Цилиндрические параболические координаты

Прямоугольная система координат ${\displaystyle \ (x,y,z)}$ 

${\displaystyle \ x={\frac {c}{2}}\left(u^{2}-v^{2}\right)}$ 

${\displaystyle \ y=cuv}$ 

${\displaystyle \ z=z}$ 

Цилиндрическая система координат ${\displaystyle \ (\rho ,\varphi ,z)}$ 

${\displaystyle \ \rho ={\frac {c}{2}}\left(u^{2}+v^{2}\right)}$ 

${\displaystyle \ \varphi =\operatorname {arctg} \left({\frac {2uv}{u^{2}-v^{2}}}\right),}$ 

${\displaystyle \ z=z}$ 

Коэффициенты Ламе в данных координатах имеют следующий вид:

${\displaystyle \ H_{u}=c{\sqrt {u^{2}+v^{2}}}}$ 

${\displaystyle \ H_{v}=c{\sqrt {u^{2}+v^{2}}}}$ 

${\displaystyle \ H_{z}=1}$ 

Градиент

${\displaystyle \operatorname {grad} F(u,v,z)={\frac {1}{c{\sqrt {u^{2}+v^{2}}}}}\left({\frac {\partial F}{\partial u}}{\vec {e}}_{u}+{\frac {\partial U}{\partial v}}{\vec {e}}_{v}\right)+{\frac {\partial F}{\partial z}}{\vec {e}}_{z}}$ 

Дивергенция

${\displaystyle \operatorname {div} {\vec {A}}(u,v,z)={\frac {1}{c(u^{2}+v^{2})}}\left[{\frac {\partial }{\partial u}}\left({\sqrt {u^{2}+v^{2}}}A_{u}\right)+{\frac {\partial }{\partial v}}\left({\sqrt {u^{2}+v^{2}}}A_{v}\right)\right]+{\frac {\partial A_{z}}{\partial z}}}$ 

Ротор

${\displaystyle \operatorname {rot} {\vec {A}}={\frac {1}{c{\sqrt {u^{2}+v^{2}}}}}\left[{\frac {\partial }{\partial v}}A_{z}-{\frac {\partial }{\partial z}}(c{\sqrt {u^{2}+v^{2}}}A_{v})\right]{\vec {e}}_{u}+{\frac {1}{c{\sqrt {u^{2}+v^{2}}}}}\left[{\frac {\partial }{\partial z}}(c{\sqrt {u^{2}+v^{2}}}A_{u})-{\frac {\partial }{\partial u}}A_{z}\right]{\vec {e}}_{v}+}$ 

${\displaystyle +{\frac {1}{c(u^{2}+v^{2})}}\left[{\frac {\partial }{\partial u}}(c{\sqrt {u^{2}+v^{2}}}A_{v})-{\frac {\partial }{\partial v}}(c{\sqrt {u^{2}+v^{2}}}A_{u})\right]{\vec {e}}_{z}}$ 

Лапласиан