Nabla in coordinate cilindriche e sferiche

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Nel calcolo vettoriale è spesso utile conoscere come esprimere \nabla in altri sistemi di coordinate diversi da quello cartesiano.

Operatore Coordinate cartesiane (x,y,z) Coordinate cilindriche (ρ,φ,z) Coordinate sferiche (r,θ,φ)
Definizione delle coordinate   \left[\begin{matrix}
    x & = & \rho\cos\phi \\
    y & = & \rho\sin\phi \\
    z & = & z \end{matrix}\right]. \left[\begin{matrix}
    x & = & r\sin\theta\cos\phi \\
    y & = & r\sin\theta\sin\phi \\
    z & = & r\cos\theta \end{matrix}\right].
\left[\begin{matrix}
    \rho & = & \sqrt{x^2 + y^2} \\
    \phi & = & \operatorname{atan2}(y, x) \\
    z & = & z \end{matrix}\right]. \left[\begin{matrix}
    r & = & \sqrt{x^2 + y^2 + z^2} \\
    \theta & = & \arccos(z / r) \\
    \phi & = & \operatorname{atan2}(y, x) \end{matrix}\right].
A Campo vettoriale \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}
Gradiente \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\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}
Divergenza \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\sin\theta}{\partial \over \partial \theta} (  A_\theta\sin\theta )
  + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}
Rotore \nabla \times \mathbf{A} \begin{matrix}
  \displaystyle({\partial A_z \over \partial y} - {\partial A_y \over \partial z}) \mathbf{\hat x} & + \\
  \displaystyle({\partial A_x \over \partial z} - {\partial A_z \over \partial x}) \mathbf{\hat y} & + \\
  \displaystyle({\partial A_y \over \partial x} - {\partial A_x \over \partial y}) \mathbf{\hat z} & \ \end{matrix} \begin{matrix}
  \displaystyle({1 \over \rho}{\partial A_z \over \partial \phi}
    - {\partial A_\phi \over \partial z}) \boldsymbol{\hat \rho} & + \\
  \displaystyle({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}) \boldsymbol{\hat \phi} & + \\
  \displaystyle{1 \over \rho}({\partial ( \rho A_\phi ) \over \partial \rho}
    - {\partial A_\rho \over \partial \phi}) \boldsymbol{\hat z} & \ \end{matrix} \begin{matrix}
  \displaystyle{1 \over r\sin\theta}({\partial \over \partial \theta} ( A_\phi\sin\theta )
    - {\partial A_\theta \over \partial \phi}) \boldsymbol{\hat r} & + \\
  \displaystyle{1 \over r}({1 \over \sin\theta}{\partial A_r \over \partial \phi}
    - {\partial \over \partial r} ( r A_\phi ) ) \boldsymbol{\hat \theta} & + \\
  \displaystyle{1 \over r}({\partial \over \partial r} ( r A_\theta )
    - {\partial A_r \over \partial \theta}) \boldsymbol{\hat \phi} & \ \end{matrix}
Laplaciano \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\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f \over \partial \theta})
  + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \phi^2}
Laplaciano di un vettore \nabla^2 \mathbf{A} \nabla^2 A_x \mathbf{\hat x} + \nabla^2 A_y \mathbf{\hat y} + \nabla^2 A_z \mathbf{\hat z} \begin{matrix}
  \displaystyle(\nabla^2 A_\rho - {A_\rho \over \rho^2}
    - {2 \over \rho^2}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat\rho} & + \\
  \displaystyle(\nabla^2 A_\phi - {A_\phi \over \rho^2}
    + {2 \over \rho^2}{\partial A_\rho \over \partial \phi}) \boldsymbol{\hat\phi} & + \\
  \displaystyle(\nabla^2 A_z ) \boldsymbol{\hat z}  & \ \end{matrix} \begin{matrix}
  (\nabla^2 A_r - {2 A_r \over r^2}
    - {2 \over r^2\sin\theta}{\partial (A_\theta \sin\theta) \over \partial\theta}
    - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat r} & + \\
  (\nabla^2 A_\theta - {A_\theta \over r^2\sin^2\theta}
    + {2 \over r^2}{\partial A_r \over \partial \theta}
    - {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat\theta} & + \\
  (\nabla^2 A_\phi - {A_\phi \over r^2\sin^2\theta}
    + {2 \over r^2\sin^2\theta}{\partial A_r \over \partial \phi}
    + {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}) \boldsymbol{\hat\phi} & \end{matrix}
Lunghezza infinitesima d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z} d\mathbf{l} = d\rho\boldsymbol{\hat \rho} + \rho d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z} d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi}
Aree infinitesime \begin{matrix}d\mathbf{S} = &dydz\mathbf{\hat x} + \\
&dxdz\mathbf{\hat y} + \\
&dxdy\mathbf{\hat z}\end{matrix} \begin{matrix}
d\mathbf{S} = & \rho d\phi dz\boldsymbol{\hat \rho} + \\
& d\rho dz\boldsymbol{\hat \phi} + \\
& \rho d\rho d\phi \mathbf{\hat z}
\end{matrix} \begin{matrix}
d\mathbf{S} = & r^2 \sin\theta d\theta d\phi \mathbf{\hat r} + \\
& r\sin\theta drd\phi \boldsymbol{\hat \theta} + \\
& rdrd\theta\boldsymbol{\hat \phi}
\end{matrix}
Volume infinitesimo dv = dxdydz dv = \rho d\rho d\phi dz dv = r^2\sin\theta drd\theta d\phi
Relazioni notevoli (valgono in tutti i sistemi di riferimento):
  • \operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 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}
  • \nabla^2 f g = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f
  • Formula di Lagrange per il prodotto vettoriale: \mathbf{A} \times (\mathbf{B} \times \mathbf{C})
  = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B})
  • \nabla\cdot(f \mathbf A)=f \nabla\cdot\mathbf A+\mathbf A\cdot\nabla f
  • \nabla\times f \mathbf A= f \nabla\times \mathbf A-\mathbf A\times \nabla f
  • \nabla ( \mathbf{A} \cdot \mathbf{B} ) 
  = ( \mathbf{A} \cdot \nabla ) \mathbf{B}
  + ( \mathbf{B} \cdot \nabla ) \mathbf{A}
  + \mathbf{A} \times ( \nabla \times \mathbf{B} )
  + \mathbf{B} \times ( \nabla \times \mathbf{A} ),
    che insieme a \mathbf{A} = \mathbf{B} = \mathbf{v} segue immediatamente la chiave per il fluido di trasformazione meccanica Weber:
    ( \mathbf{v} \cdot \nabla ) \mathbf{v} 
  = \nabla \frac{\mathbf{v}^2}{2} 
  - \mathbf{v} \times ( \nabla \times \mathbf{v} )
  • \nabla\times (\mathbf A\times\mathbf B)= \mathbf A\,(\nabla\cdot\mathbf B)  - \mathbf B\,(\nabla\cdot\mathbf A) + (\mathbf  B\cdot\nabla)\mathbf A - (\mathbf A\cdot\nabla)\mathbf B
  • \nabla\cdot(\mathbf A\times\mathbf B)=\mathbf B\cdot(\nabla\times\mathbf A)-\mathbf A\cdot(\nabla\times\mathbf B)

Nota[modifica | modifica sorgente]

  • La funzione atan2(y,x) è usata al posto di arctan(y/x) per il suo dominio. La funzione arctan(y/x) ha immagine in (-π/2, +π/2), mentre atan2(y,x) ha immagine in (-π, π].

Voci correlate[modifica | modifica sorgente]

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