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\sen\phi \\
z & = & z \end{matrix}\right. \left[\begin{matrix}
x & = & r\sen\theta\cos\phi \\
y & = & r\sen\theta\sen\phi \\
z & = & r\cos\theta \end{matrix}\right.
\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\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\sen\theta}{\partial A_\theta\sen\theta \over \partial \theta} 
+ {1 \over r\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\sen\theta}({\partial A_\phi\sen\theta \over \partial \theta}
- {\partial A_\theta \over \partial \phi}) \boldsymbol{\hat r} & + \\
({1 \over r\sen\theta}{\partial A_r \over \partial \phi}
- {1 \over r}{\partial r A_\phi \over \partial r}) \boldsymbol{\hat \theta} & + \\
{1 \over r}({\partial r A_\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\sen\theta}{\partial \over \partial \theta}(\sen\theta {\partial f \over \partial \theta}) 
+ {1 \over r^2\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 - {2 A_r \over r^2}
- {2 A_\theta\cos\theta \over r^2\sen\theta} \\ \ &
- {2 \over r^2}{\partial A_\theta \over \partial \theta}
- {2 \over r^2\sen\theta}{\partial A_\phi \over \partial \phi}) & + \\
\boldsymbol{\hat\theta} & (\Delta A_\theta - {A_\theta \over r^2\sen^2\theta} \\ \ &
+ {2 \over r^2}{\partial A_r \over \partial \theta}
- {2 \cos\theta \over r^2\sen^2\theta}{\partial A_\phi \over \partial \phi}) & + \\
\boldsymbol{\hat\phi} & (\Delta A_\phi - {A_\phi \over r^2\sen^2\theta} \\ \ &
+ {2 \over r^2\sen^2\theta}{\partial A_r \over \partial \phi}
+ {2 \cos\theta \over r^2\sen^2\theta}{\partial A_\theta \over \partial \phi}) & \ \end{matrix}
Reglas de cálculo no triviales:
  1. \operatorname{div\ grad\ }  f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f (laplaciano)
  2. \operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = 0
  3. \operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0
  4. \operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) 
= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}
  5. \Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f
  6. 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

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