# 球座標系

（重定向自球坐标系

## 定義

1. 從原點往正z-軸移動${\displaystyle r}$單位，
2. 右手定則，大拇指往y-軸指，x-軸與z-軸朝其他手指的指向旋轉${\displaystyle \theta }$角值，
3. 用右手定則，大拇指往z-軸指，x-軸與y-軸朝其他手指的指向旋轉${\displaystyle \varphi }$角值。

## 座標系變換

### 直角座標系

${\displaystyle {r}={\sqrt {x^{2}+y^{2}+z^{2}}}}$
${\displaystyle {\theta }=\arccos \left({\frac {z}{r}}\right)=\arcsin \left({\frac {\sqrt {x^{2}+y^{2}}}{r}}\right)=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)}$
${\displaystyle {\varphi }=\arccos \left({\frac {x}{r\sin \theta }}\right)=\arcsin \left({\frac {y}{r\sin \theta }}\right)=\arctan \left({\frac {y}{x}}\right)}$

1. 必須依照 ${\displaystyle (x,\ y)}$ 所處的象限來計算正確的反正切值。
2. 當 ${\displaystyle {x}=0}$ 時，判斷 ${\displaystyle {y}}$ 的值：
${\displaystyle {y}>0}$，則 ${\displaystyle {\varphi }={\frac {\pi }{2}}}$
${\displaystyle {y}<0}$，則 ${\displaystyle {\varphi }={-{\frac {\pi }{2}}}}$${\displaystyle {\frac {3\pi }{2}}}$
${\displaystyle {y}=0}$，則 ${\displaystyle {\varphi }}$ 為未定值 ( 因為 ${\displaystyle {\frac {0}{0}}}$未定式 )。

${\displaystyle x=r\sin \theta \cos \varphi }$
${\displaystyle y=r\sin \theta \sin \varphi }$
${\displaystyle z=r\cos \theta }$

### 圓柱座標系

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

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

## 球坐标系下的微积分公式

${\displaystyle h_{r}=1}$
${\displaystyle h_{\theta }=r}$
${\displaystyle h_{\varphi }=r\sin \theta }$

• 线元素是一个从${\displaystyle (r,\theta ,\varphi )}$${\displaystyle (r+\mathrm {d} r,\,\theta +\mathrm {d} \theta ,\,\varphi +\mathrm {d} \varphi )}$的无穷小位移，表示为公式：
${\displaystyle \mathrm {d} \mathbf {r} =\mathrm {d} r\,{\boldsymbol {\hat {r}}}+r\,\mathrm {d} \theta \,{\boldsymbol {\hat {\theta }}}+r\sin {\theta }\mathrm {d} \varphi \,\mathbf {\boldsymbol {\hat {\varphi }}} }$

• 面积元素1：在球面上，固定半径，天顶角从${\displaystyle \theta }$${\displaystyle \theta +\mathrm {d} \theta }$，方位角从${\displaystyle \varphi }$${\displaystyle \varphi +\mathrm {d} \varphi }$变化，公式为：
${\displaystyle \mathrm {d} S_{r}=r^{2}\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi }$
• 面积元素2：固定天顶角${\displaystyle \theta }$，其他两个变量变化，則公式为：
${\displaystyle \mathrm {d} S_{\theta }=r\,\sin \theta \,\mathrm {d} r\,\mathrm {d} \varphi }$
• 面积元素3：固定方位角${\displaystyle \varphi }$，其他两个变量变化，則公式为：
${\displaystyle \mathrm {d} S_{\varphi }=r\,\mathrm {d} r\,\mathrm {d} \theta }$
• 体积元素，徑向座標从${\displaystyle r}$${\displaystyle r+\mathrm {d} r}$，天顶角从${\displaystyle \theta }$${\displaystyle \theta +\mathrm {d} \theta }$，并且方位角从${\displaystyle \varphi }$${\displaystyle \varphi +\mathrm {d} \varphi }$的公式为：
${\displaystyle \mathrm {d} V=r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi }$

${\displaystyle \nabla f={\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\boldsymbol {\hat {\varphi }}}}$
${\displaystyle \nabla \cdot \mathbf {A} ={\frac {1}{r^{2}}}{\partial \over \partial r}\left(r^{2}A_{r}\right)+{\frac {1}{r\sin \theta }}{\partial \over \partial \theta }\left(\sin \theta A_{\theta }\right)+{\frac {1}{r\sin \theta }}{\partial A_{\varphi } \over \partial \varphi }}$
${\displaystyle \nabla \times \mathbf {A} =\displaystyle {1 \over r\sin \theta }\left({\partial \over \partial \theta }\left(A_{\varphi }\sin \theta \right)-{\partial A_{\theta } \over \partial \varphi }\right){\boldsymbol {\hat {r}}}+\displaystyle {1 \over r}\left({1 \over \sin \theta }{\partial A_{r} \over \partial \varphi }-{\partial \over \partial r}\left(rA_{\varphi }\right)\right){\boldsymbol {\hat {\theta }}}+\displaystyle {1 \over r}\left({\partial \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {\hat {\varphi }}}}$
${\displaystyle \nabla ^{2}f={1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}}$

## 地理座標系

${\displaystyle \psi =90^{\circ }-\theta }$，正值可稱北緯，負值去負號可稱南緯。

${\displaystyle \varphi \leq 180^{\circ }}$${\displaystyle \lambda =\varphi }$，正值可稱東經，
${\displaystyle \varphi \geq 180^{\circ }}$${\displaystyle \lambda =\varphi -360^{\circ }}$，負值去負號可稱西經。

${\displaystyle \tan(\psi )={\frac {b^{2}}{a^{2}}}\tan(\varphi )}$

${\displaystyle r(\psi )={\frac {b}{\sqrt {1-(e\cos \psi )^{2}}}}}$

## 引用

1. ^
2. ^ Eric W. Weisstein. Spherical Coordinates. MathWorld. 2005-10-26 [2010-01-15]. （原始内容存档于2018-06-23）.
3. ^ 引用错误：没有为名为http://mathworld.wolfram.com/SphericalCoordinates.html （页面存档备份，存于互联网档案馆）的参考文献提供内容