operatie
cartesische coördinaten (x , y , z )
cilindercoördinaten (ρ , φ , z )
bolcoördinaten (r , θ , φ ) , waar φ de azimutale en θ de polaire hoek is[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}}}