Nabla in coordinate cilindriche e sferiche

Template:F Nel calcolo vettoriale è spesso utile conoscere come esprimere $\mathrm{\nabla }$ 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		$\{\begin{array}{ccc}x& =& \rho \cos \varphi \\ y& =& \rho \sin \varphi \\ z& =& z\end{array}$	$\{\begin{array}{cccc}x& =& r\sin \theta \cos \varphi & 0⩽\theta ⩽\pi \\ y& =& r\sin \theta \sin \varphi & 0⩽\varphi <2\pi \\ z& =& r\cos \theta & 0⩽r<+\mathrm{\infty }\end{array}$
		$\{\begin{array}{ccc}\rho & =& \sqrt{x^2+y^2}\\ \varphi & =& \arctan (y/x)\\ z& =& z\end{array}$	$\{\begin{array}{ccc}r& =& \sqrt{x^2+y^2+z^2}\\ \theta & =& \arccos (z/r)\\ \varphi & =& \arctan (y/x)\end{array}$
Campo vettoriale $𝐀$	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\widehat{𝒛}$	$A_r\widehat{𝒓}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$
Gradiente $\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\widehat{𝒛}$	$\frac{\partial f}{\partial r}\widehat{𝒓}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$
Divergenza $\mathrm{\nabla }\cdot 𝐀$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial (\rho A_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial (r^2{A}_r)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$
Rotore $\mathrm{\nabla }\times 𝐀$	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\widehat{𝐱}& +\\ (\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\widehat{𝐲}& +\\ (\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\widehat{𝐳}& \end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}& +\\ (\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}& +\\ \frac{1}{\rho }(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\widehat{𝒛}& \end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\varphi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \varphi })\widehat{𝒓}& +\\ \frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial }{\partial r}(r{A}_{\varphi }))\hat{\mathit{\theta }}& +\\ \frac{1}{r}(\frac{\partial }{\partial r}(r{A}_{\theta })-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}& \end{array}$
Laplaciano ${\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}$
Laplaciano di un vettore ${\mathrm{\nabla }}^2𝐀$	${\mathrm{\nabla }}^2{A}_x\widehat{𝐱}+{\mathrm{\nabla }}^2{A}_y\widehat{𝐲}+{\mathrm{\nabla }}^2{A}_z\widehat{𝐳}$	$\begin{array}{cc}({\mathrm{\nabla }}^2{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\rho }}& +\\ ({\mathrm{\nabla }}^2{A}_{\varphi }-\frac{A_{\varphi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{\varphi }}& +\\ ({\mathrm{\nabla }}^2{A}_z)\widehat{𝒛}& \end{array}$	$\begin{array}{cc}({\mathrm{\nabla }}^2{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi })\widehat{𝒓}& +\\ ({\mathrm{\nabla }}^2{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\theta }}& +\\ ({\mathrm{\nabla }}^2{A}_{\varphi }-\frac{A_{\varphi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \varphi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{\varphi }}& \end{array}$
Lunghezza infinitesima	$d𝐥=dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$	$d𝐥=d\rho \hat{\mathit{\rho }}+\rho d\varphi \hat{\mathit{\varphi }}+dz\widehat{𝒛}$	$d𝐥=dr\widehat{𝐫}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\varphi \hat{\mathit{\varphi }}$
Area infinitesima	$\begin{array}{cc}d𝐒=& dydz\widehat{𝐱}+\\ & dxdz\widehat{𝐲}+\\ & dxdy\widehat{𝐳}\end{array}$	$\begin{array}{cc}d𝐒=& \rho d\varphi dz\hat{\mathit{\rho }}+\\ & d\rho dz\hat{\mathit{\varphi }}+\\ & \rho d\rho d\varphi \widehat{𝐳}\end{array}$	$\begin{array}{cc}d𝐒=& r^2\sin \theta d\theta d\varphi \widehat{𝐫}+\\ & r\sin \theta drd\varphi \hat{\mathit{\theta }}+\\ & rdrd\theta \hat{\mathit{\varphi }}\end{array}$
Volume infinitesimo	$dv=dxdydz$	$dv=\rho d\rho d\varphi dz$	$dv={\rho }^2\sin \theta d\rho d\theta d\varphi$

Relazioni notevoli (valgono in tutti i sistemi di riferimento)

• $\mathrm{div}\mathrm{grad}f=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f$ (Laplaciano)
• $\mathrm{rot}\mathrm{grad}f=\mathrm{\nabla }\times (\mathrm{\nabla }f)=0$
• $\mathrm{div}\mathrm{rot}𝐀=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$
• $\mathrm{rot}\mathrm{rot}𝐀=\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$
• ${\mathrm{\nabla }}^{2}fg=f{\mathrm{\nabla }}^{2}g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g{\mathrm{\nabla }}^{2}f$

Formula di Lagrange per il prodotto vettoriale: $𝐀\times (𝐁\times 𝐂)=𝐁(𝐀\cdot 𝐂)-𝐂(𝐀\cdot 𝐁)$

• $\mathrm{\nabla }\cdot (f𝐀)=f\mathrm{\nabla }\cdot 𝐀+𝐀\cdot \mathrm{\nabla }f$
• $\mathrm{\nabla }\times f𝐀=f\mathrm{\nabla }\times 𝐀-𝐀\times \mathrm{\nabla }f$
• $\mathrm{\nabla }(𝐀\cdot 𝐁)=(𝐀\cdot \mathrm{\nabla })𝐁+(𝐁\cdot \mathrm{\nabla })𝐀+𝐀\times (\mathrm{\nabla }\times 𝐁)+𝐁\times (\mathrm{\nabla }\times 𝐀),$

che insieme a $𝐀=𝐁=𝐯$ segue immediatamente Template:Senza fonte:

$(𝐯\cdot \mathrm{\nabla })𝐯=\mathrm{\nabla }\frac{{\Vert 𝐯\Vert }^2}{2}-𝐯\times (\mathrm{\nabla }\times 𝐯)$

• $\mathrm{\nabla }\times (𝐀\times 𝐁)=𝐀(\mathrm{\nabla }\cdot 𝐁)-𝐁(\mathrm{\nabla }\cdot 𝐀)+(𝐁\cdot \mathrm{\nabla })𝐀-(𝐀\cdot \mathrm{\nabla })𝐁$
• $\mathrm{\nabla }\cdot (𝐀\times 𝐁)=𝐁\cdot (\mathrm{\nabla }\times 𝐀)-𝐀\cdot (\mathrm{\nabla }\times 𝐁)$

Nota

• La funzione atan2(y,x) è usata al posto di $\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(y/x)$ per il suo dominio. La funzione $\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(y/x)$ ha immagine in $(-\pi /2,\pi /2),$ mentre $\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{2}(y,x)$ ha immagine in $(-\pi ,\pi ].$

Voci correlate

• Coordinate curvilinee
• Identità vettoriali