正交座標系

在數學裏，一個正交坐標系定義為一組正交坐標${\displaystyle \mathbf {q} =(q_{1},\ q_{2},\ q_{3},\ \dots \ q_{n})}$，其坐標曲面都以直角相交（注意：很多作者采用爱因斯坦记号对坐标标号使用上标并非表示指数）。坐標曲面定義為特定坐標${\displaystyle q_{i}}$的等值曲面，即${\displaystyle q_{i}}$為常数的曲线、曲面或超曲面。例如，三維直角坐標${\displaystyle (x,\ y,\ z)}$是一種正交坐標系，它的${\displaystyle x}$為常數，${\displaystyle y}$為常數，${\displaystyle z}$為常數的坐標曲面，都是互相以直角相交的平面，都互相垂直。正交坐标系是曲线坐标系的特殊的但极其常见的形式。

二维拋物線坐標系，其中綠色坐标曲线与紅色坐标曲线正交。

动机

正交座標時常用來解析一些出現於量子力學、流體動力學、電動力學、熱力學等等的偏微分方程。舉例而言，選擇一個恰當的的正交座標來解析氫離子${\displaystyle H_{2}\,^{-}}$ 的波函數或消防水管的噴水，也許會比用直角座標方便的多。這主要是因為恰當的正交座標能夠與一個問題的對稱性相配合，從而促使應用分離變數法來成功的解析關於這問題的方程式。分離變數法是一種數學技巧，專門用來將一個複雜的${\displaystyle n}$ 維問題變為${\displaystyle n}$ 個一維問題。很多問題都可以簡化為拉普拉斯方程或亥姆霍茲方程，這些方程式可以用很多種正交座標來分離。拉普拉斯方程可以在13个正交坐标系中分离（本文列出的14个中圆环坐标系除外），而亥姆霍茲方程可以在11个正交坐标系中分离^[1][2]。

概述

 

共形映射作用于矩形网格。注意，弯曲的网格的正交性被保留。

正交坐標的度規張量絕對沒有非對角項目。換句話說，無窮小距離的平方${\displaystyle ds^{2}}$ ，可以寫為無窮小坐標位移的平方和：

${\displaystyle ds^{2}=\sum _{i=1}^{n}\left(h_{i}dq_{i}\right)^{2}}$ ；

其中，${\displaystyle n}$ 是維數，標度因子${\displaystyle h_{i}}$ 是度規張量的對角元素${\displaystyle g_{ii}}$ 的平方根：

${\displaystyle h_{i}(\mathbf {q} )\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {g_{ii}(\mathbf {q} )}}}$ 。

這些標度因子可以用來計算一個正交坐標系的微分算子。例如，梯度、拉普拉斯算子、散度、或旋度。

在數學裏，存在有各種各樣的正交座標系。應用二維直角座標系${\displaystyle (x,\ y)}$ 的共形映射方法，可以簡易的生成這些正交座標系。一個複數${\displaystyle z=x+iy}$ 的任何全純函數${\displaystyle w=f(z)}$ ，其複值的導數，如果不等於零，則會造成一個共形映射。如果答案可以表達為${\displaystyle w=u+iv}$ ，則${\displaystyle u}$ 與${\displaystyle v}$ 的等值曲線以直角相交，就如同原本的${\displaystyle x}$ 與${\displaystyle y}$ 的等值曲線以直角相交。

三維與更高維的正交座標系可以由一個二維正交座標系生成，只要將二維正交座標往一個新的座標軸投射（形成類似圓柱座標系的座標系），或者將二維正交座標繞著其對稱軸旋轉。可是，也有一些三維正交座標系，例如橢球座標系，則不能夠用上述方法得到。更一般的正交坐标可以从一些必要的坐标曲面/曲线起步并通过考虑它们的正交轨迹线（英语：Orthogonal trajectory）而得到。

向量代数

在正交坐標系裏，內積的公式仍舊不變：

${\displaystyle \mathbf {A} \cdot \mathbf {B} =\sum _{i=1}^{n}A_{i}B_{i}}$ 。

向量微積分

從前面的距離公式，可以觀察出，一個正交坐標${\displaystyle q_{i}}$ 的無窮小改變${\displaystyle dq_{i}}$ ，其相伴的長度是${\displaystyle ds_{i}=h_{i}dq_{i}}$ 。因此，一個位移向量的全微分${\displaystyle d\mathbf {r} }$ 等於

${\displaystyle d\mathbf {r} =\sum _{i=1}^{n}h_{i}dq_{i}\mathbf {e} _{i}}$ ；

其中，${\displaystyle \mathbf {e} _{i}}$ 是垂直於${\displaystyle q_{i}}$ 等值曲面的單位向量，指向著${\displaystyle q_{i}}$ 增值最快的方向，這些單位向量形成了一個局部直角坐標系的坐標軸。

因此，向量${\displaystyle \mathbf {F} }$ 沿著周線${\displaystyle \mathbb {C} }$ 的線積分等於

${\displaystyle \int _{\mathbb {C} }\mathbf {F} \cdot d\mathbf {r} =\sum _{i=1}^{n}\int _{\mathbb {C} }F_{i}h_{i}dq_{i}}$ ；

其中，${\displaystyle F_{i}}$ 是向量${\displaystyle \mathbf {F} }$ 在單位向量${\displaystyle \mathbf {e} _{i}}$ 方向的分量：

${\displaystyle F_{i}\ {\stackrel {\mathrm {def} }{=}}\ \mathbf {e} _{i}\cdot \mathbf {F} }$ 。

類似地，一個無窮小面積元素是

${\displaystyle dA=ds_{i}ds_{j}=h_{i}h_{j}dq_{i}dq_{j},\qquad i\neq j}$ ，

一個無窮小體積元素是

${\displaystyle dV=ds_{i}ds_{j}ds_{k}=h_{i}h_{j}h_{k}dq_{i}dq_{j}dq_{k},\qquad i\neq j\neq k}$ 。

例如，向量${\displaystyle \mathbf {F} }$ 對於一個曲面${\displaystyle \mathbb {S} }$ 的曲面積分是

${\displaystyle \int _{\mathbb {S} }\mathbf {F} \cdot d\mathbf {A} =\int _{\mathbb {S} }F_{1}h_{2}h_{3}dq_{2}dq_{3}+\int _{\mathbb {S} }F_{2}h_{3}h_{1}dq_{3}dq_{1}+\int _{\mathbb {S} }F_{3}h_{1}h_{2}dq_{1}dq_{2}}$ 。

球坐標系實例

 

直角坐標${\displaystyle (x,\ y,\ z)}$ 與球坐標${\displaystyle (r,\ \theta ,\phi )}$ 的變換方程式為

${\displaystyle x=r\sin \theta \cos \phi }$ 、

${\displaystyle y=r\sin \theta \sin \phi }$ 、

${\displaystyle z=r\cos \theta }$ 。

直角坐標的全微分是

${\displaystyle dx=\sin \theta \cos \phi dr+r\cos \theta \cos \phi d\theta -r\sin \theta \sin \phi d\phi }$ 、

${\displaystyle dy=\sin \theta \sin \phi dr+r\cos \theta \sin \phi d\theta +r\sin \theta \cos \phi d\phi }$ 、

${\displaystyle dz=\cos \theta dr-r\sin \theta d\theta }$ 。

所以，無窮小距離的平方是

${\displaystyle {\begin{aligned}ds^{2}&=dx^{2}+dy^{2}+dz^{2}\\&=dr^{2}+(rd\theta )^{2}+(r\sin \theta d\phi )^{2}\\\end{aligned}}}$  。

標度因子是

${\displaystyle h_{r}=1}$ 、

${\displaystyle h_{\theta }=r}$ 、

${\displaystyle h_{\phi }=r\sin \theta }$ 。

向量${\displaystyle \mathbf {F} }$ 沿著周線${\displaystyle \mathbb {C} }$ 的線積分等於

${\displaystyle \int _{\mathbb {C} }\mathbf {F} \cdot d\mathbf {r} =\int _{\mathbb {C} }F_{r}\ dr+F_{\theta }\ rd\theta +F_{\phi }\ r\sin \theta d\phi }$ 。

向量${\displaystyle \mathbf {F} }$ 對於一個曲面${\displaystyle \mathbb {S} }$ 的曲面積分是

${\displaystyle \int _{\mathbb {S} }\mathbf {F} \cdot d\mathbf {A} =\int _{\mathbb {S} }F_{r}\ r^{2}\sin \theta d\theta d\phi +\int _{\mathbb {S} }F_{\theta }\ r\sin \theta drd\phi +\int _{\mathbb {S} }F_{\phi }\ rdrd\theta }$ 。

三維微分算子

主条目：向量分析和Nabla算子

算子	正交坐標公式
标量场的梯度	${\displaystyle \nabla \Phi ={\hat {\mathbf {e} }}_{1}{\frac {1}{h_{1}}}{\frac {\partial \Phi }{\partial q_{1}}}+{\hat {\mathbf {e} }}_{2}{\frac {1}{h_{2}}}{\frac {\partial \Phi }{\partial q_{2}}}+{\hat {\mathbf {e} }}_{3}{\frac {1}{h_{3}}}{\frac {\partial \Phi }{\partial q_{3}}}}$
向量场的散度	${\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{h_{1}h_{2}h_{3}}}\left[{\frac {\partial }{\partial q_{1}}}(F_{1}h_{2}h_{3})+{\frac {\partial }{\partial q_{2}}}(F_{2}h_{3}h_{1})+{\frac {\partial }{\partial q_{3}}}(F_{3}h_{1}h_{2})\right]}$
向量场的旋度	${\displaystyle {\begin{aligned}\nabla \times \mathbf {F} &={\frac {\mathbf {e} _{1}}{h_{2}h_{3}}}\left[{\frac {\partial }{\partial q_{2}}}\left(h_{3}F_{3}\right)-{\frac {\partial }{\partial q_{3}}}\left(h_{2}F_{2}\right)\right]+{\frac {\mathbf {e} _{2}}{h_{3}h_{1}}}\left[{\frac {\partial }{\partial q_{3}}}\left(h_{1}F_{1}\right)-{\frac {\partial }{\partial q_{1}}}\left(h_{3}F_{3}\right)\right]\\&+{\frac {\mathbf {e} _{3}}{h_{1}h_{2}}}\left[{\frac {\partial }{\partial q_{1}}}\left(h_{2}F_{2}\right)-{\frac {\partial }{\partial q_{2}}}\left(h_{1}F_{1}\right)\right]\\\end{aligned}}}$
标量场的拉普拉斯算子	${\displaystyle \nabla ^{2}\Phi ={\frac {1}{h_{1}h_{2}h_{3}}}\left[{\frac {\partial }{\partial q_{1}}}\left({\frac {h_{2}h_{3}}{h_{1}}}{\frac {\partial \Phi }{\partial q_{1}}}\right)+{\frac {\partial }{\partial q_{2}}}\left({\frac {h_{3}h_{1}}{h_{2}}}{\frac {\partial \Phi }{\partial q_{2}}}\right)+{\frac {\partial }{\partial q_{3}}}\left({\frac {h_{1}h_{2}}{h_{3}}}{\frac {\partial \Phi }{\partial q_{3}}}\right)\right]}$

上面表达式可以使用列維-奇維塔符號${\displaystyle \epsilon }$ 的更简洁形式书写，定义${\displaystyle H=h_{1}h_{2}h_{3}}$ ，并使用爱因斯坦记号，即在同时出现上标和下标的项目上求此项所有可能的总和:

算子	表达式
标量场的梯度	${\displaystyle \nabla \phi ={\frac {{\hat {\mathbf {e} }}_{k}}{h_{k}}}{\frac {\partial \phi }{\partial q^{k}}}}$
向量场的散度	${\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{H}}{\frac {\partial }{\partial q^{k}}}\left({\frac {H}{h_{k}}}F_{k}\right)}$
向量场（只3D）的旋度	${\displaystyle \left(\nabla \times \mathbf {F} \right)_{k}={\frac {h_{k}{\hat {\mathbf {e} }}_{k}}{H}}\epsilon _{ijk}{\frac {\partial }{\partial q^{i}}}\left(h_{j}F_{j}\right)}$
标量场的拉普拉斯算子	${\displaystyle \nabla ^{2}\phi ={\frac {1}{H}}{\frac {\partial }{\partial q^{k}}}\left({\frac {H}{h_{k}^{2}}}{\frac {\partial \phi }{\partial q^{k}}}\right)}$

正交坐标系表格

除了直角坐标系之外，下表列出其他常见的正交坐标系^[3]，为了简明性在坐标列中使用了区间符号。

曲线坐标 (q1, q2, q3)	从直角坐标(x, y, z)转换	缩放因子
球极坐标系 ${\displaystyle (r,\theta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi )}$	${\displaystyle {\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=1\\h_{2}&=r\\h_{3}&=r\sin \theta \end{aligned}}}$
圆柱坐标系 ${\displaystyle (\rho ,\phi ,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty )}$	${\displaystyle {\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{3}=1\\h_{2}&=\rho \end{aligned}}}$
抛物柱面坐标系 ${\displaystyle (u,v,z)\in (-\infty ,\infty )\times [0,\infty )\times (-\infty ,\infty )}$	${\displaystyle {\begin{aligned}x&={\frac {1}{2}}(u^{2}-v^{2})\\y&=uv\\z&=z\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}={\sqrt {u^{2}+v^{2}}}\\h_{3}&=1\end{aligned}}}$
抛物线坐标系 ${\displaystyle (u,v,\phi )\in [0,\infty )\times [0,\infty )\times [0,2\pi )}$	${\displaystyle {\begin{aligned}x&=uv\cos \phi \\y&=uv\sin \phi \\z&={\frac {1}{2}}(u^{2}-v^{2})\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}={\sqrt {u^{2}+v^{2}}}\\h_{3}&=uv\end{aligned}}}$
椭圆柱坐标系 ${\displaystyle (u,v,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty )}$	${\displaystyle {\begin{aligned}x&=a\cosh u\cos v\\y&=a\sinh u\sin v\\z&=z\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}u+\sin ^{2}v}}\\h_{3}&=1\end{aligned}}}$
长球面坐标系 ${\displaystyle (\xi ,\eta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi )}$	${\displaystyle {\begin{aligned}x&=a\sinh \xi \sin \eta \cos \phi \\y&=a\sinh \xi \sin \eta \sin \phi \\z&=a\cosh \xi \cos \eta \end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}\xi +\sin ^{2}\eta }}\\h_{3}&=a\sinh \xi \sin \eta \end{aligned}}}$
扁球面坐标系 ${\displaystyle (\xi ,\eta ,\phi )\in [0,\infty )\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]\times [0,2\pi )}$	${\displaystyle {\begin{aligned}x&=a\cosh \xi \cos \eta \cos \phi \\y&=a\cosh \xi \cos \eta \sin \phi \\z&=a\sinh \xi \sin \eta \end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}\xi +\sin ^{2}\eta }}\\h_{3}&=a\cosh \xi \cos \eta \end{aligned}}}$
双极圆柱坐标系 ${\displaystyle (u,v,z)\in [0,2\pi )\times (-\infty ,\infty )\times (-\infty ,\infty )}$	${\displaystyle {\begin{aligned}x&={\frac {a\sinh v}{\cosh v-\cos u}}\\y&={\frac {a\sin u}{\cosh v-\cos u}}\\z&=z\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&=1\end{aligned}}}$
圆环坐标系 ${\displaystyle (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi )}$	${\displaystyle {\begin{aligned}x&={\frac {a\sinh v\cos \phi }{\cosh v-\cos u}}\\y&={\frac {a\sinh v\sin \phi }{\cosh v-\cos u}}\\z&={\frac {a\sin u}{\cosh v-\cos u}}\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&={\frac {a\sinh v}{\cosh v-\cos u}}\end{aligned}}}$
双球坐标系 ${\displaystyle (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi )}$	${\displaystyle {\begin{aligned}x&={\frac {a\sin u\cos \phi }{\cosh v-\cos u}}\\y&={\frac {a\sin u\sin \phi }{\cosh v-\cos u}}\\z&={\frac {a\sinh v}{\cosh v-\cos u}}\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&={\frac {a\sin u}{\cosh v-\cos u}}\end{aligned}}}$
圆锥坐标系 ${\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\nu ^{2}<b^{2}<\mu ^{2}<a^{2}\\&\lambda \in [0,\infty )\end{aligned}}}$	${\displaystyle {\begin{aligned}x&={\frac {\lambda \mu \nu }{ab}}\\y&={\frac {\lambda }{a}}{\sqrt {\frac {(\mu ^{2}-a^{2})(\nu ^{2}-a^{2})}{a^{2}-b^{2}}}}\\z&={\frac {\lambda }{b}}{\sqrt {\frac {(\mu ^{2}-b^{2})(\nu ^{2}-b^{2})}{a^{2}-b^{2}}}}\end{aligned}}}$	${\displaystyle {\begin{aligned}h_{1}&=1\\h_{2}^{2}&={\frac {\lambda ^{2}(\mu ^{2}-\nu ^{2})}{(\mu ^{2}-a^{2})(b^{2}-\mu ^{2})}}\\h_{3}^{2}&={\frac {\lambda ^{2}(\mu ^{2}-\nu ^{2})}{(\nu ^{2}-a^{2})(\nu ^{2}-b^{2})}}\end{aligned}}}$
抛物面坐标系 ${\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\lambda <b^{2}<\mu <a^{2}<\nu \end{aligned}}}$	${\displaystyle {\frac {x^{2}}{q_{i}-a^{2}}}+{\frac {y^{2}}{q_{i}-b^{2}}}=2z+q_{i}}$  其中 ${\displaystyle (q_{1},q_{2},q_{3})=(\lambda ,\mu ,\nu )}$	${\displaystyle h_{i}={\frac {1}{2}}{\sqrt {\frac {(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q_{i})}}}}$
椭球坐标系 ${\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\lambda <c^{2}<b^{2}<a^{2},\\&c^{2}<\mu <b^{2}<a^{2},\\&c^{2}<b^{2}<\nu <a^{2},\end{aligned}}}$	${\displaystyle {\frac {x^{2}}{a^{2}-q_{i}}}+{\frac {y^{2}}{b^{2}-q_{i}}}+{\frac {z^{2}}{c^{2}-q_{i}}}=1}$  其中 ${\displaystyle (q_{1},q_{2},q_{3})=(\lambda ,\mu ,\nu )}$	${\displaystyle h_{i}={\frac {1}{2}}{\sqrt {\frac {(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q_{i})(c^{2}-q_{i})}}}}$

微分算子导引

梯度導引

一個函數${\displaystyle \phi }$ 的梯度朝某個方向${\displaystyle {\hat {\mathbf {n} }}}$ 的分量，等於方向導數 ${\displaystyle {\frac {d\phi }{ds}}}$ 朝${\displaystyle {\hat {\mathbf {n} }}}$ 方向的值：

${\displaystyle \nabla \Phi \cdot {\hat {\mathbf {n} }}={\frac {d\phi }{ds}}}$ ；

其中，${\displaystyle ds}$ 是朝${\displaystyle {\hat {\mathbf {n} }}}$ 方向的無窮小位移。

假若，這${\displaystyle {\hat {\mathbf {n} }}}$ 與正交坐標軸${\displaystyle {\hat {\mathbf {e} }}_{i}}$ 同方向。那麼，${\displaystyle ds=h_{i}dq_{i}}$ 。所以，函數${\displaystyle \phi }$ 的梯度朝${\displaystyle {\hat {\mathbf {e} }}_{i}}$ 的分量是${\displaystyle {\frac {\partial \phi }{h_{i}\partial q_{i}}}}$ ；也就是說，

${\displaystyle \nabla \Phi ={\hat {\mathbf {e} }}_{1}{\frac {1}{h_{1}}}{\frac {\partial \Phi }{\partial q_{1}}}+{\hat {\mathbf {e} }}_{2}{\frac {1}{h_{2}}}{\frac {\partial \Phi }{\partial q_{2}}}+{\hat {\mathbf {e} }}_{3}{\frac {1}{h_{3}}}{\frac {\partial \Phi }{\partial q_{3}}}}$ 。

散度導引

${\displaystyle \nabla \cdot \mathbf {F} =\nabla \cdot ({\hat {\mathbf {e} }}_{1}F_{1}+{\hat {\mathbf {e} }}_{2}F_{2}+{\hat {\mathbf {e} }}_{3}F_{3})}$ 。

取右手邊第一個項目，

${\displaystyle \nabla \cdot ({\hat {\mathbf {e} }}_{1}F_{1})=\nabla \cdot \left[\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{2}h_{3}}}\right)\left(h_{2}h_{3}F_{1}\right)\right]}$ 。

應用向量恆等式${\displaystyle \nabla \cdot (\mathbf {A} \phi )=\phi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot (\nabla \phi )}$ 與${\displaystyle \nabla \cdot (\nabla \phi _{1}\times \nabla \phi _{2})=0}$ ，可以得到

${\displaystyle {\begin{aligned}\nabla \cdot ({\hat {\mathbf {e} }}_{1}F_{1})&=(h_{2}h_{3}F_{1})\nabla \cdot \left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{2}h_{3}}}\right)+\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{2}h_{3}}}\right)\cdot \nabla (h_{2}h_{3}F_{1})\\&=(h_{2}h_{3}F_{1})\nabla \cdot [(\nabla q_{2})\times \nabla (q_{3})]+\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{2}h_{3}}}\right)\cdot \nabla (h_{2}h_{3}F_{1})\\&=\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{2}h_{3}}}\right)\cdot \nabla (h_{2}h_{3}F_{1})\\&={\frac {1}{h_{1}h_{2}h_{3}}}{\frac {\partial }{\partial q_{1}}}(F_{1}h_{2}h_{3})\\\end{aligned}}}$  。

總合所有項目，

${\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{h_{1}h_{2}h_{3}}}\left[{\frac {\partial }{\partial q_{1}}}(F_{1}h_{2}h_{3})+{\frac {\partial }{\partial q_{2}}}(F_{2}h_{3}h_{1})+{\frac {\partial }{\partial q_{3}}}(F_{3}h_{1}h_{2})\right]}$ 。

旋度導引

${\displaystyle \nabla \times \mathbf {F} =\nabla \times ({\hat {\mathbf {e} }}_{1}F_{1}+{\hat {\mathbf {e} }}_{2}F_{2}+{\hat {\mathbf {e} }}_{3}F_{3})}$ 。

取右手邊第一個項目，

${\displaystyle \nabla \times ({\hat {\mathbf {e} }}_{1}F_{1})=\nabla \times \left[\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{1}}}\right)\left(h_{1}F_{1}\right)\right]}$ 。

應用向量恆等式${\displaystyle \nabla \times (\mathbf {A} \phi )=\phi \nabla \times \mathbf {A} -\mathbf {A} \times (\nabla \phi )}$ ，

${\displaystyle {\begin{aligned}\nabla \times ({\hat {\mathbf {e} }}_{1}F_{1})&=(h_{1}F_{1})\nabla \times \left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{1}}}\right)-\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{1}}}\right)\times \nabla (h_{1}F_{1})\\&=(h_{1}F_{1})\nabla \times (\nabla q_{1})-\left({\frac {{\hat {\mathbf {e} }}_{1}}{h_{1}}}\right)\times \left({\frac {{\hat {\mathbf {e} }}_{2}}{h_{2}}}{\frac {\partial }{\partial q_{2}}}(h_{1}F_{1})+{\frac {{\hat {\mathbf {e} }}_{3}}{h_{3}}}{\frac {\partial }{\partial q_{3}}}(h_{1}F_{1})\right)\\\end{aligned}}}$ 。

應用向量恆等式${\displaystyle \nabla \times (\nabla \phi )=0}$ ，

${\displaystyle \nabla \times ({\hat {\mathbf {e} }}_{1}F_{1})={\frac {{\hat {\mathbf {e} }}_{2}}{h_{1}h_{3}}}{\frac {\partial }{\partial q_{3}}}(h_{1}F_{1})-{\frac {{\hat {\mathbf {e} }}_{3}}{h_{1}h_{2}}}{\frac {\partial }{\partial q_{2}}}(h_{1}F_{1})}$ 。

總合所有項目，

${\displaystyle {\begin{aligned}\nabla \times \mathbf {F} &={\frac {\mathbf {e} _{1}}{h_{2}h_{3}}}\left[{\frac {\partial }{\partial q_{2}}}\left(h_{3}F_{3}\right)-{\frac {\partial }{\partial q_{3}}}\left(h_{2}F_{2}\right)\right]+{\frac {\mathbf {e} _{2}}{h_{3}h_{1}}}\left[{\frac {\partial }{\partial q_{3}}}\left(h_{1}F_{1}\right)-{\frac {\partial }{\partial q_{1}}}\left(h_{3}F_{3}\right)\right]\\&+{\frac {\mathbf {e} _{3}}{h_{1}h_{2}}}\left[{\frac {\partial }{\partial q_{1}}}\left(h_{2}F_{2}\right)-{\frac {\partial }{\partial q_{2}}}\left(h_{1}F_{1}\right)\right]\\\end{aligned}}}$  。

拉普拉斯算子

${\displaystyle \nabla ^{2}\Phi ={\frac {1}{h_{1}h_{2}h_{3}}}\left[{\frac {\partial }{\partial q_{1}}}\left({\frac {h_{2}h_{3}}{h_{1}}}{\frac {\partial \Phi }{\partial q_{1}}}\right)+{\frac {\partial }{\partial q_{2}}}\left({\frac {h_{3}h_{1}}{h_{2}}}{\frac {\partial \Phi }{\partial q_{2}}}\right)+{\frac {\partial }{\partial q_{3}}}\left({\frac {h_{1}h_{2}}{h_{3}}}{\frac {\partial \Phi }{\partial q_{3}}}\right)\right]}$ 。

引用

1. Eric W. Weisstein. Orthogonal Coordinate System. MathWorld. [10 July 2008]. （原始内容于2014-11-12）. 
2. Morse and Feshbach 1953，Volume 1, pp. 494-523, 655-666. harvnb error: no target: CITEREFMorse_and_Feshbach1953 (help)
3. Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7

参见

• 坐标系
• 曲线坐标系
• 斜交坐标系（度規張量有非對角項目）
• 在圆柱和球坐标系中的del

參考文獻

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill, pp. 164-182。

• Morse PM and Feshbach H. (1953) Methods of Theoretical Physics, McGraw-Hill, pp. 494-523, 655-666。

• Margenau H. and Murphy GM. (1956) The Mathematics of Physics and Chemistry, 2nd. ed., Van Nostrand, pp. 172-192。