Usuario:Christian Fabregas/Gradiente, divergencia, rotacional y laplaciano en coordenadas curvilíneas

Los operadores diferenciales son de amplia aplicación en multitud de ecuaciones diferenciales, utilizadas en física e ingeniería. Pueden expresarse en diferentes coordenadas curvilíneas para sistemas que contienen esa simetría. En particular, son importantes las coordenadas cilíndricas y esféricas, simetrías muy utilizadas en las disciplinas mencionadas. Se desarrollarán todos los operadores diferenciales (gradiente, divergencia, rotacional y laplaciano) en diferentes coordenadas. Las demostraciones se llevarán a cabo mediante dos diciplinas matemáticas diferentes, el cálculo diferencial y la geometría diferencial.

>> Operadores diferenciales en coordenadas cilíndricas

* Coordenadas cilíndricas

Cambio de coordenadas cartesianas a coordenadas cilíndricas.
$x=rcos\vartheta$
$y=rsin\vartheta$
$z=z$

* Cambio de base

Cambio de la base ortonormal en coordenadas cartesianas a la base en coordenadas cilíndricas.
${\widehat {e}}_{r}=cos\vartheta {\widehat {e}}_{x}+sin\vartheta {\widehat {e}}_{y}$
${\widehat {e}}_{\vartheta }=-sin\vartheta {\widehat {e}}_{x}+cos\vartheta {\widehat {e}}_{y}$
${\widehat {e}}_{z}={\widehat {e}}_{z}$

* Derivadas de las bases en coordenadas cilíndricas

Utilizaremos las siguientes derivadas de las bases.
${\frac {\partial {\widehat {e}}_{r}}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}={\widehat {e}}_{\vartheta };{\frac {\partial {\widehat {e}}_{r}}{\partial z}}=0;{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}=-{\widehat {e}}_{r};{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}}=0;{\frac {\partial {\widehat {e}}_{z}}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }}=0;{\frac {\partial {\widehat {e}}_{z}}{\partial z}}=0;$

* Productos escalares en coordenadas cartesianas y cilíndricas

Necesitaremos los siguientes productos escalares.
${\widehat {e}}_{x}\cdot {\widehat {e}}_{x}=1;{\widehat {e}}_{y}\cdot {\widehat {e}}_{y}=1;{\widehat {e}}_{z}\cdot {\widehat {e}}_{z}=1$
${\widehat {e}}_{y}\cdot {\widehat {e}}_{x}=0;{\widehat {e}}_{x}\cdot {\widehat {e}}_{y}=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{x}=0;{\widehat {e}}_{x}\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{y}\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{y}=0$
${\widehat {e}}_{r}\cdot {\widehat {e}}_{r}=1;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }=1;{\widehat {e}}_{z}\cdot {\widehat {e}}_{z}=1$
${\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}=0;{\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }=0$

* Productos vectoriales en coordenadas cartesianas y cilíndricas

Utilizaremos los siguientes productos vectoriales.
${\widehat {e}}_{x}\times {\widehat {e}}_{x}=0;{\widehat {e}}_{y}\times {\widehat {e}}_{y}=0;{\widehat {e}}_{z}\times {\widehat {e}}_{z}=0$
${\widehat {e}}_{x}\times {\widehat {e}}_{y}={\widehat {e}}_{z};{\widehat {e}}_{y}\times {\widehat {e}}_{z}={\widehat {e}}_{x};{\widehat {e}}_{z}\times {\widehat {e}}_{x}={\widehat {e}}_{y}$
${\widehat {e}}_{y}\times {\widehat {e}}_{x}=-{\widehat {e}}_{z};{\widehat {e}}_{z}\times {\widehat {e}}_{y}=-{\widehat {e}}_{x};{\widehat {e}}_{x}\times {\widehat {e}}_{z}=-{\widehat {e}}_{y}$
${\widehat {e}}_{r}\times {\widehat {e}}_{r}=0;{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{z}\times {\widehat {e}}_{z}=0$
${\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta }={\widehat {e}}_{z};{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z}={\widehat {e}}_{r};{\widehat {e}}_{z}\times {\widehat {e}}_{r}={\widehat {e}}_{\vartheta }$
${\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r}=-{\widehat {e}}_{z};{\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta }=-{\widehat {e}}_{r};{\widehat {e}}_{r}\times {\widehat {e}}_{z}=-{\widehat {e}}_{\vartheta }$

* Factor de escala en coordenadas cilíndricas

El factor de escala nos permite expresar un vector en otras coordenadas y nos da el módulo de las bases.
$d{\overrightarrow {r}}=h_{r}dr+h_{\vartheta }d\vartheta +h_{z}dz$
${\overrightarrow {h}}=\sum |{\frac {\partial {\overrightarrow {r}}}{\partial u_{i}}}|=(|{\frac {\partial x}{\partial r}}|,|{\frac {\partial y}{\partial \vartheta }}|,|{\frac {\partial z}{\partial z}}|)=(h_{r},h_{\vartheta },h_{z})=(1,r,1)$
Podemos observar que los factores de escala también pueden expresarse en función de la métrica del espacio de la geometría de Riemann.
${\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial r}}|,|{\frac {\partial }{\partial \vartheta }}|,|{\frac {\partial }{\partial z}}|)=({\sqrt[{}]{g_{r}}},{\sqrt[{}]{g_{\vartheta }}},{\sqrt[{}]{g_{z}}})=(1,r,1)$

* Métrica o tensor métrico en coordenadas cilíndricas

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.
$g_{\alpha \beta }=g_{r}dr^{2}+g_{\vartheta }d\vartheta ^{2}+g_{z}dz^{2}=dr^{2}+r^{2}d\vartheta ^{2}+dz^{2}$
$g_{\alpha \beta }={\begin{bmatrix}g_{r}&&\\&g_{\vartheta }&\\&&g_{z}\end{bmatrix}}={\begin{bmatrix}\vert {\frac {\partial }{\partial r}}\vert \cdot |{\frac {\partial }{\partial r}}|&&\\&|{\frac {\partial }{\partial \vartheta }}|\cdot |{\frac {\partial }{\partial \vartheta }}|&\\&&|{\frac {\partial }{\partial z}}|\cdot |{\frac {\partial }{\partial z}}|\end{bmatrix}}={\begin{bmatrix}1&&\\&r^{2}&\\&&1\end{bmatrix}}$
$det(g_{\alpha \beta })=|g|=r^{2}$

* Estrella de Hodge (*) en coordenadas cilíndricas

La estrella de Hodge es un operador que actúa sobre un p-forma diferencial en un espacio de dimensión n. $*(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})$
- Para una 1-forma:
$\ast (Fdx_{\alpha })=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }}}F(dx_{\beta }\wedge dx_{\gamma }):$
$\ast (Fdr)={\frac {\sqrt[{}]{r^{2}}}{1}}F(d\vartheta \wedge dz)=rF(d\vartheta \wedge dz);\ast (Fd\vartheta )={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}F(dr\wedge dz)={\frac {1}{r}}F(dr\wedge dz);\ast (Fdz)={\frac {\sqrt[{}]{r^{2}}}{1}}F(d\vartheta \wedge dr)=rF(d\vartheta \wedge dr)$
- Para una 2-forma:
$\ast (F(dx_{\alpha }\wedge dx_{\beta }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }}}Fdx_{\gamma }:$
$\ast (F(dr\wedge d\vartheta ))={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdz={\frac {1}{r}}Fdz;\ast (F(dr\wedge dz))={\frac {\sqrt[{}]{r^{2}}}{1}}Fd\vartheta =rFd\vartheta ;$
$\ast (F(d\vartheta \wedge dr))=-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdz=-{\frac {1}{r}}Fdz;\ast (F(d\vartheta \wedge dz))={\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdr={\frac {1}{r}}Fdr;$
$\ast (F(dz\wedge dr))=-{\frac {\sqrt[{}]{r^{2}}}{1}}Fd\vartheta =-rFd\vartheta ;\ast (F(dz\wedge d\vartheta ))=-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}Fdr=-{\frac {1}{r}}Fdr;$
- Para una 3-forma:
$\ast (F(dx_{\alpha }\wedge dx_{\beta }\wedge dx_{\gamma }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }g_{\gamma \gamma }}}F:$
$\ast (F(dr\wedge d\vartheta \wedge dz))=\varepsilon {\frac {\sqrt[{}]{r^{2}}}{r^{2}}}F=\varepsilon {\frac {1}{r}}F$

* Subir y bajar índices. Coordenadas cilíndricas

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.
${\frac {\partial }{\partial x^{\alpha }}}=g_{\alpha \beta }dx^{\beta }:$
${\frac {\partial }{\partial r}}=dr;{\frac {\partial }{\partial \vartheta }}=r^{2}d\vartheta ;{\frac {\partial }{\partial z}}=dz$
$dx^{\alpha }=g^{\alpha \beta }{\frac {\partial }{\partial x^{\beta }}}:$
$dr={\frac {\partial }{\partial r}};d\vartheta ={\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }};dz={\frac {\partial }{\partial z}}$

> Gradiente en coordenadas cilíndricas. Cálculo diferencial

El gradiente se calcula aplicando el operador nabla a un campo escalar. Para determinar los factores de escala aplicamos la regla de la cadena.
${\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})f(x,y,z)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})f(x,y,z)=$
$({\widehat {e}}_{r}{\frac {\partial r}{\partial x}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {\partial \vartheta }{\partial y}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial z}{\partial z}}{\frac {\partial }{\partial z}})f(r,\vartheta ,z)={\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{|{\frac {\partial \vartheta }{\partial y}}|}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{|{\frac {\partial z}{\partial z}}|}}{\frac {\partial f}{\partial z}}{\widehat {e}}_{z}=$
${\frac {1}{h_{1}}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial z}}{\widehat {e}}_{z}={\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {\partial f}{\partial z}}{\widehat {e}}_{z}$

> Gradiente en coordenadas cilíndricas. Geometría diferencial

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.
${\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=\downarrow d(f(r,\theta ,z))=\downarrow ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \theta }}d\theta +{\frac {\partial f}{\partial \varphi }}dz)={\frac {\partial f}{\partial r}}{\frac {1}{g_{r}}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{g_{\theta }}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {1}{g_{z}}}{\frac {\partial }{\partial z}}=$
${\frac {\partial f}{\partial r}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {\partial }{\partial z}}={\frac {\partial f}{\partial r}}|{\frac {\partial }{\partial r}}|{\frac {\frac {\partial }{\partial r}}{|{\frac {\partial }{\partial r}}|}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}|{\frac {\partial }{\partial \theta }}|{\frac {\frac {\partial }{\partial \theta }}{|{\frac {\partial }{\partial \theta }}|}}+{\frac {\partial f}{\partial z}}|{\frac {\partial }{\partial z}}|{\frac {\frac {\partial }{\partial z}}{|{\frac {\partial }{\partial z}}|}}={\frac {\partial f}{\partial r}}{\frac {\widehat {\partial }}{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r}}{\frac {\widehat {\partial }}{\partial \theta }}+{\frac {\partial f}{\partial z}}{\frac {\widehat {\partial }}{\partial z}}$

> Divergencia en coordenadas cilíndricas. Cálculo diferencial

${\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot (F_{x},F_{y},F_{z})=$ $({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot (F_{x}{\widehat {e}}_{x}+F_{y}{\widehat {e}}_{y}+F_{z}{\widehat {e}}_{z})=({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{h_{z}}}{\frac {\partial }{\partial z}})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=$ $({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{1}}{\frac {\partial }{\partial z}})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=$ ${\widehat {e}}_{r}\cdot ({\frac {\partial F_{r}}{\partial r}}{\widehat {e}}_{r})+{\widehat {e}}_{r}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{r}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{z}}{\partial r}}{\widehat {e}}_{z})+{\widehat {e}}_{r}\cdot (F_{z}{\frac {\partial {\widehat {e}}_{z}}{\partial r}})+$ ${\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\vartheta }}{\partial \vartheta }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{z}}{\partial \vartheta }}{\widehat {e}}_{z})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{z}{\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }})+$ ${\widehat {e}}_{z}\cdot ({\frac {\partial F_{r}}{\partial z}}{\widehat {e}}_{r})+{\widehat {e}}_{z}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial z}})+{\widehat {e}}_{z}\cdot ({\frac {\partial F_{\vartheta }}{\partial z}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{z}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}})+{\widehat {e}}_{z}\cdot ({\frac {\partial F_{z}}{\partial z}}{\widehat {e}}_{z})+{\widehat {e}}_{z}\cdot (F_{z}{\frac {\partial {\widehat {e}}_{z}}{\partial z}})=$ ${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{r})+{\frac {F_{r}}{r}}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {F_{\vartheta }}{r}}({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r}))+{\frac {\partial F_{z}}{\partial z}}({\widehat {e}}_{z}\cdot {\widehat {e}}_{z})=$
${\frac {\partial F_{r}}{\partial r}}+{\frac {F_{r}}{r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}$

> Divergencia en coordenadas cilíndricas. Geometría diferencial

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.
$div({\overrightarrow {F}})={\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=\ast d\ast \downarrow ({\overrightarrow {F}})=\ast d\ast \downarrow (F_{x}{\frac {\widehat {\partial }}{\partial x}}+F_{y}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\ast d\ast \downarrow (F_{r}({\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial }{\partial r}})+F_{\vartheta }({\frac {1}{|{\frac {\partial y}{\partial \vartheta }}|}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{|{\frac {\partial z}{\partial z}}|}}{\frac {\partial }{\partial z}}))=$
$\ast d\ast \downarrow (F_{r}({\frac {1}{\sqrt[{}]{g_{r}}}}{\frac {\partial }{\partial r}})+F_{\vartheta }({\frac {1}{\sqrt[{}]{g_{\vartheta }}}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{\sqrt[{}]{g_{z}}}}{\frac {\partial }{\partial z}}))=\ast d\downarrow (F_{r}{\frac {\partial }{\partial r}}+F_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+F_{z}{\frac {\partial }{\partial z}})=$
$\ast d\ast (F_{r}dr+{\frac {F_{\vartheta }}{r}}(r^{2}d\vartheta )+F_{z}dz)=\ast d\ast (F_{r}dr+F_{\vartheta }rd\vartheta +F_{z}dz)=\ast d({\frac {\sqrt[{}]{r^{2}}}{1}}F_{r}(d\vartheta \wedge dz)+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}rF_{\vartheta }(dz\wedge dr)+{\frac {\sqrt[{}]{r^{2}}}{1}}F_{z}(dr\wedge d\vartheta ))=$
$\ast d(rF_{r}(d\vartheta \wedge dz)+F_{\vartheta }(dz\wedge dr)+rF_{z}(dr\wedge d\vartheta ))=\ast ({\frac {\partial (F_{r}r)}{\partial r}}(dr\wedge d\vartheta \wedge dz)+{\frac {\partial F_{\vartheta }}{\partial \vartheta }}(d\vartheta \wedge dz\wedge dr)+r{\frac {\partial F_{z}}{\partial z}}(dz\wedge dr\wedge d\vartheta ))=$
${\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{r}r)}{\partial r}}+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}r{\frac {\partial F_{z}}{\partial z}}={\frac {1}{r}}{\frac {\partial (F_{r}r)}{\partial r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}={\frac {\partial F_{r}}{\partial r}}+{\frac {F_{r}}{r}}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {\partial F_{z}}{\partial z}}$

> Rotacional en coordenadas cilíndricas. Cálculo diferencial

${\overrightarrow {\nabla }}\times {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\times (F_{x},F_{y},F_{z})=({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}},{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }},{\frac {1}{h_{z}}}{\frac {\partial }{\partial z}})\times (F_{r},F_{\vartheta },F_{z})=$
$({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{h_{z}}}{\frac {\partial }{\partial z}})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{z}}{1}}{\frac {\partial }{\partial z}})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{z}{\widehat {e}}_{z})=$
$({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{r}{\widehat {e}}_{r})+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{z}{\widehat {e}}_{z})+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{r}{\widehat {e}}_{r})+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{z}{\widehat {e}}_{z})+$
$({\widehat {e}}_{z}{\frac {\partial }{\partial z}})\times (F_{r}{\widehat {e}}_{r})+({\widehat {e}}_{z}{\frac {\partial }{\partial z}})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\widehat {e}}_{z}{\frac {\partial }{\partial z}})\times (F_{z}{\widehat {e}}_{z})=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+F_{\vartheta }({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\frac {\partial F_{z}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{z})+F_{z}({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{z}}{\partial r}})+$
${\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z})+{\frac {1}{r}}F_{z}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }})+$
${\frac {\partial F_{r}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{z}\times {\frac {\partial {\widehat {e}}_{r}}{\partial z}})+{\frac {\partial F_{\vartheta }}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta })+F_{\vartheta }({\widehat {e}}_{z}\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}})+{\frac {\partial F_{z}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{z})+F_{z}({\widehat {e}}_{z}\times {\frac {\partial {\widehat {e}}_{z}}{\partial z}})=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{z}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{z})+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+$
${\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z})+{\frac {\partial F_{r}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{z}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{z})=$
${\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{z}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{z})+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+$
${\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{z})+{\frac {\partial F_{r}}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial z}}({\widehat {e}}_{z}\times {\widehat {e}}_{\vartheta })=({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial z}}){\widehat {e}}_{r}+({\frac {\partial F_{r}}{\partial z}}-{\frac {\partial F_{z}}{\partial r}}){\widehat {e}}_{\vartheta }+({\frac {\partial F_{\vartheta }}{\partial r}}+{\frac {F_{\vartheta }}{r}}-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}){\widehat {e}}_{z}$

> Rotacional en coordenadas cilíndricas. Geometría diferencial

Para desarrollar el rotacional con geometría diferencial bajaremos índices, aplicaremos la diferencial exterior; y finalmente, aplicaremos la estrella de Hodge y subiremos índices.
$rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\widehat {\partial }}{\partial x}}+{F_{y}}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\uparrow \ast d\downarrow ({F_{r}}({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}})+{F_{\vartheta }}({\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }})+F_{z}({\frac {1}{h_{z}}}{\frac {\partial }{\partial z}}))=$
$\uparrow \ast d\downarrow ({F_{r}}{\frac {\partial }{\partial r}}+{F_{\vartheta }}({\frac {1}{r}}{\frac {\partial }{\partial \vartheta }})+F_{z}{\frac {\partial }{\partial r}})=\uparrow \ast d({F_{r}}dr+{\frac {F_{\vartheta }}{r}}r^{2}d\vartheta +F_{z}dz)=\uparrow \ast d({F_{r}}dr+F_{\vartheta }rd\vartheta +F_{z}dz)=$
$\uparrow \ast ({\frac {\partial F_{r}}{\partial \vartheta }}(d\vartheta \wedge dr)+{\frac {\partial F_{r}}{\partial z}}(dz\wedge dr)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )+{\frac {\partial (F_{\vartheta }r)}{\partial z}}(dz\wedge d\vartheta )+{\frac {\partial F_{z}}{\partial r}}(dr\wedge dz)+{\frac {\partial F_{z}}{\partial \vartheta }}(d\vartheta \wedge dz))=$
$\uparrow \ast (-{\frac {\partial F_{r}}{\partial \vartheta }}(dr\wedge d\vartheta )-{\frac {\partial F_{r}}{\partial z}}(dr\wedge dz)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )-{\frac {\partial (F_{\vartheta }r)}{\partial z}}(d\vartheta \wedge dz)+{\frac {\partial F_{z}}{\partial r}}(dr\wedge dz)+{\frac {\partial F_{z}}{\partial \vartheta }}(d\vartheta \wedge dz))=$
$\uparrow (-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{r}}{\partial \vartheta }}dz-{\sqrt[{}]{r^{2}}}{\frac {\partial F_{r}}{\partial z}}d\vartheta +{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}dz-{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}dr+{\sqrt[{}]{r^{2}}}{\frac {\partial F_{z}}{\partial r}}d\vartheta +{\frac {\sqrt[{}]{r^{2}}}{r^{2}}}{\frac {\partial F_{z}}{\partial \vartheta }}dr)=$
$\uparrow ({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}dr-{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}dr+r{\frac {\partial F_{z}}{\partial r}}d\vartheta -r{\frac {\partial F_{r}}{\partial z}}d\vartheta +{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}dz-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}dz)=$
${\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}{\frac {\partial }{\partial r}}-{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial z}}{\frac {\partial }{\partial r}}+r{\frac {\partial F_{z}}{\partial r}}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}-r{\frac {\partial F_{r}}{\partial z}}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}+{\frac {1}{r}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}{\frac {\partial }{\partial z}}-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}{\frac {\partial }{\partial z}}=$
$({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}-{\frac {\partial (F_{\vartheta })}{\partial z}}){\frac {\partial }{\partial r}}+({\frac {\partial F_{z}}{\partial r}}-{\frac {\partial F_{r}}{\partial z}}){\frac {1}{r}}|{\frac {\partial }{\partial \vartheta }}|{\frac {\frac {\partial }{\partial \vartheta }}{|{\frac {\partial }{\partial \vartheta }}|}}+{\frac {1}{r}}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\partial }{\partial z}}=$
$({\frac {1}{r}}{\frac {\partial F_{z}}{\partial \vartheta }}-{\frac {\partial (F_{\vartheta })}{\partial z}}){\frac {\widehat {\partial }}{\partial r}}+({\frac {\partial F_{z}}{\partial r}}-{\frac {\partial F_{r}}{\partial z}}){\frac {\widehat {\partial }}{\partial \vartheta }}+{\frac {1}{r}}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\widehat {\partial }}{\partial z}}$

> Laplaciano en coordenadas cilíndricas. Cálculo diferencial

$div({\overrightarrow {grad}}(f))=\nabla ^{2}(f)={\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}f)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{x}{\frac {\partial f}{\partial x}}+{\widehat {e}}_{y}{\frac {\partial f}{\partial y}}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=$
$({\widehat {e}}_{r}{\frac {1}{h_{r}}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {1}{h_{z}}}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=$
$({\widehat {e}}_{r}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=$
$({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial r}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial r}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{z}){\frac {\partial }{\partial r}}({\frac {\partial f}{\partial z}})+$
$({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial z}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial z}})+$
$({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial z}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{r}){\frac {\partial }{\partial z}}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial z}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial z}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}=$
$({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial r}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial r}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{z}){\frac {\partial }{\partial r}}({\frac {\partial f}{\partial z}})+$
$({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r})){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial z}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial z}})+$
$({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial z}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{r}){\frac {\partial }{\partial z}}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial z}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial z}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{z}\cdot {\frac {\partial {\widehat {e}}_{z}}{\partial z}}){\frac {\partial f}{\partial z}}+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}=$
$({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\widehat {e}}_{z}){\frac {\partial }{\partial r}}({\frac {\partial f}{\partial z}})+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r})){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+$
$({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{z}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial z}})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{r}){\frac {\partial }{\partial z}}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial z}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}=$
$({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{z}\cdot {\widehat {e}}_{z}){\frac {\partial ^{2}f}{\partial ^{2}z}}={\frac {\partial ^{2}f}{\partial ^{2}r}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }}+{\frac {\partial ^{2}f}{\partial ^{2}z}}$

> Laplaciano en coordenadas cilíndricas. Geometría diferencial

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.
$div({\overrightarrow {grad}}(f))=\nabla ^{2}f=\ast d\ast d(f)=\ast d\ast ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \vartheta }}d\vartheta +{\frac {\partial f}{\partial z}}dz)=$
$\ast d({\frac {\sqrt[{}]{|r^{2}|}}{1}}{\frac {\partial f}{\partial r}}(d\vartheta \wedge dz)+{\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}(dz\wedge dr)+{\frac {\sqrt[{}]{|r^{2}|}}{1}}{\frac {\partial f}{\partial z}}(dr\wedge d\vartheta ))=$
$\ast ({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})(dr\wedge d\vartheta \wedge dz)+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})(d\vartheta \wedge dz\wedge dr)+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})(dz\wedge dr\wedge d\vartheta ))=$
$\ast (({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})(dr\wedge d\vartheta \wedge dz))=$
$({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}}){\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}})=$
$({\frac {\sqrt[{}]{|r^{2}|}}{r^{2}}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}}))=$
$({\frac {1}{\sqrt[{}]{r^{2}}}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {1}{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}({\sqrt[{2}]{\frac {r^{2}}{1}}}{\frac {\partial f}{\partial z}}))={\frac {1}{r}}({\frac {\partial }{\partial r}}(r{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial z}}(r{\frac {\partial f}{\partial z}}))$

>> Operadores diferenciales en coordenadas esféricas

* Coordenadas esféricas

Cambio de coordenadas cartesianas a coordenadas esféricas.
$x=rsin\vartheta cos\varphi$
$y=rsin\vartheta sin\varphi$
$z=rcos\vartheta$

* Cambio de base

Cambio de la base ortonormal de las coordenadas cartesianas a la base en coordenadas esféricas.
${\widehat {e}}_{r}=sin\vartheta cos\varphi {\widehat {e}}_{x}+sin\vartheta sin\varphi {\widehat {e}}_{y}+cos\vartheta {\widehat {e}}_{z}$
${\widehat {e}}_{\vartheta }=cos\vartheta cos\varphi {\widehat {e}}_{x}+cos\vartheta sin\varphi {\widehat {e}}_{y}-sin\varphi {\widehat {e}}_{z}$
${\widehat {e}}_{\varphi }=-sin\varphi {\widehat {e}}_{x}+cos\varphi {\widehat {e}}_{y}$

* Derivadas de las bases en coordenadas esféricas

Para las demostraciones necesitaremos de las derivadas de las bases.
${\frac {\partial {\widehat {e}}_{r}}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}={\widehat {e}}_{\vartheta };{\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }}=sin\vartheta {\widehat {e}}_{\varphi };{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}=-{\widehat {e}}_{r};{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }}=cos\vartheta {\widehat {e}}_{\varphi };{\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}}=0;{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }}=0;{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }}=-sin\vartheta {\widehat {e}}_{r}-cos\vartheta {\widehat {e}}_{\vartheta };$

* Productos escalares en coordenadas esféricas

Para los desarrollos necesitaremos los siguientes productos escalares, de las bases en coordenadas esféricas.
${\widehat {e}}_{r}\cdot {\widehat {e}}_{r}=1;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }=1;{\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi }=1$
${\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi }=0;{\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi }=0;{\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }=0$

* Productos vectoriales en coordenadas esféricas

Necesitariemos los productos vectoriales de las bases.
${\widehat {e}}_{r}\times {\widehat {e}}_{r}=0;{\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta }={\widehat {e}}_{\varphi };{\widehat {e}}_{r}\times {\widehat {e}}_{\varphi }=-{\widehat {e}}_{\vartheta }$
${\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r}=-{\widehat {e}}_{\varphi };{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta }=0;{\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi }={\widehat {e}}_{r}$
${\widehat {e}}_{\varphi }\times {\widehat {e}}_{r}={\widehat {e}}_{\vartheta };{\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta }=-{\widehat {e}}_{r};{\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi }=0$

* Factor de escala en coordenadas esféricas

El factor de escala nos permite pasar de la base en coordenadas cartesianas la base en coordenadas esféricas.
${\overrightarrow {h}}=\sum |{\frac {\partial {\overrightarrow {r}}}{\partial u_{i}}}|=(|{\frac {\partial x}{\partial r}}|,|{\frac {\partial y}{\partial \vartheta }}|,|{\frac {\partial z}{\partial \varphi }}|)=(h_{r},h_{\vartheta },h_{z})=(1,r,rsin\vartheta )$
Podemos observar que los factores de escala también pueden expresarse en función de la métrica del espacio en la geometría de Riemann.
${\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial r}}|,|{\frac {\partial }{\partial \vartheta }}|,|{\frac {\partial }{\partial \varphi }}|)=({\sqrt[{}]{g_{r}}},{\sqrt[{}]{g_{\vartheta }}},{\sqrt[{}]{g_{\varphi }}})=(1,r,rsin\vartheta )$

* Métrica o tensor métrico en coordenadas esféricas

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.
$g_{\alpha \beta }=g_{r}dr^{2}+g_{\vartheta }d\vartheta ^{2}+g_{\varphi }d\varphi ^{2}=dr^{2}+r^{2}d\vartheta ^{2}+r^{2}sin^{2}\vartheta d\varphi ^{2}$
$g_{\alpha \beta }={\begin{bmatrix}g_{r}&&\\&g_{\vartheta }&\\&&g_{\varphi }\end{bmatrix}}={\begin{bmatrix}\vert {\frac {\partial }{\partial r}}\vert \cdot |{\frac {\partial }{\partial r}}|&&\\&|{\frac {\partial }{\partial \vartheta }}|\cdot |{\frac {\partial }{\partial \vartheta }}|&\\&&|{\frac {\partial }{\partial \varphi }}|\cdot |{\frac {\partial }{\partial \varphi }}|\end{bmatrix}}={\begin{bmatrix}1&&\\&r^{2}&\\&&r^{2}sin^{2}\vartheta \end{bmatrix}}$
$det(g_{\alpha \beta })=|g|=r^{4}sin^{2}\vartheta$

* Estrella de Hodge (*) en coordenadas esféricas

La estrella de Hodge es un operador que actúa sobre un p-forma diferencial en una dimensión n.
$*(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})$
- Para una 1-forma:
$\ast (Fdx_{\alpha })=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }}}F(dx_{\beta }\wedge dx_{\gamma }):$
$\ast (Fdr)={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{1}}F(d\vartheta \wedge d\varphi )=r^{2}sin\vartheta F(d\vartheta \wedge d\varphi );$
$\ast (Fd\vartheta )={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}F(dr\wedge d\varphi )=sin\vartheta F(dr\wedge d\varphi );$
$\ast (Fd\varphi )={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}F(d\vartheta \wedge dr)={\frac {1}{sin\vartheta }}F(d\vartheta \wedge dr)$
- Para una 2-forma:
$\ast (F(dx_{\alpha }\wedge dx_{\beta }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }}}Fdx_{\gamma }:$
$\ast (F(dr\wedge d\vartheta ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}Fd\varphi =sin\vartheta Fd\varphi ;$
$\ast (F(dr\wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}Fd\vartheta ={\frac {1}{sin\vartheta }}Fd\vartheta ;$
$\ast (F(d\vartheta \wedge dr))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}Fd\varphi =-sin\vartheta Fd\varphi ;$
$\ast (F(d\vartheta \wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}Fdr={\frac {1}{r^{2}sin\vartheta }}Fdr;$
$\ast (F(d\varphi \wedge dr))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}Fd\vartheta =-{\frac {1}{sin\vartheta }}Fd\vartheta ;$
$\ast (F(d\varphi \wedge d\vartheta ))=-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}Fdr=-{\frac {1}{r^{2}sin\vartheta }}Fdr;$
- Para una 3-forma:
$\ast (F(dx_{\alpha }\wedge dx_{\beta }\wedge dx_{\gamma }))=\varepsilon {\frac {\sqrt[{}]{|g|}}{g_{\alpha \alpha }g_{\beta \beta }g_{\gamma \gamma }}}F:$
$\ast (F(dr\wedge d\vartheta \wedge d\varphi ))=\varepsilon {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}r^{2}sin^{2}\vartheta }}F=\varepsilon {\frac {1}{r^{2}sin\vartheta }}F$

* Subir y bajar índices. Coordenadas esféricas

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.
${\frac {\partial }{\partial x^{\alpha }}}=g_{\alpha \beta }dx^{\beta }:$
${\frac {\partial }{\partial r}}=dr;{\frac {\partial }{\partial \vartheta }}=r^{2}d\vartheta ;{\frac {\partial }{\partial \varphi }}=r^{2}sin^{2}\vartheta d\varphi ;$
$dx^{\alpha }=g^{\alpha \beta }{\frac {\partial }{\partial x^{\beta }}}:$
$dr={\frac {\partial }{\partial r}};d\vartheta ={\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }};d\varphi ={\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}$

> Gradiente en coordenadas esféricas. Cálculo diferencial

${\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})f(x,y,z)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})f(x,y,z)=$
$({\widehat {e}}_{r}{\frac {\partial r}{\partial x}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {\partial \vartheta }{\partial y}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {\partial z}{\partial \varphi }}{\frac {\partial }{\partial \varphi }})f(r,\vartheta ,\varphi )={\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{|{\frac {\partial \vartheta }{\partial y}}|}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{|{\frac {\partial z}{\partial \varphi }}|}}{\frac {\partial f}{\partial \varphi }}{\widehat {e}}_{\varphi }=$
${\frac {1}{h_{1}}}{\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial \varphi }}{\widehat {e}}_{\varphi }={\frac {\partial f}{\partial r}}{\widehat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}{\widehat {e}}_{\vartheta }+{\frac {1}{rsin\theta }}{\frac {\partial f}{\partial \varphi }}{\widehat {e}}_{\varphi }$

> Gradiente en coordenadas esféricas. Geometría diferencial

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.
${\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=\downarrow d(f(r,\theta ,\varphi ))=\downarrow ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \theta }}d\theta +{\frac {\partial f}{\partial \varphi }}d\varphi )={\frac {\partial f}{\partial r}}{\frac {1}{g_{r}}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{g_{\theta }}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{g_{\varphi }}}{\frac {\partial }{\partial \varphi }}=$
${\frac {\partial f}{\partial r}}{\frac {\partial }{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{r^{2}sin^{2}\theta }}{\frac {\partial }{\partial \varphi }}={\frac {\partial f}{\partial r}}|{\frac {\partial }{\partial r}}|{\frac {\frac {\partial }{\partial r}}{|{\frac {\partial }{\partial r}}|}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r^{2}}}|{\frac {\partial }{\partial \theta }}|{\frac {\frac {\partial }{\partial \theta }}{|{\frac {\partial }{\partial \theta }}|}}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{r^{2}sin^{2}\theta }}|{\frac {\partial }{\partial \varphi }}|{\frac {\frac {\partial }{\partial \varphi }}{|{\frac {\partial }{\partial \varphi }}|}}=$
${\frac {\partial f}{\partial r}}{\frac {\widehat {\partial }}{\partial r}}+{\frac {\partial f}{\partial \theta }}{\frac {1}{r}}{\frac {\widehat {\partial }}{\partial \theta }}+{\frac {\partial f}{\partial \varphi }}{\frac {1}{rsin\theta }}{\frac {\widehat {\partial }}{\partial \varphi }}$

> Divergencia en coordenadas esféricas. Cálculo diferencial

${\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\cdot (F_{x},F_{y},F_{z})=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{\varphi }{\frac {\partial }{\partial \varphi }})\cdot (F_{x}{\widehat {e}}_{x}+F_{y}{\widehat {e}}_{y}+F_{z}{\widehat {e}}_{\varphi })=$
$({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\cdot (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=$
${\widehat {e}}_{r}\cdot ({\frac {\partial F_{r}}{\partial r}}{\widehat {e}}_{r})+{\widehat {e}}_{r}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{r}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\varphi })+{\widehat {e}}_{r}\cdot (F_{\varphi }{\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}})+$
${\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\vartheta }}{\partial \vartheta }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{\varphi }{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }})+$
${\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{r}{\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\vartheta }{\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\varphi }}{\partial \varphi }}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\varphi }{\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }})=$
${\widehat {e}}_{r}\cdot ({\frac {\partial F_{r}}{\partial r}}{\widehat {e}}_{r})+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\vartheta })+{\widehat {e}}_{r}\cdot ({\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot (F_{r}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\vartheta }}{\partial \vartheta }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot F_{\vartheta }(-{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\vartheta }}{r}}\cdot ({\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{\varphi })+$
${\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{r})+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{r})sin\vartheta {\widehat {e}}_{\varphi }+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{\vartheta })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\vartheta })cos\vartheta {\widehat {e}}_{\varphi }+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot ({\frac {\partial F_{\varphi }}{\partial \varphi }}{\widehat {e}}_{\varphi })+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}\cdot (F_{\varphi })(-sin\vartheta {\widehat {e}}_{r}-cos\vartheta {\widehat {e}}_{\vartheta })=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi })+$
${\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}(F_{r}sin\vartheta )({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+$
${\frac {1}{rsin\vartheta }}F_{\varphi })((-sin\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r})-cos\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }))=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\cdot {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })+{\frac {1}{rsin\vartheta }}(F_{r}sin\vartheta )({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta ({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })=$
${\frac {\partial F_{r}}{\partial r}}+{\frac {1}{r}}F_{r}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {1}{r}}F_{r}+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta +{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}={\frac {\partial F_{r}}{\partial r}}+{\frac {2}{r}}F_{r}+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}F_{\vartheta }cos\vartheta +{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}=$
${\frac {1}{r^{2}}}{\frac {\partial (r^{2}F_{r})}{\partial r}}+{\frac {1}{rsin\vartheta }}{\frac {\partial (F_{\vartheta }sin\vartheta )}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}$

> Divergencia en coordenadas esféricas. Geometría diferencial

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.
$div({\overrightarrow {F}})={\overrightarrow {\nabla }}\cdot ({\overrightarrow {F}})=\ast d\ast \downarrow ({\overrightarrow {F}})=\ast d\ast \downarrow (F_{x}{\frac {\widehat {\partial }}{\partial x}}+F_{y}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\ast \ d\ast \downarrow (F_{r}{\frac {1}{|{\frac {\partial x}{\partial r}}|}}{\frac {\partial }{\partial r}}+F_{\vartheta }{\frac {1}{|{\frac {\partial y}{\partial \vartheta }}|}}{\frac {\partial }{\partial \vartheta }}+F_{\varphi }{\frac {1}{|{\frac {\partial z}{\partial \varphi }}|}}{\frac {\partial }{\partial \varphi }})=$
$\ast d\ast \downarrow (F_{r}{\frac {1}{\sqrt[{}]{g_{r}}}}{\frac {\partial }{\partial r}})+F_{\vartheta }{\frac {1}{\sqrt[{}]{g_{\vartheta }}}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }{\frac {1}{\sqrt[{}]{g_{\varphi }}}}{\frac {\partial }{\partial \varphi }}))=\ast d\ast (F_{r}{\frac {1}{\sqrt[{}]{g_{r}}}}g_{r}dr+F_{\vartheta }{\frac {1}{\sqrt[{}]{g_{\vartheta }}}}g_{\vartheta }d\vartheta +F_{\varphi }{\frac {1}{\sqrt[{}]{g_{\varphi }}}}g_{\varphi }d\varphi ))=\ast d\ast (F_{r}dr+F_{\varphi }rd\vartheta +F_{\varphi }rsin\vartheta d\varphi )=$
$\ast d(F_{r}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}(d\vartheta \wedge d\varphi )+F_{\vartheta }r{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}(d\varphi \wedge dr)+F_{\varphi }rsin\vartheta {\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}(dr\wedge d\vartheta ))=$
$\ast d(F_{r}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}(d\vartheta \wedge d\varphi )+F_{\vartheta }{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r}}(d\varphi \wedge dr)+F_{\varphi }{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{rsin\vartheta }}(dr\wedge d\vartheta ))=$
$\ast ({\frac {\partial }{\partial r}}(F_{r}{\sqrt[{2}]{r^{4}sin^{2}\vartheta }})(dr\wedge d\theta \wedge d\varphi )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }{\sqrt[{2}]{r^{2}sin^{2}\vartheta }})(d\vartheta \wedge d\varphi \wedge dr)+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)(d\varphi \wedge dr\wedge d\vartheta ))=$
$\ast (({\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r))(dr\wedge d\vartheta \wedge d\varphi ))={\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)=$
${\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \varphi }}(F_{\varphi }r)=$
${\frac {1}{r^{2}sin\vartheta }}({\frac {\partial }{\partial r}}(F_{r}r^{2}sin\vartheta )+{\frac {\partial }{\partial \vartheta }}(F_{\vartheta }rsin\vartheta )+{\frac {\partial }{\partial \varphi }}(F_{\varphi }r))={\frac {1}{r^{2}}}{\frac {\partial (r^{2}F_{r})}{\partial r}}+{\frac {1}{rsin\vartheta }}{\frac {\partial (F_{\vartheta }sin\vartheta )}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}$

> Rotacional en coordenadas esféricas. Cálculo diferencial

${\overrightarrow {\nabla }}\times {\overrightarrow {F}}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\times (F_{x},F_{y},F_{z})=({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}},{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }},{\frac {1}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\times (F_{r},F_{\vartheta },F_{\varphi })=$
$({\frac {{\widehat {e}}_{r}}{h_{r}}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=({\frac {{\widehat {e}}_{r}}{1}}{\frac {\partial }{\partial r}}+{\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }}+{\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{r}{\widehat {e}}_{r}+F_{\vartheta }{\widehat {e}}_{\vartheta }+F_{\varphi }{\widehat {e}}_{\varphi })=$
$({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{r}{\widehat {e}}_{r})+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\widehat {e}}_{r}{\frac {\partial }{\partial r}})\times (F_{\varphi }{\widehat {e}}_{\varphi })+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{r}{\widehat {e}}_{r})+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\frac {{\widehat {e}}_{\vartheta }}{r}}{\frac {\partial }{\partial \vartheta }})\times (F_{\varphi }{\widehat {e}}_{\varphi })+$
$({\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{r}{\widehat {e}}_{r})+({\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{\vartheta }{\widehat {e}}_{\vartheta })+({\frac {{\widehat {e}}_{\varphi }}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\times (F_{\varphi }{\widehat {e}}_{\varphi })=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{r}}{\partial r}})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+F_{\vartheta }({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}})+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+F_{\varphi }({\widehat {e}}_{r}\times {\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}})+$
${\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }})+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}F_{\varphi }({\widehat {e}}_{\vartheta }\times {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }})+$
${\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}F_{r}({\widehat {e}}_{\varphi }\times {\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }})+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })+{\frac {1}{rsin\vartheta }}F_{\vartheta }({\widehat {e}}_{\varphi }\times {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }})+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }})=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+$
${\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}F_{r}({\widehat {e}}_{\varphi }\times (sin\vartheta {\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })+$
${\frac {1}{rsin\vartheta }}F_{\vartheta }({\widehat {e}}_{\varphi }\times (cos\vartheta {\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times (-sin\vartheta {\widehat {e}}_{r}-cos\vartheta {\widehat {e}}_{\vartheta }))=$
${\frac {\partial F_{r}}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{r})+{\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{r}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+{\frac {1}{r}}{\frac {\partial F_{\vartheta }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\vartheta })+$
${\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+F_{r}({\widehat {e}}_{\varphi }\times ({\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })+$
${\frac {cos\vartheta }{rsin\vartheta }}F_{\vartheta }({\widehat {e}}_{\varphi }\times ({\widehat {e}}_{\varphi }))+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\varphi })-{\frac {1}{r}}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})-{\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })=$
${\frac {\partial F_{\vartheta }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\vartheta })+{\frac {\partial F_{\varphi }}{\partial r}}({\widehat {e}}_{r}\times {\widehat {e}}_{\varphi })+{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{r})+{\frac {1}{r}}F_{\vartheta }({\widehat {e}}_{\vartheta }\times (-{\widehat {e}}_{r}))+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}({\widehat {e}}_{\vartheta }\times {\widehat {e}}_{\varphi })+$
${\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })-{\frac {1}{r}}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{r})-{\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }({\widehat {e}}_{\varphi }\times {\widehat {e}}_{\vartheta })=$
${\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\varphi }-{\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\vartheta }-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{\varphi }+{\frac {1}{r}}F_{\vartheta }{\widehat {e}}_{\varphi }+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{r}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{\vartheta }-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{r}-{\frac {1}{r}}F_{\varphi }{\widehat {e}}_{\vartheta }+{\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }{\widehat {e}}_{r}=$
${\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }{\widehat {e}}_{r}+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}{\widehat {e}}_{r}-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}{\widehat {e}}_{r}+{\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}{\widehat {e}}_{\vartheta }-{\frac {1}{r}}F_{\varphi }{\widehat {e}}_{\vartheta }-{\frac {\partial F_{\varphi }}{\partial r}}{\widehat {e}}_{\vartheta }+{\frac {\partial F_{\vartheta }}{\partial r}}{\widehat {e}}_{\varphi }+{\frac {1}{r}}F_{\vartheta }{\widehat {e}}_{\varphi }-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}{\widehat {e}}_{\varphi }=$
$({\frac {cos\vartheta }{rsin\vartheta }}F_{\varphi }+{\frac {1}{r}}{\frac {\partial F_{\varphi }}{\partial \vartheta }}-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\widehat {e}}_{r}+({\frac {1}{rsin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {1}{r}}F_{\varphi }-{\frac {\partial F_{\varphi }}{\partial r}}){\widehat {e}}_{\vartheta }+({\frac {\partial F_{\vartheta }}{\partial r}}+{\frac {1}{r}}F_{\vartheta }-{\frac {1}{r}}{\frac {\partial F_{r}}{\partial \vartheta }}){\widehat {e}}_{\varphi }=$
${\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\widehat {e}}_{r}+{\frac {1}{r}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (rF_{\varphi })}{\partial r}}){\widehat {e}}_{\vartheta }+{\frac {1}{r}}({\frac {\partial (rF_{\vartheta })}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\widehat {e}}_{\varphi }$

> Rotacional en coordenadas esféricas. Geometría diferencial

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.
$rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\widehat {\partial }}{\partial x}}+{F_{y}}{\frac {\widehat {\partial }}{\partial y}}+F_{z}{\frac {\widehat {\partial }}{\partial z}})=\uparrow \ast d\downarrow ({F_{r}}({\frac {1}{h_{r}}}{\frac {\partial }{\partial r}})+{F_{\vartheta }}({\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }({\frac {1}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }}))=$
$\uparrow \ast d\downarrow ({F_{r}}{\frac {\partial }{\partial r}}+{F_{\vartheta }}({\frac {1}{r}}{\frac {\partial }{\partial \vartheta }})+F_{\varphi }({\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})=\uparrow \ast d({F_{r}}dr+{\frac {F_{\vartheta }}{r}}r^{2}d\vartheta +{\frac {F_{\varphi }}{rsin\vartheta }}r^{2}sin^{2}\vartheta d\varphi )=\uparrow \ast d({F_{r}}dr+F_{\vartheta }rd\vartheta +rsin\vartheta F_{\varphi }d\varphi )=$
$\uparrow \ast ({\frac {\partial F_{r}}{\partial \vartheta }}(d\vartheta \wedge dr)+{\frac {\partial F_{r}}{\partial \varphi }}(d\varphi \wedge dr)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )+{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}(d\varphi \wedge d\vartheta )+{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}(dr\wedge d\varphi )+{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}(d\vartheta \wedge d\varphi ))=$
$\uparrow \ast (-{\frac {\partial F_{r}}{\partial \vartheta }}(dr\wedge d\vartheta )+{\frac {\partial F_{r}}{\partial \varphi }}(d\varphi \wedge dr)+{\frac {\partial (F_{\vartheta }r)}{\partial r}}(dr\wedge d\vartheta )-{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}(d\vartheta \wedge d\varphi )-{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}(d\varphi \wedge dr)+{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}(d\vartheta \wedge d\varphi ))=$
$\uparrow (-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}{\frac {\partial F_{r}}{\partial \vartheta }}d\varphi +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}d\vartheta +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}}}{\frac {\partial (F_{\vartheta }r)}{\partial r}}d\varphi -$
${\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}dr-{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{2}sin^{2}\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}d\vartheta +{\frac {\sqrt[{}]{r^{4}sin^{2}\vartheta }}{r^{4}sin^{2}\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}dr)=$
$\uparrow ({\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}dr-{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}dr+{\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}d\vartheta -{\frac {1}{sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}d\vartheta +sin\vartheta {\frac {\partial (F_{\vartheta }r)}{\partial r}}d\varphi -sin\vartheta {\frac {\partial F_{r}}{\partial \vartheta }}d\varphi )=$
${\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial \vartheta }}{\frac {\partial }{\partial r}}-{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial (F_{\vartheta }r)}{\partial \varphi }}{\frac {\partial }{\partial r}}+{\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}-{\frac {1}{sin\vartheta }}{\frac {\partial (F_{\varphi }rsin\vartheta )}{\partial r}}{\frac {1}{r^{2}}}{\frac {\partial }{\partial \vartheta }}+sin\vartheta {\frac {\partial (F_{\vartheta }r)}{\partial r}}{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}-sin\vartheta {\frac {\partial F_{r}}{\partial \vartheta }}{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial }{\partial \varphi }}=$
$({\frac {1}{rsin\vartheta }}{\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {1}{rsin\vartheta }}{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\frac {\partial }{\partial r}}+({\frac {1}{r^{2}}}{\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {1}{r^{2}}}{\frac {\partial (F_{\varphi }r)}{\partial r}}){\frac {\partial }{\partial \vartheta }}+(sin\vartheta {\frac {\partial (F_{\vartheta }r)}{\partial r}}{\frac {1}{r^{2}sin^{2}\vartheta }}-sin\vartheta {\frac {\partial F_{r}}{\partial \vartheta }}{\frac {1}{r^{2}sin^{2}\vartheta }}){\frac {\partial }{\partial \varphi }}=$
${\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (F_{\varphi }r)}{\partial r}}){\frac {\partial }{\partial \vartheta }}+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\partial }{\partial \varphi }}=$
${\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }})|{\frac {\partial }{\partial r}}|{\frac {\frac {\partial }{\partial r}}{|{\frac {\partial }{\partial r}}|}}+{\frac {1}{r^{2}}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (F_{\varphi }r)}{\partial r}})|{\frac {\partial }{\partial \vartheta }}|{\frac {\frac {\partial }{\partial \vartheta }}{|{\frac {\partial }{\partial \vartheta }}|}}+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }})|{\frac {\partial }{\partial \varphi }}|{\frac {\frac {\partial }{\partial \varphi }}{|{\frac {\partial }{\partial \varphi }}|}}=$
${\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\varphi }sin\vartheta )}{\partial \vartheta }}-{\frac {\partial F_{\vartheta }}{\partial \varphi }}){\frac {\widehat {\partial }}{\partial r}}+{\frac {1}{r}}({\frac {1}{sin\vartheta }}{\frac {\partial F_{r}}{\partial \varphi }}-{\frac {\partial (F_{\varphi }r)}{\partial r}}){\frac {\widehat {\partial }}{\partial \vartheta }}+{\frac {1}{rsin\vartheta }}({\frac {\partial (F_{\vartheta }r)}{\partial r}}-{\frac {\partial F_{r}}{\partial \vartheta }}){\frac {\widehat {\partial }}{\partial \varphi }}$

> Laplaciano en coordenadas esféricas. Cálculo diferencial

$div({\overrightarrow {grad}}(f))={\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}f)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})\cdot ({\widehat {e}}_{x}{\frac {\partial f}{\partial x}}+{\widehat {e}}_{y}{\frac {\partial f}{\partial y}}+{\widehat {e}}_{z}{\frac {\partial f}{\partial z}})=$
$({\widehat {e}}_{r}{\frac {1}{h_{r}}}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{h_{\vartheta }}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{h_{\varphi }}}{\frac {\partial }{\partial \varphi }})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})=$
$({\widehat {e}}_{r}{\frac {\partial }{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }})\cdot ({\widehat {e}}_{r}{\frac {\partial f}{\partial r}}+{\widehat {e}}_{\vartheta }{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+{\widehat {e}}_{\varphi }{\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})=$
$({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial r}}){\frac {\partial f}{\partial r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial r}}){\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\frac {\partial {\widehat {e}}_{\varphi }}{\partial r}}){\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi }){\frac {\partial }{\partial r}}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+$
$({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial \vartheta }}){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \vartheta }}){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \vartheta }}){\frac {1}{r}}{\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi }){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+$
$({\widehat {e}}_{\varphi }\cdot {\frac {\partial {\widehat {e}}_{r}}{\partial \varphi }}){\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r}){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {\partial f}{\partial r}})+({\widehat {e}}_{\varphi }\cdot {\frac {\partial {\widehat {e}}_{\vartheta }}{\partial \varphi }}){\frac {1}{r^{2}sin\vartheta }}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{\varphi }\cdot {\frac {\partial {\widehat {e}}_{\varphi }}{\partial \varphi }}){\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial f}{\partial \varphi }}+$
$({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi }){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})=$
$({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\vartheta }){\frac {\partial }{\partial r}}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})+({\widehat {e}}_{r}\cdot {\widehat {e}}_{\varphi }){\frac {\partial }{\partial r}}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{r}){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {\partial f}{\partial r}})+$
$({\widehat {e}}_{\vartheta }\cdot (-{\widehat {e}}_{r})){\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\varphi }){\frac {1}{r}}{\frac {\partial }{\partial \vartheta }}({\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }})+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })sin\vartheta {\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r}){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {\partial f}{\partial r}})+$
$({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })cos\vartheta {\frac {1}{r^{2}sin\vartheta }}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{rsin\vartheta }}{\frac {\partial }{\partial \varphi }}({\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }})-(({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{r})sin\vartheta +({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\vartheta })cos\vartheta ){\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial \varphi }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })({\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }})=$
$({\widehat {e}}_{r}\cdot {\widehat {e}}_{r}){\frac {\partial ^{2}f}{\partial ^{2}r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta }){\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\vartheta }\cdot {\widehat {e}}_{\vartheta })({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })sin\vartheta {\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })cos\vartheta {\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+({\widehat {e}}_{\varphi }\cdot {\widehat {e}}_{\varphi })({\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }})=$
${\frac {\partial ^{2}f}{\partial ^{2}r}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+sin\vartheta {\frac {1}{rsin\vartheta }}{\frac {\partial f}{\partial r}}+cos\vartheta {\frac {1}{r^{2}sin\vartheta }}{\frac {\partial f}{\partial \vartheta }}+{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }}=$
${\frac {\partial ^{2}f}{\partial ^{2}r}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+({\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial ^{2}\vartheta }})+{\frac {cos\vartheta }{sin\vartheta }}{\frac {1}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}+{\frac {1}{r^{2}sin^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial ^{2}\varphi }}=$
${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial f}{\partial r}})+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial ^{2}f}{\partial ^{2}\varphi }})$

> Laplaciano en coordenadas esféricas. Geometría diferencial

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.
$div({\overrightarrow {grad}}(f))=\nabla ^{2}f=\ast d\ast d(f)=\ast d\ast ({\frac {\partial f}{\partial r}}dr+{\frac {\partial f}{\partial \vartheta }}d\vartheta +{\frac {\partial f}{\partial \varphi }}d\varphi )=$
$\ast d({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{1}}{\frac {\partial f}{\partial r}}(d\vartheta \wedge d\varphi )+{\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{2}}}{\frac {\partial f}{\partial \vartheta }}(d\varphi \wedge dr)+{\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{2}sin^{2}\vartheta }}{\frac {\partial f}{\partial \varphi }}(dr\wedge d\vartheta ))=$
$\ast ({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})(dr\wedge d\vartheta \wedge d\varphi )+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})(d\vartheta \wedge d\varphi \wedge dr)+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})(d\varphi \wedge dr\wedge d\vartheta ))=$
$\ast (({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})(dr\wedge d\vartheta \wedge d\varphi ))=$
$({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }}){\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }})=$
$({\frac {\sqrt[{}]{|r^{4}sin^{2}\vartheta |}}{r^{4}sin^{2}\vartheta }})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }}))=$
$({\frac {1}{\sqrt[{}]{r^{4}sin^{2}\vartheta }}})({\frac {\partial }{\partial r}}({\sqrt[{2}]{\frac {r^{2}r^{2}sin^{2}\vartheta }{1}}}{\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}({\sqrt[{2}]{\frac {r^{2}sin^{2}\vartheta }{r^{2}}}}{\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\sqrt[{2}]{\frac {r^{2}}{r^{2}sin^{2}\vartheta }}}{\frac {\partial f}{\partial \varphi }}))=$
${\frac {1}{r^{2}sin\vartheta }}({\frac {\partial }{\partial r}}(r^{2}sin\vartheta {\frac {\partial f}{\partial r}})+{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {\partial }{\partial \varphi }}({\frac {1}{sin\vartheta }}{\frac {\partial f}{\partial \varphi }}))={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial f}{\partial r}})+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial ^{2}f}{\partial ^{2}\varphi }})$

>> Operadores diferenciales en coordenadas generalizadas

* Productos escalares en coordenadas generalizadas

Serán necesarios los siguientes productos escalares para coordenadas generalizadas.
${\widehat {u}}_{1}\cdot {\widehat {u}}_{1}=1;{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}=1;{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}=1$
${\widehat {u}}_{2}\cdot {\widehat {u}}_{1}=0;{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}=0;{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}=0;{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}=0;{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}=0;{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}=0$

* Productos vectoriales en coordenadas generalizadas

Utilizaremos en los desarrollos los siguientes productos vectoriales.
${\widehat {u}}_{1}\times {\widehat {u}}_{1}=0;{\widehat {u}}_{2}\times {\widehat {u}}_{2}=0;{\widehat {u}}_{3}\times {\widehat {u}}_{3}=0$
${\widehat {u}}_{1}\times {\widehat {u}}_{2}={\widehat {u}}_{3};{\widehat {u}}_{2}\times {\widehat {u}}_{3}={\widehat {u}}_{1};{\widehat {u}}_{3}\times {\widehat {u}}_{1}={\widehat {u}}_{2}$
${\widehat {u}}_{2}\times {\widehat {u}}_{1}=-{\widehat {u}}_{3};{\widehat {u}}_{3}\times {\widehat {u}}_{2}=-{\widehat {u}}_{1};{\widehat {u}}_{1}\times {\widehat {u}}_{3}=-{\widehat {u}}_{2}$

* Divergencia del rotacional

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.
${\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{n}}}\times {\frac {{\widehat {e}}_{y}}{h_{m}}})=0$
${\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{n}}}\times {\frac {{\widehat {e}}_{z}}{h_{m}}})=0$
${\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{y}}{h_{n}}}\times {\frac {{\widehat {e}}_{z}}{h_{m}}})=0$

* Factor de escala en coordenadas generalizadas

El factor de escala en cálculo diferencial nos permite pasar de unas coordenadas a otras.

${\overrightarrow {h}}=\sum |{\frac {\partial {\overrightarrow {r}}}{\partial u_{i}}}|=(|{\frac {\partial x}{\partial u_{1}}}|,|{\frac {\partial y}{\partial u_{2}}}|,|{\frac {\partial z}{\partial u_{3}}}|)=(h_{1},h_{2},h_{3})$
Podemos observar que los factores de escala también pueden expresarse en función de la métrica del espacio de la geometría de Riemann.
${\overrightarrow {h}}=\sum |{\frac {\partial }{\partial u_{i}}}|=(|{\frac {\partial }{\partial u_{1}}}|,|{\frac {|\partial }{\partial u_{2}}}|,|{\frac {\partial }{\partial u_{3}}}|)=({\sqrt[{}]{g_{11}}},{\sqrt[{}]{g_{22}}},{\sqrt[{}]{g_{33}}})$

* Métrica o tensor métrico en coordenadas generalizadas

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.
$g_{\alpha \beta }=g_{11}du_{1}^{2}+g_{22}du_{2}^{2}+g_{33}du_{3}^{2}$
$g_{\alpha \beta }={\begin{bmatrix}\vert {\frac {\partial }{\partial u_{1}}}\vert \cdot |{\frac {\partial }{\partial u_{1}}}|&&\\&|{\frac {\partial }{\partial u_{2}}}|\cdot |{\frac {\partial }{\partial u_{2}}}|&\\&&|{\frac {\partial }{\partial u_{3}}}|\cdot |{\frac {\partial }{\partial u_{3}}}|\end{bmatrix}}={\begin{bmatrix}g_{11}&&\\&g_{22}&\\&&g_{33}\end{bmatrix}}$
$det(g_{\alpha \beta })=|g|=g_{11}g_{22}g_{33}$

* Estrella de Hodge (*) en coordenadas generalizadas

La estrella de Hodge es un operador que actúa sobre un p-forma diferencial en un espacio de dimensión n.
$*(F(dx_{1}\wedge dx_{2}\wedge dx_{3}\wedge ...\wedge dx_{p})={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}...g_{pp}}}\varepsilon F(dx_{p+1}\wedge dx_{p+2}\wedge dx_{p+3}\wedge ...\wedge dx_{n-p})$
$\ast (Fdx_{1})={\frac {\sqrt[{}]{|g|}}{g_{11}}}F(dx_{2}\wedge dx_{3})$
$\ast (F(dx_{1}\wedge dx_{2}))={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}}}Fdx_{3}$
$\ast (F(dx_{1}\wedge dx_{2}\wedge dx_{3}))={\frac {\sqrt[{}]{|g|}}{g_{11}g_{22}g_{33}}}F$

* Subir y bajar índices. Coordenadas generalizadas

[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.
${\frac {\partial }{\partial x^{\alpha }}}=g_{\alpha \beta }dx^{\beta }$
$dx^{\alpha }=g^{\alpha \beta }{\frac {\partial }{\partial x^{\beta }}}$

> Gradiente en coordenadas generalizadas. Cálculo diferencial

${\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})f(x,y,z)=({\widehat {e}}_{x}{\frac {\partial }{\partial x}}+{\widehat {e}}_{y}{\frac {\partial }{\partial y}}+{\widehat {e}}_{z}{\frac {\partial }{\partial z}})f(x,y,z)=$
$({\widehat {u}}_{1}{\frac {\partial u_{1}}{\partial x}}{\frac {\partial }{\partial u_{1}}}+{\widehat {u}}_{2}{\frac {\partial u_{2}}{\partial y}}{\frac {\partial }{\partial u_{2}}}+{\widehat {u}}_{3}{\frac {\partial u_{3}}{\partial z}}{\frac {\partial }{\partial u_{3}}})f(u_{1},u_{2},u_{3})={\frac {1}{|{\frac {\partial x}{\partial u_{1}}}|}}{\frac {\partial f}{\partial u_{1}}}{\widehat {u}}_{1}+{\frac {1}{|{\frac {\partial u_{2}}{\partial y}}|}}{\frac {\partial f}{\partial u_{2}}}{\widehat {u}}_{2}+{\frac {1}{|{\frac {\partial u_{3}}{\partial z}}|}}{\frac {\partial f}{\partial u_{3}}}{\widehat {u}}_{3}=$
${\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}}{\widehat {u}}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}}{\widehat {u}}_{2}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{3}}}{\widehat {u}}_{3}$

> Gradiente en coordenadas generalizadas. Geometría diferencial

Obtendremos el gradiente de un campo escalar en geometría diferencial aplicando la diferencial exterior y posteriormente bajaremos índices.
${\overrightarrow {grad}}(f)={\overrightarrow {\nabla }}(f)=\downarrow d(f(u_{1},u_{2},u_{3}))=\downarrow ({\frac {\partial f}{\partial u_{1}}}du_{1}+{\frac {\partial f}{\partial u_{2}}}du_{2}+{\frac {\partial f}{\partial u_{3}}}du_{3})={\frac {\partial f}{\partial u_{1}}}{\frac {1}{g_{11}}}{\frac {\partial }{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{g_{22}}}{\frac {\partial }{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{g_{33}}}{\frac {\partial }{\partial u_{3}}}=$
${\frac {\partial f}{\partial u_{1}}}{\frac {1}{g_{11}}}|{\frac {\partial }{\partial u_{1}}}|{\frac {\frac {\partial }{\partial u_{1}}}{|{\frac {\partial }{\partial u_{1}}}|}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{g_{22}}}|{\frac {\partial }{\partial u_{2}}}|{\frac {\frac {\partial }{\partial u_{2}}}{|{\frac {\partial }{\partial u_{2}}}|}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{g_{33}}}|{\frac {\partial }{\partial u_{3}}}|{\frac {\frac {\partial }{\partial u_{3}}}{|{\frac {\partial }{\partial u_{3}}}|}}={\frac {\partial f}{\partial u_{1}}}{\frac {\sqrt[{}]{g_{11}}}{g_{11}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {\sqrt[{}]{g_{22}}}{g_{22}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {\sqrt[{}]{g_{33}}}{g_{33}}}{\frac {\widehat {\partial }}{\partial u_{3}}}=$
${\frac {\partial f}{\partial u_{1}}}{\frac {1}{\sqrt[{}]{g_{11}}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+{\frac {\partial f}{\partial u_{2}}}{\frac {1}{\sqrt[{}]{g_{22}}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+{\frac {\partial f}{\partial u_{3}}}{\frac {1}{\sqrt[{}]{g_{33}}}}{\frac {\widehat {\partial }}{\partial u_{3}}}$

> Divergencia en coordenadas generalizadas. Cálculo diferencial

${\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}={\overrightarrow {\nabla }}\cdot (F_{1}{\widehat {e}}_{x}+F_{2}{\widehat {e}}_{y}+F_{3}{\widehat {e}}_{z})={\overrightarrow {\nabla }}\cdot (F_{1}{\widehat {e}}_{x})+{\overrightarrow {\nabla }}\cdot (F_{2}{\widehat {e}}_{y})+{\overrightarrow {\nabla }}\cdot (F_{3}{\widehat {e}}_{z})=$
$({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{y})+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{z})=$
$({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z}))+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x}))+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y}))=$
$({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}{\overrightarrow {\nabla }}(h_{2}h_{3}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}{\overrightarrow {\nabla }}(h_{3}h_{1}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}{\overrightarrow {\nabla }}(h_{1}h_{2}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=$
$({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}})+F_{1}h_{2}h_{3}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+$
$({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}})+F_{2}h_{3}h_{1}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+$
$({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}})+F_{3}h_{1}h_{2}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=$
$({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+{\frac {F_{1}}{h_{2}h_{3}}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z})+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+{\frac {F_{2}}{h_{3}h_{1}}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+{\frac {F_{3}}{h_{1}h_{2}}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y})=$
$({\overrightarrow {\nabla }}F_{1})\cdot {\widehat {e}}_{x}+{\frac {F_{1}}{h_{2}h_{3}}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot {\widehat {e}}_{x}+({\overrightarrow {\nabla }}F_{2})\cdot {\widehat {e}}_{y}+{\frac {F_{2}}{h_{3}h_{1}}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot {\widehat {e}}_{y}+({\overrightarrow {\nabla }}F_{3})\cdot {\widehat {e}}_{z}+{\frac {F_{3}}{h_{1}h_{2}}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot {\widehat {e}}_{z}=$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(F_{1}))\cdot {\widehat {u}}_{1}+{\frac {F_{1}}{h_{2}h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{2}h_{3}))\cdot {\widehat {u}}_{1}+$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(F_{2}))\cdot {\widehat {u}}_{2}+{\frac {F_{2}}{h_{1}h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{3}))\cdot {\widehat {u}}_{2}+$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(F_{3}))\cdot {\widehat {u}}_{3}+{\frac {F_{3}}{h_{1}h_{2}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{2}))\cdot {\widehat {u}}_{3}=$
${\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}{\frac {\partial F_{1}}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}{\frac {\partial F_{1}}{\partial u_{3}}}+F_{1}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}}{h_{2}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}}{h_{2}h_{3}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{3}}})+$
${\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}{\frac {\partial F_{2}}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}{\frac {\partial F_{2}}{\partial u_{3}}}+F_{2}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}}{h_{1}h_{1}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}}{h_{1}h_{3}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{3}}})+$
${\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}{\frac {\partial F_{3}}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}{\frac {\partial F_{3}}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}{\frac {\partial F_{3}}{\partial u_{3}}}+F_{3}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}}{h_{1}h_{1}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}})=$
${\frac {1}{h_{1}}}({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {F_{1}}{h_{1}h_{2}h_{3}}}({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial F_{2}}{\partial u_{2}}}+$
${\frac {F_{2}}{h_{1}h_{2}h_{3}}}({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial F_{3}}{\partial u_{3}}}+{\frac {F_{3}}{h_{1}h_{2}h_{3}}}({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {1}{h_{1}}}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {F_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {F_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}}+{\frac {F_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {h_{2}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {F_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {h_{1}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {F_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {h_{1}h_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}}+{\frac {F_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial (F_{1}h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial (F_{2}h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial (F_{3}h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial (F_{1}h_{2}h_{3})}{\partial u_{1}}}+{\frac {\partial (F_{2}h_{1}h_{3})}{\partial u_{2}}}+{\frac {\partial (F_{3}h_{1}h_{2})}{\partial u_{3}}})$

> Divergencia en coordenadas generalizadas. Geometría diferencial

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.
$div({\overrightarrow {F}})={\overrightarrow {\nabla }}\cdot {\overrightarrow {F}}=\ast d\ast \downarrow ({\overrightarrow {F}})=\ast d\ast \downarrow (F_{x}{\widehat {e}}_{x}+F_{y}{\widehat {e}}_{y}+F_{z}{\widehat {e}}_{z})=\ast d\ast \downarrow (F_{1}{\frac {1}{|{\frac {\partial x}{\partial u_{1}}}|}}{\overrightarrow {u_{1}}}+F_{2}{\frac {1}{|{\frac {\partial y}{\partial u_{2}}}|}}{\overrightarrow {u_{2}}}+F_{3}{\frac {1}{|{\frac {\partial z}{\partial u_{3}}}|}}{\overrightarrow {u_{3}}})=$
$\ast d\ast \downarrow (F_{1}{\frac {1}{\sqrt[{2}]{g_{11}}}}{\overrightarrow {u_{1}}}+F_{2}{\frac {1}{\sqrt[{2}]{g_{22}}}}{\overrightarrow {u_{2}}}+F_{3}{\frac {1}{\sqrt[{2}]{g_{33}}}}{\overrightarrow {u_{3}}})=\ast d\ast (F_{1}{\frac {1}{\sqrt[{2}]{g_{11}}}}g_{11}du_{1}+F_{2}{\frac {1}{\sqrt[{2}]{g_{22}}}}g_{22}du_{2}+F_{3}{\frac {1}{\sqrt[{2}]{g_{33}}}}g_{33}du_{3})=$
$\ast d\ast (F_{1}{\sqrt[{2}]{g_{11}}}du_{1}+F_{2}{\sqrt[{2}]{g_{22}}}du_{2}+F_{3}{\sqrt[{2}]{g_{33}}}du_{3})=\ast d(F_{1}{\sqrt[{2}]{g_{11}}}{\frac {\sqrt[{2}]{g}}{g_{11}}}(du_{2}\wedge du_{3})+F_{2}{\sqrt[{2}]{g_{22}}}{\frac {\sqrt[{2}]{g}}{g_{22}}}(du_{3}\wedge du_{1})+F_{1}{\sqrt[{2}]{g_{33}}}{\frac {\sqrt[{2}]{g}}{g_{33}}}(du_{1}\wedge du_{2}))=$
$\ast d(F_{1}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{11}}}}(du_{2}\wedge du_{3})+F_{2}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{22}}}}(du_{3}\wedge du_{1})+F_{3}{\frac {\sqrt[{2}]{g}}{\sqrt[{2}]{g_{33}}}}(du_{1}\wedge du_{2}))=$
$\ast ({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})(du_{1}\wedge du_{2}\wedge du_{3})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})(du_{2}\wedge du_{3}\wedge du_{1})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}})(du_{3}\wedge du_{1}\wedge du_{2}))=$
$\ast (({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}}))(du_{1}\wedge du_{2}\wedge du_{3}))=$
${\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\sqrt[{2}]{g}}{g_{11}g_{22}g_{22}}}{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}})=$
${\frac {1}{\sqrt[{2}]{g_{11}g_{22}g_{22}}}}({\frac {\partial }{\partial u_{1}}}(F_{1}{\sqrt[{2}]{g_{22}g_{33}}})+{\frac {\partial }{\partial u_{2}}}(F_{2}{\sqrt[{2}]{g_{11}g_{33}}})+{\frac {\partial }{\partial u_{3}}}(F_{3}{\sqrt[{2}]{g_{22}g_{11}}}))={\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}(F_{1}h_{2}h_{3})+{\frac {\partial }{\partial u_{2}}}(F_{2}h_{1}h_{3})+{\frac {\partial }{\partial u_{3}}}(F_{3}h_{1}h_{2}))$

> Rotacional en coordenadas generalizadas. Cálculo diferencial

$rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}={\overrightarrow {\nabla }}\times (F_{1}{\widehat {e}}_{x}+F_{2}{\widehat {e}}_{y}+F_{3}{\widehat {e}}_{z})={\overrightarrow {\nabla }}\times (F_{1}{\widehat {e}}_{x})+{\overrightarrow {\nabla }}\times (F_{2}{\widehat {e}}_{y})+{\overrightarrow {\nabla }}\times (F_{3}{\widehat {e}}_{z})+$
$({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}\times {\widehat {e}}_{y})+({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}\times {\widehat {e}}_{z})=$
$({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}{\overrightarrow {\nabla }}\times (h_{1}{\frac {{\widehat {e}}_{x}}{h_{1}}})+({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}{\overrightarrow {\nabla }}\times (h_{2}{\frac {{\widehat {e}}_{y}}{h_{2}}})+({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}{\overrightarrow {\nabla }}\times (h_{3}{\frac {{\widehat {e}}_{z}}{h_{3}}})=$
$({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}(h_{1}))\times ({\frac {{\widehat {e}}_{x}}{h_{1}}}))+F_{1}h_{1}({\overrightarrow {\nabla }}\times ({\frac {{\widehat {e}}_{x}}{h_{1}}}))+$
$({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}(h_{2}))\times ({\frac {{\widehat {e}}_{y}}{h_{2}}}))+F_{2}h_{2}({\overrightarrow {\nabla }}\times ({\frac {{\widehat {e}}_{y}}{h_{2}}}))+$
$({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}(h_{3}))\times ({\frac {{\widehat {e}}_{z}}{h_{3}}}))+F_{3}h_{3}({\overrightarrow {\nabla }}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))=$
$({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+F_{1}({\overrightarrow {\nabla }}(h_{1})\times {\frac {{\widehat {e}}_{x}}{h_{1}}})+F_{1}({\overrightarrow {\nabla }}\times {\widehat {e}}_{x})+$
$({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+F_{2}({\overrightarrow {\nabla }}(h_{2})\times {\frac {{\widehat {e}}_{y}}{h_{2}}})+F_{2}({\overrightarrow {\nabla }}\times {\widehat {e}}_{y})+$
$({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+F_{3}({\overrightarrow {\nabla }}(h_{3})\times {\frac {{\widehat {e}}_{z}}{h_{3}}})+F_{3}({\overrightarrow {\nabla }}\times {\widehat {e}}_{z})=$
$({\overrightarrow {\nabla }}F_{1})\times {\widehat {e}}_{x}+{\frac {F_{1}}{h_{1}}}({\overrightarrow {\nabla }}(h_{1})\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}F_{2})\times {\widehat {e}}_{y}+{\frac {F_{2}}{h_{2}}}({\overrightarrow {\nabla }}(h_{2})\times {\widehat {e}}_{y})+({\overrightarrow {\nabla }}F_{3})\times {\widehat {e}}_{z}+{\frac {F_{3}}{h_{3}}}({\overrightarrow {\nabla }}(h_{3})\times {\widehat {e}}_{z})=$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})F_{1})\times {\widehat {u}}_{1}+{\frac {F_{1}}{h_{1}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1})\times {\widehat {u}}_{1})+$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})F_{2})\times {\widehat {u}}_{2}+{\frac {F_{2}}{h_{2}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{2})\times {\widehat {u}}_{2})+$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})F_{3})\times {\widehat {u}}_{3}+{\frac {F_{3}}{h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{3})\times {\widehat {u}}_{3})=$
$({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial F_{1}}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}})\times {\widehat {u}}_{1}+{\frac {F_{1}}{h_{1}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial (h_{1})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial (h_{1})}{\partial u_{3}}})\times {\widehat {u}}_{1})+$
$({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial F_{2}}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}})\times {\widehat {u}}_{2}+{\frac {F_{2}}{h_{2}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial (h_{2})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial (h_{2})}{\partial u_{3}}})\times {\widehat {u}}_{2})+$
$({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}})\times {\widehat {u}}_{3}+{\frac {F_{3}}{h_{3}}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial (h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial (h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{3}}})\times {\widehat {u}}_{3})=$
${\frac {1}{h_{1}}}{\frac {\partial F_{1}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{1})+{\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{1})+$
${\frac {F_{1}}{h_{1}}}({\frac {1}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{1})+{\frac {1}{h_{2}}}{\frac {\partial (h_{1})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {\partial (h_{1})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{1}))+$
${\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {\partial F_{2}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2})+$
${\frac {F_{2}}{h_{2}}}({\frac {1}{h_{1}}}{\frac {\partial (h_{2})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {\partial (h_{2})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2}))+$
${\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {\partial F_{3}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{3})+$
${\frac {F_{3}}{h_{3}}}({\frac {1}{h_{1}}}{\frac {\partial (h_{3})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {\partial (h_{3})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{3}))=$
${\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}})({\widehat {u}}_{3}\times {\widehat {u}}_{1})+{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{1})+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{1})+$
${\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2})+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{2})+{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}}({\widehat {u}}_{3}\times {\widehat {u}}_{2})+$
${\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})+{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}}({\widehat {u}}_{1}\times {\widehat {u}}_{3})+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}}({\widehat {u}}_{2}\times {\widehat {u}}_{3})=$
${\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}(-{\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}{\widehat {u}}_{2}+{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}}(-{\widehat {u}}_{3})+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}}{\widehat {u}}_{2}+$
${\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}{\widehat {u}}_{3}+{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}(-{\widehat {u}}_{1})+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}}{\widehat {u}}_{3}+{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}}(-{\widehat {u}}_{1})+$
${\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}(-{\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}{\widehat {u}}_{1}+{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}}(-{\widehat {u}}_{2})+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}}{\widehat {u}}_{1}=$
${\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}{\widehat {u}}_{1}+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}}{\widehat {u}}_{1}-{\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}{\widehat {u}}_{1}-{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}}{\widehat {u}}_{1}+$
${\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}{\widehat {u}}_{2}+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}}{\widehat {u}}_{2}-{\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}{\widehat {u}}_{2}-{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}}{\widehat {u}}_{2}+$
${\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}{\widehat {u}}_{3}+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}}{\widehat {u}}_{3}-{\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}{\widehat {u}}_{3}-{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}}{\widehat {u}}_{3}=$
$(({\frac {1}{h_{2}}}{\frac {\partial F_{3}}{\partial u_{2}}}+{\frac {1}{h_{2}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{2}}})-({\frac {1}{h_{3}}}{\frac {\partial F_{2}}{\partial u_{3}}}+{\frac {1}{h_{3}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{3}}})){\widehat {u}}_{1}+$
$(({\frac {1}{h_{3}}}{\frac {\partial F_{1}}{\partial u_{3}}}+{\frac {1}{h_{3}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{3}}})-({\frac {1}{h_{1}}}{\frac {\partial F_{3}}{\partial u_{1}}}+{\frac {1}{h_{1}}}{\frac {F_{3}}{h_{3}}}{\frac {\partial (h_{3})}{\partial u_{1}}})){\widehat {u}}_{2}+$
$(({\frac {1}{h_{1}}}{\frac {\partial F_{2}}{\partial u_{1}}}+{\frac {1}{h_{1}}}{\frac {F_{2}}{h_{2}}}{\frac {\partial (h_{2})}{\partial u_{1}}})-({\frac {1}{h_{2}}}{\frac {\partial F_{1}}{\partial u_{2}}}+{\frac {1}{h_{2}}}{\frac {F_{1}}{h_{1}}}{\frac {\partial (h_{1})}{\partial u_{2}}})){\widehat {u}}_{3}=$
$({\frac {1}{h_{2}h_{3}}}{\frac {\partial (F_{3}h_{3})}{\partial u_{2}}}-{\frac {1}{h_{2}h_{3}}}{\frac {\partial (F_{2}h_{2})}{\partial u_{3}}}){\widehat {u}}_{1}+({\frac {1}{h_{1}h_{3}}}{\frac {\partial (F_{1}h_{1})}{\partial u_{3}}}-{\frac {1}{h_{1}h_{3}}}{\frac {\partial (F_{3}h_{3})}{\partial u_{1}}}){\widehat {u}}_{2}+({\frac {1}{h_{1}h_{2}}}{\frac {\partial (F_{2}h_{2})}{\partial u_{1}}}-{\frac {1}{h_{1}h_{2}}}{\frac {\partial (F_{1}h_{1})}{\partial u_{2}}}){\widehat {u}}_{3}=$
${\frac {1}{h_{1}h_{2}h_{3}}}(({\frac {\partial (F_{3}h_{3})}{\partial u_{2}}}-{\frac {\partial (F_{2}h_{2})}{\partial u_{3}}})h_{1}{\widehat {u}}_{1}+({\frac {\partial (F_{1}h_{1})}{\partial u_{3}}}-{\frac {\partial (F_{3}h_{3})}{\partial u_{1}}})h_{2}{\widehat {u}}_{2}+({\frac {\partial (F_{2}h_{2})}{\partial u_{1}}}-{\frac {\partial (F_{1}h_{1})}{\partial u_{2}}})h_{3}{\widehat {u}}_{3})$

> Rotacional en coordenadas generalizadas. Geometría diferencial

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.
$rot({\overrightarrow {F}})={\overrightarrow {\nabla }}\times {\overrightarrow {F}}=\uparrow \ast d\downarrow ({\overrightarrow {F}})=\uparrow \ast d\downarrow ({F_{x}}{\frac {\partial }{\partial x}}+{F_{y}}{\frac {\partial }{\partial y}}+F_{z}{\frac {\partial }{\partial z}})=\uparrow \ast d\downarrow ({F_{1}}({\frac {1}{\sqrt[{}]{g_{11}}}}{\frac {\partial }{\partial u_{1}}})+F_{2}({\frac {1}{\sqrt[{}]{g_{22}}}}{\frac {\partial }{\partial u_{2}}})+F_{3}({\frac {1}{\sqrt[{}]{g_{33}}}}{\frac {\partial }{\partial u_{3}}}))=$
$\uparrow \ast d(F_{1}{\frac {1}{\sqrt[{}]{g_{11}}}}g_{11}du_{1}+F_{2}{\frac {1}{\sqrt[{}]{g_{22}}}}g_{22}du_{2}+F_{3}{\frac {1}{\sqrt[{}]{g_{33}}}}g_{33}du_{3})=\uparrow \ast d({F_{1}}{\sqrt[{2}]{g_{11}}}du_{1}+F_{2}{\sqrt[{2}]{g_{22}}}du_{2})+F_{3}{\sqrt[{2}]{g_{33}}}du_{3})=$
$\uparrow \ast ({\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{2}}}du_{2}\wedge du_{1}+{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{3}}}du_{3}\wedge du_{1}+{\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{1}}}du_{1}\wedge du_{2}+$
${\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{3}}}du_{3}\wedge du_{2}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{1}}}du_{1}\wedge du_{3})+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{2}}}du_{2}\wedge du_{3})=$
$\uparrow \ast (-{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{2}}}du_{1}\wedge du_{2}-{\frac {\partial (F_{1}{\sqrt[{2}]{g_{11}}})}{\partial u_{3}}}du_{1}\wedge du_{3}+{\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{1}}}du_{1}\wedge du_{2}-$
${\frac {\partial (F_{2}{\sqrt[{2}]{g_{22}}})}{\partial u_{3}}}du_{2}\wedge du_{3}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{1}}}du_{1}\wedge du_{3}+{\frac {\partial (F_{3}{\sqrt[{2}]{g_{33}}})}{\partial u_{2}}}du_{2}\wedge du_{3})=$
$\uparrow (-{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}{\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})du_{3}-{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})du_{2}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})du_{3}-$
${\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})du_{1}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}{\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})du_{2}+{\frac {\sqrt[{}]{\vert g\vert }}{g_{22}g_{33}}}{\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})du_{1})=$
$\uparrow ({\frac {\sqrt[{}]{\vert g\vert }}{g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}}))-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))du_{1}+$
${\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{33}}}({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))du_{2}+$
${\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}}}({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))du_{3})=$
$\uparrow ({\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}}))-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))g_{11}du_{1}+$
${\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))g_{22}du_{2}+$
${\frac {\sqrt[{}]{\vert g\vert }}{g_{11}g_{22}g_{33}}}({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))g_{33}du_{3})=$
${\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\frac {\partial }{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})){\frac {\partial }{\partial u_{2}}}+({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\frac {\partial }{\partial u_{3}}})=$
${\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}}))\vert {\frac {\partial }{\partial u_{1}}}\vert {\frac {\frac {\partial }{\partial u_{1}}}{\vert {\frac {\partial }{\partial u_{1}}}\vert }}+$
$({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}}))\vert {\frac {\partial }{\partial u_{2}}}\vert {\frac {\frac {\partial }{\partial u_{2}}}{\vert {\frac {\partial }{\partial u_{2}}}\vert }}+$
$({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}}))\vert {\frac {\partial }{\partial u_{3}}}\vert {\frac {\frac {\partial }{\partial u_{3}}}{\vert {\frac {\partial }{\partial u_{3}}}\vert }})=$
${\frac {1}{\sqrt[{}]{g_{11}g_{22}g_{33}}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\sqrt[{2}]{g_{11}}}{\frac {\widehat {\partial }}{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}{\sqrt[{2}]{g_{33}}})-{\frac {\partial }{\partial u_{3}}}(F_{1}{\sqrt[{2}]{g_{11}}})){\sqrt[{2}]{g_{22}}}{\frac {\widehat {\partial }}{\partial u_{2}}}+$
$({\frac {\partial }{\partial u_{2}}}(F_{1}{\sqrt[{2}]{g_{11}}})-{\frac {\partial }{\partial u_{1}}}(F_{2}{\sqrt[{2}]{g_{22}}})){\sqrt[{2}]{g_{33}}}){\frac {\widehat {\partial }}{\partial u_{3}}})=$
${\frac {1}{h_{1}h_{2}h_{3}}}(({\frac {\partial }{\partial u_{2}}}(F_{3}h_{3})-{\frac {\partial }{\partial u_{3}}}(F_{2}h_{2}))h_{1}{\frac {\widehat {\partial }}{\partial u_{1}}}+({\frac {\partial }{\partial u_{1}}}(F_{3}h_{3})-{\frac {\partial }{\partial u_{3}}}(F_{1}h_{1}))h_{2}{\frac {\widehat {\partial }}{\partial u_{2}}}+({\frac {\partial }{\partial u_{2}}}(F_{1}h_{1})-{\frac {\partial }{\partial u_{1}}}(F_{2}h_{2}))h_{3}{\frac {\widehat {\partial }}{\partial u_{3}}})$

> Laplaciano en coordenadas generalizadas. Cálculo diferencial

$div({\overrightarrow {grad}}(f))=\nabla ^{2}f={\overrightarrow {\nabla }}\cdot ({\overrightarrow {\nabla }}f)={\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial x}}{\widehat {e}}_{x}+{\frac {\partial f}{\partial y}}{\widehat {e}}_{y}+{\frac {\partial f}{\partial z}}{\widehat {e}}_{z})={\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial x}}{\widehat {e}}_{x})+{\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial y}}{\widehat {e}}_{y})+{\overrightarrow {\nabla }}\cdot ({\frac {\partial f}{\partial z}}{\widehat {e}}_{z})=$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{x})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{y})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}\cdot {\widehat {e}}_{z})=$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y}))=$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}{\overrightarrow {\nabla }}(h_{2}h_{3}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}{\overrightarrow {\nabla }}(h_{3}h_{1}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}{\overrightarrow {\nabla }}(h_{1}h_{2}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}})+{\frac {\partial f}{\partial x}}h_{2}h_{3}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{y}}{h_{2}}}\times {\frac {{\widehat {e}}_{z}}{h_{3}}}))+$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}})+{\frac {\partial f}{\partial y}}h_{3}h_{1}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{z}}{h_{3}}}\times {\frac {{\widehat {e}}_{x}}{h_{1}}}))+$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}})+{\frac {\partial f}{\partial z}}h_{1}h_{2}({\overrightarrow {\nabla }}\cdot ({\frac {{\widehat {e}}_{x}}{h_{1}}}\times {\frac {{\widehat {e}}_{y}}{h_{2}}}))=$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {1}{h_{2}h_{3}}}{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot ({\widehat {e}}_{y}\times {\widehat {e}}_{z})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {1}{h_{3}h_{1}}}{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot ({\widehat {e}}_{z}\times {\widehat {e}}_{x})+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {1}{h_{1}h_{2}}}{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot ({\widehat {e}}_{x}\times {\widehat {e}}_{y})=$
$({\overrightarrow {\nabla }}{\frac {\partial f}{\partial x}})\cdot {\widehat {e}}_{x}+{\frac {1}{h_{2}h_{3}}}{\frac {\partial f}{\partial x}}({\overrightarrow {\nabla }}(h_{2}h_{3}))\cdot {\widehat {e}}_{x}+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial y}})\cdot {\widehat {e}}_{y}+{\frac {1}{h_{3}h_{1}}}{\frac {\partial f}{\partial y}}({\overrightarrow {\nabla }}(h_{3}h_{1}))\cdot {\widehat {e}}_{y}+({\overrightarrow {\nabla }}{\frac {\partial f}{\partial z}})\cdot {\widehat {e}}_{z}+{\frac {1}{h_{1}h_{2}}}{\frac {\partial f}{\partial z}}({\overrightarrow {\nabla }}(h_{1}h_{2}))\cdot {\widehat {e}}_{z}=$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})({\frac {\partial f}{\partial u_{1}}}{\frac {\partial u_{1}}{\partial x}}))\cdot {\widehat {u}}_{1}+{\frac {1}{h_{2}h_{3}}}{\frac {\partial f}{\partial u_{1}}}{\frac {\partial u_{1}}{\partial x}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{2}h_{3}))\cdot {\widehat {u}}_{1}+$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})({\frac {\partial f}{\partial u_{2}}}{\frac {\partial u_{2}}{\partial y}}))\cdot {\widehat {u}}_{2}+{\frac {1}{h_{1}h_{3}}}{\frac {\partial f}{\partial u_{2}}}{\frac {\partial u_{2}}{\partial y}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{3}))\cdot {\widehat {u}}_{2}+$
$(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})({\frac {\partial f}{\partial u_{3}}}{\frac {\partial u_{3}}{\partial z}}))\cdot {\widehat {u}}_{3}+{\frac {1}{h_{1}h_{2}}}{\frac {\partial f}{\partial u_{e}}}{\frac {\partial u_{3}}{\partial z}}(({\frac {{\widehat {u}}_{1}}{h_{1}}}{\frac {\partial }{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}}{h_{2}}}{\frac {\partial }{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}}{h_{3}}}{\frac {\partial }{\partial u_{3}}})(h_{1}h_{2}))\cdot {\widehat {u}}_{3}=$
$({\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}{\frac {\partial }{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}{\frac {\partial }{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}{\frac {\partial }{\partial u_{3}}})({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{1}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{1}}{h_{2}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{1}}{h_{2}h_{3}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{3}}})+$
$({\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}{\frac {\partial }{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}{\frac {\partial }{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}{\frac {\partial }{\partial u_{3}}})({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+$
${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial f}{\partial r}})+{\frac {1}{r^{2}sin\vartheta }}{\frac {\partial }{\partial \vartheta }}(sin\vartheta {\frac {\partial f}{\partial \vartheta }})+{\frac {1}{r^{2}sin^{2}\vartheta }}({\frac {\partial ^{2}f}{\partial ^{2}\varphi }}){\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{2}}{h_{1}h_{1}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{2}}{h_{1}h_{3}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{3}}})+$
$({\frac {1}{h_{1}}}{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}{\frac {\partial }{\partial u_{1}}}+{\frac {1}{h_{2}}}{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}{\frac {\partial }{\partial u_{2}}}+{\frac {1}{h_{3}}}{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}{\frac {\partial }{\partial u_{3}}})({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}}({\frac {{\widehat {u}}_{1}\cdot {\widehat {u}}_{3}}{h_{1}h_{1}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{1}}}+{\frac {{\widehat {u}}_{2}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{2}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{2}}}+{\frac {{\widehat {u}}_{3}\cdot {\widehat {u}}_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}})=$
${\frac {1}{h_{1}}}({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial }{\partial u_{1}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})({\widehat {u}}_{1}\cdot {\widehat {u}}_{1}){\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial }{\partial u_{2}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+$
${\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})({\widehat {u}}_{2}\cdot {\widehat {u}}_{2}){\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial }{\partial u_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})({\widehat {u}}_{3}\cdot {\widehat {u}}_{3}){\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {1}{h_{1}}}{\frac {\partial }{\partial u_{1}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}}){\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {1}{h_{2}}}{\frac {\partial }{\partial u_{2}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+$
${\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}}){\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {1}{h_{3}}}{\frac {\partial }{\partial u_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}}){\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {h_{2}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial u_{1}}}({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {({\frac {1}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{2}h_{3})}{\partial u_{1}}}+{\frac {h_{1}h_{3}}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial u_{2}}}({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {({\frac {1}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{3})}{\partial u_{2}}}+{\frac {h_{1}h_{2}}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial u_{3}}}({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})+{\frac {({\frac {1}{h_{3}}}{\frac {\partial f}{\partial u_{3}}})}{h_{1}h_{2}h_{3}}}{\frac {\partial (h_{1}h_{2})}{\partial u_{3}}}=$
${\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}({\frac {\partial f}{\partial u_{1}}}{\frac {h_{2}h_{3}}{h_{1}}})+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{2}}}({\frac {\partial f}{\partial u_{2}}}{\frac {h_{1}h_{3}}{h_{2}}}))+{\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{3}}}({\frac {\partial f}{\partial u_{3}}}{\frac {h_{1}h_{2}}{h_{3}}}))=$
${\frac {1}{h_{1}h_{2}h_{3}}}({\frac {\partial }{\partial u_{1}}}({\frac {\partial f}{\partial u_{1}}}{\frac {h_{2}h_{3}}{h_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {\partial f}{\partial u_{2}}}{\frac {h_{1}h_{3}}{h_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {\partial f}{\partial u_{3}}}{\frac {h_{1}h_{2}}{h_{3}}}))$

> Laplaciano en coordenadas generalizadas. Geometría diferencial

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.
$div({\overrightarrow {grad}}(f))=\nabla ^{2}f=\ast d\ast d(f)=\ast d\ast ({\frac {\partial f}{\partial u_{1}}}du_{1}+{\frac {\partial f}{\partial u_{2}}}du_{2}+{\frac {\partial f}{\partial u_{3}}}du_{3})=$
$\ast d({\frac {\sqrt[{}]{|g|}}{g_{1}}}{\frac {\partial f}{\partial u_{1}}}(du_{2}\wedge du_{3})+{\frac {\sqrt[{}]{|g|}}{g_{2}}}{\frac {\partial f}{\partial u_{2}}}(du_{3}\wedge du_{1})+{\frac {\sqrt[{}]{|g|}}{g_{3}}}{\frac {\partial f}{\partial u_{3}}}(du_{1}\wedge du_{2}))=$
$\ast ({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})(du_{1}\wedge du_{2}\wedge du_{3})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})(du_{2}\wedge du_{3}\wedge du_{1})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})(du_{3}\wedge du_{1}\wedge du_{2}))=$
$\ast (({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})(du_{1}\wedge du_{2}\wedge du_{3}))=$
$({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}}){\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}})=$
$({\frac {\sqrt[{}]{|g|}}{g_{1}g_{2}g_{3}}})({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}}))=$
$({\frac {1}{\sqrt[{}]{g_{1}g_{2}g_{3}}}})({\frac {\partial }{\partial u_{1}}}({\sqrt[{2}]{\frac {g_{2}g_{3}}{g_{1}}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\sqrt[{2}]{\frac {g_{1}g_{3}}{g_{2}}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\sqrt[{2}]{\frac {g_{1}g_{2}}{g_{3}}}}{\frac {\partial f}{\partial u_{3}}}))=$
$({\frac {1}{h_{1}h_{2}h_{3}}})({\frac {\partial }{\partial u_{1}}}({\frac {h_{2}h_{3}}{h_{1}}}{\frac {\partial f}{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {h_{1}h_{3}}{h_{2}}}{\frac {\partial f}{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {h_{1}h_{2}}{h_{3}}}{\frac {\partial f}{\partial u_{3}}}))$

Bibliografía

Gravitation (Charles W. Miser, Kip S. Thorne)
Geometría diferencial (Carlos Ivorra Castillo)
Geometría diferencial (Fernando Chamizo Lorente)

Véase también

• Rotacional
• Gradiente
• Divergencia
• Operador laplaciano
• Nabla