Nabla in coordinate cilindriche e sferiche

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 $\triangle f = \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}(\rho {\partial f \over \partial \rho}) + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}(r^2 {\partial f \over \partial r}) + {1 \over r^2\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f \over \partial \theta}) + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \phi^2}$
Laplaciano di un vettore $\triangle \mathbf{A} = \nabla^2 \mathbf{A}$	$\triangle A_x \mathbf{\hat x} + \triangle A_y \mathbf{\hat y} + \triangle A_z \mathbf{\hat z}$	$\begin{matrix} \displaystyle(\triangle A_\rho - {A_\rho \over \rho^2} - {2 \over \rho^2}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat\rho} & + \\ \displaystyle(\triangle A_\phi - {A_\phi \over \rho^2} + {2 \over \rho^2}{\partial A_\rho \over \partial \phi}) \boldsymbol{\hat\phi} & + \\ \displaystyle(\triangle A_z ) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} (\triangle A_r - {2 A_r \over r^2} - {2 \over r^2\sin\theta}{\partial (A_\theta \sin\theta) \over \partial\theta} - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat r} & + \\ (\triangle A_\theta - {A_\theta \over r^2\sin^2\theta} + {2 \over r^2}{\partial A_r \over \partial \theta} - {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat\theta} & + \\ (\triangle A_\phi - {A_\phi \over r^2\sin^2\theta} + {2 \over r^2\sin^2\theta}{\partial A_r \over \partial \phi} + {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}) \boldsymbol{\hat\phi} & \end{matrix}$
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): $\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \triangle f$ (Laplaciano) $\operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = 0$ $\operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0$ $\operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ $\triangle f g = f \triangle g + 2 \nabla f \cdot \nabla g + g \triangle f$ 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

• 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

• Coordinate curvilinee
• Campi vettoriali in coordinate cilindriche e sferiche
• Identità vettoriali

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