正交座标系

在数学里，一个正交坐标系定义为一组正交坐标${\displaystyle \mathbf {q} =(q_{1},\ q_{2},\ q_{3},\ \dots \ q_{n})}$，其坐标曲面都以直角相交（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）。坐标曲面定义为特定坐标${\displaystyle q_{i}}$的等值曲面，即${\displaystyle q_{i}}$为常数的曲线、曲面或超曲面。例如，三维直角坐标${\displaystyle (x,\ y,\ z)}$是一种正交坐标系，它的${\displaystyle x}$为常数，${\displaystyle y}$为常数，${\displaystyle z}$为常数的坐标曲面，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。

动机

正交座标时常用来解析一些出现于量子力学、流体动力学、电动力学、热力学等等的偏微分方程。举例而言，选择一个恰当的的正交座标来解析氢离子${\displaystyle H_{2}\,^{-}}$的波函数或消防水管的喷水，也许会比用直角座标方便的多。这主要是因为恰当的正交座标能够与一个问题的对称性相配合，从而促使应用分离变数法来成功的解析关于这问题的方程式。分离变数法是一种数学技巧，专门用来将一个复杂的${\displaystyle n}$维问题变为${\displaystyle n}$个一维问题。很多问题都可以简化为拉普拉斯方程或亥姆霍兹方程，这些方程式可以用很多种正交座标来分离。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），而亥姆霍兹方程可以在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。

正交坐标的度规张量绝对没有非对角项目。换句话说，无穷小距离的平方${\displaystyle ds^{2}}$，可以写为无穷小坐标位移的平方和：

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

其中，${\displaystyle n}$是维数，标度因子${\displaystyle h_{i}}$是度规张量的对角元素${\displaystyle g_{ii}}$的平方根：

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

这些标度因子可以用来计算一个正交坐标系的微分算子。例如，梯度、拉普拉斯算子、散度、或旋度。

在数学里，存在有各种各样的正交座标系。应用二维直角座标系${\displaystyle (x,\ y)}$的共形映射方法，可以简易的生成这些正交座标系。一个复数${\displaystyle z=x+iy}$的任何全纯函数${\displaystyle w=f(z)}$，其复值的导数，如果不等于零，则会造成一个共形映射。如果答案可以表达为${\displaystyle w=u+iv}$，则${\displaystyle u}$与${\displaystyle v}$的等值曲线以直角相交，就如同原本的${\displaystyle x}$与${\displaystyle y}$的等值曲线以直角相交。

三维与更高维的正交座标系可以由一个二维正交座标系生成，只要将二维正交座标往一个新的座标轴投射（形成类似圆柱座标系的座标系），或者将二维正交座标绕著其对称轴旋转。可是，也有一些三维正交座标系，例如椭球座标系，则不能够用上述方法得到。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

在正交坐标系里，内积的公式仍旧不变：

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

向量微积分

从前面的距离公式，可以观察出，一个正交坐标${\displaystyle q_{i}}$的无穷小改变${\displaystyle dq_{i}}$，其相伴的长度是${\displaystyle ds_{i}=h_{i}dq_{i}}$。因此，一个位移向量的全微分${\displaystyle d\mathbf {r} }$等于

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

其中，${\displaystyle \mathbf {e} _{i}}$是垂直于${\displaystyle q_{i}}$等值曲面的单位向量，指向著${\displaystyle q_{i}}$增值最快的方向，这些单位向量形成了一个局部直角坐标系的坐标轴。

因此，向量${\displaystyle \mathbf {F} }$沿著周线${\displaystyle \mathbb {C} }$的线积分等于

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

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

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

类似地，一个无穷小面积元素是

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

一个无穷小体积元素是

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

例如，向量${\displaystyle \mathbf {F} }$对于一个曲面${\displaystyle \mathbb {S} }$的曲面积分是

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

球坐标系实例

直角坐标${\displaystyle (x,\ y,\ z)}$与球坐标${\displaystyle (r,\ \theta ,\phi )}$的变换方程式为

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

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

${\displaystyle z=r\cos \theta }$。

直角坐标的全微分是

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

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

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

所以，无穷小距离的平方是

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

标度因子是

${\displaystyle h_{r}=1}$、

${\displaystyle h_{\theta }=r}$、

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

向量${\displaystyle \mathbf {F} }$沿著周线${\displaystyle \mathbb {C} }$的线积分等于

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

向量${\displaystyle \mathbf {F} }$对于一个曲面${\displaystyle \mathbb {S} }$的曲面积分是

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

三维微分算子

主条目：向量分析和Nabla算子

算子	正交坐标公式
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维-奇维塔符号${\displaystyle \epsilon }$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	${\displaystyle \nabla ^{2}\phi ={\frac {1}{H}}{\frac {\partial }{\partial q^{k}}}\left({\frac {H}{h_{k}^{2}}}{\frac {\partial \phi }{\partial q^{k}}}\right)}$

二维正交坐标系表格

坐标系	复数变换 ${\displaystyle x+iy=f(u+iv)}$	${\displaystyle u}$和${\displaystyle v}$等值线的形状	注释
直角	${\displaystyle u+iv}$	直线, 直线
对数极（英语：Log-polar coordinates）	${\displaystyle \exp(u+iv)}$	圆, 直线	若${\displaystyle u=\ln r}$则为极坐标系
抛物线	${\displaystyle {\frac {1}{2}}(u+iv)^{2}}$	抛物线, 抛物线
点偶极	${\displaystyle (u+iv)^{-1}}$	圆, 圆
椭圆	${\displaystyle \cosh(u+iv)}$	椭圆, 双曲线	对于大距离看似对数极
双极	${\displaystyle \coth(u+iv)}$	圆, 圆	对于大距离看似点偶极
	${\displaystyle {\sqrt {u+iv}}}$	双曲线, 双曲线
	${\displaystyle u=x^{2}+2y^{2},\ y=vx^{2}}$	椭圆, 抛物线

直角

单极

对数极

椭圆-抛物线

抛物线

点偶极

sqrt(u+iv)

椭圆

双极

反对数极

三维正交坐标系表格

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

梯度导引

一个函数${\displaystyle \phi }$的梯度朝某个方向${\displaystyle {\hat {\mathbf {n} }}}$的分量，等于方向导数 ${\displaystyle {\frac {d\phi }{ds}}}$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${\displaystyle {\hat {\mathbf {n} }}}$方向的无穷小位移。

假若，这${\displaystyle {\hat {\mathbf {n} }}}$与正交坐标轴${\displaystyle {\hat {\mathbf {e} }}_{i}}$同方向。那么，${\displaystyle ds=h_{i}dq_{i}}$。所以，函数${\displaystyle \phi }$的梯度朝${\displaystyle {\hat {\mathbf {e} }}_{i}}$的分量是${\displaystyle {\frac {\partial \phi }{h_{i}\partial q_{i}}}}$；也就是说，

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

散度导引

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

取右手边第一个项目，

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

应用向量恒等式${\displaystyle \nabla \cdot (\mathbf {A} \phi )=\phi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot (\nabla \phi )}$与${\displaystyle \nabla \cdot (\nabla \phi _{1}\times \nabla \phi _{2})=0}$，可以得到

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

总合所有项目，

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

旋度导引

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

取右手边第一个项目，

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

应用向量恒等式${\displaystyle \nabla \times (\mathbf {A} \phi )=\phi \nabla \times \mathbf {A} -\mathbf {A} \times (\nabla \phi )}$，

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

应用向量恒等式${\displaystyle \nabla \times (\nabla \phi )=0}$，

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

总合所有项目，

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

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]. （原始内容存档于2014-11-12）.
2. Morse and Feshbach 1953，Volume 1, pp. 494-523, 655-666. harvnb模板错误: 无指向目标: CITEREFMorse_and_Feshbach1953 (帮助)
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（度规张量有非对角项目）
• 在圆柱和球坐标系中的del

参考文献

• 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。