# 正交座標系

## 概述

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

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

## 向量代数

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

## 向量微積分

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

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

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

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

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

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

### 球坐標系實例

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

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

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

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

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

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

## 二维正交坐标系表格

${\displaystyle x+iy=f(u+iv)}$

${\displaystyle u}$${\displaystyle v}$等值线的形状 注释

${\displaystyle \exp(u+iv)}$ 圆, 直线 ${\displaystyle u=\ln r}$则为极坐标系

${\displaystyle {\sqrt {u+iv}}}$ 双曲线, 双曲线
${\displaystyle u=x^{2}+2y^{2},\ y=vx^{2}}$ 椭圆, 抛物线

sqrt(u+iv)

## 三维正交坐标系表格

${\displaystyle (r,\theta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi )}$

{\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}}}

${\displaystyle (\rho ,\phi ,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty )}$

{\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}}}

${\displaystyle (u,v,z)\in (-\infty ,\infty )\times [0,\infty )\times (-\infty ,\infty )}$

{\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}}}

${\displaystyle (u,v,\phi )\in [0,\infty )\times [0,\infty )\times [0,2\pi )}$

{\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}}}

${\displaystyle (u,v,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty )}$

{\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}}}

${\displaystyle (\xi ,\eta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi )}$

{\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}}}

${\displaystyle (\xi ,\eta ,\phi )\in [0,\infty )\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]\times [0,2\pi )}$

{\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}}}

${\displaystyle (u,v,z)\in [0,2\pi )\times (-\infty ,\infty )\times (-\infty ,\infty )}$

{\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}}}

${\displaystyle (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi )}$

{\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}}}

${\displaystyle (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi )}$

{\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}}}

{\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\nu ^{2}

{\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}}}

{\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\lambda

${\displaystyle {\frac {x^{2}}{q_{i}-a^{2}}}+{\frac {y^{2}}{q_{i}-b^{2}}}=2z+q_{i}}$

${\displaystyle h_{i}={\frac {1}{2}}{\sqrt {\frac {(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q_{i})}}}}$

{\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\lambda

${\displaystyle {\frac {x^{2}}{a^{2}-q_{i}}}+{\frac {y^{2}}{b^{2}-q_{i}}}+{\frac {z^{2}}{c^{2}-q_{i}}}=1}$

${\displaystyle h_{i}={\frac {1}{2}}{\sqrt {\frac {(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q_{i})(c^{2}-q_{i})}}}}$

## 微分算子导引

### 梯度導引

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

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

### 散度導引

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

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

{\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}}}

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

### 旋度導引

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

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

{\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}}}

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

{\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.
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

## 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。