# 圆柱坐标系

（重定向自圆柱坐标）

## 定义

• ${\displaystyle \rho }$ 是 P 点与 z-轴的垂直距离。
• ${\displaystyle \phi }$ 是线 OP 在 xy-面的投影线与正 x-轴之间的夹角。
• ${\displaystyle z}$直角坐标${\displaystyle z}$ 等值。

## 坐标系变换

### 直角坐标系

${\displaystyle {\rho }={\sqrt {x^{2}+y^{2}}}}$
${\displaystyle {\phi }=\arctan \left({\frac {y}{x}}\right)}$
${\displaystyle z=z}$

${\displaystyle x=\rho \cos \phi }$
${\displaystyle y=\rho \sin \phi }$
${\displaystyle z=z}$

### 球坐标系

${\displaystyle \rho =r\sin \theta }$
${\displaystyle \phi =\phi }$
${\displaystyle z=r\cos \theta }$

${\displaystyle r={\sqrt {\rho ^{2}+z^{2}}}}$
${\displaystyle \theta =\arctan {\frac {\rho }{z}}}$
${\displaystyle \phi =\phi }$

## 圆柱坐标系下的微积分公式

${\displaystyle h_{\rho }=1}$
${\displaystyle h_{\phi }=\rho }$
${\displaystyle h_{z}=1}$

${\displaystyle \mathrm {d} \mathbf {r} =\mathrm {d} \rho \,{\boldsymbol {\hat {\rho }}}+\rho \,\mathrm {d} \varphi \,{\boldsymbol {\hat {\varphi }}}+\mathrm {d} z\,\mathbf {\hat {z}} }$

${\displaystyle \mathrm {d} S=\rho \,d\varphi \,dz}$

${\displaystyle \mathrm {d} V=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z}$

${\displaystyle \nabla ={\boldsymbol {\hat {\rho }}}{\frac {\partial }{\partial \rho }}+{\boldsymbol {\hat {\varphi }}}{\frac {1}{\rho }}{\frac {\partial }{\partial \varphi }}+\mathbf {\hat {z}} {\frac {\partial }{\partial z}}}$
${\displaystyle \nabla ^{2}\Phi ={1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial \Phi \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}\Phi \over \partial \phi ^{2}}+{\partial ^{2}\Phi \over \partial z^{2}}}$