Nabla in verschillende assenstelsels

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

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 r A_\phi \over \partial r}) \boldsymbol{\hat \theta} & + \\
{1 \over r}({\partial r A_\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 - {2 A_r \over r^2}
- {2 A_\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^2\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:
  1. \operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f (Laplaciaan)
  2. \operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = \mathbf{0}
  3. \operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0
  4. \operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) 
= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}
  5. \Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f

Afleiding[bewerken]

Cilindercoördinaten[bewerken]

Gradiënt van een scalaire functie[bewerken]

\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[bewerken]

\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)+
\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)+
\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[bewerken]


(\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[bewerken]

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[bewerken]

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^2+
\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[bewerken]

Gradiënt van een scalaire functie[bewerken]

\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 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 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 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 1r\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 1r\frac{\partial f_b}{\partial \theta}
\,

Divergentie van een vectorveld[bewerken]

\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 1r)\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 1r\cos^2(\phi)\cos^2(\theta)
+
\frac 1r\sin^2(\phi)\cos^2(\theta)
+

\frac 1r\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^2A_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[bewerken]

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 1r)\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 1r\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 1r)\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 1r\cos(\phi)
\,

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

+
A_\theta
\frac 1r\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, φ en θ:



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

Laplaciaan van een scalaire functie[bewerken]

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 1r\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[bewerken]

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 φ-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 θ-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}