>> Operadores diferenciales en coordenadas cilíndricas
editar
* Coordenadas cilíndricas
editar
Cambio de coordenadas cartesianas a coordenadas cilíndricas .
x
=
r
c
o
s
ϑ
{\displaystyle x=rcos\vartheta }
y
=
r
s
i
n
ϑ
{\displaystyle y=rsin\vartheta }
z
=
z
{\displaystyle z=z}
* Cambio de base
editar
Cambio de la base ortonormal en coordenadas cartesianas a la base en coordenadas cilíndricas .
e
^
r
=
c
o
s
ϑ
e
^
x
+
s
i
n
ϑ
e
^
y
{\displaystyle {\widehat {e}}_{r}=cos\vartheta {\widehat {e}}_{x}+sin\vartheta {\widehat {e}}_{y}}
e
^
ϑ
=
−
s
i
n
ϑ
e
^
x
+
c
o
s
ϑ
e
^
y
{\displaystyle {\widehat {e}}_{\vartheta }=-sin\vartheta {\widehat {e}}_{x}+cos\vartheta {\widehat {e}}_{y}}
e
^
z
=
e
^
z
{\displaystyle {\widehat {e}}_{z}={\widehat {e}}_{z}}
* Derivadas de las bases en coordenadas cilíndricas
editar
Utilizaremos las siguientes derivadas de las bases.
∂
e
^
r
∂
r
=
0
;
∂
e
^
r
∂
ϑ
=
e
^
ϑ
;
∂
e
^
r
∂
z
=
0
;
∂
e
^
ϑ
∂
r
=
0
;
∂
e
^
ϑ
∂
ϑ
=
−
e
^
r
;
∂
e
^
ϑ
∂
z
=
0
;
∂
e
^
z
∂
r
=
0
;
∂
e
^
z
∂
ϑ
=
0
;
∂
e
^
z
∂
z
=
0
;
{\displaystyle {\frac {\partial {\widehat {e}}_{r}}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}={\widehat {e}}_{\vartheta };{\frac {\partial {\widehat {e}}_{r}}{\partial z}}=0;{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}=-{\widehat {e}}_{r};{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}}=0;{\frac {\partial {\widehat {e}}_{z}}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }}=0;{\frac {\partial {\widehat {e}}_{z}}{\partial z}}=0;}
* Productos escalares en coordenadas cartesianas y cilíndricas
editar
Necesitaremos los siguientes productos escalares .
e
^
x
⋅
e
^
x
=
1
;
e
^
y
⋅
e
^
y
=
1
;
e
^
z
⋅
e
^
z
=
1
{\displaystyle {\widehat {e}}_{x}\cdot {\widehat {e}}_{x}=1;{\widehat {e}}_{y}\cdot {\widehat {e}}_{y}=1;{\widehat {e}}_{z}\cdot {\widehat {e}}_{z}=1}
e
^
y
⋅
e
^
x
=
0
;
e
^
x
⋅
e
^
y
=
0
;
e
^
z
⋅
e
^
x
=
0
;
e
^
x
⋅
e
^
z
=
0
;
e
^
y
⋅
e
^
z
=
0
;
e
^
z
⋅
e
^
y
=
0
{\displaystyle {\widehat {e}}_{y}\cdot {\widehat {e}}_{x}=0;{\widehat {e}}_{x}\cdot {\widehat {e}}_{y}=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{x}=0;{\widehat {e}}_{x}\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{y}\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{y}=0}
e
^
r
⋅
e
^
r
=
1
;
e
^
ϑ
⋅
e
^
ϑ
=
1
;
e
^
z
⋅
e
^
z
=
1
{\displaystyle {\widehat {e}}_{r}\cdot {\widehat {e}}_{r}=1;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }=1;{\widehat {e}}_{z}\cdot {\widehat {e}}_{z}=1}
e
^
ϑ
⋅
e
^
r
=
0
;
e
^
r
⋅
e
^
ϑ
=
0
;
e
^
z
⋅
e
^
r
=
0
;
e
^
r
⋅
e
^
z
=
0
;
e
^
ϑ
⋅
e
^
z
=
0
;
e
^
z
⋅
e
^
ϑ
=
0
{\displaystyle {\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }=0}
* Productos vectoriales en coordenadas cartesianas y cilíndricas
editar
Utilizaremos los siguientes productos vectoriales .
e
^
x
×
e
^
x
=
0
;
e
^
y
×
e
^
y
=
0
;
e
^
z
×
e
^
z
=
0
{\displaystyle {\widehat {e}}_{x}\times {\widehat {e}}_{x}=0;{\widehat {e}}_{y}\times {\widehat {e}}_{y}=0;{\widehat {e}}_{z}\times {\widehat {e}}_{z}=0}
e
^
x
×
e
^
y
=
e
^
z
;
e
^
y
×
e
^
z
=
e
^
x
;
e
^
z
×
e
^
x
=
e
^
y
{\displaystyle {\widehat {e}}_{x}\times {\widehat {e}}_{y}={\widehat {e}}_{z};{\widehat {e}}_{y}\times {\widehat {e}}_{z}={\widehat {e}}_{x};{\widehat {e}}_{z}\times {\widehat {e}}_{x}={\widehat {e}}_{y}}
e
^
y
×
e
^
x
=
−
e
^
z
;
e
^
z
×
e
^
y
=
−
e
^
x
;
e
^
x
×
e
^
z
=
−
e
^
y
{\displaystyle {\widehat {e}}_{y}\times {\widehat {e}}_{x}=-{\widehat {e}}_{z};{\widehat {e}}_{z}\times {\widehat {e}}_{y}=-{\widehat {e}}_{x};{\widehat {e}}_{x}\times {\widehat {e}}_{z}=-{\widehat {e}}_{y}}
e
^
r
×
e
^
r
=
0
;
e
^
ϑ
×
e
^
ϑ
=
0
;
e
^
z
×
e
^
z
=
0
{\displaystyle {\widehat {e}}_{r}\times {\widehat {e}}_{r}=0;{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{z}\times {\widehat {e}}_{z}=0}
e
^
r
×
e
^
ϑ
=
e
^
z
;
e
^
ϑ
×
e
^
z
=
e
^
r
;
e
^
z
×
e
^
r
=
e
^
ϑ
{\displaystyle {\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta }={\widehat {e}}_{z};{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z}={\widehat {e}}_{r};{\widehat {e}}_{z}\times {\widehat {e}}_{r}={\widehat {e}}_{\vartheta }}
e
^
ϑ
×
e
^
r
=
−
e
^
z
;
e
^
z
×
e
^
ϑ
=
−
e
^
r
;
e
^
r
×
e
^
z
=
−
e
^
ϑ
{\displaystyle {\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r}=-{\widehat {e}}_{z};{\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta }=-{\widehat {e}}_{r};{\widehat {e}}_{r}\times {\widehat {e}}_{z}=-{\widehat {e}}_{\vartheta }}
* Factor de escala en coordenadas cilíndricas
editar
El factor de escala nos permite expresar un vector en otras coordenadas y nos da el módulo de las bases.
d
r
→
=
h
r
d
r
+
h
ϑ
d
ϑ
+
h
z
d
z
{\displaystyle d{\overrightarrow {r}}=h_{r}dr+h_{\vartheta }d\vartheta +h_{z}dz}
h
→
=
∑
|
∂
r
→
∂
u
i
|
=
(
|
∂
x
∂
r
|
,
|
∂
y
∂
ϑ
|
,
|
∂
z
∂
z
|
)
=
(
h
r
,
h
ϑ
,
h
z
)
=
(
1
,
r
,
1
)
{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial {\overrightarrow {r}}}{\partial u_{i}}}|=(|{\frac {\partial x}{\partial r}}|,|{\frac {\partial y}{\partial \vartheta }}|,|{\frac {\partial z}{\partial z}}|)=(h_{r},h_{\vartheta },h_{z})=(1,r,1)}
geometría de Riemann .
h
→
=
∑
|
∂
∂
u
i
|
=
(
|
∂
∂
r
|
,
|
∂
∂
ϑ
|
,
|
∂
∂
z
|
)
=
(
g
r
,
g
ϑ
,
g
z
)
=
(
1
,
r
,
1
)
{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial r}}|,|{\frac {\partial }{\partial \vartheta }}|,|{\frac {\partial }{\partial z}}|)=({\sqrt[{}]{g_{r}}},{\sqrt[{}]{g_{\vartheta }}},{\sqrt[{}]{g_{z}}})=(1,r,1)}
* Métrica o tensor métrico en coordenadas cilíndricas
editar
La métrica (tensor métrico ) de una variedad de Riemann nos permite obtener los coeficientes de un elemento de longitud en las bases deseadas.
Tenemos la métrica de las coordenadas cilíndricas .
g
α
β
=
g
r
d
r
2
+
g
ϑ
d
ϑ
2
+
g
z
d
z
2
=
d
r
2
+
r
2
d
ϑ
2
+
d
z
2
{\displaystyle g_{\alpha \beta }=g_{r}dr^{2}+g_{\vartheta }d\vartheta ^{2}+g_{z}dz^{2}=dr^{2}+r^{2}d\vartheta ^{2}+dz^{2}}
g
α
β
=
[
g
r
g
ϑ
g
z
]
=
[
|
∂
∂
r
|
⋅
|
∂
∂
r
|
|
∂
∂
ϑ
|
⋅
|
∂
∂
ϑ
|
|
∂
∂
z
|
⋅
|
∂
∂
z
|
]
=
[
1
r
2
1
]
{\displaystyle g_{\alpha \beta }={\begin{bmatrix}g_{r}&&\\&g_{\vartheta }&\\&&g_{z}\end{bmatrix}}={\begin{bmatrix}\vert {\frac {\partial }{\partial r}}\vert \cdot |{\frac {\partial }{\partial r}}|&&\\&|{\frac {\partial }{\partial \vartheta }}|\cdot |{\frac {\partial }{\partial \vartheta }}|&\\&&|{\frac {\partial }{\partial z}}|\cdot |{\frac {\partial }{\partial z}}|\end{bmatrix}}={\begin{bmatrix}1&&\\&r^{2}&\\&&1\end{bmatrix}}}
d
e
t
(
g
α
β
)
=
|
g
|
=
r
2
{\displaystyle det(g_{\alpha \beta })=|g|=r^{2}}
* Estrella de Hodge (*) en coordenadas cilíndricas
editar
La estrella de Hodge es un operador que actúa sobre un p-forma diferencial en un espacio de dimensión n.
∗
(
F
(
d
x
1
∧
d
x
2
∧
d
x
3
∧
.
.
.
∧
d
x
p
)
=
|
g
|
g
11
g
22
g
33
.
.
.
g
p
p
ε
F
(
d
x
p
+
1
∧
d
x
p
+
2
∧
d
x
p
+
3
∧
.
.
.
∧
d
x
n
−
p
)
{\displaystyle *(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})}
∗
(
F
d
x
α
)
=
ε
|
g
|
g
α
α
F
(
d
x
β
∧
d
x
γ
)
:
{\displaystyle \ast (Fdx_{\alpha })=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }}}F(dx_{\beta }\wedge dx_{\gamma }):}
∗
(
F
d
r
)
=
r
2
1
F
(
d
ϑ
∧
d
z
)
=
r
F
(
d
ϑ
∧
d
z
)
;
∗
(
F
d
ϑ
)
=
r
2
r
2
F
(
d
r
∧
d
z
)
=
1
r
F
(
d
r
∧
d
z
)
;
∗
(
F
d
z
)
=
r
2
1
F
(
d
ϑ
∧
d
r
)
=
r
F
(
d
ϑ
∧
d
r
)
{\displaystyle \ast (Fdr)={\frac {\sqrt[{}]{r^{2}}}{1}}F(d\vartheta \wedge dz)=rF(d\vartheta \wedge dz);\ast (Fd\vartheta )={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}F(dr\wedge dz)={\frac {1}{r}}F(dr\wedge dz);\ast (Fdz)={\frac {\sqrt[{}]{r^{2}}}{1}}F(d\vartheta \wedge dr)=rF(d\vartheta \wedge dr)}
∗
(
F
(
d
x
α
∧
d
x
β
)
)
=
ε
|
g
|
g
α
α
g
β
β
F
d
x
γ
:
{\displaystyle \ast (F(dx_{\alpha }\wedge dx_{\beta }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }}}Fdx_{\gamma }:}
∗
(
F
(
d
r
∧
d
ϑ
)
)
=
r
2
r
2
F
d
z
=
1
r
F
d
z
;
∗
(
F
(
d
r
∧
d
z
)
)
=
r
2
1
F
d
ϑ
=
r
F
d
ϑ
;
{\displaystyle \ast (F(dr\wedge d\vartheta ))={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdz={\frac {1}{r}}Fdz;\ast (F(dr\wedge dz))={\frac {\sqrt[{}]{r^{2}}}{1}}Fd\vartheta =rFd\vartheta ;}
∗
(
F
(
d
ϑ
∧
d
r
)
)
=
−
r
2
r
2
F
d
z
=
−
1
r
F
d
z
;
∗
(
F
(
d
ϑ
∧
d
z
)
)
=
r
2
r
2
F
d
r
=
1
r
F
d
r
;
{\displaystyle \ast (F(d\vartheta \wedge dr))=-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdz=-{\frac {1}{r}}Fdz;\ast (F(d\vartheta \wedge dz))={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdr={\frac {1}{r}}Fdr;}
∗
(
F
(
d
z
∧
d
r
)
)
=
−
r
2
1
F
d
ϑ
=
−
r
F
d
ϑ
;
∗
(
F
(
d
z
∧
d
ϑ
)
)
=
−
r
2
r
2
F
d
r
=
−
1
r
F
d
r
;
{\displaystyle \ast (F(dz\wedge dr))=-{\frac {\sqrt[{}]{r^{2}}}{1}}Fd\vartheta =-rFd\vartheta ;\ast (F(dz\wedge d\vartheta ))=-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdr=-{\frac {1}{r}}Fdr;}
∗
(
F
(
d
x
α
∧
d
x
β
∧
d
x
γ
)
)
=
ε
|
g
|
g
α
α
g
β
β
g
γ
γ
F
:
{\displaystyle \ast (F(dx_{\alpha }\wedge dx_{\beta }\wedge dx_{\gamma }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }g_{\gamma \gamma }}}F:}
∗
(
F
(
d
r
∧
d
ϑ
∧
d
z
)
)
=
ε
r
2
r
2
F
=
ε
1
r
F
{\displaystyle \ast (F(dr\wedge d\vartheta \wedge dz))=\varepsilon {\frac {\sqrt[{}]{r^{2}}}{r^{2}}}F=\varepsilon {\frac {1}{r}}F}
* Subir y bajar índices. Coordenadas cilíndricas
editar
Subir y bajar índices nos permite pasar de la base de un espacio vectorial a la base dual . Tenemos el contravector o vector del espacio inicial y el covector o 1-forma diferencial de la base diferencial o cobase .
∂
∂
x
α
=
g
α
β
d
x
β
:
{\displaystyle {\frac {\partial }{\partial x^{\alpha }}}=g_{\alpha \beta }dx^{\beta }:}
∂
∂
r
=
d
r
;
∂
∂
ϑ
=
r
2
d
ϑ
;
∂
∂
z
=
d
z
{\displaystyle {\frac {\partial }{\partial r}}=dr;{\frac {\partial }{\partial \vartheta }}=r^{2}d\vartheta ;{\frac {\partial }{\partial z}}=dz}
d
x
α
=
g
α
β
∂
∂
x
β
:
{\displaystyle dx^{\alpha }=g^{\alpha \beta }{\frac {\partial }{\partial x^{\beta }}}:}
d
r
=
∂
∂
r
;
d
ϑ
=
1
r
2
∂
∂
ϑ
;
d
z
=
∂
∂
z
{\displaystyle dr={\frac {\partial }{\partial r}};d\vartheta ={\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }};dz={\frac {\partial }{\partial z}}}
> Gradiente en coordenadas cilíndricas. Cálculo diferencial
editar
El gradiente se calcula aplicando el operador nabla a un campo escalar . Para determinar los factores de escala aplicamos la regla de la cadena .
g
r
a
d
→
(
f
)
=
∇
→
(
f
)
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
f
(
x
,
y
,
z
)
=
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
z
∂
∂
z
)
f
(
x
,
y
,
z
)
=
{\displaystyle {\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})f(x,y,z)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})f(x,y,z)=}
(
e
^
r
∂
r
∂
x
∂
∂
r
+
e
^
ϑ
∂
ϑ
∂
y
∂
∂
ϑ
+
e
^
z
∂
z
∂
z
∂
∂
z
)
f
(
r
,
ϑ
,
z
)
=
1
|
∂
x
∂
r
|
∂
f
∂
r
e
^
r
+
1
|
∂
ϑ
∂
y
|
∂
f
∂
ϑ
e
^
ϑ
+
1
|
∂
z
∂
z
|
∂
f
∂
z
e
^
z
=
{\displaystyle ({\widehat {e}}_{r}{\frac {\partial r}{\partial x}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {\partial \vartheta }{\partial y}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial z}{\partial z}}{\frac {\partial }{\partial z}})f(r,\vartheta ,z)={\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{|{\frac {\partial \vartheta }{\partial y}}|}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{|{\frac {\partial z}{\partial z}}|}}{\frac {\partial f}{\partial z}}{\widehat {e}}_{z}=}
1
h
1
∂
f
∂
r
e
^
r
+
1
h
2
∂
f
∂
ϑ
e
^
ϑ
+
1
h
3
∂
f
∂
z
e
^
z
=
∂
f
∂
r
e
^
r
+
1
r
∂
f
∂
ϑ
e
^
ϑ
+
∂
f
∂
z
e
^
z
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial z}}{\widehat {e}}_{z}={\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {\partial f}{\partial z}}{\widehat {e}}_{z}}
> Gradiente en coordenadas cilíndricas. Geometría diferencial
editar
Para obtener el gradiente aplicamos la diferencial exterior al campo escalar 'f' y posteriormente bajamos índices. Finalmente pasaremos de la base natural en sus coordenadas a la base ortonormal .
g
r
a
d
→
(
f
)
=
∇
→
(
f
)
=↓
d
(
f
(
r
,
θ
,
z
)
)
=↓
(
∂
f
∂
r
d
r
+
∂
f
∂
θ
d
θ
+
∂
f
∂
φ
d
z
)
=
∂
f
∂
r
1
g
r
∂
∂
r
+
∂
f
∂
θ
1
g
θ
∂
∂
θ
+
∂
f
∂
z
1
g
z
∂
∂
z
=
{\displaystyle {\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=\downarrow d(f(r,\theta ,z))=\downarrow ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \theta }}d\theta +{\frac {\partial f}{\partial \varphi }}dz)={\frac {\partial f}{\partial r}}{\frac {1}{g_{r}}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{g_{\theta }}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {1}{g_{z}}}{\frac {\partial }{\partial z}}=}
∂
f
∂
r
∂
∂
r
+
∂
f
∂
θ
1
r
2
∂
∂
θ
+
∂
f
∂
z
∂
∂
z
=
∂
f
∂
r
|
∂
∂
r
|
∂
∂
r
|
∂
∂
r
|
+
∂
f
∂
θ
1
r
2
|
∂
∂
θ
|
∂
∂
θ
|
∂
∂
θ
|
+
∂
f
∂
z
|
∂
∂
z
|
∂
∂
z
|
∂
∂
z
|
=
∂
f
∂
r
∂
^
∂
r
+
∂
f
∂
θ
1
r
∂
^
∂
θ
+
∂
f
∂
z
∂
^
∂
z
{\displaystyle {\frac {\partial f}{\partial r}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {\partial }{\partial z}}={\frac {\partial f}{\partial r}}|{\frac {\partial }{\partial r}}|{\frac {\frac {\partial }{\partial r}}{|{\frac {\partial }{\partial r}}|}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}|{\frac {\partial }{\partial \theta }}|{\frac {\frac {\partial }{\partial \theta }}{|{\frac {\partial }{\partial \theta }}|}}+{\frac {\partial f}{\partial z}}|{\frac {\partial }{\partial z}}|{\frac {\frac {\partial }{\partial z}}{|{\frac {\partial }{\partial z}}|}}={\frac {\partial f}{\partial r}}{\frac {\widehat {\partial }}{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r}}{\frac {\widehat {\partial }}{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {\widehat {\partial }}{\partial z}}}
> Divergencia en coordenadas cilíndricas. Cálculo diferencial
editar
∇
→
⋅
F
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
⋅
F
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
⋅
(
F
x
,
F
y
,
F
z
)
=
{\displaystyle {\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot (F_{x},F_{y},F_{z})=}
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
z
∂
∂
z
)
⋅
(
F
x
e
^
x
+
F
y
e
^
y
+
F
z
e
^
z
)
=
(
e
^
r
h
r
∂
∂
r
+
e
^
ϑ
h
ϑ
∂
∂
ϑ
+
e
^
z
h
z
∂
∂
z
)
⋅
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
z
e
^
z
)
=
{\displaystyle ({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot (F_{x}{\widehat {e}}_{x}+F_{y}{\widehat {e}}_{y}+F_{z}{\widehat {e}}_{z})=({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{h_{z}}}{\frac {\partial }{\partial z}})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=}
(
e
^
r
1
∂
∂
r
+
e
^
ϑ
r
∂
∂
ϑ
+
e
^
z
1
∂
∂
z
)
⋅
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
z
e
^
z
)
=
{\displaystyle ({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{1}}{\frac {\partial }{\partial z}})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=}
e
^
r
⋅
(
∂
F
r
∂
r
e
^
r
)
+
e
^
r
⋅
(
F
r
∂
e
^
r
∂
r
)
+
e
^
r
⋅
(
∂
F
ϑ
∂
r
e
^
ϑ
)
+
e
^
r
⋅
(
F
ϑ
∂
e
^
ϑ
∂
r
)
+
e
^
r
⋅
(
∂
F
z
∂
r
e
^
z
)
+
e
^
r
⋅
(
F
z
∂
e
^
z
∂
r
)
+
{\displaystyle {\widehat {e}}_{r}\cdot ({\frac {\partial F_{r}}{\partial r}}{\widehat {e}}_{r})+{\widehat {e}}_{r}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{r}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{z}}{\partial r}}{\widehat {e}}_{z})+{\widehat {e}}_{r}\cdot (F_{z}{\frac {\partial {\widehat {e}}_{z}}{\partial r}})+}
e
^
ϑ
r
⋅
(
∂
F
r
∂
ϑ
e
^
r
)
+
e
^
ϑ
r
⋅
(
F
r
∂
e
^
r
∂
ϑ
)
+
e
^
ϑ
r
⋅
(
∂
F
ϑ
∂
ϑ
e
^
ϑ
)
+
e
^
ϑ
r
⋅
(
F
ϑ
∂
e
^
ϑ
∂
ϑ
)
+
e
^
ϑ
r
⋅
(
∂
F
z
∂
ϑ
e
^
z
)
+
e
^
ϑ
r
⋅
(
F
z
∂
e
^
z
∂
ϑ
)
+
{\displaystyle {\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\vartheta }}{\partial \vartheta }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{z}}{\partial \vartheta }}{\widehat {e}}_{z})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{z}{\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }})+}
e
^
z
⋅
(
∂
F
r
∂
z
e
^
r
)
+
e
^
z
⋅
(
F
r
∂
e
^
r
∂
z
)
+
e
^
z
⋅
(
∂
F
ϑ
∂
z
e
^
ϑ
)
+
e
^
z
⋅
(
F
ϑ
∂
e
^
ϑ
∂
z
)
+
e
^
z
⋅
(
∂
F
z
∂
z
e
^
z
)
+
e
^
z
⋅
(
F
z
∂
e
^
z
∂
z
)
=
{\displaystyle {\widehat {e}}_{z}\cdot ({\frac {\partial F_{r}}{\partial z}}{\widehat {e}}_{r})+{\widehat {e}}_{z}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial z}})+{\widehat {e}}_{z}\cdot ({\frac {\partial F_{\vartheta }}{\partial z}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{z}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}})+{\widehat {e}}_{z}\cdot ({\frac {\partial F_{z}}{\partial z}}{\widehat {e}}_{z})+{\widehat {e}}_{z}\cdot (F_{z}{\frac {\partial {\widehat {e}}_{z}}{\partial z}})=}
∂
F
r
∂
r
(
e
^
r
⋅
e
^
r
)
+
F
r
r
(
e
^
ϑ
⋅
e
^
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
⋅
e
^
ϑ
)
+
F
ϑ
r
(
e
^
ϑ
⋅
(
−
e
^
r
)
)
+
∂
F
z
∂
z
(
e
^
z
⋅
e
^
z
)
=
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{r})+{\frac {F_{r}}{r}}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {F_{\vartheta }}{r}}({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r}))+{\frac {\partial F_{z}}{\partial z}}({\widehat {e}}_{z}\cdot {\widehat {e}}_{z})=}
∂
F
r
∂
r
+
F
r
r
+
1
r
∂
F
ϑ
∂
ϑ
+
∂
F
z
∂
z
{\displaystyle {\frac {\partial F_{r}}{\partial r}}+{\frac {F_{r}}{r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}}
> Divergencia en coordenadas cilíndricas. Geometría diferencial
editar
Para desarrollar la divergencia con geometría diferencial bajaremos índices, aplicaremos la estrella de Hodge , posteriormente la diferencial exterior ; y finalmente, otra vez, la estrella de Hodge .
d
i
v
(
F
→
)
=
∇
→
⋅
F
→
=
∗
d
∗
↓
(
F
→
)
=
∗
d
∗
↓
(
F
x
∂
^
∂
x
+
F
y
∂
^
∂
y
+
F
z
∂
^
∂
z
)
=
∗
d
∗
↓
(
F
r
(
1
|
∂
x
∂
r
|
∂
∂
r
)
+
F
ϑ
(
1
|
∂
y
∂
ϑ
|
∂
∂
ϑ
)
+
F
z
(
1
|
∂
z
∂
z
|
∂
∂
z
)
)
=
{\displaystyle div({\overrightarrow {F}})={\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=\ast d\ast \downarrow ({\overrightarrow {F}})=\ast d\ast \downarrow (F_{x}{\frac {\widehat {\partial }}{\partial x}}+F_{y}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\ast d\ast \downarrow (F_{r}({\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial }{\partial r}})+F_{\vartheta }({\frac {1}{|{\frac {\partial y}{\partial \vartheta }}|}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{|{\frac {\partial z}{\partial z}}|}}{\frac {\partial }{\partial z}}))=}
∗
d
∗
↓
(
F
r
(
1
g
r
∂
∂
r
)
+
F
ϑ
(
1
g
ϑ
∂
∂
ϑ
)
+
F
z
(
1
g
z
∂
∂
z
)
)
=
∗
d
↓
(
F
r
∂
∂
r
+
F
ϑ
1
r
∂
∂
ϑ
+
F
z
∂
∂
z
)
=
{\displaystyle \ast d\ast \downarrow (F_{r}({\frac {1}{\sqrt[{}]{g_{r}}}}{\frac {\partial }{\partial r}})+F_{\vartheta }({\frac {1}{\sqrt[{}]{g_{\vartheta }}}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{\sqrt[{}]{g_{z}}}}{\frac {\partial }{\partial z}}))=\ast d\downarrow (F_{r}{\frac {\partial }{\partial r}}+F_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+F_{z}{\frac {\partial }{\partial z}})=}
∗
d
∗
(
F
r
d
r
+
F
ϑ
r
(
r
2
d
ϑ
)
+
F
z
d
z
)
=
∗
d
∗
(
F
r
d
r
+
F
ϑ
r
d
ϑ
+
F
z
d
z
)
=
∗
d
(
r
2
1
F
r
(
d
ϑ
∧
d
z
)
+
r
2
r
2
r
F
ϑ
(
d
z
∧
d
r
)
+
r
2
1
F
z
(
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast d\ast (F_{r}dr+{\frac {F_{\vartheta }}{r}}(r^{2}d\vartheta )+F_{z}dz)=\ast d\ast (F_{r}dr+F_{\vartheta }rd\vartheta +F_{z}dz)=\ast d({\frac {\sqrt[{}]{r^{2}}}{1}}F_{r}(d\vartheta \wedge dz)+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}rF_{\vartheta }(dz\wedge dr)+{\frac {\sqrt[{}]{r^{2}}}{1}}F_{z}(dr\wedge d\vartheta ))=}
∗
d
(
r
F
r
(
d
ϑ
∧
d
z
)
+
F
ϑ
(
d
z
∧
d
r
)
+
r
F
z
(
d
r
∧
d
ϑ
)
)
=
∗
(
∂
(
F
r
r
)
∂
r
(
d
r
∧
d
ϑ
∧
d
z
)
+
∂
F
ϑ
∂
ϑ
(
d
ϑ
∧
d
z
∧
d
r
)
+
r
∂
F
z
∂
z
(
d
z
∧
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast d(rF_{r}(d\vartheta \wedge dz)+F_{\vartheta }(dz\wedge dr)+rF_{z}(dr\wedge d\vartheta ))=\ast ({\frac {\partial (F_{r}r)}{\partial r}}(dr\wedge d\vartheta \wedge dz)+{\frac {\partial F_{\vartheta }}{\partial \vartheta }}(d\vartheta \wedge dz\wedge dr)+r{\frac {\partial F_{z}}{\partial z}}(dz\wedge dr\wedge d\vartheta ))=}
r
2
r
2
∂
(
F
r
r
)
∂
r
+
r
2
r
2
∂
F
ϑ
∂
ϑ
+
r
2
r
2
r
∂
F
z
∂
z
=
1
r
∂
(
F
r
r
)
∂
r
+
1
r
∂
F
ϑ
∂
ϑ
+
∂
F
z
∂
z
=
∂
F
r
∂
r
+
F
r
r
+
1
r
∂
F
ϑ
∂
ϑ
+
∂
F
z
∂
z
{\displaystyle {\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{r}r)}{\partial r}}+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}r{\frac {\partial F_{z}}{\partial z}}={\frac {1}{r}}{\frac {\partial (F_{r}r)}{\partial r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}={\frac {\partial F_{r}}{\partial r}}+{\frac {F_{r}}{r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}}
> Rotacional en coordenadas cilíndricas. Cálculo diferencial
editar
∇
→
×
F
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
×
(
F
x
,
F
y
,
F
z
)
=
(
1
h
r
∂
∂
r
,
1
h
ϑ
∂
∂
ϑ
,
1
h
z
∂
∂
z
)
×
(
F
r
,
F
ϑ
,
F
z
)
=
{\displaystyle {\overrightarrow {\nabla }}\times {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\times (F_{x},F_{y},F_{z})=({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}},{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }},{\frac {1}{h_{z}}}{\frac {\partial }{\partial z}})\times (F_{r},F_{\vartheta },F_{z})=}
(
e
^
r
h
r
∂
∂
r
+
e
^
ϑ
h
ϑ
∂
∂
ϑ
+
e
^
z
h
z
∂
∂
z
)
×
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
z
e
^
z
)
=
(
e
^
r
1
∂
∂
r
+
e
^
ϑ
r
∂
∂
ϑ
+
e
^
z
1
∂
∂
z
)
×
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
z
e
^
z
)
=
{\displaystyle ({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{h_{z}}}{\frac {\partial }{\partial z}})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{1}}{\frac {\partial }{\partial z}})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=}
(
e
^
r
∂
∂
r
)
×
(
F
r
e
^
r
)
+
(
e
^
r
∂
∂
r
)
×
(
F
ϑ
e
^
ϑ
)
+
(
e
^
r
∂
∂
r
)
×
(
F
z
e
^
z
)
+
(
e
^
ϑ
r
∂
∂
ϑ
)
×
(
F
r
e
^
r
)
+
(
e
^
ϑ
r
∂
∂
ϑ
)
×
(
F
ϑ
e
^
ϑ
)
+
(
e
^
ϑ
r
∂
∂
ϑ
)
×
(
F
z
e
^
z
)
+
{\displaystyle ({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{r}{\widehat {e}}_{r})+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{z}{\widehat {e}}_{z})+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{r}{\widehat {e}}_{r})+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{z}{\widehat {e}}_{z})+}
(
e
^
z
∂
∂
z
)
×
(
F
r
e
^
r
)
+
(
e
^
z
∂
∂
z
)
×
(
F
ϑ
e
^
ϑ
)
+
(
e
^
z
∂
∂
z
)
×
(
F
z
e
^
z
)
=
{\displaystyle ({\widehat {e}}_{z}{\frac {\partial }{\partial z}})\times (F_{r}{\widehat {e}}_{r})+({\widehat {e}}_{z}{\frac {\partial }{\partial z}})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\widehat {e}}_{z}{\frac {\partial }{\partial z}})\times (F_{z}{\widehat {e}}_{z})=}
∂
F
r
∂
r
(
e
^
r
×
e
^
r
)
+
F
r
(
e
^
r
×
∂
e
^
r
∂
r
)
+
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
F
ϑ
(
e
^
r
×
∂
e
^
ϑ
∂
r
)
+
∂
F
z
∂
r
(
e
^
r
×
e
^
z
)
+
F
z
(
e
^
r
×
∂
e
^
z
∂
r
)
+
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+F_{\vartheta }({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\frac {\partial F_{z}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{z})+F_{z}({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{z}}{\partial r}})+}
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
×
∂
e
^
r
∂
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
×
e
^
ϑ
)
+
1
r
F
ϑ
(
e
^
ϑ
×
∂
e
^
ϑ
∂
ϑ
)
+
1
r
∂
F
z
∂
ϑ
(
e
^
ϑ
×
e
^
z
)
+
1
r
F
z
(
e
^
ϑ
×
∂
e
^
z
∂
ϑ
)
+
{\displaystyle {\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z})+{\frac {1}{r}}F_{z}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }})+}
∂
F
r
∂
z
(
e
^
z
×
e
^
r
)
+
F
r
(
e
^
z
×
∂
e
^
r
∂
z
)
+
∂
F
ϑ
∂
z
(
e
^
z
×
e
^
ϑ
)
+
F
ϑ
(
e
^
z
×
∂
e
^
ϑ
∂
z
)
+
∂
F
z
∂
z
(
e
^
z
×
e
^
z
)
+
F
z
(
e
^
z
×
∂
e
^
z
∂
z
)
=
{\displaystyle {\frac {\partial F_{r}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{z}\times {\frac {\partial {\widehat {e}}_{r}}{\partial z}})+{\frac {\partial F_{\vartheta }}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta })+F_{\vartheta }({\widehat {e}}_{z}\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}})+{\frac {\partial F_{z}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{z})+F_{z}({\widehat {e}}_{z}\times {\frac {\partial {\widehat {e}}_{z}}{\partial z}})=}
∂
F
r
∂
r
(
e
^
r
×
e
^
r
)
+
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
∂
F
z
∂
r
(
e
^
r
×
e
^
z
)
+
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
×
∂
e
^
r
∂
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
×
e
^
ϑ
)
+
1
r
F
ϑ
(
e
^
ϑ
×
∂
e
^
ϑ
∂
ϑ
)
+
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{z}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{z})+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+}
1
r
∂
F
z
∂
ϑ
(
e
^
ϑ
×
e
^
z
)
+
∂
F
r
∂
z
(
e
^
z
×
e
^
r
)
+
∂
F
ϑ
∂
z
(
e
^
z
×
e
^
ϑ
)
+
∂
F
z
∂
z
(
e
^
z
×
e
^
z
)
=
{\displaystyle {\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z})+{\frac {\partial F_{r}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{z}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{z})=}
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
∂
F
z
∂
r
(
e
^
r
×
e
^
z
)
+
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
×
e
^
ϑ
)
+
1
r
F
ϑ
(
e
^
ϑ
×
(
−
e
^
r
)
)
+
{\displaystyle {\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{z}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{z})+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+}
1
r
∂
F
z
∂
ϑ
(
e
^
ϑ
×
e
^
z
)
+
∂
F
r
∂
z
(
e
^
z
×
e
^
r
)
+
∂
F
ϑ
∂
z
(
e
^
z
×
e
^
ϑ
)
=
(
1
r
∂
F
z
∂
ϑ
−
∂
F
ϑ
∂
z
)
e
^
r
+
(
∂
F
r
∂
z
−
∂
F
z
∂
r
)
e
^
ϑ
+
(
∂
F
ϑ
∂
r
+
F
ϑ
r
−
1
r
∂
F
r
∂
ϑ
)
e
^
z
{\displaystyle {\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z})+{\frac {\partial F_{r}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta })=({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial z}}){\widehat {e}}_{r}+({\frac {\partial F_{r}}{\partial z}}-{\frac {\partial F_{z}}{\partial r}}){\widehat {e}}_{\vartheta }+({\frac {\partial F_{\vartheta }}{\partial r}}+{\frac {F_{\vartheta }}{r}}-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}){\widehat {e}}_{z}}
> Rotacional en coordenadas cilíndricas. Geometría diferencial
editar
Para desarrollar el rotacional con geometría diferencial bajaremos índices, aplicaremos la diferencial exterior ; y finalmente, aplicaremos la estrella de Hodge y subiremos índices.
r
o
t
(
F
→
)
=
∇
→
×
F
→
=↑
∗
d
↓
(
F
→
)
=↑
∗
d
↓
(
F
x
∂
^
∂
x
+
F
y
∂
^
∂
y
+
F
z
∂
^
∂
z
)
=↑
∗
d
↓
(
F
r
(
1
h
r
∂
∂
r
)
+
F
ϑ
(
1
h
ϑ
∂
∂
ϑ
)
+
F
z
(
1
h
z
∂
∂
z
)
)
=
{\displaystyle rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\widehat {\partial }}{\partial x}}+{F_{y}}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\uparrow \ast d\downarrow ({F_{r}}({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}})+{F_{\vartheta }}({\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{h_{z}}}{\frac {\partial }{\partial z}}))=}
↑
∗
d
↓
(
F
r
∂
∂
r
+
F
ϑ
(
1
r
∂
∂
ϑ
)
+
F
z
∂
∂
r
)
=↑
∗
d
(
F
r
d
r
+
F
ϑ
r
r
2
d
ϑ
+
F
z
d
z
)
=↑
∗
d
(
F
r
d
r
+
F
ϑ
r
d
ϑ
+
F
z
d
z
)
=
{\displaystyle \uparrow \ast d\downarrow ({F_{r}}{\frac {\partial }{\partial r}}+{F_{\vartheta }}({\frac {1}{r}}{\frac {\partial }{\partial \vartheta }})+F_{z}{\frac {\partial }{\partial r}})=\uparrow \ast d({F_{r}}dr+{\frac {F_{\vartheta }}{r}}r^{2}d\vartheta +F_{z}dz)=\uparrow \ast d({F_{r}}dr+F_{\vartheta }rd\vartheta +F_{z}dz)=}
↑
∗
(
∂
F
r
∂
ϑ
(
d
ϑ
∧
d
r
)
+
∂
F
r
∂
z
(
d
z
∧
d
r
)
+
∂
(
F
ϑ
r
)
∂
r
(
d
r
∧
d
ϑ
)
+
∂
(
F
ϑ
r
)
∂
z
(
d
z
∧
d
ϑ
)
+
∂
F
z
∂
r
(
d
r
∧
d
z
)
+
∂
F
z
∂
ϑ
(
d
ϑ
∧
d
z
)
)
=
{\displaystyle \uparrow \ast ({\frac {\partial F_{r}}{\partial \vartheta }}(d\vartheta \wedge dr)+{\frac {\partial F_{r}}{\partial z}}(dz\wedge dr)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )+{\frac {\partial (F_{\vartheta }r)}{\partial z}}(dz\wedge d\vartheta )+{\frac {\partial F_{z}}{\partial r}}(dr\wedge dz)+{\frac {\partial F_{z}}{\partial \vartheta }}(d\vartheta \wedge dz))=}
↑
∗
(
−
∂
F
r
∂
ϑ
(
d
r
∧
d
ϑ
)
−
∂
F
r
∂
z
(
d
r
∧
d
z
)
+
∂
(
F
ϑ
r
)
∂
r
(
d
r
∧
d
ϑ
)
−
∂
(
F
ϑ
r
)
∂
z
(
d
ϑ
∧
d
z
)
+
∂
F
z
∂
r
(
d
r
∧
d
z
)
+
∂
F
z
∂
ϑ
(
d
ϑ
∧
d
z
)
)
=
{\displaystyle \uparrow \ast (-{\frac {\partial F_{r}}{\partial \vartheta }}(dr\wedge d\vartheta )-{\frac {\partial F_{r}}{\partial z}}(dr\wedge dz)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )-{\frac {\partial (F_{\vartheta }r)}{\partial z}}(d\vartheta \wedge dz)+{\frac {\partial F_{z}}{\partial r}}(dr\wedge dz)+{\frac {\partial F_{z}}{\partial \vartheta }}(d\vartheta \wedge dz))=}
↑
(
−
r
2
r
2
∂
F
r
∂
ϑ
d
z
−
r
2
∂
F
r
∂
z
d
ϑ
+
r
2
r
2
∂
(
F
ϑ
r
)
∂
r
d
z
−
r
2
r
2
∂
(
F
ϑ
r
)
∂
z
d
r
+
r
2
∂
F
z
∂
r
d
ϑ
+
r
2
r
2
∂
F
z
∂
ϑ
d
r
)
=
{\displaystyle \uparrow (-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{r}}{\partial \vartheta }}dz-{\sqrt[{}]{r^{2}}}{\frac {\partial F_{r}}{\partial z}}d\vartheta +{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}dz-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}dr+{\sqrt[{}]{r^{2}}}{\frac {\partial F_{z}}{\partial r}}d\vartheta +{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{z}}{\partial \vartheta }}dr)=}
↑
(
1
r
∂
F
z
∂
ϑ
d
r
−
1
r
∂
(
F
ϑ
r
)
∂
z
d
r
+
r
∂
F
z
∂
r
d
ϑ
−
r
∂
F
r
∂
z
d
ϑ
+
1
r
∂
(
F
ϑ
r
)
∂
r
d
z
−
1
r
∂
F
r
∂
ϑ
d
z
)
=
{\displaystyle \uparrow ({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}dr-{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}dr+r{\frac {\partial F_{z}}{\partial r}}d\vartheta -r{\frac {\partial F_{r}}{\partial z}}d\vartheta +{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}dz-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}dz)=}
1
r
∂
F
z
∂
ϑ
∂
∂
r
−
1
r
∂
(
F
ϑ
r
)
∂
z
∂
∂
r
+
r
∂
F
z
∂
r
1
r
2
∂
∂
ϑ
−
r
∂
F
r
∂
z
1
r
2
∂
∂
ϑ
+
1
r
∂
(
F
ϑ
r
)
∂
r
∂
∂
z
−
1
r
∂
F
r
∂
ϑ
∂
∂
z
=
{\displaystyle {\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}{\frac {\partial }{\partial r}}-{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}{\frac {\partial }{\partial r}}+r{\frac {\partial F_{z}}{\partial r}}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}-r{\frac {\partial F_{r}}{\partial z}}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}+{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}{\frac {\partial }{\partial z}}-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}{\frac {\partial }{\partial z}}=}
(
1
r
∂
F
z
∂
ϑ
−
∂
(
F
ϑ
)
∂
z
)
∂
∂
r
+
(
∂
F
z
∂
r
−
∂
F
r
∂
z
)
1
r
|
∂
∂
ϑ
|
∂
∂
ϑ
|
∂
∂
ϑ
|
+
1
r
(
∂
(
F
ϑ
r
)
∂
r
−
∂
F
r
∂
ϑ
)
∂
∂
z
=
{\displaystyle ({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}-{\frac {\partial (F_{\vartheta })}{\partial z}}){\frac {\partial }{\partial r}}+({\frac {\partial F_{z}}{\partial r}}-{\frac {\partial F_{r}}{\partial z}}){\frac {1}{r}}|{\frac {\partial }{\partial \vartheta }}|{\frac {\frac {\partial }{\partial \vartheta }}{|{\frac {\partial }{\partial \vartheta }}|}}+{\frac {1}{r}}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\partial }{\partial z}}=}
(
1
r
∂
F
z
∂
ϑ
−
∂
(
F
ϑ
)
∂
z
)
∂
^
∂
r
+
(
∂
F
z
∂
r
−
∂
F
r
∂
z
)
∂
^
∂
ϑ
+
1
r
(
∂
(
F
ϑ
r
)
∂
r
−
∂
F
r
∂
ϑ
)
∂
^
∂
z
{\displaystyle ({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}-{\frac {\partial (F_{\vartheta })}{\partial z}}){\frac {\widehat {\partial }}{\partial r}}+({\frac {\partial F_{z}}{\partial r}}-{\frac {\partial F_{r}}{\partial z}}){\frac {\widehat {\partial }}{\partial \vartheta }}+{\frac {1}{r}}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\widehat {\partial }}{\partial z}}}
> Laplaciano en coordenadas cilíndricas. Cálculo diferencial
editar
d
i
v
(
g
r
a
d
→
(
f
)
)
=
∇
2
(
f
)
=
∇
→
⋅
(
∇
→
f
)
=
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
z
∂
∂
z
)
⋅
(
e
^
x
∂
f
∂
x
+
e
^
y
∂
f
∂
y
+
e
^
z
∂
f
∂
z
)
=
{\displaystyle div({\overrightarrow {grad}}(f))=\nabla ^{2}(f)={\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}f)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{x}{\frac {\partial f}{\partial x}}+{\widehat {e}}_{y}{\frac {\partial f}{\partial y}}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=}
(
e
^
r
1
h
r
∂
∂
r
+
e
^
ϑ
1
h
ϑ
∂
∂
ϑ
+
e
^
z
1
h
z
∂
∂
z
)
⋅
(
e
^
r
∂
f
∂
r
+
e
^
ϑ
1
r
∂
f
∂
ϑ
+
e
^
z
∂
f
∂
z
)
=
{\displaystyle ({\widehat {e}}_{r}{\frac {1}{h_{r}}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {1}{h_{z}}}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=}
(
e
^
r
∂
∂
r
+
e
^
ϑ
1
r
∂
∂
ϑ
+
e
^
z
∂
∂
z
)
⋅
(
e
^
r
∂
f
∂
r
+
e
^
ϑ
1
r
∂
f
∂
ϑ
+
e
^
z
∂
f
∂
z
)
=
{\displaystyle ({\widehat {e}}_{r}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=}
(
e
^
r
⋅
∂
e
^
r
∂
r
)
∂
f
∂
r
+
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
r
⋅
∂
e
^
ϑ
∂
r
)
1
r
∂
f
∂
ϑ
+
(
e
^
r
⋅
e
^
ϑ
)
∂
∂
r
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
r
⋅
∂
e
^
z
∂
r
)
∂
f
∂
z
+
(
e
^
r
⋅
e
^
z
)
∂
∂
r
(
∂
f
∂
z
)
+
{\displaystyle ({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial r}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial r}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{z}){\frac {\partial }{\partial r}}({\frac {\partial f}{\partial z}})+}
(
e
^
ϑ
⋅
∂
e
^
r
∂
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
r
)
1
r
∂
∂
ϑ
(
∂
f
∂
r
)
+
(
e
^
ϑ
⋅
∂
e
^
ϑ
∂
ϑ
)
1
r
2
∂
f
∂
ϑ
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
(
e
^
ϑ
⋅
∂
e
^
z
∂
ϑ
)
1
r
∂
f
∂
z
+
(
e
^
ϑ
⋅
e
^
z
)
1
r
∂
∂
ϑ
(
∂
f
∂
z
)
+
{\displaystyle ({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial z}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial z}})+}
(
e
^
z
⋅
∂
e
^
r
∂
z
)
∂
f
∂
r
+
(
e
^
z
⋅
e
^
r
)
∂
∂
z
(
∂
f
∂
r
)
+
(
e
^
z
⋅
∂
e
^
ϑ
∂
z
)
1
r
∂
f
∂
ϑ
+
(
e
^
z
⋅
e
^
ϑ
)
∂
∂
z
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
z
⋅
∂
e
^
z
∂
z
)
∂
f
∂
z
+
(
e
^
z
⋅
e
^
z
)
∂
2
f
∂
2
z
=
{\displaystyle ({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial z}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{r}){\frac {\partial }{\partial z}}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial z}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial z}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}=}
(
e
^
r
⋅
∂
e
^
r
∂
r
)
∂
f
∂
r
+
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
r
⋅
∂
e
^
ϑ
∂
r
)
1
r
∂
f
∂
ϑ
+
(
e
^
r
⋅
e
^
ϑ
)
∂
∂
r
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
r
⋅
∂
e
^
z
∂
r
)
∂
f
∂
z
+
(
e
^
r
⋅
e
^
z
)
∂
∂
r
(
∂
f
∂
z
)
+
{\displaystyle ({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial r}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial r}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{z}){\frac {\partial }{\partial r}}({\frac {\partial f}{\partial z}})+}
(
e
^
ϑ
⋅
e
^
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
r
)
1
r
∂
∂
ϑ
(
∂
f
∂
r
)
+
(
e
^
ϑ
⋅
(
−
e
^
r
)
)
1
r
2
∂
f
∂
ϑ
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
(
e
^
ϑ
⋅
∂
e
^
z
∂
ϑ
)
1
r
∂
f
∂
z
+
(
e
^
ϑ
⋅
e
^
z
)
1
r
∂
∂
ϑ
(
∂
f
∂
z
)
+
{\displaystyle ({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r})){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial z}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial z}})+}
(
e
^
z
⋅
∂
e
^
r
∂
z
)
∂
f
∂
r
+
(
e
^
z
⋅
e
^
r
)
∂
∂
z
(
∂
f
∂
r
)
+
(
e
^
z
⋅
∂
e
^
ϑ
∂
z
)
1
r
∂
f
∂
ϑ
+
(
e
^
z
⋅
e
^
ϑ
)
∂
∂
z
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
z
⋅
∂
e
^
z
∂
z
)
∂
f
∂
z
+
(
e
^
z
⋅
e
^
z
)
∂
2
f
∂
2
z
=
{\displaystyle ({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial z}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{r}){\frac {\partial }{\partial z}}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial z}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial z}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}=}
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
r
⋅
e
^
ϑ
)
∂
∂
r
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
r
⋅
e
^
z
)
∂
∂
r
(
∂
f
∂
z
)
+
(
e
^
ϑ
⋅
e
^
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
r
)
1
r
∂
∂
ϑ
(
∂
f
∂
r
)
+
(
e
^
ϑ
⋅
(
−
e
^
r
)
)
1
r
∂
f
∂
ϑ
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
{\displaystyle ({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\widehat {e}}_{z}){\frac {\partial }{\partial r}}({\frac {\partial f}{\partial z}})+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r})){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+}
(
e
^
ϑ
⋅
e
^
z
)
1
r
∂
∂
ϑ
(
∂
f
∂
z
)
+
(
e
^
z
⋅
e
^
r
)
∂
∂
z
(
∂
f
∂
r
)
+
(
e
^
z
⋅
e
^
ϑ
)
∂
∂
z
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
z
⋅
e
^
z
)
∂
2
f
∂
2
z
=
{\displaystyle ({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial z}})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{r}){\frac {\partial }{\partial z}}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial z}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}=}
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
ϑ
⋅
e
^
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
(
e
^
z
⋅
e
^
z
)
∂
2
f
∂
2
z
=
∂
2
f
∂
2
r
+
1
r
∂
f
∂
r
+
1
r
2
∂
2
f
∂
2
ϑ
+
∂
2
f
∂
2
z
{\displaystyle ({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}={\frac {\partial ^{2}f}{\partial ^{2}r}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }}+{\frac {\partial ^{2}f}{\partial ^{2}z}}}
> Laplaciano en coordenadas cilíndricas. Geometría diferencial
editar
Para el cálculo del laplaciano en geometría diferencial , aplicaremos la diferencial exterior sobre el campo escalar 'f', la estrella de Hodge , otra vez la diferencial exterior y finalmente, otra vez, la estrella de Hodge .
d
i
v
(
g
r
a
d
→
(
f
)
)
=
∇
2
f
=
∗
d
∗
d
(
f
)
=
∗
d
∗
(
∂
f
∂
r
d
r
+
∂
f
∂
ϑ
d
ϑ
+
∂
f
∂
z
d
z
)
=
{\displaystyle div({\overrightarrow {grad}}(f))=\nabla ^{2}f=\ast d\ast d(f)=\ast d\ast ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \vartheta }}d\vartheta +{\frac {\partial f}{\partial z}}dz)=}
∗
d
(
|
r
2
|
1
∂
f
∂
r
(
d
ϑ
∧
d
z
)
+
|
r
2
|
r
2
∂
f
∂
ϑ
(
d
z
∧
d
r
)
+
|
r
2
|
1
∂
f
∂
z
(
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast d({\frac {\sqrt[{}]{|r^{2}|}}{1}}{\frac {\partial f}{\partial r}}(d\vartheta \wedge dz)+{\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}(dz\wedge dr)+{\frac {\sqrt[{}]{|r^{2}|}}{1}}{\frac {\partial f}{\partial z}}(dr\wedge d\vartheta ))=}
∗
(
∂
∂
r
(
r
2
1
2
∂
f
∂
r
)
(
d
r
∧
d
ϑ
∧
d
z
)
+
∂
∂
ϑ
(
1
r
2
2
∂
f
∂
ϑ
)
(
d
ϑ
∧
d
z
∧
d
r
)
+
∂
∂
z
(
r
2
1
2
∂
f
∂
z
)
(
d
z
∧
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast ({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})(dr\wedge d\vartheta \wedge dz)+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})(d\vartheta \wedge dz\wedge dr)+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})(dz\wedge dr\wedge d\vartheta ))=}
∗
(
(
∂
∂
r
(
r
2
1
2
∂
f
∂
r
)
+
∂
∂
ϑ
(
1
r
2
2
∂
f
∂
ϑ
)
+
∂
∂
z
(
r
2
1
2
∂
f
∂
z
)
(
d
r
∧
d
ϑ
∧
d
z
)
)
=
{\displaystyle \ast (({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})(dr\wedge d\vartheta \wedge dz))=}
(
|
r
2
|
r
2
)
∂
∂
r
(
r
2
1
2
∂
f
∂
r
)
+
(
|
r
2
|
r
2
)
∂
∂
ϑ
(
1
r
2
2
∂
f
∂
ϑ
)
+
(
|
r
2
|
r
2
)
∂
∂
z
(
r
2
1
2
∂
f
∂
z
)
=
{\displaystyle ({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})=}
(
|
r
2
|
r
2
)
(
∂
∂
r
(
r
2
1
2
∂
f
∂
r
)
+
∂
∂
ϑ
(
1
r
2
2
∂
f
∂
ϑ
)
+
∂
∂
z
(
r
2
1
2
∂
f
∂
z
)
)
=
{\displaystyle ({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}}))=}
(
1
r
2
)
(
∂
∂
r
(
r
2
1
2
∂
f
∂
r
)
+
∂
∂
ϑ
(
1
r
2
2
∂
f
∂
ϑ
)
+
∂
∂
z
(
r
2
1
2
∂
f
∂
z
)
)
=
1
r
(
∂
∂
r
(
r
∂
f
∂
r
)
+
∂
∂
ϑ
(
1
r
∂
f
∂
ϑ
)
+
∂
∂
z
(
r
∂
f
∂
z
)
)
{\displaystyle ({\frac {1}{\sqrt[{}]{r^{2}}}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}}))={\frac {1}{r}}({\frac {\partial }{\partial r}}(r{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}(r{\frac {\partial f}{\partial z}}))}
>> Operadores diferenciales en coordenadas esféricas
editar
* Coordenadas esféricas
editar
Cambio de coordenadas cartesianas a coordenadas esféricas .
x
=
r
s
i
n
ϑ
c
o
s
φ
{\displaystyle x=rsin\vartheta cos\varphi }
y
=
r
s
i
n
ϑ
s
i
n
φ
{\displaystyle y=rsin\vartheta sin\varphi }
z
=
r
c
o
s
ϑ
{\displaystyle z=rcos\vartheta }
* Cambio de base
editar
Cambio de la base ortonormal de las coordenadas cartesianas a la base en coordenadas esféricas .
e
^
r
=
s
i
n
ϑ
c
o
s
φ
e
^
x
+
s
i
n
ϑ
s
i
n
φ
e
^
y
+
c
o
s
ϑ
e
^
z
{\displaystyle {\widehat {e}}_{r}=sin\vartheta cos\varphi {\widehat {e}}_{x}+sin\vartheta sin\varphi {\widehat {e}}_{y}+cos\vartheta {\widehat {e}}_{z}}
e
^
ϑ
=
c
o
s
ϑ
c
o
s
φ
e
^
x
+
c
o
s
ϑ
s
i
n
φ
e
^
y
−
s
i
n
φ
e
^
z
{\displaystyle {\widehat {e}}_{\vartheta }=cos\vartheta cos\varphi {\widehat {e}}_{x}+cos\vartheta sin\varphi {\widehat {e}}_{y}-sin\varphi {\widehat {e}}_{z}}
e
^
φ
=
−
s
i
n
φ
e
^
x
+
c
o
s
φ
e
^
y
{\displaystyle {\widehat {e}}_{\varphi }=-sin\varphi {\widehat {e}}_{x}+cos\varphi {\widehat {e}}_{y}}
* Derivadas de las bases en coordenadas esféricas
editar
Para las demostraciones necesitaremos de las derivadas de las bases .
∂
e
^
r
∂
r
=
0
;
∂
e
^
r
∂
ϑ
=
e
^
ϑ
;
∂
e
^
r
∂
φ
=
s
i
n
ϑ
e
^
φ
;
∂
e
^
ϑ
∂
r
=
0
;
∂
e
^
ϑ
∂
ϑ
=
−
e
^
r
;
∂
e
^
ϑ
∂
φ
=
c
o
s
ϑ
e
^
φ
;
∂
e
^
φ
∂
r
=
0
;
∂
e
^
φ
∂
ϑ
=
0
;
∂
e
^
φ
∂
φ
=
−
s
i
n
ϑ
e
^
r
−
c
o
s
ϑ
e
^
ϑ
;
{\displaystyle {\frac {\partial {\widehat {e}}_{r}}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}={\widehat {e}}_{\vartheta };{\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }}=sin\vartheta {\widehat {e}}_{\varphi };{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}=-{\widehat {e}}_{r};{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }}=cos\vartheta {\widehat {e}}_{\varphi };{\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }}=0;{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }}=-sin\vartheta {\widehat {e}}_{r}-cos\vartheta {\widehat {e}}_{\vartheta };}
* Productos escalares en coordenadas esféricas
editar
Para los desarrollos necesitaremos los siguientes productos escalares , de las bases en coordenadas esféricas .
e
^
r
⋅
e
^
r
=
1
;
e
^
ϑ
⋅
e
^
ϑ
=
1
;
e
^
φ
⋅
e
^
φ
=
1
{\displaystyle {\widehat {e}}_{r}\cdot {\widehat {e}}_{r}=1;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }=1;{\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi }=1}
e
^
ϑ
⋅
e
^
r
=
0
;
e
^
r
⋅
e
^
ϑ
=
0
;
e
^
φ
⋅
e
^
r
=
0
;
e
^
r
⋅
e
^
φ
=
0
;
e
^
ϑ
⋅
e
^
φ
=
0
;
e
^
φ
⋅
e
^
ϑ
=
0
{\displaystyle {\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi }=0;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi }=0;{\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }=0}
* Productos vectoriales en coordenadas esféricas
editar
Necesitariemos los productos vectoriales de las bases .
e
^
r
×
e
^
r
=
0
;
e
^
r
×
e
^
ϑ
=
e
^
φ
;
e
^
r
×
e
^
φ
=
−
e
^
ϑ
{\displaystyle {\widehat {e}}_{r}\times {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta }={\widehat {e}}_{\varphi };{\widehat {e}}_{r}\times {\widehat {e}}_{\varphi }=-{\widehat {e}}_{\vartheta }}
e
^
ϑ
×
e
^
r
=
−
e
^
φ
;
e
^
ϑ
×
e
^
ϑ
=
0
;
e
^
ϑ
×
e
^
φ
=
e
^
r
{\displaystyle {\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r}=-{\widehat {e}}_{\varphi };{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi }={\widehat {e}}_{r}}
e
^
φ
×
e
^
r
=
e
^
ϑ
;
e
^
φ
×
e
^
ϑ
=
−
e
^
r
;
e
^
φ
×
e
^
φ
=
0
{\displaystyle {\widehat {e}}_{\varphi }\times {\widehat {e}}_{r}={\widehat {e}}_{\vartheta };{\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta }=-{\widehat {e}}_{r};{\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi }=0}
* Factor de escala en coordenadas esféricas
editar
El factor de escala nos permite pasar de la base en coordenadas cartesianas la base en coordenadas esféricas .
h
→
=
∑
|
∂
r
→
∂
u
i
|
=
(
|
∂
x
∂
r
|
,
|
∂
y
∂
ϑ
|
,
|
∂
z
∂
φ
|
)
=
(
h
r
,
h
ϑ
,
h
z
)
=
(
1
,
r
,
r
s
i
n
ϑ
)
{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial {\overrightarrow {r}}}{\partial u_{i}}}|=(|{\frac {\partial x}{\partial r}}|,|{\frac {\partial y}{\partial \vartheta }}|,|{\frac {\partial z}{\partial \varphi }}|)=(h_{r},h_{\vartheta },h_{z})=(1,r,rsin\vartheta )}
geometría de Riemann .
h
→
=
∑
|
∂
∂
u
i
|
=
(
|
∂
∂
r
|
,
|
∂
∂
ϑ
|
,
|
∂
∂
φ
|
)
=
(
g
r
,
g
ϑ
,
g
φ
)
=
(
1
,
r
,
r
s
i
n
ϑ
)
{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial r}}|,|{\frac {\partial }{\partial \vartheta }}|,|{\frac {\partial }{\partial \varphi }}|)=({\sqrt[{}]{g_{r}}},{\sqrt[{}]{g_{\vartheta }}},{\sqrt[{}]{g_{\varphi }}})=(1,r,rsin\vartheta )}
* Métrica o tensor métrico en coordenadas esféricas
editar
La métrica (tensor métrico ) de una variedad de Riemann nos permite obtener los coeficientes de un elemento de longitud en las bases deseadas.
En este caso tenemos la métrica de las coordenadas esféricas . Puede observarse que está muy relacionado con el factor de escala .
g
α
β
=
g
r
d
r
2
+
g
ϑ
d
ϑ
2
+
g
φ
d
φ
2
=
d
r
2
+
r
2
d
ϑ
2
+
r
2
s
i
n
2
ϑ
d
φ
2
{\displaystyle g_{\alpha \beta }=g_{r}dr^{2}+g_{\vartheta }d\vartheta ^{2}+g_{\varphi }d\varphi ^{2}=dr^{2}+r^{2}d\vartheta ^{2}+r^{2}sin^{2}\vartheta d\varphi ^{2}}
g
α
β
=
[
g
r
g
ϑ
g
φ
]
=
[
|
∂
∂
r
|
⋅
|
∂
∂
r
|
|
∂
∂
ϑ
|
⋅
|
∂
∂
ϑ
|
|
∂
∂
φ
|
⋅
|
∂
∂
φ
|
]
=
[
1
r
2
r
2
s
i
n
2
ϑ
]
{\displaystyle g_{\alpha \beta }={\begin{bmatrix}g_{r}&&\\&g_{\vartheta }&\\&&g_{\varphi }\end{bmatrix}}={\begin{bmatrix}\vert {\frac {\partial }{\partial r}}\vert \cdot |{\frac {\partial }{\partial r}}|&&\\&|{\frac {\partial }{\partial \vartheta }}|\cdot |{\frac {\partial }{\partial \vartheta }}|&\\&&|{\frac {\partial }{\partial \varphi }}|\cdot |{\frac {\partial }{\partial \varphi }}|\end{bmatrix}}={\begin{bmatrix}1&&\\&r^{2}&\\&&r^{2}sin^{2}\vartheta \end{bmatrix}}}
d
e
t
(
g
α
β
)
=
|
g
|
=
r
4
s
i
n
2
ϑ
{\displaystyle det(g_{\alpha \beta })=|g|=r^{4}sin^{2}\vartheta }
* Estrella de Hodge (*) en coordenadas esféricas
editar
La estrella de Hodge es un operador que actúa sobre un p-forma diferencial en una dimensión n.
∗
(
F
(
d
x
1
∧
d
x
2
∧
d
x
3
∧
.
.
.
∧
d
x
p
)
=
|
g
|
g
11
g
22
g
33
.
.
.
g
p
p
ε
F
(
d
x
p
+
1
∧
d
x
p
+
2
∧
d
x
p
+
3
∧
.
.
.
∧
d
x
n
−
p
)
{\displaystyle *(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})}
∗
(
F
d
x
α
)
=
ε
|
g
|
g
α
α
F
(
d
x
β
∧
d
x
γ
)
:
{\displaystyle \ast (Fdx_{\alpha })=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }}}F(dx_{\beta }\wedge dx_{\gamma }):}
∗
(
F
d
r
)
=
r
4
s
i
n
2
ϑ
1
F
(
d
ϑ
∧
d
φ
)
=
r
2
s
i
n
ϑ
F
(
d
ϑ
∧
d
φ
)
;
{\displaystyle \ast (Fdr)={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{1}}F(d\vartheta \wedge d\varphi )=r^{2}sin\vartheta F(d\vartheta \wedge d\varphi );}
∗
(
F
d
ϑ
)
=
r
4
s
i
n
2
ϑ
r
2
F
(
d
r
∧
d
φ
)
=
s
i
n
ϑ
F
(
d
r
∧
d
φ
)
;
{\displaystyle \ast (Fd\vartheta )={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}F(dr\wedge d\varphi )=sin\vartheta F(dr\wedge d\varphi );}
∗
(
F
d
φ
)
=
r
4
s
i
n
2
ϑ
r
2
s
i
n
2
ϑ
F
(
d
ϑ
∧
d
r
)
=
1
s
i
n
ϑ
F
(
d
ϑ
∧
d
r
)
{\displaystyle \ast (Fd\varphi )={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}F(d\vartheta \wedge dr)={\frac {1}{sin\vartheta }}F(d\vartheta \wedge dr)}
∗
(
F
(
d
x
α
∧
d
x
β
)
)
=
ε
|
g
|
g
α
α
g
β
β
F
d
x
γ
:
{\displaystyle \ast (F(dx_{\alpha }\wedge dx_{\beta }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }}}Fdx_{\gamma }:}
∗
(
F
(
d
r
∧
d
ϑ
)
)
=
r
4
s
i
n
2
ϑ
r
2
F
d
φ
=
s
i
n
ϑ
F
d
φ
;
{\displaystyle \ast (F(dr\wedge d\vartheta ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}Fd\varphi =sin\vartheta Fd\varphi ;}
∗
(
F
(
d
r
∧
d
φ
)
)
=
r
4
s
i
n
2
ϑ
r
2
s
i
n
2
ϑ
F
d
ϑ
=
1
s
i
n
ϑ
F
d
ϑ
;
{\displaystyle \ast (F(dr\wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}Fd\vartheta ={\frac {1}{sin\vartheta }}Fd\vartheta ;}
∗
(
F
(
d
ϑ
∧
d
r
)
)
=
−
r
4
s
i
n
2
ϑ
r
2
F
d
φ
=
−
s
i
n
ϑ
F
d
φ
;
{\displaystyle \ast (F(d\vartheta \wedge dr))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}Fd\varphi =-sin\vartheta Fd\varphi ;}
∗
(
F
(
d
ϑ
∧
d
φ
)
)
=
r
4
s
i
n
2
ϑ
r
2
r
2
s
i
n
2
ϑ
F
d
r
=
1
r
2
s
i
n
ϑ
F
d
r
;
{\displaystyle \ast (F(d\vartheta \wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}Fdr={\frac {1}{r^{2}sin\vartheta }}Fdr;}
∗
(
F
(
d
φ
∧
d
r
)
)
=
−
r
4
s
i
n
2
ϑ
r
2
s
i
n
2
ϑ
F
d
ϑ
=
−
1
s
i
n
ϑ
F
d
ϑ
;
{\displaystyle \ast (F(d\varphi \wedge dr))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}Fd\vartheta =-{\frac {1}{sin\vartheta }}Fd\vartheta ;}
∗
(
F
(
d
φ
∧
d
ϑ
)
)
=
−
r
4
s
i
n
2
ϑ
r
2
r
2
s
i
n
2
ϑ
F
d
r
=
−
1
r
2
s
i
n
ϑ
F
d
r
;
{\displaystyle \ast (F(d\varphi \wedge d\vartheta ))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}Fdr=-{\frac {1}{r^{2}sin\vartheta }}Fdr;}
∗
(
F
(
d
x
α
∧
d
x
β
∧
d
x
γ
)
)
=
ε
|
g
|
g
α
α
g
β
β
g
γ
γ
F
:
{\displaystyle \ast (F(dx_{\alpha }\wedge dx_{\beta }\wedge dx_{\gamma }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }g_{\gamma \gamma }}}F:}
∗
(
F
(
d
r
∧
d
ϑ
∧
d
φ
)
)
=
ε
r
4
s
i
n
2
ϑ
r
2
r
2
s
i
n
2
ϑ
F
=
ε
1
r
2
s
i
n
ϑ
F
{\displaystyle \ast (F(dr\wedge d\vartheta \wedge d\varphi ))=\varepsilon {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}F=\varepsilon {\frac {1}{r^{2}sin\vartheta }}F}
* Subir y bajar índices. Coordenadas esféricas
editar
Subir y bajar índices nos permite pasar de la base de un espacio vectorial a la base dual . Tenemos el contravector o vector del espacio inicial y el covector o 1-forma diferencial de la base diferencial.
∂
∂
x
α
=
g
α
β
d
x
β
:
{\displaystyle {\frac {\partial }{\partial x^{\alpha }}}=g_{\alpha \beta }dx^{\beta }:}
∂
∂
r
=
d
r
;
∂
∂
ϑ
=
r
2
d
ϑ
;
∂
∂
φ
=
r
2
s
i
n
2
ϑ
d
φ
;
{\displaystyle {\frac {\partial }{\partial r}}=dr;{\frac {\partial }{\partial \vartheta }}=r^{2}d\vartheta ;{\frac {\partial }{\partial \varphi }}=r^{2}sin^{2}\vartheta d\varphi ;}
d
x
α
=
g
α
β
∂
∂
x
β
:
{\displaystyle dx^{\alpha }=g^{\alpha \beta }{\frac {\partial }{\partial x^{\beta }}}:}
d
r
=
∂
∂
r
;
d
ϑ
=
1
r
2
∂
∂
ϑ
;
d
φ
=
1
r
2
s
i
n
2
ϑ
∂
∂
φ
{\displaystyle dr={\frac {\partial }{\partial r}};d\vartheta ={\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }};d\varphi ={\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}}
> Gradiente en coordenadas esféricas. Cálculo diferencial
editar
g
r
a
d
→
(
f
)
=
∇
→
(
f
)
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
f
(
x
,
y
,
z
)
=
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
z
∂
∂
z
)
f
(
x
,
y
,
z
)
=
{\displaystyle {\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})f(x,y,z)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})f(x,y,z)=}
(
e
^
r
∂
r
∂
x
∂
∂
r
+
e
^
ϑ
∂
ϑ
∂
y
∂
∂
ϑ
+
e
^
φ
∂
z
∂
φ
∂
∂
φ
)
f
(
r
,
ϑ
,
φ
)
=
1
|
∂
x
∂
r
|
∂
f
∂
r
e
^
r
+
1
|
∂
ϑ
∂
y
|
∂
f
∂
ϑ
e
^
ϑ
+
1
|
∂
z
∂
φ
|
∂
f
∂
φ
e
^
φ
=
{\displaystyle ({\widehat {e}}_{r}{\frac {\partial r}{\partial x}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {\partial \vartheta }{\partial y}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {\partial z}{\partial \varphi }}{\frac {\partial }{\partial \varphi }})f(r,\vartheta ,\varphi )={\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{|{\frac {\partial \vartheta }{\partial y}}|}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{|{\frac {\partial z}{\partial \varphi }}|}}{\frac {\partial f}{\partial \varphi }}{\widehat {e}}_{\varphi }=}
1
h
1
∂
f
∂
r
e
^
r
+
1
h
2
∂
f
∂
ϑ
e
^
ϑ
+
1
h
3
∂
f
∂
φ
e
^
φ
=
∂
f
∂
r
e
^
r
+
1
r
∂
f
∂
ϑ
e
^
ϑ
+
1
r
s
i
n
θ
∂
f
∂
φ
e
^
φ
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial \varphi }}{\widehat {e}}_{\varphi }={\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{rsin\theta }}{\frac {\partial f}{\partial \varphi }}{\widehat {e}}_{\varphi }}
> Gradiente en coordenadas esféricas. Geometría diferencial
editar
Para calcular el gradiente en coordenadas esféricas inicialmente aplicaremos la diferencial exterior y bajaremos índices. Finalmente pasaremos de la base natural en sus coordenadas a la base ortonormal .
g
r
a
d
→
(
f
)
=
∇
→
(
f
)
=↓
d
(
f
(
r
,
θ
,
φ
)
)
=↓
(
∂
f
∂
r
d
r
+
∂
f
∂
θ
d
θ
+
∂
f
∂
φ
d
φ
)
=
∂
f
∂
r
1
g
r
∂
∂
r
+
∂
f
∂
θ
1
g
θ
∂
∂
θ
+
∂
f
∂
φ
1
g
φ
∂
∂
φ
=
{\displaystyle {\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=\downarrow d(f(r,\theta ,\varphi ))=\downarrow ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \theta }}d\theta +{\frac {\partial f}{\partial \varphi }}d\varphi )={\frac {\partial f}{\partial r}}{\frac {1}{g_{r}}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{g_{\theta }}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{g_{\varphi }}}{\frac {\partial }{\partial \varphi }}=}
∂
f
∂
r
∂
∂
r
+
∂
f
∂
θ
1
r
2
∂
∂
θ
+
∂
f
∂
φ
1
r
2
s
i
n
2
θ
∂
∂
φ
=
∂
f
∂
r
|
∂
∂
r
|
∂
∂
r
|
∂
∂
r
|
+
∂
f
∂
θ
1
r
2
|
∂
∂
θ
|
∂
∂
θ
|
∂
∂
θ
|
+
∂
f
∂
φ
1
r
2
s
i
n
2
θ
|
∂
∂
φ
|
∂
∂
φ
|
∂
∂
φ
|
=
{\displaystyle {\frac {\partial f}{\partial r}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{r^{2}sin^{2}\theta }}{\frac {\partial }{\partial \varphi }}={\frac {\partial f}{\partial r}}|{\frac {\partial }{\partial r}}|{\frac {\frac {\partial }{\partial r}}{|{\frac {\partial }{\partial r}}|}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}|{\frac {\partial }{\partial \theta }}|{\frac {\frac {\partial }{\partial \theta }}{|{\frac {\partial }{\partial \theta }}|}}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{r^{2}sin^{2}\theta }}|{\frac {\partial }{\partial \varphi }}|{\frac {\frac {\partial }{\partial \varphi }}{|{\frac {\partial }{\partial \varphi }}|}}=}
∂
f
∂
r
∂
^
∂
r
+
∂
f
∂
θ
1
r
∂
^
∂
θ
+
∂
f
∂
φ
1
r
s
i
n
θ
∂
^
∂
φ
{\displaystyle {\frac {\partial f}{\partial r}}{\frac {\widehat {\partial }}{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r}}{\frac {\widehat {\partial }}{\partial \theta }}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{rsin\theta }}{\frac {\widehat {\partial }}{\partial \varphi }}}
> Divergencia en coordenadas esféricas. Cálculo diferencial
editar
∇
→
⋅
F
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
⋅
F
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
⋅
(
F
x
,
F
y
,
F
z
)
=
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
φ
∂
∂
φ
)
⋅
(
F
x
e
^
x
+
F
y
e
^
y
+
F
z
e
^
φ
)
=
{\displaystyle {\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot (F_{x},F_{y},F_{z})=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{\varphi }{\frac {\partial }{\partial \varphi }})\cdot (F_{x}{\widehat {e}}_{x}+F_{y}{\widehat {e}}_{y}+F_{z}{\widehat {e}}_{\varphi })=}
(
e
^
r
h
r
∂
∂
r
+
e
^
ϑ
h
ϑ
∂
∂
ϑ
+
e
^
φ
h
φ
∂
∂
φ
)
⋅
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
φ
e
^
φ
)
=
(
e
^
r
1
∂
∂
r
+
e
^
ϑ
r
∂
∂
ϑ
+
e
^
φ
r
s
i
n
ϑ
∂
∂
φ
)
⋅
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
φ
e
^
φ
)
=
{\displaystyle ({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=}
e
^
r
⋅
(
∂
F
r
∂
r
e
^
r
)
+
e
^
r
⋅
(
F
r
∂
e
^
r
∂
r
)
+
e
^
r
⋅
(
∂
F
ϑ
∂
r
e
^
ϑ
)
+
e
^
r
⋅
(
F
ϑ
∂
e
^
ϑ
∂
r
)
+
e
^
r
⋅
(
∂
F
φ
∂
r
e
^
φ
)
+
e
^
r
⋅
(
F
φ
∂
e
^
φ
∂
r
)
+
{\displaystyle {\widehat {e}}_{r}\cdot ({\frac {\partial F_{r}}{\partial r}}{\widehat {e}}_{r})+{\widehat {e}}_{r}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{r}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\varphi })+{\widehat {e}}_{r}\cdot (F_{\varphi }{\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}})+}
e
^
ϑ
r
⋅
(
∂
F
r
∂
ϑ
e
^
r
)
+
e
^
ϑ
r
⋅
(
F
r
∂
e
^
r
∂
ϑ
)
+
e
^
ϑ
r
⋅
(
∂
F
ϑ
∂
ϑ
e
^
ϑ
)
+
e
^
ϑ
r
⋅
(
F
ϑ
∂
e
^
ϑ
∂
ϑ
)
+
e
^
ϑ
r
⋅
(
∂
F
φ
∂
ϑ
e
^
φ
)
+
e
^
ϑ
r
⋅
(
F
φ
∂
e
^
φ
∂
ϑ
)
+
{\displaystyle {\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\vartheta }}{\partial \vartheta }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{\varphi }{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }})+}
e
^
φ
r
s
i
n
ϑ
⋅
(
∂
F
r
∂
φ
e
^
r
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
F
r
∂
e
^
r
∂
φ
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
∂
F
ϑ
∂
φ
e
^
ϑ
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
F
ϑ
∂
e
^
ϑ
∂
φ
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
∂
F
φ
∂
φ
e
^
φ
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
F
φ
∂
e
^
φ
∂
φ
)
=
{\displaystyle {\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\varphi }}{\partial \varphi }}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\varphi }{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }})=}
e
^
r
⋅
(
∂
F
r
∂
r
e
^
r
)
+
e
^
r
⋅
(
∂
F
ϑ
∂
r
e
^
ϑ
)
+
e
^
r
⋅
(
∂
F
φ
∂
r
e
^
φ
)
+
e
^
ϑ
r
⋅
(
∂
F
r
∂
ϑ
e
^
r
)
+
e
^
ϑ
r
⋅
(
F
r
e
^
ϑ
)
+
e
^
ϑ
r
⋅
(
∂
F
ϑ
∂
ϑ
e
^
ϑ
)
+
e
^
ϑ
r
⋅
F
ϑ
(
−
e
^
r
)
+
e
^
ϑ
r
⋅
(
∂
F
φ
∂
ϑ
e
^
φ
)
+
{\displaystyle {\widehat {e}}_{r}\cdot ({\frac {\partial F_{r}}{\partial r}}{\widehat {e}}_{r})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{r}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\vartheta }}{\partial \vartheta }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot F_{\vartheta }(-{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{\varphi })+}
e
^
φ
r
s
i
n
ϑ
⋅
(
∂
F
r
∂
φ
e
^
r
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
F
r
)
s
i
n
ϑ
e
^
φ
+
e
^
φ
r
s
i
n
ϑ
⋅
(
∂
F
ϑ
∂
φ
e
^
ϑ
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
F
ϑ
)
c
o
s
ϑ
e
^
φ
+
e
^
φ
r
s
i
n
ϑ
⋅
(
∂
F
φ
∂
φ
e
^
φ
)
+
e
^
φ
r
s
i
n
ϑ
⋅
(
F
φ
)
(
−
s
i
n
ϑ
e
^
r
−
c
o
s
ϑ
e
^
ϑ
)
=
{\displaystyle {\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{r})sin\vartheta {\widehat {e}}_{\varphi }+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\vartheta })cos\vartheta {\widehat {e}}_{\varphi }+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\varphi }}{\partial \varphi }}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\varphi })(-sin\vartheta {\widehat {e}}_{r}-cos\vartheta {\widehat {e}}_{\vartheta })=}
∂
F
r
∂
r
(
e
^
r
⋅
e
^
r
)
+
∂
F
ϑ
∂
r
(
e
^
r
⋅
e
^
ϑ
)
+
∂
F
φ
∂
r
(
e
^
r
⋅
e
^
φ
)
+
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
⋅
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
⋅
e
^
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
⋅
e
^
ϑ
)
+
1
r
F
ϑ
(
e
^
ϑ
⋅
(
−
e
^
r
)
)
+
1
r
∂
F
φ
∂
ϑ
(
e
^
ϑ
⋅
e
^
φ
)
+
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi })+}
1
r
s
i
n
ϑ
∂
F
r
∂
φ
(
e
^
φ
⋅
e
^
r
)
+
1
r
s
i
n
ϑ
(
F
r
s
i
n
ϑ
)
(
e
^
φ
⋅
e
^
φ
)
+
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
(
e
^
φ
⋅
e
^
ϑ
)
+
1
r
s
i
n
ϑ
F
ϑ
c
o
s
ϑ
(
e
^
φ
⋅
e
^
φ
)
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
(
e
^
φ
⋅
e
^
φ
)
+
{\displaystyle {\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}(F_{r}sin\vartheta )({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+}
1
r
s
i
n
ϑ
F
φ
)
(
(
−
s
i
n
ϑ
(
e
^
φ
⋅
e
^
r
)
−
c
o
s
ϑ
(
e
^
φ
⋅
e
^
ϑ
)
)
=
{\displaystyle {\frac {1}{rsin\vartheta }}F_{\varphi })((-sin\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r})-cos\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }))=}
∂
F
r
∂
r
(
e
^
r
⋅
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
⋅
e
^
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
⋅
e
^
ϑ
)
+
1
r
s
i
n
ϑ
(
F
r
s
i
n
ϑ
)
(
e
^
φ
⋅
e
^
φ
)
+
1
r
s
i
n
ϑ
F
ϑ
c
o
s
ϑ
(
e
^
φ
⋅
e
^
φ
)
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
(
e
^
φ
⋅
e
^
φ
)
=
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{rsin\vartheta }}(F_{r}sin\vartheta )({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })=}
∂
F
r
∂
r
+
1
r
F
r
+
1
r
∂
F
ϑ
∂
ϑ
+
1
r
F
r
+
1
r
s
i
n
ϑ
F
ϑ
c
o
s
ϑ
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
=
∂
F
r
∂
r
+
2
r
F
r
+
1
r
∂
F
ϑ
∂
ϑ
+
1
r
s
i
n
ϑ
F
ϑ
c
o
s
ϑ
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
=
{\displaystyle {\frac {\partial F_{r}}{\partial r}}+{\frac {1}{r}}F_{r}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {1}{r}}F_{r}+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta +{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}={\frac {\partial F_{r}}{\partial r}}+{\frac {2}{r}}F_{r}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta +{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}=}
1
r
2
∂
(
r
2
F
r
)
∂
r
+
1
r
s
i
n
ϑ
∂
(
F
ϑ
s
i
n
ϑ
)
∂
ϑ
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial (r^{2}F_{r})}{\partial r}}+{\frac {1}{rsin\vartheta }}{\frac {\partial (F_{\vartheta }sin\vartheta )}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}}
> Divergencia en coordenadas esféricas. Geometría diferencial
editar
Para desarrollar la divergencia con geometría diferencial bajaremos índices, aplicaremos la estrella de Hodge , posteriormente la diferencial exterior ; y finalmente, otra vez, la estrella de Hodge .
d
i
v
(
F
→
)
=
∇
→
⋅
(
F
→
)
=
∗
d
∗
↓
(
F
→
)
=
∗
d
∗
↓
(
F
x
∂
^
∂
x
+
F
y
∂
^
∂
y
+
F
z
∂
^
∂
z
)
=
∗
d
∗
↓
(
F
r
1
|
∂
x
∂
r
|
∂
∂
r
+
F
ϑ
1
|
∂
y
∂
ϑ
|
∂
∂
ϑ
+
F
φ
1
|
∂
z
∂
φ
|
∂
∂
φ
)
=
{\displaystyle div({\overrightarrow {F}})={\overrightarrow {\nabla }}\cdot ({\overrightarrow {F}})=\ast d\ast \downarrow ({\overrightarrow {F}})=\ast d\ast \downarrow (F_{x}{\frac {\widehat {\partial }}{\partial x}}+F_{y}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\ast \ d\ast \downarrow (F_{r}{\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial }{\partial r}}+F_{\vartheta }{\frac {1}{|{\frac {\partial y}{\partial \vartheta }}|}}{\frac {\partial }{\partial \vartheta }}+F_{\varphi }{\frac {1}{|{\frac {\partial z}{\partial \varphi }}|}}{\frac {\partial }{\partial \varphi }})=}
∗
d
∗
↓
(
F
r
1
g
r
∂
∂
r
)
+
F
ϑ
1
g
ϑ
∂
∂
ϑ
)
+
F
φ
1
g
φ
∂
∂
φ
)
)
=
∗
d
∗
(
F
r
1
g
r
g
r
d
r
+
F
ϑ
1
g
ϑ
g
ϑ
d
ϑ
+
F
φ
1
g
φ
g
φ
d
φ
)
)
=
∗
d
∗
(
F
r
d
r
+
F
φ
r
d
ϑ
+
F
φ
r
s
i
n
ϑ
d
φ
)
=
{\displaystyle \ast d\ast \downarrow (F_{r}{\frac {1}{\sqrt[{}]{g_{r}}}}{\frac {\partial }{\partial r}})+F_{\vartheta }{\frac {1}{\sqrt[{}]{g_{\vartheta }}}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }{\frac {1}{\sqrt[{}]{g_{\varphi }}}}{\frac {\partial }{\partial \varphi }}))=\ast d\ast (F_{r}{\frac {1}{\sqrt[{}]{g_{r}}}}g_{r}dr+F_{\vartheta }{\frac {1}{\sqrt[{}]{g_{\vartheta }}}}g_{\vartheta }d\vartheta +F_{\varphi }{\frac {1}{\sqrt[{}]{g_{\varphi }}}}g_{\varphi }d\varphi ))=\ast d\ast (F_{r}dr+F_{\varphi }rd\vartheta +F_{\varphi }rsin\vartheta d\varphi )=}
∗
d
(
F
r
r
4
s
i
n
2
ϑ
(
d
ϑ
∧
d
φ
)
+
F
ϑ
r
r
4
s
i
n
2
ϑ
r
2
(
d
φ
∧
d
r
)
+
F
φ
r
s
i
n
ϑ
r
4
s
i
n
2
ϑ
r
2
s
i
n
2
ϑ
(
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast d(F_{r}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}(d\vartheta \wedge d\varphi )+F_{\vartheta }r{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}(d\varphi \wedge dr)+F_{\varphi }rsin\vartheta {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}(dr\wedge d\vartheta ))=}
∗
d
(
F
r
r
4
s
i
n
2
ϑ
(
d
ϑ
∧
d
φ
)
+
F
ϑ
r
4
s
i
n
2
ϑ
r
(
d
φ
∧
d
r
)
+
F
φ
r
4
s
i
n
2
ϑ
r
s
i
n
ϑ
(
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast d(F_{r}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}(d\vartheta \wedge d\varphi )+F_{\vartheta }{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r}}(d\varphi \wedge dr)+F_{\varphi }{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{rsin\vartheta }}(dr\wedge d\vartheta ))=}
∗
(
∂
∂
r
(
F
r
r
4
s
i
n
2
ϑ
2
)
(
d
r
∧
d
θ
∧
d
φ
)
+
∂
∂
ϑ
(
F
ϑ
r
2
s
i
n
2
ϑ
2
)
(
d
ϑ
∧
d
φ
∧
d
r
)
+
∂
∂
φ
(
F
φ
r
)
(
d
φ
∧
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast ({\frac {\partial }{\partial r}}(F_{r}{\sqrt[{2}]{r^{4}sin^{2}\vartheta }})(dr\wedge d\theta \wedge d\varphi )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }{\sqrt[{2}]{r^{2}sin^{2}\vartheta }})(d\vartheta \wedge d\varphi \wedge dr)+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)(d\varphi \wedge dr\wedge d\vartheta ))=}
∗
(
(
∂
∂
r
(
F
r
r
2
s
i
n
ϑ
)
+
∂
∂
ϑ
(
F
ϑ
r
s
i
n
ϑ
)
+
∂
∂
φ
(
F
φ
r
)
)
(
d
r
∧
d
ϑ
∧
d
φ
)
)
=
r
4
s
i
n
2
ϑ
r
4
s
i
n
2
ϑ
∂
∂
r
(
F
r
r
2
s
i
n
ϑ
)
+
r
4
s
i
n
2
ϑ
r
4
s
i
n
2
ϑ
∂
∂
ϑ
(
F
ϑ
r
s
i
n
ϑ
)
+
r
4
s
i
n
2
ϑ
r
4
s
i
n
2
ϑ
∂
∂
φ
(
F
φ
r
)
=
{\displaystyle \ast (({\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r))(dr\wedge d\vartheta \wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)=}
1
r
2
s
i
n
ϑ
∂
∂
r
(
F
r
r
2
s
i
n
ϑ
)
+
1
r
2
s
i
n
ϑ
∂
∂
ϑ
(
F
ϑ
r
s
i
n
ϑ
)
+
1
r
2
s
i
n
ϑ
∂
∂
φ
(
F
φ
r
)
=
{\displaystyle {\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)=}
1
r
2
s
i
n
ϑ
(
∂
∂
r
(
F
r
r
2
s
i
n
ϑ
)
+
∂
∂
ϑ
(
F
ϑ
r
s
i
n
ϑ
)
+
∂
∂
φ
(
F
φ
r
)
)
=
1
r
2
∂
(
r
2
F
r
)
∂
r
+
1
r
s
i
n
ϑ
∂
(
F
ϑ
s
i
n
ϑ
)
∂
ϑ
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
{\displaystyle {\frac {1}{r^{2}sin\vartheta }}({\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r))={\frac {1}{r^{2}}}{\frac {\partial (r^{2}F_{r})}{\partial r}}+{\frac {1}{rsin\vartheta }}{\frac {\partial (F_{\vartheta }sin\vartheta )}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}}
> Rotacional en coordenadas esféricas. Cálculo diferencial
editar
∇
→
×
F
→
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
×
(
F
x
,
F
y
,
F
z
)
=
(
1
h
r
∂
∂
r
,
1
h
ϑ
∂
∂
ϑ
,
1
h
φ
∂
∂
φ
)
×
(
F
r
,
F
ϑ
,
F
φ
)
=
{\displaystyle {\overrightarrow {\nabla }}\times {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\times (F_{x},F_{y},F_{z})=({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}},{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }},{\frac {1}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\times (F_{r},F_{\vartheta },F_{\varphi })=}
(
e
^
r
h
r
∂
∂
r
+
e
^
ϑ
h
ϑ
∂
∂
ϑ
+
e
^
φ
h
φ
∂
∂
φ
)
×
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
φ
e
^
φ
)
=
(
e
^
r
1
∂
∂
r
+
e
^
ϑ
r
∂
∂
ϑ
+
e
^
φ
r
s
i
n
ϑ
∂
∂
φ
)
×
(
F
r
e
^
r
+
F
ϑ
e
^
ϑ
+
F
φ
e
^
φ
)
=
{\displaystyle ({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=}
(
e
^
r
∂
∂
r
)
×
(
F
r
e
^
r
)
+
(
e
^
r
∂
∂
r
)
×
(
F
ϑ
e
^
ϑ
)
+
(
e
^
r
∂
∂
r
)
×
(
F
φ
e
^
φ
)
+
(
e
^
ϑ
r
∂
∂
ϑ
)
×
(
F
r
e
^
r
)
+
(
e
^
ϑ
r
∂
∂
ϑ
)
×
(
F
ϑ
e
^
ϑ
)
+
(
e
^
ϑ
r
∂
∂
ϑ
)
×
(
F
φ
e
^
φ
)
+
{\displaystyle ({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{r}{\widehat {e}}_{r})+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{\varphi }{\widehat {e}}_{\varphi })+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{r}{\widehat {e}}_{r})+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{\varphi }{\widehat {e}}_{\varphi })+}
(
e
^
φ
r
s
i
n
ϑ
∂
∂
φ
)
×
(
F
r
e
^
r
)
+
(
e
^
φ
r
s
i
n
ϑ
∂
∂
φ
)
×
(
F
ϑ
e
^
ϑ
)
+
(
e
^
φ
r
s
i
n
ϑ
∂
∂
φ
)
×
(
F
φ
e
^
φ
)
=
{\displaystyle ({\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{r}{\widehat {e}}_{r})+({\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{\varphi }{\widehat {e}}_{\varphi })=}
∂
F
r
∂
r
(
e
^
r
×
e
^
r
)
+
F
r
(
e
^
r
×
∂
e
^
r
∂
r
)
+
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
F
ϑ
(
e
^
r
×
∂
e
^
ϑ
∂
r
)
+
∂
F
φ
∂
r
(
e
^
r
×
e
^
φ
)
+
F
φ
(
e
^
r
×
∂
e
^
φ
∂
r
)
+
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+F_{\vartheta }({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+F_{\varphi }({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}})+}
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
×
∂
e
^
r
∂
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
×
e
^
ϑ
)
+
1
r
F
ϑ
(
e
^
ϑ
×
∂
e
^
ϑ
∂
ϑ
)
+
1
r
∂
F
φ
∂
ϑ
(
e
^
ϑ
×
e
^
φ
)
+
1
r
F
φ
(
e
^
ϑ
×
∂
e
^
φ
∂
ϑ
)
+
{\displaystyle {\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}F_{\varphi }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }})+}
1
r
s
i
n
ϑ
∂
F
r
∂
φ
(
e
^
φ
×
e
^
r
)
+
1
r
s
i
n
ϑ
F
r
(
e
^
φ
×
∂
e
^
r
∂
φ
)
+
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
(
e
^
φ
×
e
^
ϑ
)
+
1
r
s
i
n
ϑ
F
ϑ
(
e
^
φ
×
∂
e
^
ϑ
∂
φ
)
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
(
e
^
φ
×
e
^
φ
)
+
1
r
s
i
n
ϑ
F
φ
(
e
^
φ
×
∂
e
^
φ
∂
φ
)
=
{\displaystyle {\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}F_{r}({\widehat {e}}_{\varphi }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }})+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })+{\frac {1}{rsin\vartheta }}F_{\vartheta }({\widehat {e}}_{\varphi }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }})+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }})=}
∂
F
r
∂
r
(
e
^
r
×
e
^
r
)
+
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
∂
F
φ
∂
r
(
e
^
r
×
e
^
φ
)
+
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
×
e
^
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
×
e
^
ϑ
)
+
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+}
1
r
F
ϑ
(
e
^
ϑ
×
(
−
e
^
r
)
)
+
1
r
∂
F
φ
∂
ϑ
(
e
^
ϑ
×
e
^
φ
)
+
1
r
s
i
n
ϑ
∂
F
r
∂
φ
(
e
^
φ
×
e
^
r
)
+
1
r
s
i
n
ϑ
F
r
(
e
^
φ
×
(
s
i
n
ϑ
e
^
φ
)
)
+
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
(
e
^
φ
×
e
^
ϑ
)
+
{\displaystyle {\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}F_{r}({\widehat {e}}_{\varphi }\times (sin\vartheta {\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })+}
1
r
s
i
n
ϑ
F
ϑ
(
e
^
φ
×
(
c
o
s
ϑ
e
^
φ
)
)
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
(
e
^
φ
×
e
^
φ
)
+
1
r
s
i
n
ϑ
F
φ
(
e
^
φ
×
(
−
s
i
n
ϑ
e
^
r
−
c
o
s
ϑ
e
^
ϑ
)
)
=
{\displaystyle {\frac {1}{rsin\vartheta }}F_{\vartheta }({\widehat {e}}_{\varphi }\times (cos\vartheta {\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times (-sin\vartheta {\widehat {e}}_{r}-cos\vartheta {\widehat {e}}_{\vartheta }))=}
∂
F
r
∂
r
(
e
^
r
×
e
^
r
)
+
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
∂
F
φ
∂
r
(
e
^
r
×
e
^
φ
)
+
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
r
(
e
^
ϑ
×
e
^
ϑ
)
+
1
r
∂
F
ϑ
∂
ϑ
(
e
^
ϑ
×
e
^
ϑ
)
+
{\displaystyle {\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+}
1
r
F
ϑ
(
e
^
ϑ
×
(
−
e
^
r
)
)
+
1
r
∂
F
φ
∂
ϑ
(
e
^
ϑ
×
e
^
φ
)
+
1
r
s
i
n
ϑ
∂
F
r
∂
φ
(
e
^
φ
×
e
^
r
)
+
F
r
(
e
^
φ
×
(
e
^
φ
)
)
+
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
(
e
^
φ
×
e
^
ϑ
)
+
{\displaystyle {\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{\varphi }\times ({\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })+}
c
o
s
ϑ
r
s
i
n
ϑ
F
ϑ
(
e
^
φ
×
(
e
^
φ
)
)
+
1
r
s
i
n
ϑ
∂
F
φ
∂
φ
(
e
^
φ
×
e
^
φ
)
−
1
r
F
φ
(
e
^
φ
×
e
^
r
)
−
c
o
s
ϑ
r
s
i
n
ϑ
F
φ
(
e
^
φ
×
e
^
ϑ
)
=
{\displaystyle {\frac {cos\vartheta }{rsin\vartheta }}F_{\vartheta }({\widehat {e}}_{\varphi }\times ({\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi })-{\frac {1}{r}}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})-{\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })=}
∂
F
ϑ
∂
r
(
e
^
r
×
e
^
ϑ
)
+
∂
F
φ
∂
r
(
e
^
r
×
e
^
φ
)
+
1
r
∂
F
r
∂
ϑ
(
e
^
ϑ
×
e
^
r
)
+
1
r
F
ϑ
(
e
^
ϑ
×
(
−
e
^
r
)
)
+
1
r
∂
F
φ
∂
ϑ
(
e
^
ϑ
×
e
^
φ
)
+
{\displaystyle {\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+}
1
r
s
i
n
ϑ
∂
F
r
∂
φ
(
e
^
φ
×
e
^
r
)
+
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
(
e
^
φ
×
e
^
ϑ
)
−
1
r
F
φ
(
e
^
φ
×
e
^
r
)
−
c
o
s
ϑ
r
s
i
n
ϑ
F
φ
(
e
^
φ
×
e
^
ϑ
)
=
{\displaystyle {\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })-{\frac {1}{r}}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})-{\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })=}
∂
F
ϑ
∂
r
e
^
φ
−
∂
F
φ
∂
r
e
^
ϑ
−
1
r
∂
F
r
∂
ϑ
e
^
φ
+
1
r
F
ϑ
e
^
φ
+
1
r
∂
F
φ
∂
ϑ
e
^
r
+
1
r
s
i
n
ϑ
∂
F
r
∂
φ
e
^
ϑ
−
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
e
^
r
−
1
r
F
φ
e
^
ϑ
+
c
o
s
ϑ
r
s
i
n
ϑ
F
φ
e
^
r
=
{\displaystyle {\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\varphi }-{\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\vartheta }-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{\varphi }+{\frac {1}{r}}F_{\vartheta }{\widehat {e}}_{\varphi }+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{r}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{\vartheta }-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{r}-{\frac {1}{r}}F_{\varphi }{\widehat {e}}_{\vartheta }+{\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }{\widehat {e}}_{r}=}
c
o
s
ϑ
r
s
i
n
ϑ
F
φ
e
^
r
+
1
r
∂
F
φ
∂
ϑ
e
^
r
−
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
e
^
r
+
1
r
s
i
n
ϑ
∂
F
r
∂
φ
e
^
ϑ
−
1
r
F
φ
e
^
ϑ
−
∂
F
φ
∂
r
e
^
ϑ
+
∂
F
ϑ
∂
r
e
^
φ
+
1
r
F
ϑ
e
^
φ
−
1
r
∂
F
r
∂
ϑ
e
^
φ
=
{\displaystyle {\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }{\widehat {e}}_{r}+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{r}-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{r}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{\vartheta }-{\frac {1}{r}}F_{\varphi }{\widehat {e}}_{\vartheta }-{\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\vartheta }+{\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\varphi }+{\frac {1}{r}}F_{\vartheta }{\widehat {e}}_{\varphi }-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{\varphi }=}
(
c
o
s
ϑ
r
s
i
n
ϑ
F
φ
+
1
r
∂
F
φ
∂
ϑ
−
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
)
e
^
r
+
(
1
r
s
i
n
ϑ
∂
F
r
∂
φ
−
1
r
F
φ
−
∂
F
φ
∂
r
)
e
^
ϑ
+
(
∂
F
ϑ
∂
r
+
1
r
F
ϑ
−
1
r
∂
F
r
∂
ϑ
)
e
^
φ
=
{\displaystyle ({\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\widehat {e}}_{r}+({\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {1}{r}}F_{\varphi }-{\frac {\partial F_{\varphi }}{\partial r}}){\widehat {e}}_{\vartheta }+({\frac {\partial F_{\vartheta }}{\partial r}}+{\frac {1}{r}}F_{\vartheta }-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}){\widehat {e}}_{\varphi }=}
1
r
s
i
n
ϑ
(
∂
(
F
φ
s
i
n
ϑ
)
∂
ϑ
−
∂
F
ϑ
∂
φ
)
e
^
r
+
1
r
(
1
s
i
n
ϑ
∂
F
r
∂
φ
−
∂
(
r
F
φ
)
∂
r
)
e
^
ϑ
+
1
r
(
∂
(
r
F
ϑ
)
∂
r
−
∂
F
r
∂
ϑ
)
e
^
φ
{\displaystyle {\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\widehat {e}}_{r}+{\frac {1}{r}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (rF_{\varphi })}{\partial r}}){\widehat {e}}_{\vartheta }+{\frac {1}{r}}({\frac {\partial (rF_{\vartheta })}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\widehat {e}}_{\varphi }}
> Rotacional en coordenadas esféricas. Geometría diferencial
editar
Para desarrollar la rotacional con geometría diferencial bajaremos índices, aplicaremos la estrella de Hodge , posteriormente la diferencial exterior ; y finalmente, otra vez, la estrella de Hodge .
r
o
t
(
F
→
)
=
∇
→
×
F
→
=↑
∗
d
↓
(
F
→
)
=↑
∗
d
↓
(
F
x
∂
^
∂
x
+
F
y
∂
^
∂
y
+
F
z
∂
^
∂
z
)
=↑
∗
d
↓
(
F
r
(
1
h
r
∂
∂
r
)
+
F
ϑ
(
1
h
ϑ
∂
∂
ϑ
)
+
F
φ
(
1
h
φ
∂
∂
φ
)
)
=
{\displaystyle rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\widehat {\partial }}{\partial x}}+{F_{y}}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\uparrow \ast d\downarrow ({F_{r}}({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}})+{F_{\vartheta }}({\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }({\frac {1}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }}))=}
↑
∗
d
↓
(
F
r
∂
∂
r
+
F
ϑ
(
1
r
∂
∂
ϑ
)
+
F
φ
(
1
r
s
i
n
ϑ
∂
∂
φ
)
=↑
∗
d
(
F
r
d
r
+
F
ϑ
r
r
2
d
ϑ
+
F
φ
r
s
i
n
ϑ
r
2
s
i
n
2
ϑ
d
φ
)
=↑
∗
d
(
F
r
d
r
+
F
ϑ
r
d
ϑ
+
r
s
i
n
ϑ
F
φ
d
φ
)
=
{\displaystyle \uparrow \ast d\downarrow ({F_{r}}{\frac {\partial }{\partial r}}+{F_{\vartheta }}({\frac {1}{r}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }({\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})=\uparrow \ast d({F_{r}}dr+{\frac {F_{\vartheta }}{r}}r^{2}d\vartheta +{\frac {F_{\varphi }}{rsin\vartheta }}r^{2}sin^{2}\vartheta d\varphi )=\uparrow \ast d({F_{r}}dr+F_{\vartheta }rd\vartheta +rsin\vartheta F_{\varphi }d\varphi )=}
↑
∗
(
∂
F
r
∂
ϑ
(
d
ϑ
∧
d
r
)
+
∂
F
r
∂
φ
(
d
φ
∧
d
r
)
+
∂
(
F
ϑ
r
)
∂
r
(
d
r
∧
d
ϑ
)
+
∂
(
F
ϑ
r
)
∂
φ
(
d
φ
∧
d
ϑ
)
+
∂
(
F
φ
r
s
i
n
ϑ
)
∂
r
(
d
r
∧
d
φ
)
+
∂
(
F
φ
r
s
i
n
ϑ
)
∂
ϑ
(
d
ϑ
∧
d
φ
)
)
=
{\displaystyle \uparrow \ast ({\frac {\partial F_{r}}{\partial \vartheta }}(d\vartheta \wedge dr)+{\frac {\partial F_{r}}{\partial \varphi }}(d\varphi \wedge dr)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )+{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}(d\varphi \wedge d\vartheta )+{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}(dr\wedge d\varphi )+{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}(d\vartheta \wedge d\varphi ))=}
↑
∗
(
−
∂
F
r
∂
ϑ
(
d
r
∧
d
ϑ
)
+
∂
F
r
∂
φ
(
d
φ
∧
d
r
)
+
∂
(
F
ϑ
r
)
∂
r
(
d
r
∧
d
ϑ
)
−
∂
(
F
ϑ
r
)
∂
φ
(
d
ϑ
∧
d
φ
)
−
∂
(
F
φ
r
s
i
n
ϑ
)
∂
r
(
d
φ
∧
d
r
)
+
∂
(
F
φ
r
s
i
n
ϑ
)
∂
ϑ
(
d
ϑ
∧
d
φ
)
)
=
{\displaystyle \uparrow \ast (-{\frac {\partial F_{r}}{\partial \vartheta }}(dr\wedge d\vartheta )+{\frac {\partial F_{r}}{\partial \varphi }}(d\varphi \wedge dr)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )-{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}(d\vartheta \wedge d\varphi )-{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}(d\varphi \wedge dr)+{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}(d\vartheta \wedge d\varphi ))=}
↑
(
−
r
4
s
i
n
2
ϑ
r
2
∂
F
r
∂
ϑ
d
φ
+
r
4
s
i
n
2
ϑ
r
2
s
i
n
2
ϑ
∂
F
r
∂
φ
d
ϑ
+
r
4
s
i
n
2
ϑ
r
2
∂
(
F
ϑ
r
)
∂
r
d
φ
−
{\displaystyle \uparrow (-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}{\frac {\partial F_{r}}{\partial \vartheta }}d\varphi +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}d\vartheta +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}d\varphi -}
r
4
s
i
n
2
ϑ
r
4
s
i
n
2
ϑ
∂
(
F
ϑ
r
)
∂
φ
d
r
−
r
4
s
i
n
2
ϑ
r
2
s
i
n
2
ϑ
∂
(
F
φ
r
s
i
n
ϑ
)
∂
r
d
ϑ
+
r
4
s
i
n
2
ϑ
r
4
s
i
n
2
ϑ
∂
(
F
φ
r
s
i
n
ϑ
)
∂
ϑ
d
r
)
=
{\displaystyle {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}dr-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}d\vartheta +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}dr)=}
↑
(
1
r
2
s
i
n
ϑ
∂
(
F
φ
r
s
i
n
ϑ
)
∂
ϑ
d
r
−
1
r
2
s
i
n
ϑ
∂
(
F
ϑ
r
)
∂
φ
d
r
+
1
s
i
n
ϑ
∂
F
r
∂
φ
d
ϑ
−
1
s
i
n
ϑ
∂
(
F
φ
r
s
i
n
ϑ
)
∂
r
d
ϑ
+
s
i
n
ϑ
∂
(
F
ϑ
r
)
∂
r
d
φ
−
s
i
n
ϑ
∂
F
r
∂
ϑ
d
φ
)
=
{\displaystyle \uparrow ({\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}dr-{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}dr+{\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}d\vartheta -{\frac {1}{sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}d\vartheta +sin\vartheta {\frac {\partial (F_{\vartheta }r)}{\partial r}}d\varphi -sin\vartheta {\frac {\partial F_{r}}{\partial \vartheta }}d\varphi )=}
1
r
2
s
i
n
ϑ
∂
(
F
φ
r
s
i
n
ϑ
)
∂
ϑ
∂
∂
r
−
1
r
2
s
i
n
ϑ
∂
(
F
ϑ
r
)
∂
φ
∂
∂
r
+
1
s
i
n
ϑ
∂
F
r
∂
φ
1
r
2
∂
∂
ϑ
−
1
s
i
n
ϑ
∂
(
F
φ
r
s
i
n
ϑ
)
∂
r
1
r
2
∂
∂
ϑ
+
s
i
n
ϑ
∂
(
F
ϑ
r
)
∂
r
1
r
2
s
i
n
2
ϑ
∂
∂
φ
−
s
i
n
ϑ
∂
F
r
∂
ϑ
1
r
2
s
i
n
2
ϑ
∂
∂
φ
=
{\displaystyle {\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}{\frac {\partial }{\partial r}}-{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}{\frac {\partial }{\partial r}}+{\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}-{\frac {1}{sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}+sin\vartheta {\frac {\partial (F_{\vartheta }r)}{\partial r}}{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}-sin\vartheta {\frac {\partial F_{r}}{\partial \vartheta }}{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}=}
(
1
r
s
i
n
ϑ
∂
(
F
φ
s
i
n
ϑ
)
∂
ϑ
−
1
r
s
i
n
ϑ
∂
F
ϑ
∂
φ
)
∂
∂
r
+
(
1
r
2
1
s
i
n
ϑ
∂
F
r
∂
φ
−
1
r
2
∂
(
F
φ
r
)
∂
r
)
∂
∂
ϑ
+
(
s
i
n
ϑ
∂
(
F
ϑ
r
)
∂
r
1
r
2
s
i
n
2
ϑ
−
s
i
n
ϑ
∂
F
r
∂
ϑ
1
r
2
s
i
n
2
ϑ
)
∂
∂
φ
=
{\displaystyle ({\frac {1}{rsin\vartheta }}{\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\frac {\partial }{\partial r}}+({\frac {1}{r^{2}}}{\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {1}{r^{2}}}{\frac {\partial (F_{\varphi }r)}{\partial r}}){\frac {\partial }{\partial \vartheta }}+(sin\vartheta {\frac {\partial (F_{\vartheta }r)}{\partial r}}{\frac {1}{r^{2}sin^{2}\vartheta }}-sin\vartheta {\frac {\partial F_{r}}{\partial \vartheta }}{\frac {1}{r^{2}sin^{2}\vartheta }}){\frac {\partial }{\partial \varphi }}=}
1
r
s
i
n
ϑ
(
∂
(
F
φ
s
i
n
ϑ
)
∂
ϑ
−
∂
F
ϑ
∂
φ
)
∂
∂
r
+
1
r
2
(
1
s
i
n
ϑ
∂
F
r
∂
φ
−
∂
(
F
φ
r
)
∂
r
)
∂
∂
ϑ
+
1
r
2
s
i
n
2
ϑ
(
∂
(
F
ϑ
r
)
∂
r
−
∂
F
r
∂
ϑ
)
∂
∂
φ
=
{\displaystyle {\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (F_{\varphi }r)}{\partial r}}){\frac {\partial }{\partial \vartheta }}+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\partial }{\partial \varphi }}=}
1
r
s
i
n
ϑ
(
∂
(
F
φ
s
i
n
ϑ
)
∂
ϑ
−
∂
F
ϑ
∂
φ
)
|
∂
∂
r
|
∂
∂
r
|
∂
∂
r
|
+
1
r
2
(
1
s
i
n
ϑ
∂
F
r
∂
φ
−
∂
(
F
φ
r
)
∂
r
)
|
∂
∂
ϑ
|
∂
∂
ϑ
|
∂
∂
ϑ
|
+
1
r
2
s
i
n
2
ϑ
(
∂
(
F
ϑ
r
)
∂
r
−
∂
F
r
∂
ϑ
)
|
∂
∂
φ
|
∂
∂
φ
|
∂
∂
φ
|
=
{\displaystyle {\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }})|{\frac {\partial }{\partial r}}|{\frac {\frac {\partial }{\partial r}}{|{\frac {\partial }{\partial r}}|}}+{\frac {1}{r^{2}}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (F_{\varphi }r)}{\partial r}})|{\frac {\partial }{\partial \vartheta }}|{\frac {\frac {\partial }{\partial \vartheta }}{|{\frac {\partial }{\partial \vartheta }}|}}+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }})|{\frac {\partial }{\partial \varphi }}|{\frac {\frac {\partial }{\partial \varphi }}{|{\frac {\partial }{\partial \varphi }}|}}=}
1
r
s
i
n
ϑ
(
∂
(
F
φ
s
i
n
ϑ
)
∂
ϑ
−
∂
F
ϑ
∂
φ
)
∂
^
∂
r
+
1
r
(
1
s
i
n
ϑ
∂
F
r
∂
φ
−
∂
(
F
φ
r
)
∂
r
)
∂
^
∂
ϑ
+
1
r
s
i
n
ϑ
(
∂
(
F
ϑ
r
)
∂
r
−
∂
F
r
∂
ϑ
)
∂
^
∂
φ
{\displaystyle {\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\frac {\widehat {\partial }}{\partial r}}+{\frac {1}{r}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (F_{\varphi }r)}{\partial r}}){\frac {\widehat {\partial }}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\widehat {\partial }}{\partial \varphi }}}
> Laplaciano en coordenadas esféricas. Cálculo diferencial
editar
d
i
v
(
g
r
a
d
→
(
f
)
)
=
∇
→
⋅
(
∇
→
f
)
=
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
z
∂
∂
z
)
⋅
(
e
^
x
∂
f
∂
x
+
e
^
y
∂
f
∂
y
+
e
^
z
∂
f
∂
z
)
=
{\displaystyle div({\overrightarrow {grad}}(f))={\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}f)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{x}{\frac {\partial f}{\partial x}}+{\widehat {e}}_{y}{\frac {\partial f}{\partial y}}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=}
(
e
^
r
1
h
r
∂
∂
r
+
e
^
ϑ
1
h
ϑ
∂
∂
ϑ
+
e
^
φ
1
h
φ
∂
∂
φ
)
⋅
(
e
^
r
∂
f
∂
r
+
e
^
ϑ
1
r
∂
f
∂
ϑ
+
e
^
φ
1
r
s
i
n
ϑ
∂
f
∂
φ
)
=
{\displaystyle ({\widehat {e}}_{r}{\frac {1}{h_{r}}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})=}
(
e
^
r
∂
∂
r
+
e
^
ϑ
1
r
∂
∂
ϑ
+
e
^
φ
1
r
s
i
n
ϑ
∂
∂
φ
)
⋅
(
e
^
r
∂
f
∂
r
+
e
^
ϑ
1
r
∂
f
∂
ϑ
+
e
^
φ
1
r
s
i
n
ϑ
∂
f
∂
φ
)
=
{\displaystyle ({\widehat {e}}_{r}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})=}
(
e
^
r
⋅
∂
e
^
r
∂
r
)
∂
f
∂
r
+
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
r
⋅
∂
e
^
ϑ
∂
r
)
1
r
∂
f
∂
ϑ
+
(
e
^
r
⋅
e
^
ϑ
)
∂
∂
r
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
r
⋅
∂
e
^
φ
∂
r
)
1
r
s
i
n
ϑ
∂
f
∂
φ
+
(
e
^
r
⋅
e
^
φ
)
∂
∂
r
(
1
r
s
i
n
ϑ
∂
f
∂
φ
)
+
{\displaystyle ({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial r}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}}){\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi }){\frac {\partial }{\partial r}}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+}
(
e
^
ϑ
⋅
∂
e
^
r
∂
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
r
)
1
r
∂
∂
ϑ
(
∂
f
∂
r
)
+
(
e
^
ϑ
⋅
∂
e
^
ϑ
∂
ϑ
)
1
r
2
∂
f
∂
ϑ
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
(
e
^
ϑ
⋅
∂
e
^
φ
∂
ϑ
)
1
r
1
r
s
i
n
ϑ
∂
f
∂
φ
+
(
e
^
ϑ
⋅
e
^
φ
)
1
r
∂
∂
ϑ
(
1
r
s
i
n
ϑ
∂
f
∂
φ
)
+
{\displaystyle ({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }}){\frac {1}{r}}{\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi }){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+}
(
e
^
φ
⋅
∂
e
^
r
∂
φ
)
1
r
s
i
n
ϑ
∂
f
∂
r
+
(
e
^
φ
⋅
e
^
r
)
1
r
s
i
n
ϑ
∂
∂
φ
(
∂
f
∂
r
)
+
(
e
^
φ
⋅
∂
e
^
ϑ
∂
φ
)
1
r
2
s
i
n
ϑ
∂
f
∂
ϑ
+
(
e
^
φ
⋅
e
^
ϑ
)
1
r
s
i
n
ϑ
∂
∂
φ
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
φ
⋅
∂
e
^
φ
∂
φ
)
1
r
2
s
i
n
2
ϑ
∂
f
∂
φ
+
{\displaystyle ({\widehat {e}}_{\varphi }\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }}){\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r}){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\varphi }\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }}){\frac {1}{r^{2}sin\vartheta }}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{\varphi }\cdot {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }}){\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial f}{\partial \varphi }}+}
(
e
^
φ
⋅
e
^
φ
)
1
r
s
i
n
ϑ
∂
∂
φ
(
1
r
s
i
n
ϑ
∂
f
∂
φ
)
=
{\displaystyle ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi }){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})=}
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
r
⋅
e
^
ϑ
)
∂
∂
r
(
1
r
∂
f
∂
ϑ
)
+
(
e
^
r
⋅
e
^
φ
)
∂
∂
r
(
1
r
s
i
n
ϑ
∂
f
∂
φ
)
+
(
e
^
ϑ
⋅
e
^
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
r
)
1
r
∂
∂
ϑ
(
∂
f
∂
r
)
+
{\displaystyle ({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi }){\frac {\partial }{\partial r}}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+}
(
e
^
ϑ
⋅
(
−
e
^
r
)
)
1
r
2
∂
f
∂
ϑ
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
(
e
^
ϑ
⋅
e
^
φ
)
1
r
∂
∂
ϑ
(
1
r
s
i
n
ϑ
∂
f
∂
φ
)
+
(
e
^
φ
⋅
e
^
φ
)
s
i
n
ϑ
1
r
s
i
n
ϑ
∂
f
∂
r
+
(
e
^
φ
⋅
e
^
r
)
1
r
s
i
n
ϑ
∂
∂
φ
(
∂
f
∂
r
)
+
{\displaystyle ({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r})){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi }){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })sin\vartheta {\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r}){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {\partial f}{\partial r}})+}
(
e
^
φ
⋅
e
^
φ
)
c
o
s
ϑ
1
r
2
s
i
n
ϑ
∂
f
∂
ϑ
+
(
e
^
φ
⋅
e
^
ϑ
)
1
r
s
i
n
ϑ
∂
∂
φ
(
1
r
∂
f
∂
ϑ
)
−
(
(
e
^
φ
⋅
e
^
r
)
s
i
n
ϑ
+
(
e
^
φ
⋅
e
^
ϑ
)
c
o
s
ϑ
)
1
r
s
i
n
ϑ
∂
f
∂
φ
+
(
e
^
φ
⋅
e
^
φ
)
(
1
r
2
s
i
n
2
ϑ
∂
2
f
∂
2
φ
)
=
{\displaystyle ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })cos\vartheta {\frac {1}{r^{2}sin\vartheta }}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})-(({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r})sin\vartheta +({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta })cos\vartheta ){\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })({\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }})=}
(
e
^
r
⋅
e
^
r
)
∂
2
f
∂
2
r
+
(
e
^
ϑ
⋅
e
^
ϑ
)
1
r
∂
f
∂
r
+
(
e
^
ϑ
⋅
e
^
ϑ
)
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
(
e
^
φ
⋅
e
^
φ
)
s
i
n
ϑ
1
r
s
i
n
ϑ
∂
f
∂
r
+
(
e
^
φ
⋅
e
^
φ
)
c
o
s
ϑ
1
r
2
∂
f
∂
ϑ
+
(
e
^
φ
⋅
e
^
φ
)
(
1
r
2
s
i
n
2
ϑ
∂
2
f
∂
2
φ
)
=
{\displaystyle ({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })sin\vartheta {\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })cos\vartheta {\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })({\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }})=}
∂
2
f
∂
2
r
+
1
r
∂
f
∂
r
+
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
s
i
n
ϑ
1
r
s
i
n
ϑ
∂
f
∂
r
+
c
o
s
ϑ
1
r
2
s
i
n
ϑ
∂
f
∂
ϑ
+
1
r
2
s
i
n
2
ϑ
∂
2
f
∂
2
φ
=
{\displaystyle {\frac {\partial ^{2}f}{\partial ^{2}r}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+sin\vartheta {\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+cos\vartheta {\frac {1}{r^{2}sin\vartheta }}{\frac {\partial f}{\partial \vartheta }}+{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }}=}
∂
2
f
∂
2
r
+
2
r
∂
f
∂
r
+
(
1
r
2
∂
2
f
∂
2
ϑ
)
+
c
o
s
ϑ
s
i
n
ϑ
1
r
2
∂
f
∂
ϑ
+
1
r
2
s
i
n
2
ϑ
∂
2
f
∂
2
φ
=
{\displaystyle {\frac {\partial ^{2}f}{\partial ^{2}r}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+{\frac {cos\vartheta }{sin\vartheta }}{\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }}=}
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
s
i
n
ϑ
∂
∂
ϑ
(
s
i
n
ϑ
∂
f
∂
ϑ
)
+
1
r
2
s
i
n
2
ϑ
(
∂
2
f
∂
2
φ
)
{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial f}{\partial r}})+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial ^{2}f}{\partial ^{2}\varphi }})}
> Laplaciano en coordenadas esféricas. Geometría diferencial
editar
Para hallar el laplaciano en coordenadas esféricas, inicialmente aplicaremos la diferencial al campo escalar 'f, después la estrella de Hodge , otra vez la diferencial y otra vez la estrella de Hodge.
d
i
v
(
g
r
a
d
→
(
f
)
)
=
∇
2
f
=
∗
d
∗
d
(
f
)
=
∗
d
∗
(
∂
f
∂
r
d
r
+
∂
f
∂
ϑ
d
ϑ
+
∂
f
∂
φ
d
φ
)
=
{\displaystyle div({\overrightarrow {grad}}(f))=\nabla ^{2}f=\ast d\ast d(f)=\ast d\ast ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \vartheta }}d\vartheta +{\frac {\partial f}{\partial \varphi }}d\varphi )=}
∗
d
(
|
r
4
s
i
n
2
ϑ
|
1
∂
f
∂
r
(
d
ϑ
∧
d
φ
)
+
|
r
4
s
i
n
2
ϑ
|
r
2
∂
f
∂
ϑ
(
d
φ
∧
d
r
)
+
|
r
4
s
i
n
2
ϑ
|
r
2
s
i
n
2
ϑ
∂
f
∂
φ
(
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast d({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{1}}{\frac {\partial f}{\partial r}}(d\vartheta \wedge d\varphi )+{\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}(d\varphi \wedge dr)+{\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{2}sin^{2}\vartheta }}{\frac {\partial f}{\partial \varphi }}(dr\wedge d\vartheta ))=}
∗
(
∂
∂
r
(
r
2
r
2
s
i
n
2
ϑ
1
2
∂
f
∂
r
)
(
d
r
∧
d
ϑ
∧
d
φ
)
+
∂
∂
ϑ
(
r
2
s
i
n
2
ϑ
r
2
2
∂
f
∂
ϑ
)
(
d
ϑ
∧
d
φ
∧
d
r
)
+
∂
∂
φ
(
r
2
r
2
s
i
n
2
ϑ
2
∂
f
∂
φ
)
(
d
φ
∧
d
r
∧
d
ϑ
)
)
=
{\displaystyle \ast ({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})(dr\wedge d\vartheta \wedge d\varphi )+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})(d\vartheta \wedge d\varphi \wedge dr)+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})(d\varphi \wedge dr\wedge d\vartheta ))=}
∗
(
(
∂
∂
r
(
r
2
r
2
s
i
n
2
ϑ
1
2
∂
f
∂
r
)
+
∂
∂
ϑ
(
r
2
s
i
n
2
ϑ
r
2
2
∂
f
∂
ϑ
)
+
∂
∂
φ
(
r
2
r
2
s
i
n
2
ϑ
2
∂
f
∂
φ
)
(
d
r
∧
d
ϑ
∧
d
φ
)
)
=
{\displaystyle \ast (({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})(dr\wedge d\vartheta \wedge d\varphi ))=}
(
|
r
4
s
i
n
2
ϑ
|
r
4
s
i
n
2
ϑ
)
∂
∂
r
(
r
2
r
2
s
i
n
2
ϑ
1
2
∂
f
∂
r
)
+
(
|
r
4
s
i
n
2
ϑ
|
r
4
s
i
n
2
ϑ
)
∂
∂
ϑ
(
r
2
s
i
n
2
ϑ
r
2
2
∂
f
∂
ϑ
)
+
(
|
r
4
s
i
n
2
ϑ
|
r
4
s
i
n
2
ϑ
)
∂
∂
φ
(
r
2
r
2
s
i
n
2
ϑ
2
∂
f
∂
φ
)
=
{\displaystyle ({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})=}
(
|
r
4
s
i
n
2
ϑ
|
r
4
s
i
n
2
ϑ
)
(
∂
∂
r
(
r
2
r
2
s
i
n
2
ϑ
1
2
∂
f
∂
r
)
+
∂
∂
ϑ
(
r
2
s
i
n
2
ϑ
r
2
2
∂
f
∂
ϑ
)
+
∂
∂
φ
(
r
2
r
2
s
i
n
2
ϑ
2
∂
f
∂
φ
)
)
=
{\displaystyle ({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }}))=}
(
1
r
4
s
i
n
2
ϑ
)
(
∂
∂
r
(
r
2
r
2
s
i
n
2
ϑ
1
2
∂
f
∂
r
)
+
∂
∂
ϑ
(
r
2
s
i
n
2
ϑ
r
2
2
∂
f
∂
ϑ
)
+
∂
∂
φ
(
r
2
r
2
s
i
n
2
ϑ
2
∂
f
∂
φ
)
)
=
{\displaystyle ({\frac {1}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }}))=}
1
r
2
s
i
n
ϑ
(
∂
∂
r
(
r
2
s
i
n
ϑ
∂
f
∂
r
)
+
∂
∂
ϑ
(
s
i
n
ϑ
∂
f
∂
ϑ
)
+
∂
∂
φ
(
1
s
i
n
ϑ
∂
f
∂
φ
)
)
=
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
s
i
n
ϑ
∂
∂
ϑ
(
s
i
n
ϑ
∂
f
∂
ϑ
)
+
1
r
2
s
i
n
2
ϑ
(
∂
2
f
∂
2
φ
)
{\displaystyle {\frac {1}{r^{2}sin\vartheta }}({\frac {\partial }{\partial r}}(r^{2}sin\vartheta {\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\frac {1}{sin\vartheta }}{\frac {\partial f}{\partial \varphi }}))={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial f}{\partial r}})+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial ^{2}f}{\partial ^{2}\varphi }})}
>> Operadores diferenciales en coordenadas generalizadas
editar
* Productos escalares en coordenadas generalizadas
editar
Serán necesarios los siguientes productos escalares para coordenadas generalizadas .
u
^
1
⋅
u
^
1
=
1
;
u
^
2
⋅
u
^
2
=
1
;
u
^
3
⋅
u
^
3
=
1
{\displaystyle {\widehat {u}}_{1}\cdot {\widehat {u}}_{1}=1;{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}=1;{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}=1}
u
^
2
⋅
u
^
1
=
0
;
u
^
1
⋅
u
^
2
=
0
;
u
^
3
⋅
u
^
1
=
0
;
u
^
1
⋅
u
^
3
=
0
;
u
^
2
⋅
u
^
3
=
0
;
u
^
3
⋅
u
^
2
=
0
{\displaystyle {\widehat {u}}_{2}\cdot {\widehat {u}}_{1}=0;{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}=0;{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}=0;{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}=0;{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}=0;{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}=0}
* Productos vectoriales en coordenadas generalizadas
editar
Utilizaremos en los desarrollos los siguientes productos vectoriales .
u
^
1
×
u
^
1
=
0
;
u
^
2
×
u
^
2
=
0
;
u
^
3
×
u
^
3
=
0
{\displaystyle {\widehat {u}}_{1}\times {\widehat {u}}_{1}=0;{\widehat {u}}_{2}\times {\widehat {u}}_{2}=0;{\widehat {u}}_{3}\times {\widehat {u}}_{3}=0}
u
^
1
×
u
^
2
=
u
^
3
;
u
^
2
×
u
^
3
=
u
^
1
;
u
^
3
×
u
^
1
=
u
^
2
{\displaystyle {\widehat {u}}_{1}\times {\widehat {u}}_{2}={\widehat {u}}_{3};{\widehat {u}}_{2}\times {\widehat {u}}_{3}={\widehat {u}}_{1};{\widehat {u}}_{3}\times {\widehat {u}}_{1}={\widehat {u}}_{2}}
u
^
2
×
u
^
1
=
−
u
^
3
;
u
^
3
×
u
^
2
=
−
u
^
1
;
u
^
1
×
u
^
3
=
−
u
^
2
{\displaystyle {\widehat {u}}_{2}\times {\widehat {u}}_{1}=-{\widehat {u}}_{3};{\widehat {u}}_{3}\times {\widehat {u}}_{2}=-{\widehat {u}}_{1};{\widehat {u}}_{1}\times {\widehat {u}}_{3}=-{\widehat {u}}_{2}}
* Divergencia del rotacional
editar
En las demostraciones generalizadas en cálculo diferencial necesitaremos estas identidades para los factores de escala h_1, h_2 i h_3.
La divergencia del rotacional vale cero.
∇
→
⋅
(
e
^
x
h
n
×
e
^
y
h
m
)
=
0
{\displaystyle {\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{n}}}\times {\frac {{\widehat {e}}_{y}}{h_{m}}})=0}
∇
→
⋅
(
e
^
x
h
n
×
e
^
z
h
m
)
=
0
{\displaystyle {\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{n}}}\times {\frac {{\widehat {e}}_{z}}{h_{m}}})=0}
∇
→
⋅
(
e
^
y
h
n
×
e
^
z
h
m
)
=
0
{\displaystyle {\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{y}}{h_{n}}}\times {\frac {{\widehat {e}}_{z}}{h_{m}}})=0}
* Factor de escala en coordenadas generalizadas
editar
El factor de escala en cálculo diferencial nos permite pasar de unas coordenadas a otras.
h
→
=
∑
|
∂
r
→
∂
u
i
|
=
(
|
∂
x
∂
u
1
|
,
|
∂
y
∂
u
2
|
,
|
∂
z
∂
u
3
|
)
=
(
h
1
,
h
2
,
h
3
)
{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial {\overrightarrow {r}}}{\partial u_{i}}}|=(|{\frac {\partial x}{\partial u_{1}}}|,|{\frac {\partial y}{\partial u_{2}}}|,|{\frac {\partial z}{\partial u_{3}}}|)=(h_{1},h_{2},h_{3})}
geometría de Riemann .
h
→
=
∑
|
∂
∂
u
i
|
=
(
|
∂
∂
u
1
|
,
|
|
∂
∂
u
2
|
,
|
∂
∂
u
3
|
)
=
(
g
11
,
g
22
,
g
33
)
{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial u_{1}}}|,|{\frac {|\partial }{\partial u_{2}}}|,|{\frac {\partial }{\partial u_{3}}}|)=({\sqrt[{}]{g_{11}}},{\sqrt[{}]{g_{22}}},{\sqrt[{}]{g_{33}}})}
* Métrica o tensor métrico en coordenadas generalizadas
editar
La métrica (tensor métrico ) de una variedad de Riemann nos permite obtener los coeficientes de un elemento de longitud en las bases deseadas.
En este caso tenemos la métrica de las coordenadas esféricas .
g
α
β
=
g
11
d
u
1
2
+
g
22
d
u
2
2
+
g
33
d
u
3
2
{\displaystyle g_{\alpha \beta }=g_{11}du_{1}^{2}+g_{22}du_{2}^{2}+g_{33}du_{3}^{2}}
g
α
β
=
[
|
∂
∂
u
1
|
⋅
|
∂
∂
u
1
|
|
∂
∂
u
2
|
⋅
|
∂
∂
u
2
|
|
∂
∂
u
3
|
⋅
|
∂
∂
u
3
|
]
=
[
g
11
g
22
g
33
]
{\displaystyle g_{\alpha \beta }={\begin{bmatrix}\vert {\frac {\partial }{\partial u_{1}}}\vert \cdot |{\frac {\partial }{\partial u_{1}}}|&&\\&|{\frac {\partial }{\partial u_{2}}}|\cdot |{\frac {\partial }{\partial u_{2}}}|&\\&&|{\frac {\partial }{\partial u_{3}}}|\cdot |{\frac {\partial }{\partial u_{3}}}|\end{bmatrix}}={\begin{bmatrix}g_{11}&&\\&g_{22}&\\&&g_{33}\end{bmatrix}}}
d
e
t
(
g
α
β
)
=
|
g
|
=
g
11
g
22
g
33
{\displaystyle det(g_{\alpha \beta })=|g|=g_{11}g_{22}g_{33}}
* Estrella de Hodge (*) en coordenadas generalizadas
editar
La estrella de Hodge es un operador que actúa sobre un p-forma diferencial en un espacio de dimensión n.
∗
(
F
(
d
x
1
∧
d
x
2
∧
d
x
3
∧
.
.
.
∧
d
x
p
)
=
|
g
|
g
11
g
22
g
33
.
.
.
g
p
p
ε
F
(
d
x
p
+
1
∧
d
x
p
+
2
∧
d
x
p
+
3
∧
.
.
.
∧
d
x
n
−
p
)
{\displaystyle *(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})}
∗
(
F
d
x
1
)
=
|
g
|
g
11
F
(
d
x
2
∧
d
x
3
)
{\displaystyle \ast (Fdx_{1})={\frac {\sqrt[{}]{|g|}}{g_{11}}}F(dx_{2}\wedge dx_{3})}
∗
(
F
(
d
x
1
∧
d
x
2
)
)
=
|
g
|
g
11
g
22
F
d
x
3
{\displaystyle \ast (F(dx_{1}\wedge dx_{2}))={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}}}Fdx_{3}}
∗
(
F
(
d
x
1
∧
d
x
2
∧
d
x
3
)
)
=
|
g
|
g
11
g
22
g
33
F
{\displaystyle \ast (F(dx_{1}\wedge dx_{2}\wedge dx_{3}))={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}}}F}
* Subir y bajar índices. Coordenadas generalizadas
editar
[Ley de subir o bajar índices (tensores)| Subir y bajar índices]] nos permite pasar de la base de un espacio vectorial a la base dual . Tenemos el contravector o vector del espacio inicial y el covector o 1-forma diferencial de la base diferencial.
∂
∂
x
α
=
g
α
β
d
x
β
{\displaystyle {\frac {\partial }{\partial x^{\alpha }}}=g_{\alpha \beta }dx^{\beta }}
d
x
α
=
g
α
β
∂
∂
x
β
{\displaystyle dx^{\alpha }=g^{\alpha \beta }{\frac {\partial }{\partial x^{\beta }}}}
> Gradiente en coordenadas generalizadas. Cálculo diferencial
editar
g
r
a
d
→
(
f
)
=
∇
→
(
f
)
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
f
(
x
,
y
,
z
)
=
(
e
^
x
∂
∂
x
+
e
^
y
∂
∂
y
+
e
^
z
∂
∂
z
)
f
(
x
,
y
,
z
)
=
{\displaystyle {\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})f(x,y,z)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})f(x,y,z)=}
(
u
^
1
∂
u
1
∂
x
∂
∂
u
1
+
u
^
2
∂
u
2
∂
y
∂
∂
u
2
+
u
^
3
∂
u
3
∂
z
∂
∂
u
3
)
f
(
u
1
,
u
2
,
u
3
)
=
1
|
∂
x
∂
u
1
|
∂
f
∂
u
1
u
^
1
+
1
|
∂
u
2
∂
y
|
∂
f
∂
u
2
u
^
2
+
1
|
∂
u
3
∂
z
|
∂
f
∂
u
3
u
^
3
=
{\displaystyle ({\widehat {u}}_{1}{\frac {\partial u_{1}}{\partial x}}{\frac {\partial }{\partial u_{1}}}+{\widehat {u}}_{2}{\frac {\partial u_{2}}{\partial y}}{\frac {\partial }{\partial u_{2}}}+{\widehat {u}}_{3}{\frac {\partial u_{3}}{\partial z}}{\frac {\partial }{\partial u_{3}}})f(u_{1},u_{2},u_{3})={\frac {1}{|{\frac {\partial x}{\partial u_{1}}}|}}{\frac {\partial f}{\partial u_{1}}}{\widehat {u}}_{1}+{\frac {1}{|{\frac {\partial u_{2}}{\partial y}}|}}{\frac {\partial f}{\partial u_{2}}}{\widehat {u}}_{2}+{\frac {1}{|{\frac {\partial u_{3}}{\partial z}}|}}{\frac {\partial f}{\partial u_{3}}}{\widehat {u}}_{3}=}
1
h
1
∂
f
∂
u
1
u
^
1
+
1
h
2
∂
f
∂
u
2
u
^
2
+
1
h
2
∂
f
∂
u
3
u
^
3
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}}{\widehat {u}}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}}{\widehat {u}}_{2}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{3}}}{\widehat {u}}_{3}}
> Gradiente en coordenadas generalizadas. Geometría diferencial
editar
Obtendremos el gradiente de un campo escalar en geometría diferencial aplicando la diferencial exterior y posteriormente bajaremos índices.
g
r
a
d
→
(
f
)
=
∇
→
(
f
)
=↓
d
(
f
(
u
1
,
u
2
,
u
3
)
)
=↓
(
∂
f
∂
u
1
d
u
1
+
∂
f
∂
u
2
d
u
2
+
∂
f
∂
u
3
d
u
3
)
=
∂
f
∂
u
1
1
g
11
∂
∂
u
1
+
∂
f
∂
u
2
1
g
22
∂
∂
u
2
+
∂
f
∂
u
3
1
g
33
∂
∂
u
3
=
{\displaystyle {\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=\downarrow d(f(u_{1},u_{2},u_{3}))=\downarrow ({\frac {\partial f}{\partial u_{1}}}du_{1}+{\frac {\partial f}{\partial u_{2}}}du_{2}+{\frac {\partial f}{\partial u_{3}}}du_{3})={\frac {\partial f}{\partial u_{1}}}{\frac {1}{g_{11}}}{\frac {\partial }{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{g_{22}}}{\frac {\partial }{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{g_{33}}}{\frac {\partial }{\partial u_{3}}}=}
∂
f
∂
u
1
1
g
11
|
∂
∂
u
1
|
∂
∂
u
1
|
∂
∂
u
1
|
+
∂
f
∂
u
2
1
g
22
|
∂
∂
u
2
|
∂
∂
u
2
|
∂
∂
u
2
|
+
∂
f
∂
u
3
1
g
33
|
∂
∂
u
3
|
∂
∂
u
3
|
∂
∂
u
3
|
=
∂
f
∂
u
1
g
11
g
11
∂
^
∂
u
1
+
∂
f
∂
u
2
g
22
g
22
∂
^
∂
u
2
+
∂
f
∂
u
3
g
33
g
33
∂
^
∂
u
3
=
{\displaystyle {\frac {\partial f}{\partial u_{1}}}{\frac {1}{g_{11}}}|{\frac {\partial }{\partial u_{1}}}|{\frac {\frac {\partial }{\partial u_{1}}}{|{\frac {\partial }{\partial u_{1}}}|}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{g_{22}}}|{\frac {\partial }{\partial u_{2}}}|{\frac {\frac {\partial }{\partial u_{2}}}{|{\frac {\partial }{\partial u_{2}}}|}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{g_{33}}}|{\frac {\partial }{\partial u_{3}}}|{\frac {\frac {\partial }{\partial u_{3}}}{|{\frac {\partial }{\partial u_{3}}}|}}={\frac {\partial f}{\partial u_{1}}}{\frac {\sqrt[{}]{g_{11}}}{g_{11}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {\sqrt[{}]{g_{22}}}{g_{22}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {\sqrt[{}]{g_{33}}}{g_{33}}}{\frac {\widehat {\partial }}{\partial u_{3}}}=}
∂
f
∂
u
1
1
g
11
∂
^
∂
u
1
+
∂
f
∂
u
2
1
g
22
∂
^
∂
u
2
+
∂
f
∂
u
3
1
g
33
∂
^
∂
u
3
{\displaystyle {\frac {\partial f}{\partial u_{1}}}{\frac {1}{\sqrt[{}]{g_{11}}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{\sqrt[{}]{g_{22}}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{\sqrt[{}]{g_{33}}}}{\frac {\widehat {\partial }}{\partial u_{3}}}}
> Divergencia en coordenadas generalizadas. Cálculo diferencial
editar
∇
→
⋅
F
→
=
∇
→
⋅
(
F
1
e
^
x
+
F
2
e
^
y
+
F
3
e
^
z
)
=
∇
→
⋅
(
F
1
e
^
x
)
+
∇
→
⋅
(
F
2
e
^
y
)
+
∇
→
⋅
(
F
3
e
^
z
)
=
{\displaystyle {\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}={\overrightarrow {\nabla }}\cdot (F_{1}{\widehat {e}}_{x}+F_{2}{\widehat {e}}_{y}+F_{3}{\widehat {e}}_{z})={\overrightarrow {\nabla }}\cdot (F_{1}{\widehat {e}}_{x})+{\overrightarrow {\nabla }}\cdot (F_{2}{\widehat {e}}_{y})+{\overrightarrow {\nabla }}\cdot (F_{3}{\widehat {e}}_{z})=}
(
∇
→
F
1
)
⋅
e
^
x
+
F
1
(
∇
→
⋅
e
^
x
)
+
(
∇
→
F
2
)
⋅
e
^
y
+
F
2
(
∇
→
⋅
e
^
y
)
+
(
∇
→
F
3
)
⋅
e
^
z
+
F
3
(
∇
→
⋅
e
^
z
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{y})+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{z})=}
(
∇
→
F
1
)
⋅
e
^
x
+
F
1
(
∇
→
⋅
(
e
^
y
×
e
^
z
)
)
+
(
∇
→
F
2
)
⋅
e
^
y
+
F
2
(
∇
→
⋅
(
e
^
z
×
e
^
x
)
)
+
(
∇
→
F
3
)
⋅
e
^
z
+
F
3
(
∇
→
⋅
(
e
^
x
×
e
^
y
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z}))+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x}))+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y}))=}
(
∇
→
F
1
)
⋅
e
^
x
+
F
1
∇
→
(
h
2
h
3
⋅
(
e
^
y
h
2
×
e
^
z
h
3
)
)
+
(
∇
→
F
2
)
⋅
e
^
y
+
F
2
∇
→
(
h
3
h
1
⋅
(
e
^
z
h
3
×
e
^
x
h
1
)
)
+
(
∇
→
F
3
)
⋅
e
^
z
+
F
3
∇
→
(
h
1
h
2
⋅
(
e
^
x
h
1
×
e
^
y
h
2
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}{\overrightarrow {\nabla }}(h_{2}h_{3}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}{\overrightarrow {\nabla }}(h_{3}h_{1}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}{\overrightarrow {\nabla }}(h_{1}h_{2}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=}
(
∇
→
F
1
)
⋅
e
^
x
+
F
1
(
∇
→
(
h
2
h
3
)
)
⋅
(
e
^
y
h
2
×
e
^
z
h
3
)
+
F
1
h
2
h
3
(
∇
→
⋅
(
e
^
y
h
2
×
e
^
z
h
3
)
)
+
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}})+F_{1}h_{2}h_{3}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+}
(
∇
→
F
2
)
⋅
e
^
y
+
F
2
(
∇
→
(
h
3
h
1
)
)
⋅
(
e
^
z
h
3
×
e
^
x
h
1
)
+
F
2
h
3
h
1
(
∇
→
⋅
(
e
^
z
h
3
×
e
^
x
h
1
)
)
+
{\displaystyle ({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}})+F_{2}h_{3}h_{1}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+}
(
∇
→
F
3
)
⋅
e
^
z
+
F
3
(
∇
→
(
h
1
h
2
)
)
⋅
(
e
^
x
h
1
×
e
^
y
h
2
)
+
F
3
h
1
h
2
(
∇
→
⋅
(
e
^
x
h
1
×
e
^
y
h
2
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}})+F_{3}h_{1}h_{2}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=}
(
∇
→
F
1
)
⋅
e
^
x
+
F
1
h
2
h
3
(
∇
→
(
h
2
h
3
)
)
⋅
(
e
^
y
×
e
^
z
)
+
(
∇
→
F
2
)
⋅
e
^
y
+
F
2
h
3
h
1
(
∇
→
(
h
3
h
1
)
)
⋅
(
e
^
z
×
e
^
x
)
+
(
∇
→
F
3
)
⋅
e
^
z
+
F
3
h
1
h
2
(
∇
→
(
h
1
h
2
)
)
⋅
(
e
^
x
×
e
^
y
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+{\frac {F_{1}}{h_{2}h_{3}}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z})+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+{\frac {F_{2}}{h_{3}h_{1}}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+{\frac {F_{3}}{h_{1}h_{2}}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y})=}
(
∇
→
F
1
)
⋅
e
^
x
+
F
1
h
2
h
3
(
∇
→
(
h
2
h
3
)
)
⋅
e
^
x
+
(
∇
→
F
2
)
⋅
e
^
y
+
F
2
h
3
h
1
(
∇
→
(
h
3
h
1
)
)
⋅
e
^
y
+
(
∇
→
F
3
)
⋅
e
^
z
+
F
3
h
1
h
2
(
∇
→
(
h
1
h
2
)
)
⋅
e
^
z
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+{\frac {F_{1}}{h_{2}h_{3}}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot {\widehat {e}}_{x}+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+{\frac {F_{2}}{h_{3}h_{1}}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot {\widehat {e}}_{y}+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+{\frac {F_{3}}{h_{1}h_{2}}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot {\widehat {e}}_{z}=}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
F
1
)
)
⋅
u
^
1
+
F
1
h
2
h
3
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
2
h
3
)
)
⋅
u
^
1
+
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(F_{1}))\cdot {\widehat {u}}_{1}+{\frac {F_{1}}{h_{2}h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{2}h_{3}))\cdot {\widehat {u}}_{1}+}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
F
2
)
)
⋅
u
^
2
+
F
2
h
1
h
3
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
1
h
3
)
)
⋅
u
^
2
+
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(F_{2}))\cdot {\widehat {u}}_{2}+{\frac {F_{2}}{h_{1}h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{3}))\cdot {\widehat {u}}_{2}+}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
F
3
)
)
⋅
u
^
3
+
F
3
h
1
h
2
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
1
h
2
)
)
⋅
u
^
3
=
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(F_{3}))\cdot {\widehat {u}}_{3}+{\frac {F_{3}}{h_{1}h_{2}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{2}))\cdot {\widehat {u}}_{3}=}
1
h
1
u
^
1
⋅
u
^
1
∂
F
1
∂
u
1
+
1
h
2
u
^
2
⋅
u
^
1
∂
F
1
∂
u
2
+
1
h
3
u
^
3
⋅
u
^
1
∂
F
1
∂
u
3
+
F
1
(
u
^
1
⋅
u
^
1
h
1
h
2
h
3
∂
(
h
2
h
3
)
∂
u
1
+
u
^
2
⋅
u
^
1
h
2
h
2
h
3
∂
(
h
2
h
3
)
∂
u
2
+
u
^
3
⋅
u
^
1
h
2
h
3
h
3
∂
(
h
2
h
3
)
∂
u
3
)
+
{\displaystyle {\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}{\frac {\partial F_{1}}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}{\frac {\partial F_{1}}{\partial u_{3}}}+F_{1}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}}{h_{2}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}}{h_{2}h_{3}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{3}}})+}
1
h
1
u
^
1
⋅
u
^
2
∂
F
2
∂
u
1
+
1
h
2
u
^
2
⋅
u
^
2
∂
F
2
∂
u
2
+
1
h
3
u
^
3
⋅
u
^
2
∂
F
2
∂
u
3
+
F
2
(
u
^
1
⋅
u
^
2
h
1
h
1
h
3
∂
(
h
1
h
3
)
∂
u
1
+
u
^
2
⋅
u
^
2
h
1
h
2
h
3
∂
(
h
1
h
3
)
∂
u
2
+
u
^
3
⋅
u
^
2
h
1
h
3
h
3
∂
(
h
1
h
3
)
∂
u
3
)
+
{\displaystyle {\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}{\frac {\partial F_{2}}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}{\frac {\partial F_{2}}{\partial u_{3}}}+F_{2}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}}{h_{1}h_{1}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}}{h_{1}h_{3}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{3}}})+}
1
h
1
u
^
1
⋅
u
^
3
∂
F
3
∂
u
1
+
1
h
2
u
^
2
⋅
u
^
3
∂
F
3
∂
u
2
+
1
h
3
u
^
3
⋅
u
^
3
∂
F
3
∂
u
3
+
F
3
(
u
^
1
⋅
u
^
3
h
1
h
1
h
2
∂
(
h
1
h
2
)
∂
u
1
+
u
^
2
⋅
u
^
3
h
1
h
2
h
2
∂
(
h
1
h
2
)
∂
u
2
+
u
^
3
⋅
u
^
3
h
1
h
2
h
3
∂
(
h
1
h
2
)
∂
u
3
)
=
{\displaystyle {\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}{\frac {\partial F_{3}}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}{\frac {\partial F_{3}}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}{\frac {\partial F_{3}}{\partial u_{3}}}+F_{3}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}}{h_{1}h_{1}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}})=}
1
h
1
(
u
^
1
⋅
u
^
1
)
∂
F
1
∂
u
1
+
F
1
h
1
h
2
h
3
(
u
^
1
⋅
u
^
1
)
∂
(
h
2
h
3
)
∂
u
1
+
1
h
2
(
u
^
2
⋅
u
^
2
)
∂
F
2
∂
u
2
+
{\displaystyle {\frac {1}{h_{1}}}({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {F_{1}}{h_{1}h_{2}h_{3}}}({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial F_{2}}{\partial u_{2}}}+}
F
2
h
1
h
2
h
3
(
u
^
2
⋅
u
^
2
)
∂
(
h
1
h
3
)
∂
u
2
+
1
h
3
(
u
^
3
⋅
u
^
3
)
∂
F
3
∂
u
3
+
F
3
h
1
h
2
h
3
(
u
^
3
⋅
u
^
3
)
∂
(
h
1
h
2
)
∂
u
3
=
{\displaystyle {\frac {F_{2}}{h_{1}h_{2}h_{3}}}({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial F_{3}}{\partial u_{3}}}+{\frac {F_{3}}{h_{1}h_{2}h_{3}}}({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=}
1
h
1
∂
F
1
∂
u
1
+
F
1
h
1
h
2
h
3
∂
(
h
2
h
3
)
∂
u
1
+
1
h
2
∂
F
2
∂
u
2
+
F
2
h
1
h
2
h
3
∂
(
h
1
h
3
)
∂
u
2
+
1
h
3
∂
F
3
∂
u
3
+
F
3
h
1
h
2
h
3
∂
(
h
1
h
2
)
∂
u
3
=
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {F_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {F_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}}+{\frac {F_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=}
h
2
h
3
h
1
h
2
h
3
∂
F
1
∂
u
1
+
F
1
h
1
h
2
h
3
∂
(
h
2
h
3
)
∂
u
1
+
h
1
h
3
h
1
h
2
h
3
∂
F
2
∂
u
2
+
F
2
h
1
h
2
h
3
∂
(
h
1
h
3
)
∂
u
2
+
h
1
h
2
h
1
h
2
h
3
∂
F
3
∂
u
3
+
F
3
h
1
h
2
h
3
∂
(
h
1
h
2
)
∂
u
3
=
{\displaystyle {\frac {h_{2}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {F_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {h_{1}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {F_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {h_{1}h_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}}+{\frac {F_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=}
1
h
1
h
2
h
3
∂
(
F
1
h
2
h
3
)
∂
u
1
+
1
h
1
h
2
h
3
∂
(
F
2
h
1
h
3
)
∂
u
2
+
1
h
1
h
2
h
3
∂
(
F
3
h
1
h
2
)
∂
u
3
=
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial (F_{1}h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial (F_{2}h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial (F_{3}h_{1}h_{2})}{\partial u_{3}}}=}
1
h
1
h
2
h
3
(
∂
(
F
1
h
2
h
3
)
∂
u
1
+
∂
(
F
2
h
1
h
3
)
∂
u
2
+
∂
(
F
3
h
1
h
2
)
∂
u
3
)
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial (F_{1}h_{2}h_{3})}{\partial u_{1}}}+{\frac {\partial (F_{2}h_{1}h_{3})}{\partial u_{2}}}+{\frac {\partial (F_{3}h_{1}h_{2})}{\partial u_{3}}})}
> Divergencia en coordenadas generalizadas. Geometría diferencial
editar
Para obtener la divergencia de un campo vectorial , bajaremos índices, aplicaremos la estrella de Hodge , aplicaremos la diferencial exterior y volveremos a aplicar la estrella de Hodge .
d
i
v
(
F
→
)
=
∇
→
⋅
F
→
=
∗
d
∗
↓
(
F
→
)
=
∗
d
∗
↓
(
F
x
e
^
x
+
F
y
e
^
y
+
F
z
e
^
z
)
=
∗
d
∗
↓
(
F
1
1
|
∂
x
∂
u
1
|
u
1
→
+
F
2
1
|
∂
y
∂
u
2
|
u
2
→
+
F
3
1
|
∂
z
∂
u
3
|
u
3
→
)
=
{\displaystyle div({\overrightarrow {F}})={\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=\ast d\ast \downarrow ({\overrightarrow {F}})=\ast d\ast \downarrow (F_{x}{\widehat {e}}_{x}+F_{y}{\widehat {e}}_{y}+F_{z}{\widehat {e}}_{z})=\ast d\ast \downarrow (F_{1}{\frac {1}{|{\frac {\partial x}{\partial u_{1}}}|}}{\overrightarrow {u_{1}}}+F_{2}{\frac {1}{|{\frac {\partial y}{\partial u_{2}}}|}}{\overrightarrow {u_{2}}}+F_{3}{\frac {1}{|{\frac {\partial z}{\partial u_{3}}}|}}{\overrightarrow {u_{3}}})=}
∗
d
∗
↓
(
F
1
1
g
11
2
u
1
→
+
F
2
1
g
22
2
u
2
→
+
F
3
1
g
33
2
u
3
→
)
=
∗
d
∗
(
F
1
1
g
11
2
g
11
d
u
1
+
F
2
1
g
22
2
g
22
d
u
2
+
F
3
1
g
33
2
g
33
d
u
3
)
=
{\displaystyle \ast d\ast \downarrow (F_{1}{\frac {1}{\sqrt[{2}]{g_{11}}}}{\overrightarrow {u_{1}}}+F_{2}{\frac {1}{\sqrt[{2}]{g_{22}}}}{\overrightarrow {u_{2}}}+F_{3}{\frac {1}{\sqrt[{2}]{g_{33}}}}{\overrightarrow {u_{3}}})=\ast d\ast (F_{1}{\frac {1}{\sqrt[{2}]{g_{11}}}}g_{11}du_{1}+F_{2}{\frac {1}{\sqrt[{2}]{g_{22}}}}g_{22}du_{2}+F_{3}{\frac {1}{\sqrt[{2}]{g_{33}}}}g_{33}du_{3})=}
∗
d
∗
(
F
1
g
11
2
d
u
1
+
F
2
g
22
2
d
u
2
+
F
3
g
33
2
d
u
3
)
=
∗
d
(
F
1
g
11
2
g
2
g
11
(
d
u
2
∧
d
u
3
)
+
F
2
g
22
2
g
2
g
22
(
d
u
3
∧
d
u
1
)
+
F
1
g
33
2
g
2
g
33
(
d
u
1
∧
d
u
2
)
)
=
{\displaystyle \ast d\ast (F_{1}{\sqrt[{2}]{g_{11}}}du_{1}+F_{2}{\sqrt[{2}]{g_{22}}}du_{2}+F_{3}{\sqrt[{2}]{g_{33}}}du_{3})=\ast d(F_{1}{\sqrt[{2}]{g_{11}}}{\frac {\sqrt[{2}]{g}}{g_{11}}}(du_{2}\wedge du_{3})+F_{2}{\sqrt[{2}]{g_{22}}}{\frac {\sqrt[{2}]{g}}{g_{22}}}(du_{3}\wedge du_{1})+F_{1}{\sqrt[{2}]{g_{33}}}{\frac {\sqrt[{2}]{g}}{g_{33}}}(du_{1}\wedge du_{2}))=}
∗
d
(
F
1
g
2
g
11
2
(
d
u
2
∧
d
u
3
)
+
F
2
g
2
g
22
2
(
d
u
3
∧
d
u
1
)
+
F
3
g
2
g
33
2
(
d
u
1
∧
d
u
2
)
)
=
{\displaystyle \ast d(F_{1}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{11}}}}(du_{2}\wedge du_{3})+F_{2}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{22}}}}(du_{3}\wedge du_{1})+F_{3}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{33}}}}(du_{1}\wedge du_{2}))=}
∗
(
∂
∂
u
1
(
F
1
g
22
g
33
2
)
(
d
u
1
∧
d
u
2
∧
d
u
3
)
+
∂
∂
u
2
(
F
2
g
11
g
33
2
)
(
d
u
2
∧
d
u
3
∧
d
u
1
)
+
∂
∂
u
3
(
F
3
g
22
g
11
2
)
(
d
u
3
∧
d
u
1
∧
d
u
2
)
)
=
{\displaystyle \ast ({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})(du_{1}\wedge du_{2}\wedge du_{3})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})(du_{2}\wedge du_{3}\wedge du_{1})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}})(du_{3}\wedge du_{1}\wedge du_{2}))=}
∗
(
(
∂
∂
u
1
(
F
1
g
22
g
33
2
)
+
∂
∂
u
2
(
F
2
g
11
g
33
2
)
+
∂
∂
u
3
(
F
3
g
22
g
11
2
)
)
(
d
u
1
∧
d
u
2
∧
d
u
3
)
)
=
{\displaystyle \ast (({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}}))(du_{1}\wedge du_{2}\wedge du_{3}))=}
g
2
g
11
g
22
g
22
∂
∂
u
1
(
F
1
g
22
g
33
2
)
+
g
2
g
11
g
22
g
22
∂
∂
u
2
(
F
2
g
11
g
33
2
)
+
g
2
g
11
g
22
g
22
∂
∂
u
3
(
F
3
g
22
g
11
2
)
=
{\displaystyle {\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}})=}
1
g
11
g
22
g
22
2
(
∂
∂
u
1
(
F
1
g
22
g
33
2
)
+
∂
∂
u
2
(
F
2
g
11
g
33
2
)
+
∂
∂
u
3
(
F
3
g
22
g
11
2
)
)
=
1
h
1
h
2
h
3
(
∂
∂
u
1
(
F
1
h
2
h
3
)
+
∂
∂
u
2
(
F
2
h
1
h
3
)
+
∂
∂
u
3
(
F
3
h
1
h
2
)
)
{\displaystyle {\frac {1}{\sqrt[{2}]{g_{11}g_{22}g_{22}}}}({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}}))={\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}(F_{1}h_{2}h_{3})+{\frac {\partial }{\partial u_{2}}}(F_{2}h_{1}h_{3})+{\frac {\partial }{\partial u_{3}}}(F_{3}h_{1}h_{2}))}
> Rotacional en coordenadas generalizadas. Cálculo diferencial
editar
r
o
t
(
F
→
)
=
∇
→
×
F
→
=
∇
→
×
(
F
1
e
^
x
+
F
2
e
^
y
+
F
3
e
^
z
)
=
∇
→
×
(
F
1
e
^
x
)
+
∇
→
×
(
F
2
e
^
y
)
+
∇
→
×
(
F
3
e
^
z
)
+
{\displaystyle rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}={\overrightarrow {\nabla }}\times (F_{1}{\widehat {e}}_{x}+F_{2}{\widehat {e}}_{y}+F_{3}{\widehat {e}}_{z})={\overrightarrow {\nabla }}\times (F_{1}{\widehat {e}}_{x})+{\overrightarrow {\nabla }}\times (F_{2}{\widehat {e}}_{y})+{\overrightarrow {\nabla }}\times (F_{3}{\widehat {e}}_{z})+}
(
∇
→
F
1
)
×
e
^
x
+
F
1
(
∇
→
×
e
^
x
)
+
(
∇
→
F
2
)
×
e
^
y
+
F
2
(
∇
→
×
e
^
y
)
+
(
∇
→
F
3
)
×
e
^
z
+
F
3
(
∇
→
×
e
^
z
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}\times {\widehat {e}}_{y})+({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}\times {\widehat {e}}_{z})=}
(
∇
→
F
1
)
×
e
^
x
+
F
1
∇
→
×
(
h
1
e
^
x
h
1
)
+
(
∇
→
F
2
)
×
e
^
y
+
F
2
∇
→
×
(
h
2
e
^
y
h
2
)
+
(
∇
→
F
3
)
×
e
^
z
+
F
3
∇
→
×
(
h
3
e
^
z
h
3
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}{\overrightarrow {\nabla }}\times (h_{1}{\frac {{\widehat {e}}_{x}}{h_{1}}})+({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}{\overrightarrow {\nabla }}\times (h_{2}{\frac {{\widehat {e}}_{y}}{h_{2}}})+({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}{\overrightarrow {\nabla }}\times (h_{3}{\frac {{\widehat {e}}_{z}}{h_{3}}})=}
(
∇
→
F
1
)
×
e
^
x
+
F
1
(
∇
→
(
h
1
)
)
×
(
e
^
x
h
1
)
)
+
F
1
h
1
(
∇
→
×
(
e
^
x
h
1
)
)
+
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}(h_{1}))\times ({\frac {{\widehat {e}}_{x}}{h_{1}}}))+F_{1}h_{1}({\overrightarrow {\nabla }}\times ({\frac {{\widehat {e}}_{x}}{h_{1}}}))+}
(
∇
→
F
2
)
×
e
^
y
+
F
2
(
∇
→
(
h
2
)
)
×
(
e
^
y
h
2
)
)
+
F
2
h
2
(
∇
→
×
(
e
^
y
h
2
)
)
+
{\displaystyle ({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}(h_{2}))\times ({\frac {{\widehat {e}}_{y}}{h_{2}}}))+F_{2}h_{2}({\overrightarrow {\nabla }}\times ({\frac {{\widehat {e}}_{y}}{h_{2}}}))+}
(
∇
→
F
3
)
×
e
^
z
+
F
3
(
∇
→
(
h
3
)
)
×
(
e
^
z
h
3
)
)
+
F
3
h
3
(
∇
→
×
e
^
z
h
3
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}(h_{3}))\times ({\frac {{\widehat {e}}_{z}}{h_{3}}}))+F_{3}h_{3}({\overrightarrow {\nabla }}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))=}
(
∇
→
F
1
)
×
e
^
x
+
F
1
(
∇
→
(
h
1
)
×
e
^
x
h
1
)
+
F
1
(
∇
→
×
e
^
x
)
+
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}(h_{1})\times {\frac {{\widehat {e}}_{x}}{h_{1}}})+F_{1}({\overrightarrow {\nabla }}\times {\widehat {e}}_{x})+}
(
∇
→
F
2
)
×
e
^
y
+
F
2
(
∇
→
(
h
2
)
×
e
^
y
h
2
)
+
F
2
(
∇
→
×
e
^
y
)
+
{\displaystyle ({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}(h_{2})\times {\frac {{\widehat {e}}_{y}}{h_{2}}})+F_{2}({\overrightarrow {\nabla }}\times {\widehat {e}}_{y})+}
(
∇
→
F
3
)
×
e
^
z
+
F
3
(
∇
→
(
h
3
)
×
e
^
z
h
3
)
+
F
3
(
∇
→
×
e
^
z
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}(h_{3})\times {\frac {{\widehat {e}}_{z}}{h_{3}}})+F_{3}({\overrightarrow {\nabla }}\times {\widehat {e}}_{z})=}
(
∇
→
F
1
)
×
e
^
x
+
F
1
h
1
(
∇
→
(
h
1
)
×
e
^
x
)
+
(
∇
→
F
2
)
×
e
^
y
+
F
2
h
2
(
∇
→
(
h
2
)
×
e
^
y
)
+
(
∇
→
F
3
)
×
e
^
z
+
F
3
h
3
(
∇
→
(
h
3
)
×
e
^
z
)
=
{\displaystyle ({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+{\frac {F_{1}}{h_{1}}}({\overrightarrow {\nabla }}(h_{1})\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+{\frac {F_{2}}{h_{2}}}({\overrightarrow {\nabla }}(h_{2})\times {\widehat {e}}_{y})+({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+{\frac {F_{3}}{h_{3}}}({\overrightarrow {\nabla }}(h_{3})\times {\widehat {e}}_{z})=}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
F
1
)
×
u
^
1
+
F
1
h
1
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
1
)
×
u
^
1
)
+
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})F_{1})\times {\widehat {u}}_{1}+{\frac {F_{1}}{h_{1}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1})\times {\widehat {u}}_{1})+}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
F
2
)
×
u
^
2
+
F
2
h
2
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
2
)
×
u
^
2
)
+
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})F_{2})\times {\widehat {u}}_{2}+{\frac {F_{2}}{h_{2}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{2})\times {\widehat {u}}_{2})+}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
F
3
)
×
u
^
3
+
F
3
h
3
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
3
)
×
u
^
3
)
=
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})F_{3})\times {\widehat {u}}_{3}+{\frac {F_{3}}{h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{3})\times {\widehat {u}}_{3})=}
(
u
^
1
h
1
∂
F
1
∂
u
1
+
u
^
2
h
2
∂
F
1
∂
u
2
+
u
^
3
h
3
∂
F
1
∂
u
3
)
×
u
^
1
+
F
1
h
1
(
(
u
^
1
h
1
∂
(
h
1
)
∂
u
1
+
u
^
2
h
2
∂
(
h
1
)
∂
u
2
+
u
^
3
h
3
∂
(
h
1
)
∂
u
3
)
×
u
^
1
)
+
{\displaystyle ({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}})\times {\widehat {u}}_{1}+{\frac {F_{1}}{h_{1}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial (h_{1})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial (h_{1})}{\partial u_{3}}})\times {\widehat {u}}_{1})+}
(
u
^
1
h
1
∂
F
2
∂
u
1
+
u
^
2
h
2
∂
F
2
∂
u
2
+
u
^
3
h
3
∂
F
2
∂
u
3
)
×
u
^
2
+
F
2
h
2
(
(
u
^
1
h
1
∂
(
h
2
)
∂
u
1
+
u
^
2
h
2
∂
(
h
2
)
∂
u
2
+
u
^
3
h
3
∂
(
h
2
)
∂
u
3
)
×
u
^
2
)
+
{\displaystyle ({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}})\times {\widehat {u}}_{2}+{\frac {F_{2}}{h_{2}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial (h_{2})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial (h_{2})}{\partial u_{3}}})\times {\widehat {u}}_{2})+}
(
u
^
1
h
1
∂
F
3
∂
u
1
+
u
^
2
h
2
∂
F
3
∂
u
2
+
u
^
3
h
3
∂
F
3
∂
u
3
)
×
u
^
3
+
F
3
h
3
(
(
u
^
1
h
1
∂
(
h
3
)
∂
u
1
+
u
^
2
h
2
∂
(
h
3
)
∂
u
2
+
u
^
3
h
3
∂
(
h
3
)
∂
u
3
)
×
u
^
3
)
=
{\displaystyle ({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}})\times {\widehat {u}}_{3}+{\frac {F_{3}}{h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial (h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial (h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{3}}})\times {\widehat {u}}_{3})=}
1
h
1
∂
F
1
∂
u
1
(
u
^
1
×
u
^
1
)
+
1
h
2
∂
F
1
∂
u
2
(
u
^
2
×
u
^
1
)
+
1
h
3
∂
F
1
∂
u
3
(
u
^
3
×
u
^
1
)
+
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{1}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{1})+{\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{1})+}
F
1
h
1
(
1
h
1
∂
(
h
1
)
∂
u
1
(
u
^
1
×
u
^
1
)
+
1
h
2
∂
(
h
1
)
∂
u
2
(
u
^
2
×
u
^
1
)
+
1
h
3
∂
(
h
1
)
∂
u
3
(
u
^
3
×
u
^
1
)
)
+
{\displaystyle {\frac {F_{1}}{h_{1}}}({\frac {1}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{1})+{\frac {1}{h_{2}}}{\frac {\partial (h_{1})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {\partial (h_{1})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{1}))+}
1
h
1
∂
F
2
∂
u
1
(
u
^
1
×
u
^
2
)
+
1
h
2
∂
F
2
∂
u
2
(
u
^
2
×
u
^
2
)
+
1
h
3
∂
F
2
∂
u
3
(
u
^
3
×
u
^
2
)
+
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {\partial F_{2}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2})+}
F
2
h
2
(
1
h
1
∂
(
h
2
)
∂
u
1
(
u
^
1
×
u
^
2
)
+
1
h
2
∂
(
h
2
)
∂
u
2
(
u
^
2
×
u
^
2
)
+
1
h
3
∂
(
h
2
)
∂
u
3
(
u
^
3
×
u
^
2
)
)
+
{\displaystyle {\frac {F_{2}}{h_{2}}}({\frac {1}{h_{1}}}{\frac {\partial (h_{2})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {\partial (h_{2})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2}))+}
1
h
1
∂
F
3
∂
u
1
(
u
^
1
×
u
^
3
)
+
1
h
2
∂
F
3
∂
u
2
(
u
^
2
×
u
^
3
)
+
1
h
3
∂
F
3
∂
u
3
(
u
^
3
×
u
^
3
)
+
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{3})+}
F
3
h
3
(
1
h
1
∂
(
h
3
)
∂
u
1
(
u
^
1
×
u
^
3
)
+
1
h
2
∂
(
h
3
)
∂
u
2
(
u
^
2
×
u
^
3
)
+
1
h
3
∂
(
h
3
)
∂
u
3
(
u
^
3
×
u
^
3
)
)
=
{\displaystyle {\frac {F_{3}}{h_{3}}}({\frac {1}{h_{1}}}{\frac {\partial (h_{3})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {\partial (h_{3})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{3}))=}
1
h
2
∂
F
1
∂
u
2
(
u
^
2
×
u
^
1
)
+
1
h
3
∂
F
1
∂
u
3
)
(
u
^
3
×
u
^
1
)
+
1
h
2
F
1
h
1
∂
(
h
1
)
∂
u
2
(
u
^
2
×
u
^
1
)
+
1
h
3
F
1
h
1
∂
(
h
1
)
∂
u
3
(
u
^
3
×
u
^
1
)
+
{\displaystyle {\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}})({\widehat {u}}_{3}\times {\widehat {u}}_{1})+{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{1})+}
1
h
1
∂
F
2
∂
u
1
(
u
^
1
×
u
^
2
)
+
1
h
3
∂
F
2
∂
u
3
(
u
^
3
×
u
^
2
)
+
1
h
1
F
2
h
2
∂
(
h
2
)
∂
u
1
(
u
^
1
×
u
^
2
)
+
1
h
3
F
2
h
2
∂
(
h
2
)
∂
u
3
(
u
^
3
×
u
^
2
)
+
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2})+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2})+}
1
h
1
∂
F
2
∂
u
1
(
u
^
1
×
u
^
3
)
+
1
h
2
∂
F
3
∂
u
2
(
u
^
2
×
u
^
3
)
+
1
h
1
F
3
h
3
∂
(
h
3
)
∂
u
1
(
u
^
1
×
u
^
3
)
+
1
h
2
F
3
h
3
∂
(
h
3
)
∂
u
2
(
u
^
2
×
u
^
3
)
=
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})+{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})=}
1
h
2
∂
F
1
∂
u
2
(
−
u
^
3
)
+
1
h
3
∂
F
1
∂
u
3
u
^
2
+
1
h
2
F
1
h
1
∂
(
h
1
)
∂
u
2
(
−
u
^
3
)
+
1
h
3
F
1
h
1
∂
(
h
1
)
∂
u
3
u
^
2
+
{\displaystyle {\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}(-{\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}{\widehat {u}}_{2}+{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}}(-{\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}}{\widehat {u}}_{2}+}
1
h
1
∂
F
2
∂
u
1
u
^
3
+
1
h
3
∂
F
2
∂
u
3
(
−
u
^
1
)
+
1
h
1
F
2
h
2
∂
(
h
2
)
∂
u
1
u
^
3
+
1
h
3
F
2
h
2
∂
(
h
2
)
∂
u
3
(
−
u
^
1
)
+
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}{\widehat {u}}_{3}+{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}(-{\widehat {u}}_{1})+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}}{\widehat {u}}_{3}+{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}}(-{\widehat {u}}_{1})+}
1
h
1
∂
F
3
∂
u
1
(
−
u
^
2
)
+
1
h
2
∂
F
3
∂
u
2
u
^
1
+
1
h
1
F
3
h
3
∂
(
h
3
)
∂
u
1
(
−
u
^
2
)
+
1
h
2
F
3
h
3
∂
(
h
3
)
∂
u
2
u
^
1
=
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}(-{\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}{\widehat {u}}_{1}+{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}}(-{\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}}{\widehat {u}}_{1}=}
1
h
2
∂
F
3
∂
u
2
u
^
1
+
1
h
2
F
3
h
3
∂
(
h
3
)
∂
u
2
u
^
1
−
1
h
3
∂
F
2
∂
u
3
u
^
1
−
1
h
3
F
2
h
2
∂
(
h
2
)
∂
u
3
u
^
1
+
{\displaystyle {\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}{\widehat {u}}_{1}+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}}{\widehat {u}}_{1}-{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}{\widehat {u}}_{1}-{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}}{\widehat {u}}_{1}+}
1
h
3
∂
F
1
∂
u
3
u
^
2
+
1
h
3
F
1
h
1
∂
(
h
1
)
∂
u
3
u
^
2
−
1
h
1
∂
F
3
∂
u
1
u
^
2
−
1
h
1
F
3
h
3
∂
(
h
3
)
∂
u
1
u
^
2
+
{\displaystyle {\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}{\widehat {u}}_{2}+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}}{\widehat {u}}_{2}-{\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}{\widehat {u}}_{2}-{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}}{\widehat {u}}_{2}+}
1
h
1
∂
F
2
∂
u
1
u
^
3
+
1
h
1
F
2
h
2
∂
(
h
2
)
∂
u
1
u
^
3
−
1
h
2
∂
F
1
∂
u
2
u
^
3
−
1
h
2
F
1
h
1
∂
(
h
1
)
∂
u
2
u
^
3
=
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}{\widehat {u}}_{3}+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}}{\widehat {u}}_{3}-{\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}{\widehat {u}}_{3}-{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}}{\widehat {u}}_{3}=}
(
(
1
h
2
∂
F
3
∂
u
2
+
1
h
2
F
3
h
3
∂
(
h
3
)
∂
u
2
)
−
(
1
h
3
∂
F
2
∂
u
3
+
1
h
3
F
2
h
2
∂
(
h
2
)
∂
u
3
)
)
u
^
1
+
{\displaystyle (({\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}})-({\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}+{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}})){\widehat {u}}_{1}+}
(
(
1
h
3
∂
F
1
∂
u
3
+
1
h
3
F
1
h
1
∂
(
h
1
)
∂
u
3
)
−
(
1
h
1
∂
F
3
∂
u
1
+
1
h
1
F
3
h
3
∂
(
h
3
)
∂
u
1
)
)
u
^
2
+
{\displaystyle (({\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}})-({\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}+{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}})){\widehat {u}}_{2}+}
(
(
1
h
1
∂
F
2
∂
u
1
+
1
h
1
F
2
h
2
∂
(
h
2
)
∂
u
1
)
−
(
1
h
2
∂
F
1
∂
u
2
+
1
h
2
F
1
h
1
∂
(
h
1
)
∂
u
2
)
)
u
^
3
=
{\displaystyle (({\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}})-({\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}+{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}})){\widehat {u}}_{3}=}
(
1
h
2
h
3
∂
(
F
3
h
3
)
∂
u
2
−
1
h
2
h
3
∂
(
F
2
h
2
)
∂
u
3
)
u
^
1
+
(
1
h
1
h
3
∂
(
F
1
h
1
)
∂
u
3
−
1
h
1
h
3
∂
(
F
3
h
3
)
∂
u
1
)
u
^
2
+
(
1
h
1
h
2
∂
(
F
2
h
2
)
∂
u
1
−
1
h
1
h
2
∂
(
F
1
h
1
)
∂
u
2
)
u
^
3
=
{\displaystyle ({\frac {1}{h_{2}h_{3}}}{\frac {\partial (F_{3}h_{3})}{\partial u_{2}}}-{\frac {1}{h_{2}h_{3}}}{\frac {\partial (F_{2}h_{2})}{\partial u_{3}}}){\widehat {u}}_{1}+({\frac {1}{h_{1}h_{3}}}{\frac {\partial (F_{1}h_{1})}{\partial u_{3}}}-{\frac {1}{h_{1}h_{3}}}{\frac {\partial (F_{3}h_{3})}{\partial u_{1}}}){\widehat {u}}_{2}+({\frac {1}{h_{1}h_{2}}}{\frac {\partial (F_{2}h_{2})}{\partial u_{1}}}-{\frac {1}{h_{1}h_{2}}}{\frac {\partial (F_{1}h_{1})}{\partial u_{2}}}){\widehat {u}}_{3}=}
1
h
1
h
2
h
3
(
(
∂
(
F
3
h
3
)
∂
u
2
−
∂
(
F
2
h
2
)
∂
u
3
)
h
1
u
^
1
+
(
∂
(
F
1
h
1
)
∂
u
3
−
∂
(
F
3
h
3
)
∂
u
1
)
h
2
u
^
2
+
(
∂
(
F
2
h
2
)
∂
u
1
−
∂
(
F
1
h
1
)
∂
u
2
)
h
3
u
^
3
)
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}(({\frac {\partial (F_{3}h_{3})}{\partial u_{2}}}-{\frac {\partial (F_{2}h_{2})}{\partial u_{3}}})h_{1}{\widehat {u}}_{1}+({\frac {\partial (F_{1}h_{1})}{\partial u_{3}}}-{\frac {\partial (F_{3}h_{3})}{\partial u_{1}}})h_{2}{\widehat {u}}_{2}+({\frac {\partial (F_{2}h_{2})}{\partial u_{1}}}-{\frac {\partial (F_{1}h_{1})}{\partial u_{2}}})h_{3}{\widehat {u}}_{3})}
> Rotacional en coordenadas generalizadas. Geometría diferencial
editar
Para calcular el rotacional en geometría diferencial bajaremos índices, aplicaremos la diferencial exterior , posteriormente la estrella de Hodge y subiremos índices para dar el vector resultante. Finalmente pasaremos de la base natural a la base ortonormal .
r
o
t
(
F
→
)
=
∇
→
×
F
→
=↑
∗
d
↓
(
F
→
)
=↑
∗
d
↓
(
F
x
∂
∂
x
+
F
y
∂
∂
y
+
F
z
∂
∂
z
)
=↑
∗
d
↓
(
F
1
(
1
g
11
∂
∂
u
1
)
+
F
2
(
1
g
22
∂
∂
u
2
)
+
F
3
(
1
g
33
∂
∂
u
3
)
)
=
{\displaystyle rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\partial }{\partial x}}+{F_{y}}{\frac {\partial }{\partial y}}+F_{z}{\frac {\partial }{\partial z}})=\uparrow \ast d\downarrow ({F_{1}}({\frac {1}{\sqrt[{}]{g_{11}}}}{\frac {\partial }{\partial u_{1}}})+F_{2}({\frac {1}{\sqrt[{}]{g_{22}}}}{\frac {\partial }{\partial u_{2}}})+F_{3}({\frac {1}{\sqrt[{}]{g_{33}}}}{\frac {\partial }{\partial u_{3}}}))=}
↑
∗
d
(
F
1
1
g
11
g
11
d
u
1
+
F
2
1
g
22
g
22
d
u
2
+
F
3
1
g
33
g
33
d
u
3
)
=↑
∗
d
(
F
1
g
11
2
d
u
1
+
F
2
g
22
2
d
u
2
)
+
F
3
g
33
2
d
u
3
)
=
{\displaystyle \uparrow \ast d(F_{1}{\frac {1}{\sqrt[{}]{g_{11}}}}g_{11}du_{1}+F_{2}{\frac {1}{\sqrt[{}]{g_{22}}}}g_{22}du_{2}+F_{3}{\frac {1}{\sqrt[{}]{g_{33}}}}g_{33}du_{3})=\uparrow \ast d({F_{1}}{\sqrt[{2}]{g_{11}}}du_{1}+F_{2}{\sqrt[{2}]{g_{22}}}du_{2})+F_{3}{\sqrt[{2}]{g_{33}}}du_{3})=}
↑
∗
(
∂
(
F
1
g
11
2
)
∂
u
2
d
u
2
∧
d
u
1
+
∂
(
F
1
g
11
2
)
∂
u
3
d
u
3
∧
d
u
1
+
∂
(
F
2
g
22
2
)
∂
u
1
d
u
1
∧
d
u
2
+
{\displaystyle \uparrow \ast ({\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{2}}}du_{2}\wedge du_{1}+{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{3}}}du_{3}\wedge du_{1}+{\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{1}}}du_{1}\wedge du_{2}+}
∂
(
F
2
g
22
2
)
∂
u
3
d
u
3
∧
d
u
2
+
∂
(
F
3
g
33
2
)
∂
u
1
d
u
1
∧
d
u
3
)
+
∂
(
F
3
g
33
2
)
∂
u
2
d
u
2
∧
d
u
3
)
=
{\displaystyle {\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{3}}}du_{3}\wedge du_{2}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{1}}}du_{1}\wedge du_{3})+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{2}}}du_{2}\wedge du_{3})=}
↑
∗
(
−
∂
(
F
1
g
11
2
)
∂
u
2
d
u
1
∧
d
u
2
−
∂
(
F
1
g
11
2
)
∂
u
3
d
u
1
∧
d
u
3
+
∂
(
F
2
g
22
2
)
∂
u
1
d
u
1
∧
d
u
2
−
{\displaystyle \uparrow \ast (-{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{2}}}du_{1}\wedge du_{2}-{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{3}}}du_{1}\wedge du_{3}+{\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{1}}}du_{1}\wedge du_{2}-}
∂
(
F
2
g
22
2
)
∂
u
3
d
u
2
∧
d
u
3
+
∂
(
F
3
g
33
2
)
∂
u
1
d
u
1
∧
d
u
3
+
∂
(
F
3
g
33
2
)
∂
u
2
d
u
2
∧
d
u
3
)
=
{\displaystyle {\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{3}}}du_{2}\wedge du_{3}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{1}}}du_{1}\wedge du_{3}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{2}}}du_{2}\wedge du_{3})=}
↑
(
−
|
g
|
g
11
g
22
∂
∂
u
2
(
F
1
g
11
2
)
d
u
3
−
|
g
|
g
11
g
33
∂
∂
u
3
(
F
1
g
11
2
)
d
u
2
+
|
g
|
g
11
g
22
∂
∂
u
1
(
F
2
g
22
2
)
d
u
3
−
{\displaystyle \uparrow (-{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}{\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})du_{3}-{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})du_{2}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})du_{3}-}
|
g
|
g
11
g
33
∂
∂
u
3
(
F
2
g
22
2
)
d
u
1
+
|
g
|
g
11
g
33
∂
∂
u
1
(
F
3
g
33
2
)
d
u
2
+
|
g
|
g
22
g
33
∂
∂
u
2
(
F
3
g
33
2
)
d
u
1
)
=
{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})du_{1}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})du_{2}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{22}g_{33}}}{\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})du_{1})=}
↑
(
|
g
|
g
22
g
33
(
∂
∂
u
2
(
F
3
g
33
2
)
)
−
∂
∂
u
3
(
F
2
g
22
2
)
)
d
u
1
+
{\displaystyle \uparrow ({\frac {\sqrt[{}]{\vert g\vert }}{g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}}))-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))du_{1}+}
|
g
|
g
11
g
33
(
∂
∂
u
1
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
1
g
11
2
)
)
d
u
2
+
{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))du_{2}+}
|
g
|
g
11
g
22
(
∂
∂
u
2
(
F
1
g
11
2
)
−
∂
∂
u
1
(
F
2
g
22
2
)
)
d
u
3
)
=
{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))du_{3})=}
↑
(
|
g
|
g
11
g
22
g
33
(
∂
∂
u
2
(
F
3
g
33
2
)
)
−
∂
∂
u
3
(
F
2
g
22
2
)
)
g
11
d
u
1
+
{\displaystyle \uparrow ({\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}}))-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))g_{11}du_{1}+}
|
g
|
g
11
g
22
g
33
(
∂
∂
u
1
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
1
g
11
2
)
)
g
22
d
u
2
+
{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))g_{22}du_{2}+}
|
g
|
g
11
g
22
g
33
(
∂
∂
u
2
(
F
1
g
11
2
)
−
∂
∂
u
1
(
F
2
g
22
2
)
)
g
33
d
u
3
)
=
{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))g_{33}du_{3})=}
1
g
11
g
22
g
33
(
(
∂
∂
u
2
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
2
g
22
2
)
)
∂
∂
u
1
+
(
∂
∂
u
1
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
1
g
11
2
)
)
∂
∂
u
2
+
(
∂
∂
u
2
(
F
1
g
11
2
)
−
∂
∂
u
1
(
F
2
g
22
2
)
)
∂
∂
u
3
)
=
{\displaystyle {\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\frac {\partial }{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})){\frac {\partial }{\partial u_{2}}}+({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\frac {\partial }{\partial u_{3}}})=}
1
g
11
g
22
g
33
(
(
∂
∂
u
2
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
2
g
22
2
)
)
|
∂
∂
u
1
|
∂
∂
u
1
|
∂
∂
u
1
|
+
{\displaystyle {\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))\vert {\frac {\partial }{\partial u_{1}}}\vert {\frac {\frac {\partial }{\partial u_{1}}}{\vert {\frac {\partial }{\partial u_{1}}}\vert }}+}
(
∂
∂
u
1
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
1
g
11
2
)
)
|
∂
∂
u
2
|
∂
∂
u
2
|
∂
∂
u
2
|
+
{\displaystyle ({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))\vert {\frac {\partial }{\partial u_{2}}}\vert {\frac {\frac {\partial }{\partial u_{2}}}{\vert {\frac {\partial }{\partial u_{2}}}\vert }}+}
(
∂
∂
u
2
(
F
1
g
11
2
)
−
∂
∂
u
1
(
F
2
g
22
2
)
)
|
∂
∂
u
3
|
∂
∂
u
3
|
∂
∂
u
3
|
)
=
{\displaystyle ({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))\vert {\frac {\partial }{\partial u_{3}}}\vert {\frac {\frac {\partial }{\partial u_{3}}}{\vert {\frac {\partial }{\partial u_{3}}}\vert }})=}
1
g
11
g
22
g
33
(
(
∂
∂
u
2
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
2
g
22
2
)
)
g
11
2
∂
^
∂
u
1
+
(
∂
∂
u
1
(
F
3
g
33
2
)
−
∂
∂
u
3
(
F
1
g
11
2
)
)
g
22
2
∂
^
∂
u
2
+
{\displaystyle {\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\sqrt[{2}]{g_{11}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})){\sqrt[{2}]{g_{22}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+}
(
∂
∂
u
2
(
F
1
g
11
2
)
−
∂
∂
u
1
(
F
2
g
22
2
)
)
g
33
2
)
∂
^
∂
u
3
)
=
{\displaystyle ({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\sqrt[{2}]{g_{33}}}){\frac {\widehat {\partial }}{\partial u_{3}}})=}
1
h
1
h
2
h
3
(
(
∂
∂
u
2
(
F
3
h
3
)
−
∂
∂
u
3
(
F
2
h
2
)
)
h
1
∂
^
∂
u
1
+
(
∂
∂
u
1
(
F
3
h
3
)
−
∂
∂
u
3
(
F
1
h
1
)
)
h
2
∂
^
∂
u
2
+
(
∂
∂
u
2
(
F
1
h
1
)
−
∂
∂
u
1
(
F
2
h
2
)
)
h
3
∂
^
∂
u
3
)
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}h_{3})-{\frac {\partial }{\partial u_{3}}}(F_{2}h_{2}))h_{1}{\frac {\widehat {\partial }}{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}h_{3})-{\frac {\partial }{\partial u_{3}}}(F_{1}h_{1}))h_{2}{\frac {\widehat {\partial }}{\partial u_{2}}}+({\frac {\partial }{\partial u_{2}}}(F_{1}h_{1})-{\frac {\partial }{\partial u_{1}}}(F_{2}h_{2}))h_{3}{\frac {\widehat {\partial }}{\partial u_{3}}})}
> Laplaciano en coordenadas generalizadas. Cálculo diferencial
editar
d
i
v
(
g
r
a
d
→
(
f
)
)
=
∇
2
f
=
∇
→
⋅
(
∇
→
f
)
=
∇
→
⋅
(
∂
f
∂
x
e
^
x
+
∂
f
∂
y
e
^
y
+
∂
f
∂
z
e
^
z
)
=
∇
→
⋅
(
∂
f
∂
x
e
^
x
)
+
∇
→
⋅
(
∂
f
∂
y
e
^
y
)
+
∇
→
⋅
(
∂
f
∂
z
e
^
z
)
=
{\displaystyle div({\overrightarrow {grad}}(f))=\nabla ^{2}f={\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}f)={\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial x}}{\widehat {e}}_{x}+{\frac {\partial f}{\partial y}}{\widehat {e}}_{y}+{\frac {\partial f}{\partial z}}{\widehat {e}}_{z})={\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial x}}{\widehat {e}}_{x})+{\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial y}}{\widehat {e}}_{y})+{\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial z}}{\widehat {e}}_{z})=}
(
∇
→
∂
f
∂
x
)
⋅
e
^
x
+
∂
f
∂
x
(
∇
→
⋅
e
^
x
)
+
(
∇
→
∂
f
∂
y
)
⋅
e
^
y
+
∂
f
∂
y
(
∇
→
⋅
e
^
y
)
+
(
∇
→
∂
f
∂
z
)
⋅
e
^
z
+
∂
f
∂
z
(
∇
→
⋅
e
^
z
)
=
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{x})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{y})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{z})=}
(
∇
→
∂
f
∂
x
)
⋅
e
^
x
+
∂
f
∂
x
(
∇
→
⋅
(
e
^
y
×
e
^
z
)
)
+
(
∇
→
∂
f
∂
y
)
⋅
e
^
y
+
∂
f
∂
y
(
∇
→
⋅
(
e
^
z
×
e
^
x
)
)
+
(
∇
→
∂
f
∂
z
)
⋅
e
^
z
+
∂
f
∂
z
(
∇
→
⋅
(
e
^
x
×
e
^
y
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y}))=}
(
∇
→
∂
f
∂
x
)
⋅
e
^
x
+
∂
f
∂
x
∇
→
(
h
2
h
3
⋅
(
e
^
y
h
2
×
e
^
z
h
3
)
)
+
(
∇
→
∂
f
∂
y
)
⋅
e
^
y
+
∂
f
∂
y
∇
→
(
h
3
h
1
⋅
(
e
^
z
h
3
×
e
^
x
h
1
)
)
+
(
∇
→
∂
f
∂
z
)
⋅
e
^
z
+
∂
f
∂
z
∇
→
(
h
1
h
2
⋅
(
e
^
x
h
1
×
e
^
y
h
2
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}{\overrightarrow {\nabla }}(h_{2}h_{3}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}{\overrightarrow {\nabla }}(h_{3}h_{1}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}{\overrightarrow {\nabla }}(h_{1}h_{2}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=}
(
∇
→
∂
f
∂
x
)
⋅
e
^
x
+
∂
f
∂
x
(
∇
→
(
h
2
h
3
)
)
⋅
(
e
^
y
h
2
×
e
^
z
h
3
)
+
∂
f
∂
x
h
2
h
3
(
∇
→
⋅
(
e
^
y
h
2
×
e
^
z
h
3
)
)
+
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}})+{\frac {\partial f}{\partial x}}h_{2}h_{3}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+}
(
∇
→
∂
f
∂
y
)
⋅
e
^
y
+
∂
f
∂
y
(
∇
→
(
h
3
h
1
)
)
⋅
(
e
^
z
h
3
×
e
^
x
h
1
)
+
∂
f
∂
y
h
3
h
1
(
∇
→
⋅
(
e
^
z
h
3
×
e
^
x
h
1
)
)
+
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}})+{\frac {\partial f}{\partial y}}h_{3}h_{1}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+}
(
∇
→
∂
f
∂
z
)
⋅
e
^
z
+
∂
f
∂
z
(
∇
→
(
h
1
h
2
)
)
⋅
(
e
^
x
h
1
×
e
^
y
h
2
)
+
∂
f
∂
z
h
1
h
2
(
∇
→
⋅
(
e
^
x
h
1
×
e
^
y
h
2
)
)
=
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}})+{\frac {\partial f}{\partial z}}h_{1}h_{2}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=}
(
∇
→
∂
f
∂
x
)
⋅
e
^
x
+
1
h
2
h
3
∂
f
∂
x
(
∇
→
(
h
2
h
3
)
)
⋅
(
e
^
y
×
e
^
z
)
+
(
∇
→
∂
f
∂
y
)
⋅
e
^
y
+
1
h
3
h
1
∂
f
∂
y
(
∇
→
(
h
3
h
1
)
)
⋅
(
e
^
z
×
e
^
x
)
+
(
∇
→
∂
f
∂
z
)
⋅
e
^
z
+
1
h
1
h
2
∂
f
∂
z
(
∇
→
(
h
1
h
2
)
)
⋅
(
e
^
x
×
e
^
y
)
=
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {1}{h_{2}h_{3}}}{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {1}{h_{3}h_{1}}}{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {1}{h_{1}h_{2}}}{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y})=}
(
∇
→
∂
f
∂
x
)
⋅
e
^
x
+
1
h
2
h
3
∂
f
∂
x
(
∇
→
(
h
2
h
3
)
)
⋅
e
^
x
+
(
∇
→
∂
f
∂
y
)
⋅
e
^
y
+
1
h
3
h
1
∂
f
∂
y
(
∇
→
(
h
3
h
1
)
)
⋅
e
^
y
+
(
∇
→
∂
f
∂
z
)
⋅
e
^
z
+
1
h
1
h
2
∂
f
∂
z
(
∇
→
(
h
1
h
2
)
)
⋅
e
^
z
=
{\displaystyle ({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {1}{h_{2}h_{3}}}{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot {\widehat {e}}_{x}+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {1}{h_{3}h_{1}}}{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot {\widehat {e}}_{y}+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {1}{h_{1}h_{2}}}{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot {\widehat {e}}_{z}=}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
∂
f
∂
u
1
∂
u
1
∂
x
)
)
⋅
u
^
1
+
1
h
2
h
3
∂
f
∂
u
1
∂
u
1
∂
x
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
2
h
3
)
)
⋅
u
^
1
+
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})({\frac {\partial f}{\partial u_{1}}}{\frac {\partial u_{1}}{\partial x}}))\cdot {\widehat {u}}_{1}+{\frac {1}{h_{2}h_{3}}}{\frac {\partial f}{\partial u_{1}}}{\frac {\partial u_{1}}{\partial x}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{2}h_{3}))\cdot {\widehat {u}}_{1}+}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
∂
f
∂
u
2
∂
u
2
∂
y
)
)
⋅
u
^
2
+
1
h
1
h
3
∂
f
∂
u
2
∂
u
2
∂
y
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
1
h
3
)
)
⋅
u
^
2
+
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})({\frac {\partial f}{\partial u_{2}}}{\frac {\partial u_{2}}{\partial y}}))\cdot {\widehat {u}}_{2}+{\frac {1}{h_{1}h_{3}}}{\frac {\partial f}{\partial u_{2}}}{\frac {\partial u_{2}}{\partial y}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{3}))\cdot {\widehat {u}}_{2}+}
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
∂
f
∂
u
3
∂
u
3
∂
z
)
)
⋅
u
^
3
+
1
h
1
h
2
∂
f
∂
u
e
∂
u
3
∂
z
(
(
u
^
1
h
1
∂
∂
u
1
+
u
^
2
h
2
∂
∂
u
2
+
u
^
3
h
3
∂
∂
u
3
)
(
h
1
h
2
)
)
⋅
u
^
3
=
{\displaystyle (({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})({\frac {\partial f}{\partial u_{3}}}{\frac {\partial u_{3}}{\partial z}}))\cdot {\widehat {u}}_{3}+{\frac {1}{h_{1}h_{2}}}{\frac {\partial f}{\partial u_{e}}}{\frac {\partial u_{3}}{\partial z}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{2}))\cdot {\widehat {u}}_{3}=}
(
1
h
1
u
^
1
⋅
u
^
1
∂
∂
u
1
+
1
h
2
u
^
2
⋅
u
^
1
∂
∂
u
2
+
1
h
3
u
^
3
⋅
u
^
1
∂
∂
u
3
)
(
1
h
1
∂
f
∂
u
1
)
+
1
h
1
∂
f
∂
u
1
(
u
^
1
⋅
u
^
1
h
1
h
2
h
3
∂
(
h
2
h
3
)
∂
u
1
+
u
^
2
⋅
u
^
1
h
2
h
2
h
3
∂
(
h
2
h
3
)
∂
u
2
+
u
^
3
⋅
u
^
1
h
2
h
3
h
3
∂
(
h
2
h
3
)
∂
u
3
)
+
{\displaystyle ({\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}{\frac {\partial }{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}{\frac {\partial }{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}{\frac {\partial }{\partial u_{3}}})({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}}{h_{2}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}}{h_{2}h_{3}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{3}}})+}
(
1
h
1
u
^
1
⋅
u
^
2
∂
∂
u
1
+
1
h
2
u
^
2
⋅
u
^
2
∂
∂
u
2
+
1
h
3
u
^
3
⋅
u
^
2
∂
∂
u
3
)
(
1
h
2
∂
f
∂
u
2
)
+
{\displaystyle ({\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}{\frac {\partial }{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}{\frac {\partial }{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}{\frac {\partial }{\partial u_{3}}})({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+}
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
s
i
n
ϑ
∂
∂
ϑ
(
s
i
n
ϑ
∂
f
∂
ϑ
)
+
1
r
2
s
i
n
2
ϑ
(
∂
2
f
∂
2
φ
)
1
h
2
∂
f
∂
u
2
(
u
^
1
⋅
u
^
2
h
1
h
1
h
3
∂
(
h
1
h
3
)
∂
u
1
+
u
^
2
⋅
u
^
2
h
1
h
2
h
3
∂
(
h
1
h
3
)
∂
u
2
+
u
^
3
⋅
u
^
2
h
1
h
3
h
3
∂
(
h
1
h
3
)
∂
u
3
)
+
{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial f}{\partial r}})+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial ^{2}f}{\partial ^{2}\varphi }}){\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}}{h_{1}h_{1}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}}{h_{1}h_{3}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{3}}})+}
(
1
h
1
u
^
1
⋅
u
^
3
∂
∂
u
1
+
1
h
2
u
^
2
⋅
u
^
3
∂
∂
u
2
+
1
h
3
u
^
3
⋅
u
^
3
∂
∂
u
3
)
(
1
h
3
∂
f
∂
u
3
)
+
1
h
3
∂
f
∂
u
3
(
u
^
1
⋅
u
^
3
h
1
h
1
h
2
∂
(
h
1
h
2
)
∂
u
1
+
u
^
2
⋅
u
^
3
h
1
h
2
h
2
∂
(
h
1
h
2
)
∂
u
2
+
u
^
3
⋅
u
^
3
h
1
h
2
h
3
∂
(
h
1
h
2
)
∂
u
3
)
=
{\displaystyle ({\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}{\frac {\partial }{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}{\frac {\partial }{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}{\frac {\partial }{\partial u_{3}}})({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}}{h_{1}h_{1}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}})=}
1
h
1
(
u
^
1
⋅
u
^
1
)
∂
∂
u
1
(
1
h
1
∂
f
∂
u
1
)
+
1
h
1
h
2
h
3
(
1
h
1
∂
f
∂
u
1
)
(
u
^
1
⋅
u
^
1
)
∂
(
h
2
h
3
)
∂
u
1
+
1
h
2
(
u
^
2
⋅
u
^
2
)
∂
∂
u
2
(
1
h
2
∂
f
∂
u
2
)
+
{\displaystyle {\frac {1}{h_{1}}}({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial }{\partial u_{1}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial }{\partial u_{2}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+}
1
h
1
h
2
h
3
(
1
h
2
∂
f
∂
u
2
)
(
u
^
2
⋅
u
^
2
)
∂
(
h
1
h
3
)
∂
u
2
+
1
h
3
(
u
^
3
⋅
u
^
3
)
∂
∂
u
3
(
1
h
3
∂
f
∂
u
3
)
+
1
h
1
h
2
h
3
(
1
h
3
∂
f
∂
u
3
)
(
u
^
3
⋅
u
^
3
)
∂
(
h
1
h
2
)
∂
u
3
=
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial }{\partial u_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=}
1
h
1
∂
∂
u
1
(
1
h
1
∂
f
∂
u
1
)
+
1
h
1
h
2
h
3
(
1
h
1
∂
f
∂
u
1
)
∂
(
h
2
h
3
)
∂
u
1
+
1
h
2
∂
∂
u
2
(
1
h
2
∂
f
∂
u
2
)
+
{\displaystyle {\frac {1}{h_{1}}}{\frac {\partial }{\partial u_{1}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}}){\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\frac {\partial }{\partial u_{2}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+}
1
h
1
h
2
h
3
(
1
h
2
∂
f
∂
u
2
)
∂
(
h
1
h
3
)
∂
u
2
+
1
h
3
∂
∂
u
3
(
1
h
3
∂
f
∂
u
3
)
+
1
h
1
h
2
h
3
(
1
h
3
∂
f
∂
u
3
)
∂
(
h
1
h
2
)
∂
u
3
=
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}}){\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\frac {\partial }{\partial u_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}}){\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=}
h
2
h
3
h
1
h
2
h
3
∂
∂
u
1
(
1
h
1
∂
f
∂
u
1
)
+
(
1
h
1
∂
f
∂
u
1
)
h
1
h
2
h
3
∂
(
h
2
h
3
)
∂
u
1
+
h
1
h
3
h
1
h
2
h
3
∂
∂
u
2
(
1
h
2
∂
f
∂
u
2
)
+
(
1
h
2
∂
f
∂
u
2
)
h
1
h
2
h
3
∂
(
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{\displaystyle {\frac {h_{2}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial u_{1}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {h_{1}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial u_{2}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {h_{1}h_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial u_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=}
1
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=
{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}({\frac {\partial f}{\partial u_{1}}}{\frac {h_{2}h_{3}}{h_{1}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{2}}}({\frac {\partial f}{\partial u_{2}}}{\frac {h_{1}h_{3}}{h_{2}}}))+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{3}}}({\frac {\partial f}{\partial u_{3}}}{\frac {h_{1}h_{2}}{h_{3}}}))=}
1
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{\displaystyle {\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}({\frac {\partial f}{\partial u_{1}}}{\frac {h_{2}h_{3}}{h_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {\partial f}{\partial u_{2}}}{\frac {h_{1}h_{3}}{h_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {\partial f}{\partial u_{3}}}{\frac {h_{1}h_{2}}{h_{3}}}))}
> Laplaciano en coordenadas generalizadas. Geometría diferencial
editar
Para calcular el laplaciano en geometría diferencial aplicaremos la diferencial exterior , la estrella de Hodge y otra vez la diferencial exterior y la estrella de Hodge.
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=
{\displaystyle div({\overrightarrow {grad}}(f))=\nabla ^{2}f=\ast d\ast d(f)=\ast d\ast ({\frac {\partial f}{\partial u_{1}}}du_{1}+{\frac {\partial f}{\partial u_{2}}}du_{2}+{\frac {\partial f}{\partial u_{3}}}du_{3})=}
∗
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{\displaystyle \ast d({\frac {\sqrt[{}]{|g|}}{g_{1}}}{\frac {\partial f}{\partial u_{1}}}(du_{2}\wedge du_{3})+{\frac {\sqrt[{}]{|g|}}{g_{2}}}{\frac {\partial f}{\partial u_{2}}}(du_{3}\wedge du_{1})+{\frac {\sqrt[{}]{|g|}}{g_{3}}}{\frac {\partial f}{\partial u_{3}}}(du_{1}\wedge du_{2}))=}
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{\displaystyle \ast ({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})(du_{1}\wedge du_{2}\wedge du_{3})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})(du_{2}\wedge du_{3}\wedge du_{1})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})(du_{3}\wedge du_{1}\wedge du_{2}))=}
∗
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{\displaystyle \ast (({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})(du_{1}\wedge du_{2}\wedge du_{3}))=}
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{\displaystyle ({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})=}
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{\displaystyle ({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}})({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}}))=}
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{\displaystyle ({\frac {1}{\sqrt[{}]{g_{1}g_{2}g_{3}}}})({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}}))=}
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{\displaystyle ({\frac {1}{h_{1}h_{2}h_{3}}})({\frac {\partial }{\partial u_{1}}}({\frac {h_{2}h_{3}}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {h_{1}h_{3}}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {h_{1}h_{2}}{h_{3}}}{\frac {\partial f}{\partial u_{3}}}))}
Bibliografía
editar
Gravitation (Charles W. Miser, Kip S. Thorne)
Véase también
editar
![{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial r}}|,|{\frac {\partial }{\partial \vartheta }}|,|{\frac {\partial }{\partial z}}|)=({\sqrt[{}]{g_{r}}},{\sqrt[{}]{g_{\vartheta }}},{\sqrt[{}]{g_{z}}})=(1,r,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2159c18eecac7187a4e4449e423c5d201c627226)
![{\displaystyle *(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fea7249aced849a6435ae6b5621638e0e999c5)
![{\displaystyle \ast (Fdx_{\alpha })=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }}}F(dx_{\beta }\wedge dx_{\gamma }):}](https://wikimedia.org/api/rest_v1/media/math/render/svg/734e67476a671b907b2705142d3e19201d533ad6)
![{\displaystyle \ast (Fdr)={\frac {\sqrt[{}]{r^{2}}}{1}}F(d\vartheta \wedge dz)=rF(d\vartheta \wedge dz);\ast (Fd\vartheta )={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}F(dr\wedge dz)={\frac {1}{r}}F(dr\wedge dz);\ast (Fdz)={\frac {\sqrt[{}]{r^{2}}}{1}}F(d\vartheta \wedge dr)=rF(d\vartheta \wedge dr)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6971a3db130344dbbe8f9364f7619e0a71c019)
![{\displaystyle \ast (F(dx_{\alpha }\wedge dx_{\beta }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }}}Fdx_{\gamma }:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0d15e707a8f8fdabb4782f745b7bb720421d28)
![{\displaystyle \ast (F(dr\wedge d\vartheta ))={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdz={\frac {1}{r}}Fdz;\ast (F(dr\wedge dz))={\frac {\sqrt[{}]{r^{2}}}{1}}Fd\vartheta =rFd\vartheta ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64efa0e985842b6950e4edb6e656d3f787299a5c)
![{\displaystyle \ast (F(d\vartheta \wedge dr))=-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdz=-{\frac {1}{r}}Fdz;\ast (F(d\vartheta \wedge dz))={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdr={\frac {1}{r}}Fdr;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef39ffba464fc70b7a79df530194461498ae57f)
![{\displaystyle \ast (F(dz\wedge dr))=-{\frac {\sqrt[{}]{r^{2}}}{1}}Fd\vartheta =-rFd\vartheta ;\ast (F(dz\wedge d\vartheta ))=-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdr=-{\frac {1}{r}}Fdr;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a43f328dfd67115213cde3ab667bee0b2b3839a6)
![{\displaystyle \ast (F(dx_{\alpha }\wedge dx_{\beta }\wedge dx_{\gamma }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }g_{\gamma \gamma }}}F:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbba69c59f8aaa0b2d2ec56ffae472c7169719d4)
![{\displaystyle \ast (F(dr\wedge d\vartheta \wedge dz))=\varepsilon {\frac {\sqrt[{}]{r^{2}}}{r^{2}}}F=\varepsilon {\frac {1}{r}}F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58585c2f23fc4d65a878ae918926640d17f32d0b)
![{\displaystyle \ast d\ast \downarrow (F_{r}({\frac {1}{\sqrt[{}]{g_{r}}}}{\frac {\partial }{\partial r}})+F_{\vartheta }({\frac {1}{\sqrt[{}]{g_{\vartheta }}}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{\sqrt[{}]{g_{z}}}}{\frac {\partial }{\partial z}}))=\ast d\downarrow (F_{r}{\frac {\partial }{\partial r}}+F_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+F_{z}{\frac {\partial }{\partial z}})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a41b8e8b8b4257a628a8b21178c069436cbf31c7)
![{\displaystyle \ast d\ast (F_{r}dr+{\frac {F_{\vartheta }}{r}}(r^{2}d\vartheta )+F_{z}dz)=\ast d\ast (F_{r}dr+F_{\vartheta }rd\vartheta +F_{z}dz)=\ast d({\frac {\sqrt[{}]{r^{2}}}{1}}F_{r}(d\vartheta \wedge dz)+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}rF_{\vartheta }(dz\wedge dr)+{\frac {\sqrt[{}]{r^{2}}}{1}}F_{z}(dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f339f290a32d6693d07b017c6ed8ec9cecabb3)
![{\displaystyle {\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{r}r)}{\partial r}}+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}r{\frac {\partial F_{z}}{\partial z}}={\frac {1}{r}}{\frac {\partial (F_{r}r)}{\partial r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}={\frac {\partial F_{r}}{\partial r}}+{\frac {F_{r}}{r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6ccbb3933412ac5df3cec621fb9b9405ffc18c)
![{\displaystyle \uparrow (-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{r}}{\partial \vartheta }}dz-{\sqrt[{}]{r^{2}}}{\frac {\partial F_{r}}{\partial z}}d\vartheta +{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}dz-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}dr+{\sqrt[{}]{r^{2}}}{\frac {\partial F_{z}}{\partial r}}d\vartheta +{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{z}}{\partial \vartheta }}dr)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5398888823dfcdf01d73e11bb70dfa2d8158e60)
![{\displaystyle \ast d({\frac {\sqrt[{}]{|r^{2}|}}{1}}{\frac {\partial f}{\partial r}}(d\vartheta \wedge dz)+{\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}(dz\wedge dr)+{\frac {\sqrt[{}]{|r^{2}|}}{1}}{\frac {\partial f}{\partial z}}(dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19be4ae0414d762480b759294cacb3afec752ce)
![{\displaystyle \ast ({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})(dr\wedge d\vartheta \wedge dz)+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})(d\vartheta \wedge dz\wedge dr)+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})(dz\wedge dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/383900ae3b80f9c4b9a5a350ab96c2822e513f75)
![{\displaystyle \ast (({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})(dr\wedge d\vartheta \wedge dz))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/020bcfd8258c5852e3ba4cae651fbf2761a72d29)
![{\displaystyle ({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8c9cc9a1371d4c23602a006479bb861f5a9140)
![{\displaystyle ({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7def0dc71b033b87ffea3b4ea3b800a5be31bb68)
![{\displaystyle ({\frac {1}{\sqrt[{}]{r^{2}}}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}}))={\frac {1}{r}}({\frac {\partial }{\partial r}}(r{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}(r{\frac {\partial f}{\partial z}}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fafbbc6a56595ef8835a395edf1d3c86205af98c)
![{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial r}}|,|{\frac {\partial }{\partial \vartheta }}|,|{\frac {\partial }{\partial \varphi }}|)=({\sqrt[{}]{g_{r}}},{\sqrt[{}]{g_{\vartheta }}},{\sqrt[{}]{g_{\varphi }}})=(1,r,rsin\vartheta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca204bbb49eb4627d617466190b92e24cabce65e)
![{\displaystyle \ast (Fdr)={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{1}}F(d\vartheta \wedge d\varphi )=r^{2}sin\vartheta F(d\vartheta \wedge d\varphi );}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5703eb95376f146a96bc2c3fb8a3cc68e4939598)
![{\displaystyle \ast (Fd\vartheta )={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}F(dr\wedge d\varphi )=sin\vartheta F(dr\wedge d\varphi );}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52024edd8cc407e2c17abf221dbda38e520ace7c)
![{\displaystyle \ast (Fd\varphi )={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}F(d\vartheta \wedge dr)={\frac {1}{sin\vartheta }}F(d\vartheta \wedge dr)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/815a54e033e8d5fe4c98ef0725b147926fad1a8d)
![{\displaystyle \ast (F(dr\wedge d\vartheta ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}Fd\varphi =sin\vartheta Fd\varphi ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f2049c87cc04d927dcc909fddf9d85701f2f5a)
![{\displaystyle \ast (F(dr\wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}Fd\vartheta ={\frac {1}{sin\vartheta }}Fd\vartheta ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87ce2025a025feea420dda296e36edbb5e8eca8d)
![{\displaystyle \ast (F(d\vartheta \wedge dr))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}Fd\varphi =-sin\vartheta Fd\varphi ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f671861c848654b9681892ac8d926925e32bb5b)
![{\displaystyle \ast (F(d\vartheta \wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}Fdr={\frac {1}{r^{2}sin\vartheta }}Fdr;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9800df951f81a995493d5529395f172e6f73060b)
![{\displaystyle \ast (F(d\varphi \wedge dr))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}Fd\vartheta =-{\frac {1}{sin\vartheta }}Fd\vartheta ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3883dc4c7cee2327d04f7071843fae3f64788)
![{\displaystyle \ast (F(d\varphi \wedge d\vartheta ))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}Fdr=-{\frac {1}{r^{2}sin\vartheta }}Fdr;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc86f9425026fbbc1ac7d2cd0781768a1b5e8d0)
![{\displaystyle \ast (F(dr\wedge d\vartheta \wedge d\varphi ))=\varepsilon {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}F=\varepsilon {\frac {1}{r^{2}sin\vartheta }}F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/307969bea729f16e21a9288c2cc03a27e31f1049)
![{\displaystyle \ast d\ast \downarrow (F_{r}{\frac {1}{\sqrt[{}]{g_{r}}}}{\frac {\partial }{\partial r}})+F_{\vartheta }{\frac {1}{\sqrt[{}]{g_{\vartheta }}}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }{\frac {1}{\sqrt[{}]{g_{\varphi }}}}{\frac {\partial }{\partial \varphi }}))=\ast d\ast (F_{r}{\frac {1}{\sqrt[{}]{g_{r}}}}g_{r}dr+F_{\vartheta }{\frac {1}{\sqrt[{}]{g_{\vartheta }}}}g_{\vartheta }d\vartheta +F_{\varphi }{\frac {1}{\sqrt[{}]{g_{\varphi }}}}g_{\varphi }d\varphi ))=\ast d\ast (F_{r}dr+F_{\varphi }rd\vartheta +F_{\varphi }rsin\vartheta d\varphi )=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c90ac6149447fb57c6546a35c2050ecac029258)
![{\displaystyle \ast d(F_{r}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}(d\vartheta \wedge d\varphi )+F_{\vartheta }r{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}(d\varphi \wedge dr)+F_{\varphi }rsin\vartheta {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}(dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd90d0433d6e51e96fa81853da8138f8808a5646)
![{\displaystyle \ast d(F_{r}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}(d\vartheta \wedge d\varphi )+F_{\vartheta }{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r}}(d\varphi \wedge dr)+F_{\varphi }{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{rsin\vartheta }}(dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edf20cc6bfd7a719b74144261e29410822063074)
![{\displaystyle \ast ({\frac {\partial }{\partial r}}(F_{r}{\sqrt[{2}]{r^{4}sin^{2}\vartheta }})(dr\wedge d\theta \wedge d\varphi )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }{\sqrt[{2}]{r^{2}sin^{2}\vartheta }})(d\vartheta \wedge d\varphi \wedge dr)+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)(d\varphi \wedge dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c70ebaa42695baea75cd9c4bad5ea785e0dbbf0f)
![{\displaystyle \ast (({\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r))(dr\wedge d\vartheta \wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b90f584b3df9b195975f13cdf448f64fd8890911)
![{\displaystyle \uparrow (-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}{\frac {\partial F_{r}}{\partial \vartheta }}d\varphi +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}d\vartheta +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}d\varphi -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d79c34d68a7897f30014e602111cafa21ea79f2c)
![{\displaystyle {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}dr-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}d\vartheta +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}dr)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be11366bebc6a1cac7eaef5fb9540e1b086f985)
![{\displaystyle \ast d({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{1}}{\frac {\partial f}{\partial r}}(d\vartheta \wedge d\varphi )+{\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}(d\varphi \wedge dr)+{\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{2}sin^{2}\vartheta }}{\frac {\partial f}{\partial \varphi }}(dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7402d7b91f249815f7d4dfec23bb39ef53347cb)
![{\displaystyle \ast ({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})(dr\wedge d\vartheta \wedge d\varphi )+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})(d\vartheta \wedge d\varphi \wedge dr)+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})(d\varphi \wedge dr\wedge d\vartheta ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/079942c318e6ebbb4ea24aeb164fed41c3463f29)
![{\displaystyle \ast (({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})(dr\wedge d\vartheta \wedge d\varphi ))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7161ff1798284cd555bce8a0cc194c5b6a123e)
![{\displaystyle ({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a4c6e86851689f9c1af5277195c1bbc527738b)
![{\displaystyle ({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffaae08dc95692e75ff6dc4c68a61a7577623c17)
![{\displaystyle ({\frac {1}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6efb171fa73ebdb0273a2403542128d07c417a)
![{\displaystyle {\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial u_{1}}}|,|{\frac {|\partial }{\partial u_{2}}}|,|{\frac {\partial }{\partial u_{3}}}|)=({\sqrt[{}]{g_{11}}},{\sqrt[{}]{g_{22}}},{\sqrt[{}]{g_{33}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18c64a44a1ef381a1724fbb17c07945acfe622d0)
![{\displaystyle \ast (Fdx_{1})={\frac {\sqrt[{}]{|g|}}{g_{11}}}F(dx_{2}\wedge dx_{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3b647093f0ba319d03f560b616197d173429a7)
![{\displaystyle \ast (F(dx_{1}\wedge dx_{2}))={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}}}Fdx_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7147e22d2216c0a35f5cf8d94cb8fe19f9385313)
![{\displaystyle \ast (F(dx_{1}\wedge dx_{2}\wedge dx_{3}))={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}}}F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b40459aa892be84a0e327fe5fb10d8fdf133eaa)
![{\displaystyle {\frac {\partial f}{\partial u_{1}}}{\frac {1}{g_{11}}}|{\frac {\partial }{\partial u_{1}}}|{\frac {\frac {\partial }{\partial u_{1}}}{|{\frac {\partial }{\partial u_{1}}}|}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{g_{22}}}|{\frac {\partial }{\partial u_{2}}}|{\frac {\frac {\partial }{\partial u_{2}}}{|{\frac {\partial }{\partial u_{2}}}|}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{g_{33}}}|{\frac {\partial }{\partial u_{3}}}|{\frac {\frac {\partial }{\partial u_{3}}}{|{\frac {\partial }{\partial u_{3}}}|}}={\frac {\partial f}{\partial u_{1}}}{\frac {\sqrt[{}]{g_{11}}}{g_{11}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {\sqrt[{}]{g_{22}}}{g_{22}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {\sqrt[{}]{g_{33}}}{g_{33}}}{\frac {\widehat {\partial }}{\partial u_{3}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/434379e3f988f53b4728fc646f06bf020683c1c4)
![{\displaystyle {\frac {\partial f}{\partial u_{1}}}{\frac {1}{\sqrt[{}]{g_{11}}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{\sqrt[{}]{g_{22}}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{\sqrt[{}]{g_{33}}}}{\frac {\widehat {\partial }}{\partial u_{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6751af088b87836c638aa509ce612e3a093a4938)
![{\displaystyle \ast d\ast \downarrow (F_{1}{\frac {1}{\sqrt[{2}]{g_{11}}}}{\overrightarrow {u_{1}}}+F_{2}{\frac {1}{\sqrt[{2}]{g_{22}}}}{\overrightarrow {u_{2}}}+F_{3}{\frac {1}{\sqrt[{2}]{g_{33}}}}{\overrightarrow {u_{3}}})=\ast d\ast (F_{1}{\frac {1}{\sqrt[{2}]{g_{11}}}}g_{11}du_{1}+F_{2}{\frac {1}{\sqrt[{2}]{g_{22}}}}g_{22}du_{2}+F_{3}{\frac {1}{\sqrt[{2}]{g_{33}}}}g_{33}du_{3})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32dd0223b4029aa5ecc7058c6e430c1900ea38e9)
![{\displaystyle \ast d\ast (F_{1}{\sqrt[{2}]{g_{11}}}du_{1}+F_{2}{\sqrt[{2}]{g_{22}}}du_{2}+F_{3}{\sqrt[{2}]{g_{33}}}du_{3})=\ast d(F_{1}{\sqrt[{2}]{g_{11}}}{\frac {\sqrt[{2}]{g}}{g_{11}}}(du_{2}\wedge du_{3})+F_{2}{\sqrt[{2}]{g_{22}}}{\frac {\sqrt[{2}]{g}}{g_{22}}}(du_{3}\wedge du_{1})+F_{1}{\sqrt[{2}]{g_{33}}}{\frac {\sqrt[{2}]{g}}{g_{33}}}(du_{1}\wedge du_{2}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dadd3a61fbf3a1526000276d5b7f7e1575cf2ba)
![{\displaystyle \ast d(F_{1}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{11}}}}(du_{2}\wedge du_{3})+F_{2}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{22}}}}(du_{3}\wedge du_{1})+F_{3}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{33}}}}(du_{1}\wedge du_{2}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a413a7b671c55d35613b3f969724b2d793fabe2)
![{\displaystyle \ast ({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})(du_{1}\wedge du_{2}\wedge du_{3})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})(du_{2}\wedge du_{3}\wedge du_{1})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}})(du_{3}\wedge du_{1}\wedge du_{2}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e29b2a1fb3a9de154abe4a9793a726923aa107b9)
![{\displaystyle \ast (({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}}))(du_{1}\wedge du_{2}\wedge du_{3}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88283c3a11541b0a4b1c9ca3467a4fd444091af4)
![{\displaystyle {\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44da7eb9dde96af7fa6e9e599fd3da2583843667)
![{\displaystyle {\frac {1}{\sqrt[{2}]{g_{11}g_{22}g_{22}}}}({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}}))={\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}(F_{1}h_{2}h_{3})+{\frac {\partial }{\partial u_{2}}}(F_{2}h_{1}h_{3})+{\frac {\partial }{\partial u_{3}}}(F_{3}h_{1}h_{2}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d8ad5e482b53c82d53a7656c1e9276b13d1f5e)
![{\displaystyle rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\partial }{\partial x}}+{F_{y}}{\frac {\partial }{\partial y}}+F_{z}{\frac {\partial }{\partial z}})=\uparrow \ast d\downarrow ({F_{1}}({\frac {1}{\sqrt[{}]{g_{11}}}}{\frac {\partial }{\partial u_{1}}})+F_{2}({\frac {1}{\sqrt[{}]{g_{22}}}}{\frac {\partial }{\partial u_{2}}})+F_{3}({\frac {1}{\sqrt[{}]{g_{33}}}}{\frac {\partial }{\partial u_{3}}}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf0bedbaa71e1f8752c54b0fb44714995f3db97)
![{\displaystyle \uparrow \ast d(F_{1}{\frac {1}{\sqrt[{}]{g_{11}}}}g_{11}du_{1}+F_{2}{\frac {1}{\sqrt[{}]{g_{22}}}}g_{22}du_{2}+F_{3}{\frac {1}{\sqrt[{}]{g_{33}}}}g_{33}du_{3})=\uparrow \ast d({F_{1}}{\sqrt[{2}]{g_{11}}}du_{1}+F_{2}{\sqrt[{2}]{g_{22}}}du_{2})+F_{3}{\sqrt[{2}]{g_{33}}}du_{3})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44d08fb0854b5c40cc6c3e15f5554748443adc22)
![{\displaystyle \uparrow \ast ({\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{2}}}du_{2}\wedge du_{1}+{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{3}}}du_{3}\wedge du_{1}+{\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{1}}}du_{1}\wedge du_{2}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547db06657bce0b05797dbaeaad0b105f652877c)
![{\displaystyle {\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{3}}}du_{3}\wedge du_{2}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{1}}}du_{1}\wedge du_{3})+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{2}}}du_{2}\wedge du_{3})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0140f702df70bdcda88671746a212c1c6243608c)
![{\displaystyle \uparrow \ast (-{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{2}}}du_{1}\wedge du_{2}-{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{3}}}du_{1}\wedge du_{3}+{\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{1}}}du_{1}\wedge du_{2}-}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be0fb2b25c4fa1a54f42c42477789edcec420386)
![{\displaystyle {\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{3}}}du_{2}\wedge du_{3}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{1}}}du_{1}\wedge du_{3}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{2}}}du_{2}\wedge du_{3})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1d58f55085ed56ea1e9ca447d727d6f6c9ab8b)
![{\displaystyle \uparrow (-{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}{\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})du_{3}-{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})du_{2}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})du_{3}-}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd85526a0354c9b4ce30ad03d95235d8275d42f3)
![{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})du_{1}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})du_{2}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{22}g_{33}}}{\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})du_{1})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdccf0e4b6b91b4eafe771525f1a5581fafe0b25)
![{\displaystyle \uparrow ({\frac {\sqrt[{}]{\vert g\vert }}{g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}}))-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))du_{1}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6a86b510b19245a2dc17c3f9a6781c2c699fa4)
![{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))du_{2}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5913f44ac674cb1184e08b16bcb77f457598f17b)
![{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))du_{3})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acbb0d8549ef278a988498a1d5c84d4ceb65fe84)
![{\displaystyle \uparrow ({\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}}))-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))g_{11}du_{1}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69f95993818d2222864685a33cca94530b1c82d8)
![{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))g_{22}du_{2}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e125a9fee550fb930b2d15734d08ca5483b4bbd)
![{\displaystyle {\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))g_{33}du_{3})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aedcb361fcb31d4ddb4b3500e3ced8268fc4ea43)
![{\displaystyle {\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\frac {\partial }{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})){\frac {\partial }{\partial u_{2}}}+({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\frac {\partial }{\partial u_{3}}})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7c0142ce346e15ceecc36588d7be5f64bf090b)
![{\displaystyle {\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))\vert {\frac {\partial }{\partial u_{1}}}\vert {\frac {\frac {\partial }{\partial u_{1}}}{\vert {\frac {\partial }{\partial u_{1}}}\vert }}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8577aad275a4dd08c25c12aad6258afb79310ab2)
![{\displaystyle ({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))\vert {\frac {\partial }{\partial u_{2}}}\vert {\frac {\frac {\partial }{\partial u_{2}}}{\vert {\frac {\partial }{\partial u_{2}}}\vert }}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a27c86811408fe8e2425abd5680c267755162d0)
![{\displaystyle ({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))\vert {\frac {\partial }{\partial u_{3}}}\vert {\frac {\frac {\partial }{\partial u_{3}}}{\vert {\frac {\partial }{\partial u_{3}}}\vert }})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1079b325c5ee1bddc4537ca7f2fdb7acf2bec0a1)
![{\displaystyle {\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\sqrt[{2}]{g_{11}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})){\sqrt[{2}]{g_{22}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdceca9636950b984e74dffd8a2e2241338e889)
![{\displaystyle ({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\sqrt[{2}]{g_{33}}}){\frac {\widehat {\partial }}{\partial u_{3}}})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4878be7ae63d665ef288b018b9d90c7f6f9b25)
![{\displaystyle \ast d({\frac {\sqrt[{}]{|g|}}{g_{1}}}{\frac {\partial f}{\partial u_{1}}}(du_{2}\wedge du_{3})+{\frac {\sqrt[{}]{|g|}}{g_{2}}}{\frac {\partial f}{\partial u_{2}}}(du_{3}\wedge du_{1})+{\frac {\sqrt[{}]{|g|}}{g_{3}}}{\frac {\partial f}{\partial u_{3}}}(du_{1}\wedge du_{2}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084e328184e16fe3e2172a03181c6f2ba2c1b5be)
![{\displaystyle \ast ({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})(du_{1}\wedge du_{2}\wedge du_{3})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})(du_{2}\wedge du_{3}\wedge du_{1})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})(du_{3}\wedge du_{1}\wedge du_{2}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/172d6ab1042f61b38efeaf54b894d9b4dd66f205)
![{\displaystyle \ast (({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})(du_{1}\wedge du_{2}\wedge du_{3}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06f3772e94de4cb1d3248cd5b231eab00bd754e5)
![{\displaystyle ({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06eb013e03d475c78b02332d1f463438564de9ad)
![{\displaystyle ({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}})({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da77e4c5dc93570b34aea6f0790caab97c6f2337)
![{\displaystyle ({\frac {1}{\sqrt[{}]{g_{1}g_{2}g_{3}}}})({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}}))=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdafbd5b598bd5acd7efc60ef6354c3d91e6333e)
