Оператор набла в различных системах координат

Здесь приведён список векторных дифференциальных операторов в некоторых системах координат.

Таблица операторов

Здесь используются стандартные физические обозначения. Для сферических координат, $\theta$ обозначает угол между осью z и радиус-вектором точки. $\phi$ — угол между проекцией радиус-вектора на плоскость x-y и осью x.

Запись оператора Гамильтона в различных системах координат

Оператор	Декартовы координаты (x, y, z)	Цилиндрические координаты (ρ,φ,z)	Сферические координаты (r,θ,φ)	Параболические координаты (σ,τ,z)
Формулы преобразования координат	${\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\operatorname {arctg} (y/x)\\z&=&z\end{matrix}}$	${\begin{matrix}x&=&\rho \cos \phi \\y&=&\rho \sin \phi \\z&=&z\end{matrix}}$	${\begin{matrix}x&=&r\sin \theta \cos \phi \\y&=&r\sin \theta \sin \phi \\z&=&r\cos \theta \end{matrix}}$	${\begin{matrix}x&=&\sigma \tau \\y&=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}$
	${\begin{matrix}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\operatorname {arctg} {\left({\frac {\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\right)}\\\phi &=&\operatorname {arctg} (y/x)\\\end{matrix}}$	${\begin{matrix}r&=&{\sqrt {\rho ^{2}+z^{2}}}\\\theta &=&\operatorname {arctg} {(\rho /z)}\\\phi &=&\phi \end{matrix}}$	${\begin{matrix}\rho &=&r\sin(\theta )\\\phi &=&\phi \\z&=&r\cos(\theta )\end{matrix}}$	${\begin{matrix}\rho \cos \phi &=&\sigma \tau \\\rho \sin \phi &=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}$
Радиус-вектор произвольной точки	$x\mathbf {\hat {x}} +y\mathbf {\hat {y}} +z\mathbf {\hat {z}}$	$\rho {\boldsymbol {\hat {\rho }}}+z{\boldsymbol {\hat {z}}}$	$r{\boldsymbol {\hat {r}}}$	?
Связь единичных векторов	${\begin{matrix}{\boldsymbol {\hat {\rho }}}&=&{\frac {x}{\rho }}\mathbf {\hat {x}} +{\frac {y}{\rho }}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\phi }}}&=&-{\frac {y}{\rho }}\mathbf {\hat {x}} +{\frac {x}{\rho }}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}\mathbf {\hat {x}} &=&\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=&\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}\mathbf {\hat {x}} &=&\sin \theta \cos \phi {\boldsymbol {\hat {r}}}+\cos \theta \cos \phi {\boldsymbol {\hat {\theta }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=&\sin \theta \sin \phi {\boldsymbol {\hat {r}}}+\cos \theta \sin \phi {\boldsymbol {\hat {\theta }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\\\end{matrix}}$	${\begin{matrix}{\boldsymbol {\hat {\sigma }}}&=&{\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} -{\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\tau }}}&=&{\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} +{\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}$
	${\begin{matrix}\mathbf {\hat {r}} &=&{\frac {x\mathbf {\hat {x}} +y\mathbf {\hat {y}} +z\mathbf {\hat {z}} }{r}}\\{\boldsymbol {\hat {\theta }}}&=&{\frac {xz\mathbf {\hat {x}} +yz\mathbf {\hat {y}} -\rho ^{2}\mathbf {\hat {z}} }{r\rho }}\\{\boldsymbol {\hat {\phi }}}&=&{\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\rho }}\end{matrix}}$	${\begin{matrix}\mathbf {\hat {r}} &=&{\frac {\rho }{r}}{\boldsymbol {\hat {\rho }}}+{\frac {z}{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\theta }}}&=&{\frac {z}{r}}{\boldsymbol {\hat {\rho }}}-{\frac {\rho }{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\phi }}}&=&{\boldsymbol {\hat {\phi }}}\end{matrix}}$	${\begin{matrix}{\boldsymbol {\hat {\rho }}}&=&\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&=&{\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\\\end{matrix}}$	${\begin{matrix}\end{matrix}}$
Векторное поле $\mathbf{A}$	$A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}$	$A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}$	$A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}$	$A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }{\boldsymbol {\hat {z}}}$
Градиент $\nabla f$	${\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y} + {\partial f \over \partial z}\mathbf{\hat z}$	${\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z}$	${\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}$	${\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\boldsymbol {\hat {\tau }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}}$
Дивергенция $\nabla \cdot \mathbf{A}$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$	${\frac {1}{\sigma ^{2}+\tau ^{2}}}{\partial A_{\sigma } \over \partial \sigma }+{\frac {1}{\sigma ^{2}+\tau ^{2}}}{\partial A_{\tau } \over \partial \tau }+{\partial A_{z} \over \partial z}$
Ротор $\nabla \times \mathbf{A}$	${\begin{matrix}\left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right)\mathbf {\hat {x}} &+\\\left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right)\mathbf {\hat {y}} &+\\\left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right)\mathbf {\hat {z}} &\ \end{matrix}}$	${\begin{matrix}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\boldsymbol {\hat {\rho }}}&+\\\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {\hat {\phi }}}&+\\{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}$	${\begin{matrix}{1 \over r\sin \theta }\left({\partial \over \partial \theta }\left(A_{\phi }\sin \theta \right)-{\partial A_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {r}}}&+\\{1 \over r}\left({1 \over \sin \theta }{\partial A_{r} \over \partial \phi }-{\partial \over \partial r}\left(rA_{\phi }\right)\right){\boldsymbol {\hat {\theta }}}&+\\{1 \over r}\left({\partial \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {\hat {\phi }}}&\ \end{matrix}}$	${\begin{matrix}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial A_{z} \over \partial \tau }-{\partial A_{\tau } \over \partial z}\right){\boldsymbol {\hat {\sigma }}}&-\\\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial A_{z} \over \partial \sigma }-{\partial A_{\sigma } \over \partial z}\right){\boldsymbol {\hat {\tau }}}&+\\{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\partial \left(sA_{\phi }\right) \over \partial s}-{\partial A_{s} \over \partial \phi }\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}$
Оператор Лапласа $\Delta f = \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$	${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}$
Лапласиан (англ.) векторной функции $\Delta \mathbf{A} = \nabla^2 \mathbf{A}$	$\Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z}$	${\begin{matrix}\left(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {\rho }}}&+\\\left(\Delta A_{\phi }-{A_{\phi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \phi }\right){\boldsymbol {\hat {\phi }}}&+\\\left(\Delta A_{z}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}$	${\begin{matrix}\left(\Delta A_{r}-{2A_{r} \over r^{2}}-{2 \over r^{2}\sin \theta }{\partial \left(A_{\theta }\sin \theta \right) \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {r}}}&+\\\left(\Delta A_{\theta }-{A_{\theta } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {\theta }}}&+\\\left(\Delta A_{\phi }-{A_{\phi } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}\sin \theta }{\partial A_{r} \over \partial \phi }+{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {\phi }}}&\end{matrix}}$	?
Элемент длины	$d\mathbf {l} =dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}}$	$d\mathbf {l} =d\rho {\boldsymbol {\hat {\rho }}}+\rho d\phi {\boldsymbol {\hat {\phi }}}+dz{\boldsymbol {\hat {z}}}$	$d\mathbf {l} =dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\phi {\boldsymbol {\hat {\phi }}}$	$d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma {\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}d\tau {\boldsymbol {\hat {\tau }}}+dz{\boldsymbol {\hat {z}}}$
Элемент ориентированной площади	${\begin{matrix}d\mathbf {S} =&dy\,dz\,\mathbf {\hat {x}} +\\&dx\,dz\,\mathbf {\hat {y}} +\\&dx\,dy\,\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}d\mathbf {S} =&\rho \,d\phi \,dz\,{\boldsymbol {\hat {\rho }}}+\\&d\rho \,dz\,{\boldsymbol {\hat {\phi }}}+\\&\rho \,d\rho d\phi \,\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}d\mathbf {S} =&r^{2}\sin \theta \,d\theta \,d\phi \,\mathbf {\hat {r}} +\\&r\sin \theta \,dr\,d\phi \,{\boldsymbol {\hat {\theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {\hat {\phi }}}\end{matrix}}$	${\begin{matrix}d\mathbf {S} =&{\sqrt {\sigma ^{2}+\tau ^{2}}},d\tau \,dz\,{\boldsymbol {\hat {\sigma }}}+\\&{\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma \,dz\,{\boldsymbol {\hat {\tau }}}+\\&\sigma ^{2}+\tau ^{2}d\sigma ,d\tau \,\mathbf {\hat {z}} \end{matrix}}$
Элемент объёма	$d\tau =dx\,dy\,dz\,$	$d\tau =\rho \,d\rho \,d\phi \,dz\,$	$d\tau =r^{2}\sin \theta \,dr\,d\theta \,d\phi \,$	$d\tau =\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,$

Некоторые свойства

Не равные нулю операторы второго порядка:

1. $\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f$ (Оператор Лапласа)
2. $\operatorname {rot\ grad\ } f=\nabla \times (\nabla f)=0$
3. $\operatorname {div\ rot\ } \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$
4. $\operatorname {rot\ rot\ } \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$

(используя формулу Лагранжа для двойного векторного произведения)

1. $\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$

См. также

• Ортогональные координаты
• Криволинейные координаты
• Коэффициенты Ламе
• Метод координат
• Векторные поля в цилиндрической и сферической системах координат (англ.)

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Эта страница использует содержимое раздела Википедии на русском языке. Оригинальная статья находится по адресу: Оператор набла в различных системах координат. Список первоначальных авторов статьи можно посмотреть в истории правок. Эта статья так же, как и статья, размещённая в Википедии, доступна на условиях CC-BY-SA .

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