Дифференциальные операторы в различных системах координат

Общее выражение для оператора ∇ в произвольной системе ортогональных координат можно записать так:

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

где "${\displaystyle \circ }$" - любой из трех значков, соответствующих действию оператора ∇:

• " " - градиент;
• " · " - дивергенция;
• " × " - ротор.

Элементы ${\displaystyle \partial r_{m}}$ в этой записи соответствуют элементам радиус-вектора в соответствующей системе координат:

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

Иначе говоря, первым действием является взятие частной производной ${\displaystyle {\partial \mathbf {A} \over \partial r_{m}}}$ по проекции радиус-вектора от всего вектора ${\displaystyle \mathbf {A} }$ (с учетом производных орт в данной системе координат), и лишь потом умножение (простое для градиента, скалярное для дивергенции и векторное для ротора) орта направления на ${\displaystyle {\partial \mathbf {A} \over \partial r_{m}}}$.

При этом достаточно знать выражения:

• в цилиндрических координатах: ${\displaystyle {\partial i_{\rho } \over \partial \varphi }=i_{\varphi }}$ и ${\displaystyle {\partial i_{\varphi } \over \partial \varphi }=-i_{\rho }}$;
• в сферических координатах: ${\displaystyle {\partial i_{r} \over \partial \theta }=i_{\theta }}$, ${\displaystyle {\partial i_{\theta } \over \partial \theta }=-i_{r}}$, ${\displaystyle {\partial i_{r} \over \partial \varphi }=i_{\varphi }\sin \theta }$, ${\displaystyle {\partial i_{\theta } \over \partial \varphi }=i_{\varphi }\cos \theta }$ и ${\displaystyle {\partial i_{\varphi } \over \partial \varphi }=-i_{r}\sin \theta -i_{\theta }\cos \theta }$.

Например: в приведенной ниже таблице запись дивергенции в цилиндрических координатах получена следующим образом:

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

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

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

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

Запись оператора Гамильтона в различных системах координат

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

Выражения для операторов второго порядка:

1. ${\displaystyle \mathrm {div\ grad} \;f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}$ (Оператор Лапласа)
2. ${\displaystyle \mathrm {rot\ grad} \;f=\nabla \times (\nabla f)=0}$
3. ${\displaystyle \mathrm {div\ rot} \;\mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0}$
4. ${\displaystyle \mathrm {rot\ rot} \;\mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\Delta \mathbf {A} }$ (используя формулу Лагранжа для двойного векторного произведения)
5. ${\displaystyle \Delta (fg)=\mathrm {div\ grad} \ (fg)=\mathrm {div} \ (f\mathrm {grad} \ g+g\mathrm {grad} \ f)=f\Delta g+2\mathrm {grad} \ f\cdot \mathrm {grad} \ g+g\Delta f}$

• Оператор Д’Аламбера
• Ортогональные координаты
• Криволинейные координаты
• Метод координат
• Векторные поля в цилиндрической и сферической системах координат