )}}\cos(\phi )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta )\right){\frac {1}{r}}\sin(\phi )\cos(\theta )

{\frac {\partial A_{y}}{\partial x}}={\frac {\partial A_{y}}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial A_{y}}{\partial \phi }}{\frac {\partial \phi }{\partial x}}+{\frac {\partial A_{y}}{\partial \theta }}{\frac {\partial \theta }{\partial x}}=

{\frac {\partial }{\partial r}}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right)\cos(\phi )\sin(\theta )+

{\frac {\partial }{\partial \phi }}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right)(-{\frac {1}{r\sin(\theta )}})\sin(\phi )+

{\frac {\partial }{\partial \theta }}\left(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta )\right){\frac {1}{r}}\cos(\phi )\cos(\theta )

Daaruit volgt:

(\nabla \times A)_{z}={\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}=

{\frac {\partial A_{r}}{\partial \phi }}{\frac {1}{r}}-A_{\phi }{\frac {1}{r\sin(\theta )}}-{\frac {\partial A_{\phi }}{\partial r}}\sin(\theta )-{\frac {\partial A_{\phi }}{\partial \theta }}{\frac {1}{r}}\cos(\theta )+{\frac {\partial A_{\theta }}{\partial \phi }}{\frac {1}{r\sin(\theta )}}\cos(\theta )

Transformeren naar $r,\,\phi$ en $\theta$ :

(\nabla \times A)_{r}=(\nabla \times A)_{x}\cos(\phi )\sin(\theta )+(\nabla \times A)_{y}\sin(\phi )\sin(\theta )+(\nabla \times A)_{z}\cos(\theta )=

A_{\phi }{\frac {1}{r\sin(\theta )}}\cos(\theta )+{\frac {1}{r}}{\frac {\partial A_{\phi }}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \phi }}{\frac {1}{r\sin(\theta )}}

(\nabla \times A)_{\phi }=-(\nabla \times A)_{x}\sin(\phi )+(\nabla \times A)_{y}\cos(\phi )=

{\frac {\partial A_{\theta }}{\partial r}}-{\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}+{\frac {1}{r}}A_{\theta }

(\nabla \times A)_{\theta }=(\nabla \times A)_{x}\cos(\phi )\cos(\theta )+(\nabla \times A)_{y}\sin(\phi )\cos(\theta )-(\nabla \times A)_{z}\sin(\theta )=

{\frac {1}{r}}A_{\phi }-{\frac {1}{r\sin(\theta )}}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {\partial A_{\phi }}{\partial r}}

Laplaciaan van een scalaire functie

Uit het bovenstaande volgt voor de laplaciaan van een scalaire functie $f$ :

\Delta f=\nabla \cdot \nabla f={\frac {1}{r^{2}}}{\frac {\partial (r^{2}{\frac {\partial f_{b}}{\partial r}})}{\partial r}}+{\frac {1}{r\sin(\theta )}}{\frac {\partial ({\frac {1}{r\sin(\theta )}}{\frac {\partial f_{b}}{\partial \phi }})}{\partial \phi }}+{\frac {1}{r\sin(\theta )}}{\frac {\partial (\sin(\theta ){\frac {1}{r}}{\frac {\partial f_{b}}{\partial \theta }})}{\partial \theta }}=

{\frac {1}{r^{2}}}{\frac {\partial (r^{2}{\frac {\partial f_{b}}{\partial r}})}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}f_{b}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial (\sin(\theta ){\frac {\partial f_{b}}{\partial \theta }})}{\partial \theta }}

Laplaciaan van een vectorveld

Uit het bovenstaande volgt in bolcoördinaten voor de laplaciaan van een vectorveld $A$ :

(\Delta A)_{r}=(\Delta A)_{x}\cos(\phi )\sin(\theta )+(\Delta A)_{y}\sin(\phi )\sin(\theta )+(\Delta A)_{z}\cos(\theta )=

\Delta A_{x}\cos(\phi )\sin(\theta )+\Delta A_{y}\sin(\phi )\sin(\theta )+\Delta A_{z}\cos(\theta )=

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{x})\cos(\phi )\sin(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{y})\sin(\phi )\sin(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{z})\cos(\theta )=

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta ))\cos(\phi )\sin(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta ))\sin(\phi )\sin(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{r}\cos(\theta )-A_{\theta }\sin(\theta ))\cos(\theta )=

{\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial A_{r}}{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}A_{r}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial A_{r}}{\partial \theta }}}{\partial \theta }}-{\frac {2}{r^{2}}}A_{r}-{\frac {2}{r^{2}\sin(\theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}-{\frac {2\cos(\theta )}{r^{2}\sin(\theta )}}A_{\theta }-{\frac {2}{r^{2}}}{\frac {\partial A_{\theta }}{\partial \theta }}=

\Delta A_{r}-{\frac {2}{r^{2}}}A_{r}-{\frac {2\cos(\theta )}{r^{2}\sin(\theta )}}A_{\theta }-{\frac {2}{r^{2}}}{\frac {\partial A_{\theta }}{\partial \theta }}-{\frac {2}{r^{2}\sin(\theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}

En analoog voor de $\phi$ -component:

(\Delta A)_{\phi }=-(\Delta A)_{x}\sin(\phi )+(\Delta A)_{y}\cos(\phi )=-\Delta A_{x}\sin(\phi )+\Delta A_{y}\cos(\phi )=

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(-A_{x})\sin(\phi )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{y})\cos(\phi )=

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(-A_{r}\cos(\phi )\sin(\theta )+A_{\phi }\sin(\phi )-A_{\theta }\cos(\phi )\cos(\theta ))\sin(\phi )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta ))\cos(\phi )=

{\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial A_{\phi }}{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}A_{\phi }}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial A_{\phi }}{\partial \theta }}}{\partial \theta }}-{\frac {A_{\phi }}{(r\sin(\theta ))^{2}}}+{\frac {2}{(r\sin(\theta ))^{2}}}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos(\theta )}{(r\sin(\theta ))^{2}}}{\frac {\partial A_{\theta }}{\partial \phi }}=

\Delta A_{\phi }-{\frac {A_{\phi }}{(r\sin(\theta ))^{2}}}+{\frac {2}{(r\sin(\theta ))^{2}}}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos(\theta )}{(r\sin(\theta ))^{2}}}{\frac {\partial A_{\theta }}{\partial \phi }}

En analoog voor de $\theta$ -component:

(\Delta A)_{\theta }=(\Delta A)_{x}\cos(\phi )\cos(\theta )+(\Delta A)_{y}\sin(\phi )\cos(\theta )-(\Delta A)_{z}\sin(\theta )=

\Delta A_{x}\cos(\phi )\cos(\theta )+\Delta A_{y}\sin(\phi )\cos(\theta )-\Delta A_{z}\sin(\theta )=

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{x})\cos(\phi )\cos(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{y})\sin(\phi )\cos(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(-A_{z})\sin(\theta )=

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{r}\cos(\phi )\sin(\theta )-A_{\phi }\sin(\phi )+A_{\theta }\cos(\phi )\cos(\theta ))\cos(\phi )\cos(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(A_{r}\sin(\phi )\sin(\theta )+A_{\phi }\cos(\phi )+A_{\theta }\sin(\phi )\cos(\theta ))\sin(\phi )\cos(\theta )+

\left({\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial }{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial }{\partial \theta }}}{\partial \theta }}\right)(-A_{r}\cos(\theta )+A_{\theta }\sin(\theta ))\sin(\theta )=

{\frac {1}{r^{2}}}{\frac {\partial r^{2}{\frac {\partial A_{\theta }}{\partial r}}}{\partial r}}+{\frac {1}{(r\sin(\theta ))^{2}}}{\frac {\partial ^{2}A_{\theta }}{\partial \phi ^{2}}}+{\frac {1}{r^{2}\sin(\theta )}}{\frac {\partial \sin(\theta ){\frac {\partial A_{\theta }}{\partial \theta }}}{\partial \theta }}-{\frac {A_{\theta }}{(r\sin(\theta ))^{2}}}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos(\theta )}{(r\sin(\theta ))^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}=

\Delta A_{\theta }-{\frac {A_{\theta }}{(r\sin(\theta ))^{2}}}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos(\theta )}{(r\sin(\theta ))^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}

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