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Dit is een lijst van enkele formules uit vectoranalyse voor het werken met veelvoorkomende kromlijnige coördinatenstelsels: cartesische coördinaten , cilindercoördinaten , bolcoördinaten .
conversies tussen cartesische, cilinder- en bolcoördinaten[1]
van
cartesisch
cylindrisch
bol
naar
cartesisch
x
=
x
y
=
y
z
=
z
{\displaystyle {\begin{aligned}x&=x\\y&=y\\z&=z\end{aligned}}}
x
=
ρ
cos
φ
y
=
ρ
sin
φ
z
=
z
{\displaystyle {\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{aligned}}}
x
=
r
sin
θ
cos
φ
y
=
r
sin
θ
sin
φ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\sin \theta \sin \varphi \\z&=r\cos \theta \end{aligned}}}
cylindrisch
ρ
=
x
2
+
y
2
φ
=
arctan
(
y
x
)
z
=
z
{\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\varphi &=\arctan \left({\frac {y}{x}}\right)\\z&=z\end{aligned}}}
ρ
=
ρ
φ
=
φ
z
=
z
{\displaystyle {\begin{aligned}\rho &=\rho \\\varphi &=\varphi \\z&=z\end{aligned}}}
ρ
=
r
sin
θ
φ
=
φ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}\rho &=r\sin \theta \\\varphi &=\varphi \\z&=r\cos \theta \end{aligned}}}
bol
r
=
x
2
+
y
2
+
z
2
θ
=
arctan
(
x
2
+
y
2
z
)
φ
=
arctan
(
y
x
)
{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)\\\varphi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}}
r
=
ρ
2
+
z
2
θ
=
arctan
(
ρ
z
)
φ
=
φ
{\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {\left({\frac {\rho }{z}}\right)}\\\varphi &=\varphi \end{aligned}}}
r
=
r
φ
=
φ
θ
=
θ
{\displaystyle {\begin{aligned}r&=r\\\varphi &=\varphi \\\theta &=\theta \\\end{aligned}}}
conversies tussen eenheidsvectoren in cartesische, cilindrische en bolcoördinaten in termen van bestemmingscoördinaten[1]
cartesische
cilindrische
bol
cartesische
x
^
=
cos
φ
ρ
^
−
sin
φ
φ
^
y
^
=
sin
φ
ρ
^
+
cos
φ
φ
^
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\cos \varphi {\hat {\boldsymbol {\rho }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {y} }}&=\sin \varphi {\hat {\boldsymbol {\rho }}}+\cos \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
x
^
=
sin
θ
cos
φ
r
^
+
cos
θ
cos
φ
θ
^
−
sin
φ
φ
^
y
^
=
sin
θ
sin
φ
r
^
+
cos
θ
sin
φ
θ
^
+
cos
φ
φ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \varphi {\hat {\mathbf {r} }}+\cos \theta \cos \varphi {\hat {\boldsymbol {\theta }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \varphi {\hat {\mathbf {r} }}+\cos \theta \sin \varphi {\hat {\boldsymbol {\theta }}}+\cos \varphi {\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}}
cilindrische
ρ
^
=
x
x
^
+
y
y
^
x
2
+
y
2
φ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
ρ
^
=
sin
θ
r
^
+
cos
θ
θ
^
φ
^
=
φ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}}
bol
r
^
=
x
x
^
+
y
y
^
+
z
z
^
x
2
+
y
2
+
z
2
θ
^
=
(
x
x
^
+
y
y
^
)
z
−
(
x
2
+
y
2
)
z
^
x
2
+
y
2
+
z
2
x
2
+
y
2
φ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {\left(x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}\right)z-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}}
r
^
=
ρ
ρ
^
+
z
z
^
ρ
2
+
z
2
θ
^
=
z
ρ
^
−
ρ
z
^
ρ
2
+
z
2
φ
^
=
φ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}}
conversies tussen eenheidsvectoren in cartesische, cilindrische en bolcoördinaten in termen van oorsprongscoördinaten
cartesische
cilindrische
bol
cartesische
x
^
=
x
ρ
^
−
y
φ
^
x
2
+
y
2
y
^
=
y
ρ
^
+
x
φ
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x{\hat {\boldsymbol {\rho }}}-y{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {y} }}&={\frac {y{\hat {\boldsymbol {\rho }}}+x{\hat {\boldsymbol {\varphi }}}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
x
^
=
x
(
x
2
+
y
2
r
^
+
z
θ
^
)
−
y
x
2
+
y
2
+
z
2
φ
^
x
2
+
y
2
x
2
+
y
2
+
z
2
y
^
=
y
(
x
2
+
y
2
r
^
+
z
θ
^
)
+
x
x
2
+
y
2
+
z
2
φ
^
x
2
+
y
2
x
2
+
y
2
+
z
2
z
^
=
z
r
^
−
x
2
+
y
2
θ
^
x
2
+
y
2
+
z
2
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&={\frac {x\left({\sqrt {x^{2}+y^{2}}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)-y{\sqrt {x^{2}+y^{2}+z^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\mathbf {y} }}&={\frac {y\left({\sqrt {x^{2}+y^{2}}}{\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}\right)+x{\sqrt {x^{2}+y^{2}+z^{2}}}{\hat {\boldsymbol {\varphi }}}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r} }}-{\sqrt {x^{2}+y^{2}}}{\hat {\boldsymbol {\theta }}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\end{aligned}}}
cilindrische
ρ
^
=
cos
φ
x
^
+
sin
φ
y
^
φ
^
=
−
sin
φ
x
^
+
cos
φ
y
^
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
ρ
^
=
ρ
r
^
+
z
θ
^
ρ
2
+
z
2
φ
^
=
φ
^
z
^
=
z
r
^
−
ρ
θ
^
ρ
2
+
z
2
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {\rho {\hat {\mathbf {r} }}+z{\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&={\frac {z{\hat {\mathbf {r} }}-\rho {\hat {\boldsymbol {\theta }}}}{\sqrt {\rho ^{2}+z^{2}}}}\end{aligned}}}
bol
r
^
=
sin
θ
(
cos
φ
x
^
+
sin
φ
y
^
)
+
cos
θ
z
^
θ
^
=
cos
θ
(
cos
φ
x
^
+
sin
φ
y
^
)
−
sin
θ
z
^
φ
^
=
−
sin
φ
x
^
+
cos
φ
y
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\end{aligned}}}
r
^
=
sin
θ
ρ
^
+
cos
θ
z
^
θ
^
=
cos
θ
ρ
^
−
sin
θ
z
^
φ
^
=
φ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta {\hat {\boldsymbol {\rho }}}+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta {\hat {\boldsymbol {\rho }}}-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}}
Tabel met nabla-operator in cartesische, cilindrische en bolcoördinaten
operatie
cartesische coördinaten (x , y , z )
cilindercoördinaten (ρ , φ , z )
bolcoördinaten (r , θ , φ ) , waar φ de azimutale en θ [2]
vectorveld A
A
x
x
^
+
A
y
y
^
+
A
z
z
^
{\displaystyle A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}}
A
ρ
ρ
^
+
A
φ
φ
^
+
A
z
z
^
{\displaystyle A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}+A_{z}{\hat {\mathbf {z} }}}
A
r
r
^
+
A
θ
θ
^
+
A
φ
φ
^
{\displaystyle A_{r}{\hat {\mathbf {r} }}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}}
gradiënt ∇f [1]
∂
f
∂
x
x
^
+
∂
f
∂
y
y
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∂
f
∂
ρ
ρ
^
+
1
ρ
∂
f
∂
φ
φ
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∂
f
∂
r
r
^
+
1
r
∂
f
∂
θ
θ
^
+
1
r
sin
θ
∂
f
∂
φ
φ
^
{\displaystyle {\partial f \over \partial r}{\hat {\mathbf {r} }}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}}
divergentie ∇ ⋅ A [1]
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
φ
∂
φ
+
∂
A
z
∂
z
{\displaystyle {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}}
1
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
∂
θ
(
A
θ
sin
θ
)
+
1
r
sin
θ
∂
A
φ
∂
φ
{\displaystyle {1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\varphi } \over \partial \varphi }}
rotatie ∇ × A [1]
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
x
^
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
y
^
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
(
1
ρ
∂
A
z
∂
φ
−
∂
A
φ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
φ
^
+
1
ρ
(
∂
(
ρ
A
φ
)
∂
ρ
−
∂
A
ρ
∂
φ
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\{}+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\varphi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
1
r
sin
θ
(
∂
∂
θ
(
A
φ
sin
θ
)
−
∂
A
θ
∂
φ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
φ
−
∂
∂
r
(
r
A
φ
)
)
θ
^
+
1
r
(
∂
∂
r
(
r
A
θ
)
−
∂
A
r
∂
θ
)
φ
^
{\displaystyle {\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\{}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi }\right)\right)&{\hat {\boldsymbol {\theta }}}\\{}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
laplace-operator ∇2 f ≡ ∆f [1]
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
{\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
ρ
∂
∂
ρ
(
ρ
∂
f
∂
ρ
)
+
1
ρ
2
∂
2
f
∂
φ
2
+
∂
2
f
∂
z
2
{\displaystyle {1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
φ
2
{\displaystyle {1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}}
vector Laplaciaan ∇2 A ≡ ∆A
∇
2
A
x
x
^
+
∇
2
A
y
y
^
+
∇
2
A
z
z
^
{\displaystyle \nabla ^{2}A_{x}{\hat {\mathbf {x} }}+\nabla ^{2}A_{y}{\hat {\mathbf {y} }}+\nabla ^{2}A_{z}{\hat {\mathbf {z} }}}
(
∇
2
A
ρ
−
A
ρ
ρ
2
−
2
ρ
2
∂
A
φ
∂
φ
)
ρ
^
+
(
∇
2
A
φ
−
A
φ
ρ
2
+
2
ρ
2
∂
A
ρ
∂
φ
)
φ
^
+
∇
2
A
z
z
^
{\displaystyle {\begin{aligned}{\mathopen {}}\left(\nabla ^{2}A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\rho }}}\\+{\mathopen {}}\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\varphi }}}\\{}+\nabla ^{2}A_{z}&{\hat {\mathbf {z} }}\end{aligned}}}
(
∇
2
A
r
−
2
A
r
r
2
−
2
r
2
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
−
2
r
2
sin
θ
∂
A
φ
∂
φ
)
r
^
+
(
∇
2
A
θ
−
A
θ
r
2
sin
2
θ
+
2
r
2
∂
A
r
∂
θ
−
2
cos
θ
r
2
sin
2
θ
∂
A
φ
∂
φ
)
θ
^
+
(
∇
2
A
φ
−
A
φ
r
2
sin
2
θ
+
2
r
2
sin
θ
∂
A
r
∂
φ
+
2
cos
θ
r
2
sin
2
θ
∂
A
θ
∂
φ
)
φ
^
{\displaystyle {\begin{aligned}\left(\nabla ^{2}A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\+\left(\nabla ^{2}A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
div
grad
f
≡
∇
⋅
∇
f
≡
∇
2
f
{\displaystyle \operatorname {div} \operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f}
rot
grad
f
≡
∇
×
∇
f
=
0
{\displaystyle \operatorname {rot} \operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }
div
rot
A
≡
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \operatorname {div} \operatorname {rot} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}
rot
rot
A
≡
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \operatorname {rot} \operatorname {rot} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
∇
2
(
f
g
)
=
f
∇
2
g
+
2
∇
f
⋅
∇
g
+
g
∇
2
f
{\displaystyle \nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f}
Voetnoten
↑ a b c d e f DJ Griffiths. Introduction to Electrodynamics, 2012. isbn 978-0-321-85656-2
↑ Deze pagina gebruikt
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