Differentiella operatörer i olika koordinatsystem

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Här är en lista över vektordifferentialoperatorer i olika koordinatsystem .

Allmänt uttryck

Det allmänna uttrycket för operatorn ∇ som verkar på vektorfältet A i ett godtyckligt system av ortogonala koordinater kan skrivas på följande sätt:

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

där " " är någon av de tre ikonerna som motsvarar operatörens åtgärd ∇: $\circ$

" " - gradient;
"·" - divergens;
"×" - rotor.

Elementen i denna post motsvarar elementen i radievektorn i motsvarande koordinatsystem: ${\displaystyle \partial r_{m))$

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

Med andra ord, den första åtgärden är att ta den partiella derivatan med avseende på projektionen av radievektorn för hela vektorn (med hänsyn till derivatorna av enhetsvektorerna i det givna koordinatsystemet), och först därefter multiplicera (enkelt för gradienten, skalär för divergensen och vektorn för rotorn) för enhetsvektorn för riktningen med . ${\partial \mathbf {A} \over \partial r_{m))$ $\mathbf {A}$ ${\partial \mathbf {A} \over \partial r_{m))$

Det räcker att känna till uttrycken:

i cylindriska koordinater: och ; ${\displaystyle {\partial i_{\rho } \over \partial \varphi }=i_{\varphi ))$ ${\displaystyle {\partial i_{\varphi } \over \partial \varphi }=-i_{\rho ))$
i sfäriska koordinater: , , , och . ${\displaystyle {\partial i_{r} \over \partial \theta }=i_{\theta ))$ ${\displaystyle {\partial i_{\theta } \over \partial \theta }=-i_{r))$ ${\partial i_{r} \over \partial \varphi }=i_{\varphi }\sin \theta$ ${\partial i_{\theta } \over \partial \varphi }=i_{\varphi }\cos \theta$ ${\partial i_{\varphi } \over \partial \varphi }=-i_{r}\sin \theta -i_{\theta }\cos \theta$

Till exempel: i tabellen nedan erhålls registreringen av divergens i cylindriska koordinater enligt följande:

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

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

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

Operatörstabell

Standard fysisk notation används här. För sfäriska koordinater betecknar θ vinkeln mellan z -axeln och punktens radievektor , φ är vinkeln mellan projektionen av radievektorn på xy -planet och x -axeln .

Registrering av Hamilton-operatören i olika koordinatsystem

Operatör	Rektangulära koordinater ( x, y, z )	Cylindriska koordinater ( ρ, φ, z )	Sfäriska koordinater ( r , θ, φ )	Paraboliska koordinater ( σ, τ, z )
Koordinattransformationsformler	${\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\varphi &=&\operatörsnamn {arctg}(y/x)\\z&=&z \end{matris}}$	${\begin{matrix}x&=&\rho \cos \varphi \\y&=&\rho \sin \varphi \\z&=&z\end{matris))$	${\begin{matrix}x&=&r\sin \theta \cos \varphi \\y&=&r\sin \theta \sin \varphi \\z&=&r\cos \theta \end{matris))$	${\begin{matrix}x&=&\sigma \tau \\y&=&{\frac {1}{2}}\left(\tau ^{{2}}-\sigma ^{{2}}\right )\\z&=&z\end{matris}}$
Koordinattransformationsformler	${\begin{matrix}r&=&{\sqrt {{{x}^{{2}}}+{{y}^{{2}}}+{{z}^{{2}}}}} \\\theta &=&\arccos \left(z/r\right)\\\varphi &=&\operatörsnamn {arctg}(y/x)\\\end{matris}}$	${\begin{matrix}r&=&{\sqrt {\rho ^{2}+z^{2}}}\\\theta &=&\operatörsnamn {arctg}{(\rho /z)}\\\ varphi &=&\varphi \end{matris}}$	${\begin{matrix}\rho &=&r\sin {\theta }\\\varphi &=&\varphi \\z&=&r\cos {\theta }\end{matris))$	${\begin{matrix}\rho \cos \varphi &=&\sigma \tau \\\rho \sin \varphi &=&{\frac {1}{2}}\left(\tau ^{{2} }-\sigma ^{{2}}\right)\\z&=&z\end{matris}}$
Radievektor för en godtycklig punkt	$x{\mathbf {{\hat x}}}+y{\mathbf {{\hat y}}}+z{\mathbf {{\hat z}}}$	$\rho {\boldsymbol {{\hat \rho }}}+z{\boldsymbol {{\hat z}}}$	$r{\boldsymbol {{\hat r))}$	${\frac {1}{2}}{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}\sigma {\boldsymbol {{\hat \sigma }}}+{\ frac {1}{2}}{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}\tau {\boldsymbol {{\hat \tau }}}+z{\mathbf {{\hat z))}$
Anslutning av enhetsvektorer	${\begin{matrix}{\boldsymbol {{\hat \rho }}}&=&{\frac {x}{\rho }}{\mathbf {{\hat x}}}+{\frac {y} {\rho }}{\mathbf {{\hat y}}}\\{\boldsymbol {{\hat \varphi }}}&=&-{\frac {y}{\rho }}{\mathbf {{ \hat x}}}+{\frac {x}{\rho }}{\mathbf {{\hat y}}}\\{\mathbf {{\hat z}}}&=&{\mathbf {{ \hat z}}}\end{matris}}$	${\begin{matrix}{\mathbf {{\hat x}}}&=&\cos \varphi {\boldsymbol {{\hat \rho }}}-\sin \varphi {\boldsymbol {{\hat \varphi ))}\\{\mathbf {{\hat y}}}&=&\sin \varphi {\boldsymbol {{\hat \rho }}}+\cos \varphi {\boldsymbol {{\hat \varphi } ))\\{\mathbf {{\hat z}}}&=&{\mathbf {{\hat z}}}\end{matris}}$	${\begin{matrix}{\mathbf {{\hat x))}&=&\sin \theta \cos \varphi {\boldsymbol {{\hat r))}+\cos \theta \cos \varphi {\ fetsymbol {{\hat \theta }}}-\sin \varphi {\boldsymbol {{\hat \varphi }}}\\{\mathbf {{\hat y}}}&=&\sin \theta \sin \ varphi {\boldsymbol {{\hat r}}}+\cos \theta \sin \varphi {\boldsymbol {{\hat \theta }}}+\cos \varphi {\boldsymbol {{\hat \varphi }}} \\{\mathbf {{\hat z}}}&=&\cos \theta {\boldsymbol {{\hat r}}}-\sin \theta {\boldsymbol {{\hat \theta }}}\\ \end{matris}}$	${\begin{matrix}{\boldsymbol {{\hat \sigma }}}&=&{\frac {\tau }{{\sqrt {\tau ^{2}+\sigma ^{2))))} {\mathbf {{\hat x}}}-{\frac {\sigma }{{\sqrt {\tau ^{2}+\sigma ^{2}}}}}{\mathbf {{\hat y} ))\\{\boldsymbol {{\hat \tau }}}&=&{\frac {\sigma }({\sqrt {\tau ^{2}+\sigma ^{2))))}{\ mathbf {{\hat x}}}+{\frac {\tau }{{\sqrt {\tau ^{2}+\sigma ^{2}}}}}{\mathbf {{\hat y}}} \\{\mathbf {{\hat z}}}&=&{\mathbf {{\hat z}}}\end{matris}}$
Anslutning av enhetsvektorer	${\begin{matrix}{\mathbf {{\hat r}}}&=&{\frac {x{\mathbf {{\hat x}}}+y{\mathbf {{\hat y}}}+ z{\mathbf {{\hat z}}}}{r}}\\{\boldsymbol {{\hat \theta }}}&=&{\frac {xz{\mathbf {{\hat x}}} +yz{\mathbf {{\hat y}}}-\rho ^{2}{\mathbf {{\hat z}}}}{r\rho }}\\{\boldsymbol {{\hat \varphi } }}&=&{\frac {-y{\mathbf {{\hat x}}}+x{\mathbf {{\hat y}}}}{\rho }}\end{matrix}}$	${\begin{matrix}{\mathbf {{\hat r}}}&=&{\frac {\rho }{r}}{\boldsymbol {{\hat \rho }}}+{\frac {z} {r}}{\mathbf {{\hat z}}}\\{\boldsymbol {{\hat \theta }}}&=&{\frac {z}{r}}{\boldsymbol {{\hat \ rho ))}-{\frac {\rho }{r)){\mathbf ({\hat z))}\\{\boldsymbol ({\hat \varphi ))}&=&{\boldsymbol {{\ hat\varphi}}}\end{matris}}$	${\begin{matrix}{\boldsymbol {{\hat \rho }}}&=&\sin \theta {\mathbf {{\hat r}}}+\cos \theta {\boldsymbol {{\hat \theta ))}\\{\boldsymbol {{\hat \varphi }}}&=&{\boldsymbol {{\hat \varphi }}}\\{\mathbf {{\hat z}}}&=&\cos \theta {\mathbf {{\hat r}}}-\sin \theta {\boldsymbol {{\hat \theta }}}\\\end{matris}}$	.
vektor fält $\mathbf {A}$	$A_{x}{\mathbf {{\hat x}}}+A_{y}{\mathbf {{\hat y}}}+A_{z}{\mathbf {{\hat z}}}$	$A_{\rho }{\boldsymbol {{\hat \rho }}}+A_{\varphi }{\boldsymbol {{\hat \varphi }}}+A_{z}{\boldsymbol {{\hat z}} }$	$A_{r}{\boldsymbol {{\hat r}}}+A_{\theta }{\boldsymbol {{\hat \theta }}}+A_{\varphi }{\boldsymbol {{\hat \varphi }} }$	$A_{\sigma }{\boldsymbol {{\hat \sigma }}}+A_{\tau }{\boldsymbol {{\hat \tau }}}+A_{\varphi }{\boldsymbol {{\hat z} }}$
Lutning $\nabla f$	${\partial f \over \partial x}{\mathbf {{\hat x}}}+{\partial f \over \partial y}{\mathbf {{\hat y}}}+{\partial f \over \partial z}{\mathbf {{\hat z))}$	${\partial f \over \partial \rho }{\boldsymbol {{\hat \rho }}}+{1 \over \rho }{\partial f \over \partial \varphi }{\boldsymbol {{\hat \ varphi ))}+{\partial f \over \partial z}{\boldsymbol {{\hat z))}$	${\partial f \over \partial r}{\boldsymbol {{\hat r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {{\hat \theta }} }+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\boldsymbol {{\hat \varphi }}}$	${\frac {1}{{\sqrt {\sigma ^{{2}}+\tau ^{{2)))))){\partial f \over \partial \sigma }{\boldsymbol {{\hat \sigma }}}+{\frac {1}{{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}}}{\partial f \over \partial \tau }{ \boldsymbol {{\hat \tau }}}+{\partial f \over \partial z}{\boldsymbol {{\hat z}}}$
Divergens $\nabla \cdot {\mathbf {A}}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\varphi } \over \partial \varphi }$	${\frac {1}{\sigma ^{{2}}+\tau ^{{2}}}}{\partial A_{\sigma } \over \partial \sigma }+{\frac {1}{\ sigma ^{{2}}+\tau ^{{2}}}}{\partial A_{\tau } \over \partial \tau }+{\partial A_{z} \over \partial z}$
Rotor $\nabla \times {\mathbf {A}}$	${\begin{matrix}\left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right){\mathbf {{\hat x))} &+\\\left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right){\mathbf {{\hat y))}&+ \\\left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right){\mathbf ({\hat z))}&\ \end {matris}}$	${\begin{matrix}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi } }{\partial z}}\right){\boldsymbol {{\hat \rho }}}&+\\\left({\frac {\partial A_{\rho }}{\partial z}}-{\ frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {{\hat \varphi }}}&+\\{\frac {1}{\rho }}\left({ \frac {\partial (\rho A_{\varphi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\boldsymbol {{ \hat z}}}&\ \end{matris}}$	${\begin{matris}{1 \over r\sin \theta }\left({\partial \over \partial \theta}\left(A_{\varphi}\sin \theta \right)-{\partial A_{ \theta } \over \partial \varphi }\right){\boldsymbol {{\hat r))}&+\\{1 \over r}\left({1 \over \sin \theta }{\partial A_ {r} \over \partial \varphi }-{\partial \over \partial r}\left(rA_{\varphi }\right)\right){\boldsymbol {{\hat \theta }}}&+\\ {1 \over r}\left({\partial \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {{\hat \varphi }}}&\ \end{matris}}$	${\begin{matrix}\left({\frac {1}{{\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}}}{\partial A_{z} \över \partial \tau }-{\partial A_{\tau } \over \partial z}\right){\boldsymbol {{\hat \sigma }}}&-\\\left({\frac {1}{{ \sqrt {\sigma ^{{2}}+\tau ^{{2}}}}}}{\partial A_{z} \over \partial \sigma }-{\partial A_{\sigma } \over \ partiell z}\right){\boldsymbol {{\hat \tau }}}&+\\{\frac {1}({\sqrt {\sigma ^{{2}}+\tau ^{{2}} }}}}\left({\partial \left(sA_{\varphi }\right) \over \partial s}-{\partial A_{s} \over \partial \varphi }\right){\boldsymbol {{ \hat z}}}&\ \end{matris}}$
Laplace-operatör $\Delta f=\nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2)){\ partiell ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \ över r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta}\right)\!+\ !{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2))$	${\frac {1}{\sigma ^{{2}}+\tau ^{{2}}}}\left({\frac {\partial ^{{2}}f}{\partial \sigma ^{ {2))))+{\frac {\partial ^{{2}}f}{\partial \tau ^{{2}}}}\right)+{\frac {\partial ^{{2}} f}{\partial z^{{2))))$
Laplace vektor operatör $\Delta \mathbf {A}$	${\begin{matrix}\Delta A_{x}\mathbf {\hat {x)) +\Delta A_{y}\mathbf {\hat {y)) +\Delta A_{z}\mathbf { \hat {z}} =\\{\biggl (}{\partial ^{2}A_{x} \over \partial x^{2}}+{\partial ^{2}A_{x} \over \ partiell y^{2}}+{\partial ^{2}A_{x} \over \partial z^{2}}{\biggr )}\mathbf {\hat {x}} +\\{\biggl ( }{\partial ^{2}A_{y} \over \partial x^{2}}+{\partial ^{2}A_{y} \over \partial y^{2}}+{\partial ^{ 2}A_{y} \over \partial z^{2}}{\biggr )}\mathbf {\hat {y}} +\\{\biggl (}{\partial ^{2}A_{z} \ över \partial x^{2}}+{\partial ^{2}A_{z} \over \partial y^{2}}+{\partial ^{2}A_{z} \over \partial z^{ 2}}{\biggr )}\mathbf {\hat {z}} \ \end{matris}}$	${\begin{matrix}\left(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\ varphi } \over \partial \varphi }\right){\boldsymbol {{\hat \rho }}}&+\\\left(\Delta A_{\varphi }-{A_{\varphi } \over \rho ^ {2))+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \varphi }\right){\boldsymbol {{\hat \varphi }}}&+\\ \left(\Delta A_{z}\right){\boldsymbol {{\hat z}}}&\ \end{matris}}$	${\begin{matrix}\left(\Delta A_{r}-{2A_{r} \over r^{2}}-{2 \over r^{2}\sin \theta }{\partial \left( A_{\theta }\sin \theta \right) \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\varphi } \over \partial \varphi }\ höger){\boldsymbol {{\hat r}}}&+\\\left(\Delta A_{\theta }-{A_{\theta } \över r^{2}\sin ^{2}\theta } +{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\sin ^{2}\theta }{\ partiell A_{\varphi } \over \partial \varphi }\right){\boldsymbol {{\hat \theta ))}&+\\\left(\Delta A_{\varphi }-{A_{\varphi } \ över r^{2}\sin ^{2}\theta }+{2 \över r^{2}\sin \theta }{\partial A_{r} \over \partial \varphi }+{2\cos \ theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \varphi }\right){\boldsymbol {{\hat \varphi }}}&\end {matris}}$	?
Längdelement	$d{\mathbf {l}}=dx{\mathbf {{\hat x}}}+dy{\mathbf {{\hat y}}}+dz{\mathbf {{\hat z}}}$	$d{\mathbf {l}}=d\rho {\boldsymbol {{\hat \rho }}}+\rho d\varphi {\boldsymbol {{\hat \varphi }}}+dz{\boldsymbol {{\ hatt z}}}$	$d{\mathbf {l}}=dr{\mathbf {{\hat r}}}+rd\theta {\boldsymbol {{\hat \theta }}}+r\sin \theta d\varphi {\boldsymbol { {\hat \varphi ))}$	$d{\mathbf {l}}={\sqrt {\sigma ^{{2}}+\tau ^{{2}}}}d\sigma {\boldsymbol {{\hat \sigma }}}+{\ sqrt {\sigma ^{{2}}+\tau ^{{2}}}}d\tau {\boldsymbol {{\hat \tau }}}+dz{\boldsymbol {{\hat z}}}$
Orienterat områdeselement	${\begin{matrix}d{\mathbf {S}}=&dy\,dz\,{\mathbf {{\hat x}}}+\\&dx\,dz\,{\mathbf {{\hat y} ))+\\&dx\,dy\,{\mathbf {{\hat z))}\end{matris))$	${\begin{matrix}d{\mathbf {S}}=&\rho \,d\varphi \,dz\,{\boldsymbol {{\hat \rho }}}+\\&d\rho \,dz\ ,{\boldsymbol {{\hat \varphi }}}+\\&\rho \,d\rho d\varphi \,{\mathbf {{\hat z}}}\end{matris}}$	${\begin{matrix}d{\mathbf {S}}=&r^{2}\sin \theta \,d\theta \,d\varphi \,{\mathbf {{\hat r}}}+\\ &r\sin \theta \,dr\,d\varphi \,{\boldsymbol {{\hat \theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {{\hat \varphi }}}\end{matris}}$	${\begin{matrix}d\mathbf {S} =&{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz\,{\boldsymbol { \hat {\sigma }}}+\\&{\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma \,dz\,{\boldsymbol {\hat {\tau }} }+\\&\sigma ^{2}+\tau ^{2}d\sigma \,d\tau \,\mathbf {\hat {z)) \end{matris))$
Volymelement	$dV=dx\,dy\,dz$	$dV=\rho \,d\rho \,d\varphi \,dz$	$dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi$	$dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz$

Vissa egenskaper

Uttryck för andra ordningens operatorer:

${\mathrm {div\ grad}}\;f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ ( Laplace operatör )
${\mathrm {rot\ grad}}\;f=\nabla \times (\nabla f)=0$
$\mathrm {grad\ div} \;\mathbf {A} =\nabla \cdot (\nabla \cdot \mathbf {A} )$
${\mathrm {div\ rot}}\;{\mathbf {A}}=\nabla \cdot (\nabla \times {\mathbf {A}})=0$
$\mathrm {rot\ rot} \;\mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\Delta \mathbf {A}$ (med Lagranges formel för dubbelkorsprodukten )
$\Delta (fg)=\mathrm {div\ grad} \ (fg)=\mathrm {div} \ (f\mathrm {grad} \ g+g\mathrm {grad} \ f)=f\Delta g+2\mathrm {grad} \ f\cdot \mathrm {grad} \ g+g\Delta f$

Se även

D'Alembert operatör
Ortogonala koordinater
Krökta koordinater
Koordinatmetod
Vektorfält i cylindriska och sfäriska koordinatsystem

Differentialkalkyl

Main

privata vyer

Differentialoperatorer
( i olika koordinater )

Första beställning	Nabla-operatör Lutning Divergens Rotor Helmholtzian
andra beställning	Elliptisk operatör Laplace-operatör Laplace vektor operatör D'Alembert operatör Laplace-Beltrami operatör
högre order	Hypoelliptisk operatör

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