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 		  [ x = ρ cos ⁡ ϕ y = ρ sen ⁡ ϕ z = z {\displaystyle \left[{\begin{matrix}x&=&\rho \cos \phi \\y&=&\rho \operatorname {sen} \phi \\z&=&z\end{matrix}}\right.} [ x = r sen ⁡ θ cos ⁡ ϕ y = r sen ⁡ θ sen ⁡ ϕ z = r cos ⁡ θ {\displaystyle \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.}
[ ρ = x 2 + y 2 ϕ = arctan ⁡ ( y / x ) z = z {\displaystyle \left[{\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\arctan(y/x)\\z&=&z\end{matrix}}\right.} [ r = x 2 + y 2 + z 2 θ = arctan ⁡ ( ( x 2 + y 2 ) / z ) ϕ = arctan ⁡ ( y / x ) {\displaystyle \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.}


A {\displaystyle \mathbf {A} } A x x ^ + A y y ^ + A z z ^ {\displaystyle A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} } A ρ ρ ^ + A ϕ ϕ ^ + A z z ^ {\displaystyle A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}} A r r ^ + A θ θ ^ + A ϕ ϕ ^ {\displaystyle A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}}
∇ f {\displaystyle \nabla f} ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ {\displaystyle {\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}} } ∂ f ∂ ρ ρ ^ + 1 ρ ∂ f ∂ ϕ ϕ ^ + ∂ f ∂ z z ^ {\displaystyle {\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}}}} ∂ f ∂ r r ^ + 1 r ∂ f ∂ θ θ ^ + 1 r sen ⁡ θ ∂ f ∂ ϕ ϕ ^ {\displaystyle {\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 }}}}


∇ ⋅ A {\displaystyle \nabla \cdot \mathbf {A} } ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z {\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}} 1 ρ ∂ ρ A ρ ∂ ρ + 1 ρ ∂ A ϕ ∂ ϕ + ∂ A z ∂ z {\displaystyle {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 r 2 ∂ r 2 A r ∂ r + 1 r sen ⁡ θ ∂ A θ sen ⁡ θ ∂ θ + 1 r sen ⁡ θ ∂ A ϕ ∂ ϕ {\displaystyle {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 }}


∇ × A {\displaystyle \nabla \times \mathbf {A} } ( ∂ A z ∂ y − ∂ A y ∂ z ) x ^ + ( ∂ A x ∂ z − ∂ A z ∂ x ) y ^ + ( ∂ A y ∂ x − ∂ A x ∂ y ) z ^   {\displaystyle {\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}}} ( 1 ρ ∂ A z ∂ ϕ − ∂ A ϕ ∂ z ) ρ ^ + ( ∂ A ρ ∂ z − ∂ A z ∂ ρ ) ϕ ^ + 1 ρ ( ∂ ρ A ϕ ∂ ρ − ∂ A ρ ∂ ϕ ) z ^   {\displaystyle {\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}}} 1 r sen ⁡ θ ( ∂ A ϕ sen ⁡ θ ∂ θ − ∂ A θ ∂ ϕ ) r ^ + ( 1 r sen ⁡ θ ∂ A r ∂ ϕ − 1 r ∂ r A ϕ ∂ r ) θ ^ + 1 r ( ∂ r A θ ∂ r − ∂ A r ∂ θ ) ϕ ^   {\displaystyle {\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}}}


Δ f = ∇ 2 f {\displaystyle \Delta f=\nabla ^{2}f} ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 {\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}} 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ ϕ 2 + ∂ 2 f ∂ z 2 {\displaystyle {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 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sen ⁡ θ ∂ ∂ θ ( sen ⁡ θ ∂ f ∂ θ ) + 1 r 2 sen 2 ⁡ θ ∂ 2 f ∂ ϕ 2 {\displaystyle {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}}}


Δ A = ∇ 2 A {\displaystyle \Delta \mathbf {A} =\nabla ^{2}\mathbf {A} } x ^ Δ A x + y ^ Δ A y + z ^ Δ A z {\displaystyle \mathbf {\hat {x}} \Delta A_{x}+\mathbf {\hat {y}} \Delta A_{y}+\mathbf {\hat {z}} \Delta A_{z}} ρ ^ ( Δ A ρ − A ρ ρ 2 − 2 ρ 2 ∂ A ϕ ∂ ϕ ) + ϕ ^ ( Δ A ϕ − A ϕ ρ 2 + 2 ρ 2 ∂ A ρ ∂ ϕ ) + z ^ Δ A z   {\displaystyle {\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}}} r ^ ( Δ A r − 2 A r r 2 − 2 A θ cos ⁡ θ r 2 sen ⁡ θ   − 2 r 2 ∂ A θ ∂ θ − 2 r 2 sen ⁡ θ ∂ A ϕ ∂ ϕ ) + θ ^ ( Δ A θ − A θ r 2 sen 2 ⁡ θ   + 2 r 2 ∂ A r ∂ θ − 2 cos ⁡ θ r 2 sen 2 ⁡ θ ∂ A ϕ ∂ ϕ ) + ϕ ^ ( Δ A ϕ − A ϕ r 2 sen 2 ⁡ θ   + 2 r 2 sen 2 ⁡ θ ∂ A r ∂ ϕ + 2 cos ⁡ θ r 2 sen 2 ⁡ θ ∂ A θ ∂ ϕ )   {\displaystyle {\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:
  1. d i v   g r a d   ⁡ f = ∇ ⋅ ( ∇ f ) = ∇ 2 f = Δ f {\displaystyle \operatorname {div\ grad\ } f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f} (laplaciano)
  2. r o t   g r a d   ⁡ f = ∇ × ( ∇ f ) = 0 {\displaystyle \operatorname {rot\ grad\ } f=\nabla \times (\nabla f)=0}
  3. d i v   r o t   ⁡ A = ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \operatorname {div\ rot\ } \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0}
  4. r o t   r o t   ⁡ A = ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \operatorname {rot\ rot\ } \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
  5. Δ f g = f Δ g + 2 ∇ f ⋅ ∇ g + g Δ f {\displaystyle \Delta fg=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f}
  6. Fórmula de Lagrange para el producto vectorial:
    A × ( B × C ) = B ( A ⋅ C ) − C ( A ⋅ B ) {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )}


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