U vektorskoj analizi i teoriji polja, rotor ili rotacija (rot, eng. curl ) je veličina koja odražava svojstva vektorskoga polja u prostoru. Najviše se primjenjuje u fizici , pogotovo u elektromagnetizmu i hidrodinamici .
Definicija
Shematski prikaz uz definiciju rotacije vektorskoga polja Pogledajmo linijski integral vektorskog polja
W
→
{\displaystyle {\overrightarrow {W}
C
{\displaystyle C}
S
{\displaystyle S}
C
1
+
C
2
=
C
{\displaystyle C_{1}+C_{2}=C}
S
{\displaystyle S}
∮
W
→
d
S
→
=
∫
C
W
→
d
S
→
=
∑
i
=
1
N
∫
C
i
W
→
d
S
→
i
.
{\displaystyle \oint {\overrightarrow {W}d{\vec {S}=\int \limits _{C}{\overrightarrow {W}d{\vec {S}=\sum _{i=1}^{N}\int \limits _{C_{i}{\overrightarrow {W}d{\vec {S}_{i}.}
Uzmimo sada omjer te vrijednosti i infinitezimalno malog dijela površine
A
i
{\displaystyle A_{i}
C
i
{\displaystyle C_{i}
N
↦
∞
{\displaystyle N\mapsto \infty }
A
i
↦
0
{\displaystyle A_{i}\mapsto 0}
graničnu vrijednost koja predstavlja skalarnu veličinu pridruženu određenoj točki prostora, pa je
stoga možemo smatrati komponentom vektora . Pomnožimo li dati izraz s vektorom normale
n
^
{\displaystyle {\hat {n}
rotacje ili
rotora vektorskog polja:
n
^
⋅
rot
W
→
=
d
e
f
.
lim
A
i
→
0
∫
C
i
W
→
d
S
→
A
i
=
lim
Δ
S
→
0
∮
W
→
d
S
→
Δ
S
.
{\displaystyle {\hat {n}\cdot {\mbox{rot}{\overrightarrow {W}{\stackrel {def.}{=}\lim _{A_{i}\rightarrow 0}{\frac {\int \limits _{C_{i}{\overrightarrow {W}d{\vec {S}{A_{i}=\lim _{\Delta S\rightarrow 0}{\frac {\oint {\overrightarrow {W}d{\vec {S}{\Delta S}.}
Svojstva i pretpostavke
Nije nužno da ploha omeđena krvuljom koju promatramo leži u ravnini , traži se jedino da ta ploha nema singularnosti .
Nadalje,
pretpostavlja se da se vektor normale
n
^
{\displaystyle {\hat {n}
Divergencija , također invarijanta vektorskog
polja.
Rotor u kartezijevu sustavu
Shematski prikaz uz definiciju rotacije vektorskoga polja Kako bismo izveli izraz za rotor u kartezijevu sustavu, napravimo integraciju po rubu
pravokutnika paralelnog s
x
O
y
{\displaystyle xOy}
n
^
=
z
^
{\displaystyle {\hat {n}={\hat {z}
∮
W
→
d
S
→
=
∫
C
1
W
→
d
S
→
+
∫
C
2
W
→
d
S
→
+
∫
C
3
W
→
d
S
→
+
∫
C
4
W
→
d
S
→
=
{\displaystyle \oint {\overrightarrow {W}d{\vec {S}=\int \limits _{C_{1}{\overrightarrow {W}d{\vec {S}+\int \limits _{C_{2}{\overrightarrow {W}d{\vec {S}+\int \limits _{C_{3}{\overrightarrow {W}d{\vec {S}+\int \limits _{C_{4}{\overrightarrow {W}d{\vec {S}=}
=
∫
C
1
W
x
(
x
,
y
0
,
z
0
)
d
x
+
∫
C
2
W
y
(
x
0
+
Δ
x
,
y
,
z
0
)
d
y
−
{\displaystyle =\int \limits _{C_{1}W_{x}(x,y_{0},z_{0})dx+\int \limits _{C_{2}W_{y}(x_{0}+\Delta x,y,z_{0})dy-}
−
∫
C
3
W
x
(
x
,
y
0
+
Δ
y
,
z
0
)
d
x
−
∫
C
4
W
y
(
x
0
,
y
,
z
0
)
d
y
=
{\displaystyle -\int \limits _{C_{3}W_{x}(x,y_{0}+\Delta y,z_{0})dx-\int \limits _{C_{4}W_{y}(x_{0},y,z_{0})dy=}
=
∫
[
W
x
(
x
,
y
0
,
z
0
)
−
W
x
(
x
,
y
0
+
Δ
y
,
z
0
)
]
d
x
+
{\displaystyle =\int {\Bigl [}W_{x}(x,y_{0},z_{0})-W_{x}(x,y_{0}+\Delta y,z_{0}){\Bigr ]}dx+}
+
∫
[
W
y
(
x
0
+
Δ
x
,
y
,
z
0
)
−
W
y
(
x
0
,
y
,
z
0
)
]
d
y
=
{\displaystyle +\int {\Bigl [}W_{y}(x_{0}+\Delta x,y,z_{0})-W_{y}(x_{0},y,z_{0}){\Bigr ]}dy=}
=
∂
W
y
∂
x
⋅
Δ
x
Δ
y
−
∂
W
x
∂
y
⋅
Δ
x
Δ
y
=
{\displaystyle ={\frac {\partial W_{y}{\partial x}\cdot \Delta x\Delta y-{\frac {\partial W_{x}{\partial y}\cdot \Delta x\Delta y=}
=
Δ
S
⋅
(
∂
W
y
∂
x
−
∂
W
x
∂
y
)
.
{\displaystyle =\Delta S\cdot {\Bigl (}{\frac {\partial W_{y}{\partial x}-{\frac {\partial W_{x}{\partial y}{\Bigr )}.}
Uvršatavanjem u
definiciju rotacije, te potpunom analogijom, imamo:
z
^
⋅
rot
W
→
=
lim
Δ
S
→
0
∮
W
→
d
S
→
Δ
S
=
lim
Δ
S
→
0
(
∂
W
y
∂
x
−
∂
W
x
∂
y
)
Δ
S
Δ
S
=
(
∂
W
y
∂
x
−
∂
W
x
∂
y
)
=
(
rot
W
→
)
z
.
{\displaystyle {\hat {z}\cdot {\mbox{rot}{\overrightarrow {W}=\lim _{\Delta S\rightarrow 0}{\frac {\oint {\overrightarrow {W}d{\vec {S}{\Delta S}=\lim _{\Delta S\rightarrow 0}{\frac {\Bigl (}{\frac {\partial W_{y}{\partial x}-{\frac {\partial W_{x}{\partial y}{\Bigr )}\Delta S}{\Delta S}={\Bigl (}{\frac {\partial W_{y}{\partial x}-{\frac {\partial W_{x}{\partial y}{\Bigr )}=({\mbox{rot}{\overrightarrow {W})_{z}.}
(
rot
W
→
)
x
=
(
∂
W
z
∂
y
−
∂
W
y
∂
z
)
{\displaystyle ({\mbox{rot}{\overrightarrow {W})_{x}={\Bigl (}{\frac {\partial W_{z}{\partial y}-{\frac {\partial W_{y}{\partial z}{\Bigr )}
(
rot
W
→
)
y
=
(
∂
W
x
∂
z
−
∂
W
z
∂
x
)
{\displaystyle ({\mbox{rot}{\overrightarrow {W})_{y}={\Bigl (}{\frac {\partial W_{x}{\partial z}-{\frac {\partial W_{z}{\partial x}{\Bigr )}
(
rot
W
→
)
z
=
(
∂
W
y
∂
x
−
∂
W
x
∂
y
)
{\displaystyle ({\mbox{rot}{\overrightarrow {W})_{z}={\Bigl (}{\frac {\partial W_{y}{\partial x}-{\frac {\partial W_{x}{\partial y}{\Bigr )}
rot
W
→
=
x
^
(
∂
W
z
∂
y
−
∂
W
y
∂
z
)
+
y
^
(
∂
W
x
∂
z
−
∂
W
z
∂
x
)
+
z
^
(
∂
W
y
∂
x
−
∂
W
x
∂
y
)
.
{\displaystyle {\mbox{rot}{\overrightarrow {W}={\hat {x}{\Bigl (}{\frac {\partial W_{z}{\partial y}-{\frac {\partial W_{y}{\partial z}{\Bigr )}+{\hat {y}{\Bigl (}{\frac {\partial W_{x}{\partial z}-{\frac {\partial W_{z}{\partial x}{\Bigr )}+{\hat {z}{\Bigl (}{\frac {\partial W_{y}{\partial x}-{\frac {\partial W_{x}{\partial y}{\Bigr )}.}
Očito u danoj
fomuli možemo prepoznati simbolički zapisanu determinantu :
rot
W
→
=
|
x
^
y
^
z
^
∂
∂
x
∂
∂
y
∂
∂
z
W
x
W
y
W
z
|
.
{\displaystyle {\mbox{rot}{\overrightarrow {W}=\left|{\begin{array}{ccc}\displaystyle {\hat {x}&\displaystyle {\hat {y}&\displaystyle {\hat {z}\\\displaystyle {\frac {\partial }{\partial x}&\displaystyle {\frac {\partial }{\partial y}&\displaystyle {\frac {\partial }{\partial z}\\\displaystyle {W_{x}&\displaystyle {W_{y}&\displaystyle {W_{z}\end{array}\right|.}
Nadalje, očito je
rot
W
→
=
(
x
^
∂
∂
x
+
y
^
∂
∂
y
+
z
^
∂
∂
z
)
×
(
x
^
W
x
+
y
^
W
y
+
z
^
W
z
)
=
∇
→
×
W
→
,
{\displaystyle {\mbox{rot}{\overrightarrow {W}={\Bigl (}{\hat {x}{\frac {\partial }{\partial x}+{\hat {y}{\frac {\partial }{\partial y}+{\hat {z}{\frac {\partial }{\partial z}{\Bigr )}\times ({\hat {x}W_{x}+{\hat {y}W_{y}+{\hat {z}W_{z})={\vec {\nabla }\times {\overrightarrow {W},}
pa
rot
W
→
{\displaystyle {\mbox{rot}{\overrightarrow {W}
∇
→
×
W
→
{\displaystyle {\vec {\nabla }\times {\overrightarrow {W}
∇
→
{\displaystyle {\vec {\nabla }
Hamiltonov operator.
Rotacija i Stokesov teorem
Za rotaciju vrijedi Stokesov teorem
∫
S
rot
W
→
⋅
d
A
→
=
∫
C
W
→
⋅
d
S
→
.
{\displaystyle \int \limits _{S}{\mbox{rot}{\overrightarrow {W}\cdot d{\vec {A}=\int \limits _{C}{\overrightarrow {W}\cdot d{\vec {S}.}
Izrazi za rotaciju u drugim koordinatnim sustavima
|
(
rot
W
→
)
ρ
|
=
1
ρ
∂
W
z
∂
φ
−
∂
W
φ
∂
z
{\displaystyle |({\mbox{rot}{\overrightarrow {W})_{\rho }|={\frac {1}{\rho }{\frac {\partial W_{z}{\partial \varphi }-{\frac {\partial W_{\varphi }{\partial z}
|
(
rot
W
→
)
φ
|
=
∂
W
ρ
∂
z
−
∂
W
z
∂
ρ
{\displaystyle |({\mbox{rot}{\overrightarrow {W})_{\varphi }|={\frac {\partial W_{\rho }{\partial z}-{\frac {\partial W_{z}{\partial \rho }
|
(
rot
W
→
)
z
|
=
1
ρ
∂
∂
ρ
(
ρ
W
φ
)
−
1
ρ
∂
W
ρ
∂
φ
{\displaystyle |({\mbox{rot}{\overrightarrow {W})_{z}|={\frac {1}{\rho }{\frac {\partial }{\partial \rho }(\rho W_{\varphi })-{\frac {1}{\rho }{\frac {\partial W_{\rho }{\partial \varphi }
rot
W
→
=
[
1
ρ
∂
W
z
∂
φ
−
∂
W
φ
∂
z
]
ρ
^
+
[
∂
W
ρ
∂
z
−
∂
W
z
∂
ρ
]
φ
^
+
[
1
ρ
∂
∂
ρ
(
ρ
W
φ
)
−
1
ρ
∂
W
ρ
∂
φ
]
z
^
{\displaystyle {\mbox{rot}{\overrightarrow {W}={\biggl [}{\frac {1}{\rho }{\frac {\partial W_{z}{\partial \varphi }-{\frac {\partial W_{\varphi }{\partial z}{\biggr ]}{\hat {\rho }+{\biggl [}{\frac {\partial W_{\rho }{\partial z}-{\frac {\partial W_{z}{\partial \rho }{\biggr ]}{\hat {\varphi }+{\biggl [}{\frac {1}{\rho }{\frac {\partial }{\partial \rho }(\rho W_{\varphi })-{\frac {1}{\rho }{\frac {\partial W_{\rho }{\partial \varphi }{\biggr ]}{\hat {z}
|
(
rot
W
→
)
r
|
=
1
r
sin
ϑ
∂
∂
ϑ
(
W
φ
sin
ϑ
)
−
1
r
sin
ϑ
∂
W
ϑ
∂
φ
{\displaystyle |({\mbox{rot}{\overrightarrow {W})_{r}|={\frac {1}{r\sin \vartheta }{\frac {\partial }{\partial \vartheta }(W_{\varphi }\sin \vartheta )-{\frac {1}{r\sin \vartheta }{\frac {\partial W_{\vartheta }{\partial \varphi }
|
(
rot
W
→
)
ϑ
|
=
1
r
sin
ϑ
∂
W
r
∂
φ
−
1
r
∂
∂
r
(
r
W
φ
)
{\displaystyle |({\mbox{rot}{\overrightarrow {W})_{\vartheta }|={\frac {1}{r\sin \vartheta }{\frac {\partial W_{r}{\partial \varphi }-{\frac {1}{r}{\frac {\partial }{\partial r}(rW_{\varphi })}
|
(
rot
W
→
)
φ
|
=
1
r
∂
∂
r
(
r
W
ϑ
)
−
1
r
∂
W
r
∂
ϑ
{\displaystyle |({\mbox{rot}{\overrightarrow {W})_{\varphi }|={\frac {1}{r}{\frac {\partial }{\partial r}(rW_{\vartheta })-{\frac {1}{r}{\frac {\partial W_{r}{\partial \vartheta }
rot
W
→
=
[
1
r
sin
ϑ
∂
∂
ϑ
(
W
φ
sin
ϑ
)
−
1
r
sin
ϑ
∂
W
ϑ
∂
φ
]
r
^
+
[
1
r
sin
ϑ
∂
W
r
∂
φ
−
1
r
∂
∂
r
(
r
W
φ
)
]
ϑ
^
+
[
1
r
∂
∂
r
(
r
W
ϑ
)
−
1
r
∂
W
r
∂
ϑ
]
φ
^
.
{\displaystyle {\mbox{rot}{\overrightarrow {W}={\biggl [}{\frac {1}{r\sin \vartheta }{\frac {\partial }{\partial \vartheta }(W_{\varphi }\sin \vartheta )-{\frac {1}{r\sin \vartheta }{\frac {\partial W_{\vartheta }{\partial \varphi }{\biggr ]}{\hat {r}+{\biggl [}{\frac {1}{r\sin \vartheta }{\frac {\partial W_{r}{\partial \varphi }-{\frac {1}{r}{\frac {\partial }{\partial r}(rW_{\varphi }){\biggr ]}{\hat {\vartheta }+{\biggl [}{\frac {1}{r}{\frac {\partial }{\partial r}(rW_{\vartheta })-{\frac {1}{r}{\frac {\partial W_{r}{\partial \vartheta }{\biggr ]}{\hat {\varphi }.}
Rotacija i algebarske operacije
Neka su dana vektorska polja
u
→
{\displaystyle {\vec {u}
v
→
{\displaystyle {\vec {v}
skalar
U
{\displaystyle U}
f
(
U
)
{\displaystyle f(U)}
r
→
{\displaystyle {\vec {r}
rot
(
u
→
+
v
→
)
=
rot
u
→
+
rot
v
→
{\displaystyle {\textrm {rot}({\vec {u}+{\vec {v})={\textrm {rot}{\vec {u}+{\textrm {rot}{\vec {v}
rot
(
U
⋅
v
→
)
=
U
⋅
rot
v
→
−
v
→
×
grad
U
{\displaystyle {\textrm {rot}(U\cdot {\vec {v})=U\cdot {\textrm {rot}{\vec {v}-{\vec {v}\times {\mbox{grad}U}
rot
[
f
(
U
)
⋅
v
→
]
=
f
(
U
)
⋅
rot
v
→
−
v
→
×
f
U
′
(
U
)
grad
U
{\displaystyle {\textrm {rot}[f(U)\cdot {\vec {v}]=f(U)\cdot {\textrm {rot}{\vec {v}-{\vec {v}\times f_{U}^{'}(U){\textrm {grad}U}
rot
r
→
=
0
{\displaystyle {\textrm {rot}{\vec {r}=0}
rot
(
u
→
×
v
→
)
=
u
→
div
v
→
−
v
→
div
u
→
+
(
v
→
⋅
∇
→
)
u
→
−
(
u
→
⋅
∇
→
)
v
→
{\displaystyle {\mbox{rot}({\vec {u}\times {\vec {v})={\vec {u}{\mbox{div}{\vec {v}-{\vec {v}{\mbox{div}{\vec {u}+({\vec {v}\cdot {\vec {\nabla }){\vec {u}-({\vec {u}\cdot {\vec {\nabla }){\vec {v}
grad
(
u
→
⋅
v
→
)
=
v
→
×
rot
u
→
+
u
→
×
rot
v
→
+
(
v
→
⋅
∇
→
)
u
→
+
(
u
→
⋅
∇
→
)
v
→
{\displaystyle {\mbox{grad}({\vec {u}\cdot {\vec {v})={\vec {v}\times {\mbox{rot}{\vec {u}+{\vec {u}\times {\mbox{rot}{\vec {v}+({\vec {v}\cdot {\vec {\nabla }){\vec {u}+({\vec {u}\cdot {\vec {\nabla }){\vec {v}
div
(
u
→
×
v
→
)
=
v
→
rot
u
→
−
u
→
rot
v
→
.
{\displaystyle {\mbox{div}({\vec {u}\times {\vec {v})={\vec {v}{\mbox{rot}{\vec {u}-{\vec {u}{\mbox{rot}{\vec {v}.}
Primjeri
Rotor elektrostaskog polja točkastog naboja ,
E
→
=
1
4
π
ε
0
q
r
3
r
→
{\displaystyle {\overrightarrow {E}={\frac {1}{4\pi \varepsilon _{0}{\frac {q}{r^{3}{\vec {r}
rot
E
→
=
rot
(
1
4
π
ε
0
q
r
3
r
→
)
=
(
2.
)
q
4
π
ε
0
r
3
rot
r
→
−
r
→
×
grad
q
4
π
ε
0
r
3
=
−
3
q
4
π
ε
0
r
4
r
→
r
×
r
→
=
[
r
→
×
r
→
=
0
]
=
0.
{\displaystyle {\mbox{rot}{\overrightarrow {E}={\mbox{rot}{\Bigl (}{\frac {1}{4\pi \varepsilon _{0}{\frac {q}{r^{3}{\vec {r}{\Bigr )}{\stackrel {(2.)}{=}{\frac {q}{4\pi \varepsilon _{0}r^{3}{\mbox{rot}{\vec {r}-{\vec {r}\times {\mbox{grad}{\frac {q}{4\pi \varepsilon _{0}r^{3}=-{\frac {3q}{4\pi \varepsilon _{0}r^{4}{\frac {\vec {r}{r}\times {\vec {r}=[{\vec {r}\times {\vec {r}=0]=0.}
Rotor vektorskog polja obodne kružne brzine ,
v
→
=
ω
→
×
r
→
{\displaystyle {\vec {v}={\vec {\omega }\times {\vec {r}
Shematski prikaz uz rotaciju polja obodne brzine
v
→
=
ω
→
×
r
→
=
|
x
^
y
^
z
^
ω
x
ω
y
ω
z
x
y
z
|
=
x
^
(
z
ω
y
−
y
ω
z
)
+
y
^
(
x
ω
z
−
z
ω
x
)
+
z
^
(
y
ω
x
−
x
ω
y
)
;
{\displaystyle {\vec {v}={\vec {\omega }\times {\vec {r}=\left|{\begin{array}{ccc}{\hat {x}&{\hat {y}&{\hat {z}\\\omega _{x}&\omega _{y}&\omega _{z}\\x&y&z\end{array}\right|={\hat {x}(z\omega _{y}-y\omega _{z})+{\hat {y}(x\omega _{z}-z\omega _{x})+{\hat {z}(y\omega _{x}-x\omega _{y});}
rot
v
→
=
|
x
^
y
^
z
^
∂
∂
x
∂
∂
y
∂
∂
z
(
z
ω
y
−
y
ω
z
)
(
x
ω
z
−
z
ω
x
)
(
y
ω
x
−
x
ω
y
)
|
=
2
ω
x
x
^
+
2
ω
y
y
^
+
2
ω
z
z
^
=
2
ω
→
.
{\displaystyle {\mbox{rot}{\vec {v}=\left|{\begin{array}{ccc}{\hat {x}&{\hat {y}&{\hat {z}\\{\frac {\partial }{\partial x}&{\frac {\partial }{\partial y}&{\frac {\partial }{\partial z}\\(z\omega _{y}-y\omega _{z})&(x\omega _{z}-z\omega _{x})&(y\omega _{x}-x\omega _{y})\end{array}\right|=2\omega _{x}{\hat {x}+2\omega _{y}{\hat {y}+2\omega _{z}{\hat {z}=2{\vec {\omega }.}
Odatle se lako mogu iščitati komponente kutne brzine :
ω
x
=
1
2
(
∂
v
z
∂
y
−
∂
v
y
∂
z
)
{\displaystyle \omega _{x}={\frac {1}{2}{\Bigl (}{\frac {\partial v_{z}{\partial y}-{\frac {\partial v_{y}{\partial z}{\Bigr )}
ω
y
=
1
2
(
∂
v
x
∂
z
−
∂
v
z
∂
x
)
{\displaystyle \omega _{y}={\frac {1}{2}{\Bigl (}{\frac {\partial v_{x}{\partial z}-{\frac {\partial v_{z}{\partial x}{\Bigr )}
ω
z
=
1
2
(
∂
v
y
∂
x
−
∂
v
x
∂
y
)
.
{\displaystyle \omega _{z}={\frac {1}{2}{\Bigl (}{\frac {\partial v_{y}{\partial x}-{\frac {\partial v_{x}{\partial y}{\Bigr )}.}
Na ovom primjeru primijetimo: vektor brzine
v
→
{\displaystyle {\vec {v}
rot
v
→
{\displaystyle {\mbox{rot}{\vec {v}
rotor polarnog vektora je aksijalni vektor, a rotor aksijalnog vektora je polarni vektor.
Vezani pojmovi
Vanjske veze
![{\displaystyle =\int {\Bigl [}W_{x}(x,y_{0},z_{0})-W_{x}(x,y_{0}+\Delta y,z_{0}){\Bigr ]}dx+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2251aff11c9411916c051ffb0ad813c2b7cd3e10)
![{\displaystyle +\int {\Bigl [}W_{y}(x_{0}+\Delta x,y,z_{0})-W_{y}(x_{0},y,z_{0}){\Bigr ]}dy=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15bd52da214e0df4c0ad7efacc4ad6d6c2855a20)
![{\displaystyle {\mbox{rot}{\overrightarrow {W}={\biggl [}{\frac {1}{\rho }{\frac {\partial W_{z}{\partial \varphi }-{\frac {\partial W_{\varphi }{\partial z}{\biggr ]}{\hat {\rho }+{\biggl [}{\frac {\partial W_{\rho }{\partial z}-{\frac {\partial W_{z}{\partial \rho }{\biggr ]}{\hat {\varphi }+{\biggl [}{\frac {1}{\rho }{\frac {\partial }{\partial \rho }(\rho W_{\varphi })-{\frac {1}{\rho }{\frac {\partial W_{\rho }{\partial \varphi }{\biggr ]}{\hat {z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49fefc775d65dd25d7ec1b8f7aad59a69c33cf2e)
![{\displaystyle {\mbox{rot}{\overrightarrow {W}={\biggl [}{\frac {1}{r\sin \vartheta }{\frac {\partial }{\partial \vartheta }(W_{\varphi }\sin \vartheta )-{\frac {1}{r\sin \vartheta }{\frac {\partial W_{\vartheta }{\partial \varphi }{\biggr ]}{\hat {r}+{\biggl [}{\frac {1}{r\sin \vartheta }{\frac {\partial W_{r}{\partial \varphi }-{\frac {1}{r}{\frac {\partial }{\partial r}(rW_{\varphi }){\biggr ]}{\hat {\vartheta }+{\biggl [}{\frac {1}{r}{\frac {\partial }{\partial r}(rW_{\vartheta })-{\frac {1}{r}{\frac {\partial W_{r}{\partial \vartheta }{\biggr ]}{\hat {\varphi }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba47136ca928b5f8ecdad19c3ab150d3c73ec333)
![{\displaystyle {\textrm {rot}[f(U)\cdot {\vec {v}]=f(U)\cdot {\textrm {rot}{\vec {v}-{\vec {v}\times f_{U}^{'}(U){\textrm {grad}U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1820ae51935e8046a1d48e42a7a216a78442098)
![{\displaystyle {\mbox{rot}{\overrightarrow {E}={\mbox{rot}{\Bigl (}{\frac {1}{4\pi \varepsilon _{0}{\frac {q}{r^{3}{\vec {r}{\Bigr )}{\stackrel {(2.)}{=}{\frac {q}{4\pi \varepsilon _{0}r^{3}{\mbox{rot}{\vec {r}-{\vec {r}\times {\mbox{grad}{\frac {q}{4\pi \varepsilon _{0}r^{3}=-{\frac {3q}{4\pi \varepsilon _{0}r^{4}{\frac {\vec {r}{r}\times {\vec {r}=[{\vec {r}\times {\vec {r}=0]=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de75b5c57ece87650c3ef250f865fb8799ba8b5f)
