Differentiaalioperaattorit eri koordinaattijärjestelmissä

Kokeneet kirjoittajat eivät ole vielä tarkistaneet sivun nykyistä versiota, ja se voi poiketa merkittävästi 3.10.2020 tarkistetusta versiosta . tarkastukset vaativat 5 muokkausta .

Tässä on luettelo vektoridifferentiaalioperaattoreista eri koordinaattijärjestelmissä .

Yleinen lauseke

Yleinen lauseke operaattorille ∇ , joka vaikuttaa vektorikenttään A mielivaltaisessa ortogonaalisten koordinaattien järjestelmässä, voidaan kirjoittaa seuraavasti:

$\nabla \circ \mathbf {A} =\sum _{m}i_{m}\circ {\partial \mathbf {A} \over \partial r_{m}}=i_{1}\circ { \partial \mathbf {A} \over \partial r_{1}}+i_{2}\circ {\partial \mathbf {A} \over \partial r_{2}}+i_{3}\circ {\partial \mathbf {A} \over \partial r_{3}}$ ,

jossa " " on mikä tahansa kolmesta kuvakkeesta, jotka vastaavat operaattorin ∇ toimintaa: ${\näyttötyyli \circ }$

" " - gradientti;
"·" - ero;
"×" - roottori.

Tämän merkinnän elementit vastaavat sädevektorin elementtejä vastaavassa koordinaattijärjestelmässä: ${\displaystyle \partial r_{m))$

$d\mathbf {r} =\sum _{m}i_{m}\partial r_{m}=i_{1}\partial r_{1}+i_{2}\partial r_{2}+i_ {3}\osittainen r_{3}$

Toisin sanoen ensimmäinen toimenpide on ottaa osaderivaata suhteessa koko vektorin sädevektorin projektioon (ottaen huomioon yksikkövektorien derivaatat tietyssä koordinaatistossa) ja vasta sitten kertoa (yksinkertaista gradientti, skalaari divergentille ja vektori roottorille) suunnan yksikkövektorin . ${\partial \mathbf {A} \over \partial r_{m))$ $\mathbf {A}$ ${\partial \mathbf {A} \over \partial r_{m))$

Riittää, kun tietää ilmaukset:

sylinterimäisinä koordinaatteina: ja ; ${\displaystyle {\partial i_{\rho } \over \partial \varphi }=i_{\varphi ))$ ${\displaystyle {\partial i_{\varphi } \over \partial \varphi }=-i_{\rho ))$
pallokoordinaateissa: , , , ja . ${\displaystyle {\partial i_{r} \over \partial \theta }=i_{\theta ))$ ${\displaystyle {\partial i_{\theta } \over \partial \theta }=-i_{r))$ ${\partial i_{r} \over \partial \varphi }=i_{\varphi }\sin \theta$ ${\partial i_{\theta } \over \partial \varphi }=i_{\varphi }\cos \theta$ ${\partial i_{\varphi } \over \partial \varphi }=-i_{r}\sin \theta -i_{\theta }\cos \theta$

Esimerkiksi: alla olevassa taulukossa sylinterimäisten koordinaattien erot saadaan seuraavasti:

$\nabla \cdot \mathbf {A} =i_{\rho }\cdot {\partial \over \partial \rho }(i_{\rho }A_{\rho }+i_{\varphi }A_{\ varphi }+i_{z}A_{z})+{1 \over \rho }i_{\varphi }\cdot {\partial \over \partial \varphi }(i_{\rho }A_{\rho }+i_ {\varphi }A_{\varphi }+i_{z}A_{z})+i_{z}\cdot {\partial \over \partial z}(i_{\rho }A_{\rho }+i_{\ varphi }A_{\varphi }+i_{z}A_{z})=$

$={\partial A_{\rho } \over \partial \rho }+{\biggl (}{1 \over \rho }i_{\varphi }\cdot {\partial i_{\rho } \over \ osittainen \varphi }A_{\rho }{\biggr )}+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z }={\biggl (}{\partial A_{\rho } \over \partial \rho }+{A_{\rho } \over \rho }{\biggr )}+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}=$

$={1 \over \rho }{\partial (\rho A_{\rho }) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}$

Ohjauspöytä

Tässä käytetään standardia fyysistä merkintää. Pallomaisissa koordinaateissa θ tarkoittaa z -akselin ja pisteen sädevektorin välistä kulmaa, φ on kulma sädevektorin xy -tasolle projektion ja x - akselin välillä .

Hamiltonin operaattorin tallentaminen erilaisiin koordinaattijärjestelmiin

Operaattori	Suorakulmaiset koordinaatit ( x, y, z )	Sylinterimäiset koordinaatit ( ρ, φ, z )	Pallokoordinaatit ( r , θ, φ )	Paraboliset koordinaatit ( σ, τ, z )
Koordinaattien muunnoskaavat	${\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\varphi &=&\operaattorin nimi {arctg}(y/x)\\z&=&z \end{matrix}}$	${\begin{matrix}x&=&\rho \cos \varphi \\y&=&\rho \sin \varphi \\z&=&z\end{matrix))$	${\begin{matrix}x&=&r\sin \theta \cos \varphi \\y&=&r\sin \theta \sin \varphi \\z&=&r\cos \theta \end{matrix))$	${\begin{matrix}x&=&\sigma \tau \\y&=&{\frac {1}{2}}\left(\tau ^{{2}}-\sigma ^{{2}}\oikea )\\z&=&z\end{matriisi}}$
Koordinaattien muunnoskaavat	${\begin{matrix}r&=&{\sqrt {{{x}^{{2}}}+{{y}^{{2}}}+{{z}^{{2}}}}} \\\theta &=&\arccos \left(z/r\right)\\\varphi &=&\operaattorin nimi {arctg}(y/x)\\\end{matrix}}$	${\begin{matrix}r&=&{\sqrt {\rho ^{2}+z^{2}}}\\\theta &=&\operaattorin nimi {arctg}{(\rho /z)}\\\ varphi &=&\varphi \end{matrix}}$	${\begin{matrix}\rho &=&r\sin {\theta }\\\varphi &=&\varphi \\z&=&r\cos {\theta }\end{matrix))$	${\begin{matrix}\rho \cos \varphi &=&\sigma \tau \\\rho \sin \varphi &=&{\frac {1}{2}}\left(\tau ^{{2} }-\sigma ^{{2}}\oikea)\\z&=&z\end{matriisi}}$
Mielivaltaisen pisteen sädevektori	$x{\mathbf {{\hat x}}}+y{\mathbf {{\hat y}}}+z{\mathbf {{\hat z}}}$	$\rho {\boldsymbol {{\hat \rho }}}+z{\boldsymbol {{\hat z}}}$	$r{\boldsymbol {{\hat r))}$	${\frac {1}{2}}{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}\sigma {\boldsymbol {{\hat \sigma }}}+{\ frac {1}{2}}{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}\tau {\boldsymbol {{\hat \tau }}}+z{\mathbf {{\hattu z))}$
Yksikkövektoreiden kytkentä	${\begin{matrix}{\boldsymbol {{\hat \rho }}}&=&{\frac {x}{\rho }}{\mathbf {{\hat x}}}+{\frac {y} {\rho }}{\mathbf {{\hat y}}}\\{\boldsymbol {{\hat \varphi }}}&=&-{\frac {y}{\rho }}{\mathbf {{ \hat x}}}+{\frac {x}{\rho }}{\mathbf {{\hat y}}}\\{\mathbf {{\hat z}}}&=&{\mathbf {{ \hat z}}}\end{matrix}}$	${\begin{matrix}{\mathbf {{\hat x}}}&=&\cos \varphi {\boldsymbol {{\hat \rho }}}-\sin \varphi {\boldsymbol {{\hat \varphi ))}\\{\mathbf {{\hat y}}}&=&\sin \varphi {\boldsymbol {{\hat \rho }}}+\cos \varphi {\boldsymbol {{\hat \varphi } ))\\{\mathbf {{\hat z}}}&=&{\mathbf {{\hat z}}}\end{matrix}}$	${\begin{matrix}{\mathbf {{\hat x))}&=&\sin \theta \cos \varphi {\boldsymbol {{\hat r))}+\cos \theta \cos \varphi {\ lihavoitu symboli {{\hat \theta }}}-\sin \varphi {\boldsymbol {{\hat \varphi }}}\\{\mathbf {{\hat y}}}&=&\sin \theta \sin \ varphi {\boldsymbol {{\hat r}}}+\cos \theta \sin \varphi {\boldsymbol {{\hat \theta }}}+\cos \varphi {\boldsymbol {{\hat \varphi }}} \\{\mathbf {{\hat z}}}&=&\cos \theta {\boldsymbol {{\hat r}}}-\sin \theta {\boldsymbol {{\hat \theta }}}\\ \end{matrix}}$	${\begin{matrix}{\boldsymbol {{\hat \sigma }}}&=&{\frac {\tau }{{\sqrt {\tau ^{2}+\sigma ^{2))))} {\mathbf {{\hat x}}}-{\frac {\sigma }{{\sqrt {\tau ^{2}+\sigma ^{2}}}}{\mathbf {{\hat y} ))\\{\boldsymbol {{\hat \tau }}}&=&{\frac {\sigma }({\sqrt {\tau ^{2}+\sigma ^{2))))}{\ mathbf {{\hat x}}}+{\frac {\tau }{{\sqrt {\tau ^{2}+\sigma ^{2}}}}}{\mathbf {{\hat y}}} \\{\mathbf {{\hat z}}}&=&{\mathbf {{\hat z}}}\end{matrix}}$
Yksikkövektoreiden kytkentä	${\begin{matrix}{\mathbf {{\hat r}}}&=&{\frac {x{\mathbf {{\hat x}}}+y{\mathbf {{\hat y}}}+ z{\mathbf {{\hat z}}}}{r}}\\{\boldsymbol {{\hat \theta }}}&=&{\frac {xz{\mathbf {{\hat x}}} +yz{\mathbf {{\hat y}}}-\rho ^{2}{\mathbf {{\hat z}}}}{r\rho }}\\{\boldsymbol {{\hat \varphi } }}&=&{\frac {-y{\mathbf {{\hat x}}}+x{\mathbf {{\hat y}}}}{\rho }}\end{matrix}}$	${\begin{matrix}{\mathbf {{\hat r}}}&=&{\frac {\rho }{r}}{\boldsymbol {{\hat \rho }}}+{\frac {z} {r}}{\mathbf {{\hat z}}}\\{\boldsymbol {{\hat \theta }}}&=&{\frac {z}{r}}{\boldsymbol {{\hat \ rho ))}-{\frac {\rho }{r)){\mathbf ({\hat z))}\\{\boldsymbol ({\hat \varphi ))}&=&{\boldsymbol {{\ hat\varphi}}}\end{matrix}}$	${\begin{matrix}{\boldsymbol {{\hat \rho }}}&=&\sin \theta {\mathbf {{\hat r}}}+\cos \theta {\boldsymbol {{\hat \theta ))}\\{\boldsymbol {{\hat \varphi }}}&=&{\boldsymbol {{\hat \varphi }}}\\{\mathbf {{\hat z}}}&=&\cos \theta {\mathbf {{\hat r}}}-\sin \theta {\boldsymbol {{\hat \theta }}}\\\end{matrix}}$	.
vektorikenttä $\mathbf {A}$	$A_{x}{\mathbf {{\hat x}}}+A_{y}{\mathbf {{\hat y}}}+A_{z}{\mathbf {{\hat z}}}$	$A_{\rho }{\boldsymbol {{\hat \rho }}}+A_{\varphi }{\boldsymbol {{\hat \varphi }}}+A_{z}{\boldsymbol {{\hat z}} }$	$A_{r}{\boldsymbol {{\hat r}}}+A_{\theta }{\boldsymbol {{\hat \theta }}}+A_{\varphi }{\boldsymbol {{\hat \varphi }} }$	$A_{\sigma }{\boldsymbol {{\hat \sigma }}}+A_{\tau }{\boldsymbol {{\hat \tau }}}+A_{\varphi }{\boldsymbol {{\hat z} }}$
Kaltevuus $\nabla f$	${\partial f \over \partial x}{\mathbf {{\hat x}}}+{\partial f \over \partial y}{\mathbf {{\hat y}}}+{\partial f \over \partial z}{\mathbf {{\hat z))}$	${\partial f \over \partial \rho }{\boldsymbol {{\hat \rho }}}+{1 \over \rho }{\partial f \over \partial \varphi }{\boldsymbol {{\hat \ varphi ))}+{\partial f \over \partial z}{\boldsymbol {{\hat z))}$	${\partial f \over \partial r}{\boldsymbol {{\hat r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {{\hat \theta }} }+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\boldsymbol {{\hat \varphi }}}$	${\frac {1}{{\sqrt {\sigma ^{{2}}+\tau ^{{2)))))){\partial f \over \partial \sigma }{\boldsymbol {{\hattu \sigma }}}+{\frac {1}{{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}}{\partial f \over \partial \tau }{ \boldsymbol {{\hat \tau }}}+{\partial f \over \partial z}{\boldsymbol {{\hat z}}}$
Eroaminen $\nabla \cdot {\mathbf {A}}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\varphi } \over \partial \varphi }$	${\frac {1}{\sigma ^{{2}}+\tau ^{{2}}}}{\partial A_{\sigma } \over \partial \sigma }+{\frac {1}{\ sigma ^{{2}}+\tau ^{{2}}}}{\partial A_{\tau } \over \partial \tau }+{\partial A_{z} \over \partial z}$
Roottori $\nabla \times {\mathbf {A}}$	${\begin{matrix}\left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right){\mathbf {{\hat x))} &+\\\left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right){\mathbf {{\hat y))}&+ \\\left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right){\mathbf ({\hat z))}&\ \end {matriisi}}$	${\begin{matrix}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi } }{\partial z}}\right){\boldsymbol {{\hat \rho }}}&+\\\left({\frac {\partial A_{\rho }}{\partial z}}-{\ frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {{\hat \varphi }}}&+\\{\frac {1}{\rho }}\left({ \frac {\partial (\rho A_{\varphi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\boldsymbol {{ \hat z}}}&\ \end{matrix}}$	${\begin{matrix}{1 \over r\sin \theta }\left({\partial \over \partial \theta }\left(A_{\varphi }\sin \theta \right)-{\partial A_{ \theta } \over \partial \varphi }\right){\boldsymbol {{\hat r))}&+\\{1 \over r}\left({1 \over \sin \theta }{\partial A_ {r} \over \partial \varphi }-{\partial \over \partial r}\left(rA_{\varphi }\right)\right){\boldsymbol {{\hat \theta }}}&+\\ {1 \over r}\left({\partial \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {{\hattu \varphi }}}&\ \end{matrix}}$	${\begin{matrix}\left({\frac {1}{{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}}{\partial A_{z} \over \partial \tau }-{\partial A_{\tau } \over \partial z}\right){\boldsymbol {{\hat \sigma }}}&-\\\left({\frac {1}{{ \sqrt {\sigma ^{{2}}+\tau ^{{2}}}}}}{\partial A_{z} \over \partial \sigma }-{\partial A_{\sigma } \over \ osittainen z}\oikea){\boldsymbol {{\hat \tau }}}&+\\{\frac {1}({\sqrt {\sigma ^{{2}}+\tau ^{{2}} }}}}\left({\partial \left(sA_{\varphi }\right) \over \partial s}-{\partial A_{s} \over \partial \varphi }\right){\boldsymbol {{ \hat z}}}&\ \end{matrix}}$
Laplacen operaattori $\Delta f=\nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2)){\ osittainen ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \ r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\ !{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2))$	${\frac {1}{\sigma ^{{2}}+\tau ^{{2}}}}\left({\frac {\partial ^{{2}}f}{\partial \sigma ^{ {2))))+{\frac {\partial ^{{2}}f}{\partial \tau ^{{2}}}}\right)+{\frac {\partial ^{{2}} f}{\osittainen z^{{2))))$
Laplace-vektorioperaattori $\Delta \mathbf {A}$	${\begin{matrix}\Delta A_{x}\mathbf {\hat {x)) +\Delta A_{y}\mathbf {\hat {y)) +\Delta A_{z}\mathbf { \hat {z}} =\\{\biggl (}{\partial ^{2}A_{x} \over \partial x^{2}}+{\partial ^{2}A_{x} \over \ osittainen y^{2}}+{\osittais ^{2}A_{x} \over \partial z^{2}}{\biggr )}\mathbf {\hat {x}} +\\{\biggl ( }{\partial ^{2}A_{y} \over \partial x^{2}}+{\partial ^{2}A_{y} \over \partial y^{2}}+{\partial ^{ 2}A_{y} \over \partial z^{2}}{\biggr )}\mathbf {\hat {y}} +\\{\biggl (}{\osittainen ^{2}A_{z} \ yli \partial x^{2}}+{\partial ^{2}A_{z} \over \partial y^{2}}+{\partial ^{2}A_{z} \over \partial z^{ 2}}{\biggr )}\mathbf {\hat {z}} \ \end{matrix}}$	${\begin{matrix}\left(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\ varphi } \over \partial \varphi }\right){\boldsymbol {{\hat \rho }}}&+\\\left(\Delta A_{\varphi }-{A_{\varphi } \over \rho ^ {2))+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \varphi }\right){\boldsymbol {{\hat \varphi }}}&+\\ \left(\Delta A_{z}\right){\boldsymbol {{\hat z}}}&\ \end{matrix}}$	${\begin{matrix}\left(\Delta A_{r}-{2A_{r} \over r^{2}}-{2 \over r^{2}\sin \theta }{\partial \left( A_{\theta }\sin \theta \right) \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\varphi } \over \partial \varphi }\ oikea){\boldsymbol {{\hat r}}}&+\\\vasen(\Delta A_{\theta }-{A_{\theta } \over r^{2}\sin ^{2}\theta } +{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\sin ^{2}\theta }{\ osittainen A_{\varphi } \over \partial \varphi }\right){\boldsymbol {{\hat \theta ))}&+\\\left(\Delta A_{\varphi }-{A_{\varphi } \ yli r^{2}\sin ^{2}\theta }+{2 \over r^{2}\sin \theta }{\partial A_{r} \over \partial \varphi }+{2\cos \ theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \varphi }\right){\boldsymbol {{\hat \varphi }}}&\end {matriisi}}$	?
Pituuselementti	$d{\mathbf {l}}=dx{\mathbf {{\hat x}}}+dy{\mathbf {{\hat y}}}+dz{\mathbf {{\hat z}}}$	$d{\mathbf {l}}=d\rho {\boldsymbol {{\hat \rho }}}+\rho d\varphi {\boldsymbol {{\hat \varphi }}}+dz{\boldsymbol {{\ hattu z}}}$	$d{\mathbf {l}}=dr{\mathbf {{\hat r}}}+rd\theta {\boldsymbol {{\hat \theta }}}+r\sin \theta d\varphi {\boldsymbol { {\hattu \varphi ))}$	$d{\mathbf {l}}={\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}d\sigma {\boldsymbol {{\hat \sigma }}}+{\ sqrt {\sigma ^{{2}}+\tau ^{{2}}}}d\tau {\boldsymbol {{\hat \tau }}}+dz{\boldsymbol {{\hat z}}}$
Orientoitu alueelementti	${\begin{matrix}d{\mathbf {S}}=&dy\,dz\,{\mathbf {{\hat x}}}+\\&dx\,dz\,{\mathbf {{\hat y} ))+\\&dx\,dy\,{\mathbf {{\hat z))}\end{matriisi))$	${\begin{matrix}d{\mathbf {S}}=&\rho \,d\varphi \,dz\,{\boldsymbol {{\hat \rho }}}+\\&d\rho \,dz\ ,{\boldsymbol {{\hat \varphi }}}+\\&\rho \,d\rho d\varphi \,{\mathbf {{\hat z}}}\end{matrix}}$	${\begin{matrix}d{\mathbf {S}}=&r^{2}\sin \theta \,d\theta \,d\varphi \,{\mathbf {{\hat r}}}+\\ &r\sin \theta \,dr\,d\varphi \,{\boldsymbol {{\hat \theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {{\hat \varphi }}}\end{matrix}}$	${\begin{matrix}d\mathbf {S} =&{\sqrt {\sigma ^{2}+\tau ^{2))}\,d\tau \,dz\,{\boldsymbol { \hat {\sigma }}}+\\&{\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma \,dz\,{\boldsymbol {\hat {\tau }} }+\\&\sigma ^{2}+\tau ^{2}d\sigma \,d\tau \,\mathbf {\hat {z)) \end{matrix))$
Äänenvoimakkuuden elementti	$dV=dx\,dy\,dz$	$dV=\rho \,d\rho \,d\varphi \,dz$	$dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi$	$dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz$

Jotkut ominaisuudet

Toisen asteen operaattoreiden lausekkeet:

${\mathrm {div\ grad}}\;f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ ( Laplace - operaattori )
${\mathrm {rot\ grad}}\;f=\nabla \times (\nabla f)=0$
$\mathrm {grad\ div} \;\mathbf {A} =\nabla \cdot (\nabla \cdot \mathbf {A} )$
${\mathrm {div\ rot}}\;{\mathbf {A}}=\nabla \cdot (\nabla \times {\mathbf {A}})=0$
$\mathrm {rot\ rot} \;\mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\Delta \mathbf {A}$ (käyttäen Lagrangen kaavaa kaksoisristituotteelle )
$\Delta (fg)=\mathrm {div\ grad} \ (fg)=\mathrm {div} \ (f\mathrm {grad} \ g+g\mathrm {grad} \ f)=f\Delta g+2\mathrm {grad} \ f\cdot \mathrm {grad} \ g+g\Delta f$

Katso myös

D'Alembert-operaattori
Ortogonaaliset koordinaatit
Käyräviivaiset koordinaatit
Koordinaattimenetelmä
Vektorikentät lieriömäisissä ja pallomaisissa koordinaattijärjestelmissä

Differentiaalilaskenta

Main

yksityiset näkymät

Differentiaalioperaattorit
( eri koordinaateissa )

Ensimmäinen tilaus	Nabla operaattori Kaltevuus Eroaminen Roottori Helmholtzian
toinen tilaus	Elliptinen operaattori Laplacen operaattori Laplace-vektorioperaattori D'Alembert-operaattori Laplace-Beltrami operaattori
korkeammat tilaukset	Hypoelliptinen operaattori

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