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== Таблица операторов ==
Формулы не копируются
Здесь используются стандартные физические обозначения. Для сферических координат, {{math|θ}} обозначает угол между осью ''{{math|z}}'' и [[радиус-вектор]]ом точки, {{math|φ}} — угол между проекцией радиус-вектора на плоскость ''{{math|x-y}}'' и осью ''{{math|x}}''.
{| class="wikitable"
<caption>Запись [[Оператор набла|оператора Гамильтона]] в различных системах координат</caption>
! Оператор
! [[Прямоугольная система координат|Прямоугольные координаты]] <br />(''{{math|x, y, z}}'')
! [[Цилиндрические координаты]] <br />({{math|ρ, φ, ''z''}})
! [[Сферические координаты]] <br />({{math|''r'', θ, φ}})
! [[Параболические координаты]] <br />({{math|σ, τ, ''z''}})
|-
| rowspan="2" | Формулы преобразования координат
| <math>\begin{matrix}
\rho & = & \sqrt{x^2+y^2} \\
\varphi & = & \operatorname{arctg}(y/x) \\
z & = & z \end{matrix}</math>
| <math>\begin{matrix}
x & = & \rho\cos\varphi \\
y & = & \rho\sin\varphi \\
z & = & z \end{matrix}</math>
| <math>\begin{matrix}
x & = & r\sin\theta\cos\varphi \\
y & = & r\sin\theta\sin\varphi \\
z & = & r\cos\theta \end{matrix}</math>
| <math>\begin{matrix}
x & = & \sigma \tau\\
y & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z & = & z \end{matrix}</math>
|-
| <math>\begin{matrix}
r & = & \sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}} \\
\theta & = & \arccos \left( z/r \right) \\
\varphi & = & \operatorname{arctg}(y/x) \\
\end{matrix}</math>
| <math>\begin{matrix}
r & = & \sqrt{\rho^2 + z^2} \\
\theta & = & \operatorname{arctg}{(\rho/z)}\\
\varphi & = & \varphi \end{matrix}</math>
| <math>\begin{matrix}
\rho & = & r\sin{\theta} \\
\varphi & = & \varphi\\
z & = & r\cos{\theta} \end{matrix}</math>
| <math>\begin{matrix}
\rho\cos\varphi & = & \sigma \tau\\
\rho\sin\varphi & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z & = & z \end{matrix}</math>
|-
| [[Радиус-вектор]] произвольной точки
| <math>x\mathbf{\hat x} + y\mathbf{\hat y} + z\mathbf{\hat z}</math>
| <math>\rho\boldsymbol{\hat \rho} + z\boldsymbol{\hat z}</math>
| <math>r\boldsymbol{\hat r}</math>
| <math>\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}</math>
|-
|rowspan="2"| Связь [[Единичный вектор|единичных векторов]]
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
|-
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
| .
|-
| [[Векторное поле]] <math>\mathbf{A}</math>
| <math>A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}</math>
| <math>A_\rho\boldsymbol{\hat \rho} + A_\varphi\boldsymbol{\hat \varphi} + A_z\boldsymbol{\hat z}</math>
| <math>A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\varphi\boldsymbol{\hat \varphi}</math>
| <math>A_\sigma\boldsymbol{\hat \sigma} + A_\tau\boldsymbol{\hat \tau} + A_\varphi\boldsymbol{\hat z}</math>
|-
| [[Градиент]] <math>\nabla f</math>
| <math>{\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y}
+ {\partial f \over \partial z}\mathbf{\hat z}</math>
| <math>{\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}</math>
| <math>{\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}</math>
| <math> \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}</math>
|-
| [[Дивергенция]] <math>\nabla \cdot \mathbf{A}</math>
| <math>{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math>
| <math>{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}</math>
| <math>{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}</math>
| <math> \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}</math>
|-
| [[Ротор (математика)|Ротор]] <math>\nabla \times \mathbf{A}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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( r A_\varphi \right) \right) \boldsymbol{\hat \theta} & + \\
{1 \over r}\left({\partial \over \partial r} \left( r A_\theta \right)
- {\partial A_r \over \partial \theta}\right) \boldsymbol{\hat \varphi} & \ \end{matrix}</math>
| <math>\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( s A_\varphi \right) \over \partial s}
- {\partial A_s \over \partial \varphi}\right) \boldsymbol{\hat z} & \ \end{matrix}</math>
|-
| [[Оператор Лапласа]] <math>\Delta f = \nabla^2 f</math>
| <math>{\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}</math>
| <math>{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}</math>
| <math>{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}</math>
| <math> \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}}
</math>
|-
| [[Векторный оператор Лапласа]] <math>\Delta \mathbf{A}</math>
| <math>\Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z} </math>
| <math>\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}</math>
| <math>\begin{matrix}
\left(\Delta A_r - {2 A_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}</math>
| ?
|-
| Элемент длины
| <math>d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z}</math>
| <math>d\mathbf{l} = d\rho\boldsymbol{\hat\rho} + \rho d\varphi\boldsymbol{\hat\varphi} + dz\boldsymbol{\hat z}</math>
| <math>d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\varphi\boldsymbol{\hat \varphi}</math>
| <math>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}</math>
|-
| Элемент ориентированной площади
| <math>\begin{matrix}d\mathbf{S} = &dy\,dz\,\mathbf{\hat x} + \\
&dx\,dz\,\mathbf{\hat y} + \\
&dx\,dy\,\mathbf{\hat z}\end{matrix}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
|-
| Элемент объёма
| <math>d\tau = dx\,dy\,dz</math>
| <math>d\tau = \rho\, d\rho\, d\varphi\, dz</math>
| <math>d\tau = r^2\sin\theta \,dr\,d\theta\, d\varphi</math>
| <math>d\tau = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,</math>
|}
== Некоторые свойства ==
Выражения для операторов второго порядка:
# <math>\mathrm{div\ grad}\; f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f</math> ([[Оператор Лапласа]])
# <math>\mathrm{rot\ grad}\; f = \nabla \times (\nabla f) = 0</math>
# <math>\mathrm{div\ rot}\; \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
# <math>\mathrm{rot\ rot}\; \mathbf{A} = \nabla \times (\nabla \times \mathbf{A})
= \nabla (\nabla \cdot \mathbf{A}) - \Delta \mathbf{A}</math>
(используя формулу Лагранжа для [[Двойное векторное произведение|двойного векторного произведения]])
# <math>\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f</math>
== См. также ==
| 