Anexo:Tabla en coordenadas cilíndricas y esféricas

Esta es una lista de algunas fórmulas de cálculo vectorial de empleo corriente trabajando con varios sistemas de coordenadas.

Operación	coordenadas cartesianas (x,y,z)	coordenadas cilíndricas (ρ,φ,z)	coordenadas esféricas (r,θ,φ)
Definición de las coordenadas		$\left[{\begin{matrix}x&=&\rho \cos \phi \\y&=&\rho \operatorname {sen} \phi \\z&=&z\end{matrix}}\right.$	$\left[{\begin{matrix}x&=&r\operatorname {sen} \theta \cos \phi \\y&=&r\operatorname {sen} \theta \operatorname {sen} \phi \\z&=&r\cos \theta \end{matrix}}\right.$
Definición de las coordenadas		$\left[{\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\arctan(y/x)\\z&=&z\end{matrix}}\right.$	$\left[{\begin{matrix}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\arctan(({\sqrt {x^{2}+y^{2}}})/z)\\\phi &=&\arctan(y/x)\end{matrix}}\right.$
$\mathbf {A}$	$A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$
$\nabla f$	${\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}}$	${\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\operatorname {sen} \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}$
$\nabla \cdot \mathbf {A}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${1 \over \rho }{\partial \rho A_{\rho } \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial r^{2}A_{r} \over \partial r}+{1 \over r\operatorname {sen} \theta }{\partial A_{\theta }\operatorname {sen} \theta \over \partial \theta }+{1 \over r\operatorname {sen} \theta }{\partial A_{\phi } \over \partial \phi }$
$\nabla \times \mathbf {A}$	${\begin{matrix}\left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right)\mathbf {\hat {x}} &+\\\left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right)\mathbf {\hat {y}} &+\\\left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right)\mathbf {\hat {z}} &\ \end{matrix}}$	${\begin{matrix}\left({1 \over \rho }{\partial A_{z} \over \partial \phi }-{\partial A_{\phi } \over \partial z}\right){\boldsymbol {\hat {\rho }}}&+\\\left({\partial A_{\rho } \over \partial z}-{\partial A_{z} \over \partial \rho }\right){\boldsymbol {\hat {\phi }}}&+\\{1 \over \rho }({\partial \rho A_{\phi } \over \partial \rho }-{\partial A_{\rho } \over \partial \phi }){\boldsymbol {\hat {z}}}&\ \end{matrix}}$	${\begin{matrix}{1 \over r\operatorname {sen} \theta }({\partial A_{\phi }\operatorname {sen} \theta \over \partial \theta }-{\partial A_{\theta } \over \partial \phi }){\boldsymbol {\hat {r}}}&+\\({1 \over r\operatorname {sen} \theta }{\partial A_{r} \over \partial \phi }-{1 \over r}{\partial rA_{\phi } \over \partial r}){\boldsymbol {\hat {\theta }}}&+\\{1 \over r}({\partial rA_{\theta } \over \partial r}-{\partial A_{r} \over \partial \theta }){\boldsymbol {\hat {\phi }}}&\ \end{matrix}}$
$\Delta f=\nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }(\rho {\partial f \over \partial \rho })+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}(r^{2}{\partial f \over \partial r})+{1 \over r^{2}\operatorname {sen} \theta }{\partial \over \partial \theta }(\operatorname {sen} \theta {\partial f \over \partial \theta })+{1 \over r^{2}\operatorname {sen} ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$
$\Delta \mathbf {A} =\nabla ^{2}\mathbf {A}$	$\mathbf {\hat {x}} \Delta A_{x}+\mathbf {\hat {y}} \Delta A_{y}+\mathbf {\hat {z}} \Delta A_{z}$	${\begin{matrix}{\boldsymbol {\hat {\rho }}}(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\phi } \over \partial \phi })&+\\{\boldsymbol {\hat {\phi }}}(\Delta A_{\phi }-{A_{\phi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \phi })&+\\{\boldsymbol {\hat {z}}}\Delta A_{z}&\ \end{matrix}}$	${\begin{matrix}{\boldsymbol {\hat {r}}}&(\Delta A_{r}-{2A_{r} \over r^{2}}-{2A_{\theta }\cos \theta \over r^{2}\operatorname {sen} \theta }\\\ &-{2 \over r^{2}}{\partial A_{\theta } \over \partial \theta }-{2 \over r^{2}\operatorname {sen} \theta }{\partial A_{\phi } \over \partial \phi })&+\\{\boldsymbol {\hat {\theta }}}&(\Delta A_{\theta }-{A_{\theta } \over r^{2}\operatorname {sen} ^{2}\theta }\\\ &+{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\operatorname {sen} ^{2}\theta }{\partial A_{\phi } \over \partial \phi })&+\\{\boldsymbol {\hat {\phi }}}&(\Delta A_{\phi }-{A_{\phi } \over r^{2}\operatorname {sen} ^{2}\theta }\\\ &+{2 \over r^{2}\operatorname {sen} ^{2}\theta }{\partial A_{r} \over \partial \phi }+{2\cos \theta \over r^{2}\operatorname {sen} ^{2}\theta }{\partial A_{\theta } \over \partial \phi })&\ \end{matrix}}$
Reglas de cálculo no triviales: $\operatorname {div\ grad\ } f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ (laplaciano) $\operatorname {rot\ grad\ } f=\nabla \times (\nabla f)=0$ $\operatorname {div\ rot\ } \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {rot\ rot\ } \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ $\Delta fg=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$ Fórmula de Lagrange para el producto vectorial: $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )$

Véase también

Coordenadas curvilíneas
Campos vectoriales en coordenadas cilíndricas y esféricas

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