在數學裏，一個正交坐標系定義為一組正交坐標 q = ( q 1 , q 2 , q 3 , … q n ) {\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：很多作者採用愛因斯坦記號對坐標標號使用上標並非表示指數）。坐標曲面定義為特定坐標 q i {\displaystyle q_{i}} 的等值曲面，即 q i {\displaystyle q_{i}} 為常數的曲線、曲面或超曲面。例如，三維直角坐標 ( x , y , z ) {\displaystyle (x,\ y,\ z)} 是一種正交坐標系，它的 x {\displaystyle x} 為常數， y {\displaystyle y} 為常數， z {\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。正交坐標系是曲線坐標系的特殊的但極其常見的形式。 Remove ads 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 H 2 − {\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量法來成功的解析關於這問題的方程式。分離變量法是一種數學技巧，專門用來將一個複雜的 n {\displaystyle n} 維問題變為 n {\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個正交坐標系中分離（本文列出的14個中圓環坐標系除外），而亥姆霍茲方程可以在11個正交坐標系中分離[1][2]。 Remove ads 共形映射作用於矩形網格。注意，彎曲的網格的正交性被保留。 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 d s 2 {\displaystyle ds^{2}} ，可以寫為無窮小坐標位移的平方和： d s 2 = ∑ i = 1 n ( h i d q i ) 2 {\displaystyle ds^{2}=\sum _{i=1}^{n}\left(h_{i}dq_{i}\right)^{2}} ； 其中， n {\displaystyle n} 是維數，標度因子 h i {\displaystyle h_{i}} 是度規張量的對角元素 g i i {\displaystyle g_{ii}} 的平方根： h i ( q ) = d e f g i i ( q ) {\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 ( x , y ) {\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 z = x + i y {\displaystyle z=x+iy} 的任何全純函數 w = f ( z ) {\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 w = u + i v {\displaystyle w=u+iv} ，則 u {\displaystyle u} 與 v {\displaystyle v} 的等值曲線以直角相交，就如同原本的 x {\displaystyle x} 與 y {\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着其對稱軸旋轉。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。更一般的正交坐標可以從一些必要的坐標曲面/曲線起步並通過考慮它們的正交軌跡線（英語：Orthogonal trajectory）而得到。 Remove ads 在正交坐標系裏，內積的公式仍舊不變： A ⋅ B = ∑ i = 1 n A i B i {\displaystyle \mathbf {A} \cdot \mathbf {B} =\sum _{i=1}^{n}A_{i}B_{i}} 。 從前面的距離公式，可以觀察出，一個正交坐標 q i {\displaystyle q_{i}} 的無窮小改變 d q i {\displaystyle dq_{i}} ，其相伴的長度是 d s i = h i d q i {\displaystyle ds_{i}=h_{i}dq_{i}} 。因此，一個位移向量的全微分 d r {\displaystyle d\mathbf {r} } 等於 d r = ∑ i = 1 n h i d q i e i {\displaystyle d\mathbf {r} =\sum _{i=1}^{n}h_{i}dq_{i}\mathbf {e} _{i}} ； 其中， e i {\displaystyle \mathbf {e} _{i}} 是垂直於 q i {\displaystyle q_{i}} 等值曲面的單位向量，指向着 q i {\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 F {\displaystyle \mathbf {F} } 沿着周線 C {\displaystyle \mathbb {C} } 的線積分等於 ∫ C F ⋅ d r = ∑ i = 1 n ∫ C F i h i d q i {\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， F i {\displaystyle F_{i}} 是向量 F {\displaystyle \mathbf {F} } 在單位向量 e i {\displaystyle \mathbf {e} _{i}} 方向的分量： F i = d e f e i ⋅ F {\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 d A = d s i d s j = h i h j d q i d q j , i ≠ j {\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 d V = d s i d s j d s k = h i h j h k d q i d q j d q k , i ≠ j ≠ k {\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 F {\displaystyle \mathbf {F} } 對於一個曲面 S {\displaystyle \mathbb {S} } 的曲面積分是 ∫ S F ⋅ d A = ∫ S F 1 h 2 h 3 d q 2 d q 3 + ∫ S F 2 h 3 h 1 d q 3 d q 1 + ∫ S F 3 h 1 h 2 d q 1 d q 2 {\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}} 。 Remove ads球坐標系實例 直角坐標 ( x , y , z ) {\displaystyle (x,\ y,\ z)} 與球坐標 ( r , θ , ϕ ) {\displaystyle (r,\ \theta ,\phi )} 的變換方程式為 x = r sin θ cos ϕ {\displaystyle x=r\sin \theta \cos \phi } 、 y = r sin θ sin ϕ {\displaystyle y=r\sin \theta \sin \phi } 、 z = r cos θ {\displaystyle z=r\cos \theta } 。 直角坐標的全微分是 d x = sin θ cos ϕ d r + r cos θ cos ϕ d θ − r sin θ sin ϕ d ϕ {\displaystyle dx=\sin \theta \cos \phi dr+r\cos \theta \cos \phi d\theta -r\sin \theta \sin \phi d\phi } 、 d y = sin θ sin ϕ d r + r cos θ sin ϕ d θ + r sin θ cos ϕ d ϕ {\displaystyle dy=\sin \theta \sin \phi dr+r\cos \theta \sin \phi d\theta +r\sin \theta \cos \phi d\phi } 、 d z = cos θ d r − r sin θ d θ {\displaystyle dz=\cos \theta dr-r\sin \theta d\theta } 。 所以，無窮小距離的平方是 d s 2 = d x 2 + d y 2 + d z 2 = d r 2 + ( r d θ ) 2 + ( r sin θ d ϕ ) 2 {\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 h r = 1 {\displaystyle h_{r}=1} 、 h θ = r {\displaystyle h_{\theta }=r} 、 h ϕ = r sin θ {\displaystyle h_{\phi }=r\sin \theta } 。 向量 F {\displaystyle \mathbf {F} } 沿着周線 C {\displaystyle \mathbb {C} } 的線積分等於 ∫ C F ⋅ d r = ∫ C F r d r + F θ r d θ + F ϕ r sin θ d ϕ {\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 F {\displaystyle \mathbf {F} } 對於一個曲面 S {\displaystyle \mathbb {S} } 的曲面積分是 ∫ S F ⋅ d A = ∫ S F r r 2 sin θ d θ d ϕ + ∫ S F θ r sin θ d r d ϕ + ∫ S F ϕ r d r d θ {\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 } 。 Remove ads 主條目：向量分析和Nabla算子 更多資訊 , ... 算子 正交坐標公式 純量場的梯度 ∇ Φ = e ^ 1 1 h 1 ∂ Φ ∂ q 1 + e ^ 2 1 h 2 ∂ Φ ∂ q 2 + e ^ 3 1 h 3 ∂ Φ ∂ q 3 {\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}}}} 向量場的散度 ∇ ⋅ F = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ( F 1 h 2 h 3 ) + ∂ ∂ q 2 ( F 2 h 3 h 1 ) + ∂ ∂ q 3 ( F 3 h 1 h 2 ) ] {\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]} 向量場的旋度 ∇ × F = e 1 h 2 h 3 [ ∂ ∂ q 2 ( h 3 F 3 ) − ∂ ∂ q 3 ( h 2 F 2 ) ] + e 2 h 3 h 1 [ ∂ ∂ q 3 ( h 1 F 1 ) − ∂ ∂ q 1 ( h 3 F 3 ) ] + e 3 h 1 h 2 [ ∂ ∂ q 1 ( h 2 F 2 ) − ∂ ∂ 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}}} 純量場的拉普拉斯算子 ∇ 2 Φ = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ( h 2 h 3 h 1 ∂ Φ ∂ q 1 ) + ∂ ∂ q 2 ( h 3 h 1 h 2 ∂ Φ ∂ q 2 ) + ∂ ∂ q 3 ( h 1 h 2 h 3 ∂ Φ ∂ q 3 ) ] {\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]} 關閉 上面表達式可以使用列維-奇維塔符號 ϵ {\displaystyle \epsilon } 的更簡潔形式書寫，定義 H = h 1 h 2 h 3 {\displaystyle H=h_{1}h_{2}h_{3}} ，並使用愛因斯坦記號，即在同時出現上標和下標的項目上求此項所有可能的總和: 更多資訊 , ... 算子 表達式 純量場的梯度 ∇ ϕ = e ^ k h k ∂ ϕ ∂ q k {\displaystyle \nabla \phi ={\frac {{\hat {\mathbf {e} }}_{k}}{h_{k}}}{\frac {\partial \phi }{\partial q^{k}}}} 向量場的散度 ∇ ⋅ F = 1 H ∂ ∂ q k ( H h k F k ) {\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{H}}{\frac {\partial }{\partial q^{k}}}\left({\frac {H}{h_{k}}}F_{k}\right)} 向量場（只3D）的旋度 ( ∇ × F ) k = h k e ^ k H ϵ i j k ∂ ∂ q i ( h j F j ) {\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)} 純量場的拉普拉斯算子 ∇ 2 ϕ = 1 H ∂ ∂ q k ( H h k 2 ∂ ϕ ∂ q k ) {\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)} 關閉 更多資訊 , ... 坐標系 複數轉換 x + i y = f ( u + i v ) {\displaystyle x+iy=f(u+iv)} u {\displaystyle u} 和 v {\displaystyle v} 等值線的形狀 註釋 直角 u + i v {\displaystyle u+iv} 直線, 直線 對數極（英語：Log-polar coordinates） exp ( u + i v ) {\displaystyle \exp(u+iv)} 圓, 直線 若 u = ln r {\displaystyle u=\ln r} 則為極坐標系 拋物線 1 2 ( u + i v ) 2 {\displaystyle {\frac {1}{2}}(u+iv)^{2}} 拋物線, 拋物線 點偶極 ( u + i v ) − 1 {\displaystyle (u+iv)^{-1}} 圓, 圓 橢圓 cosh ( u + i v ) {\displaystyle \cosh(u+iv)} 橢圓, 雙曲線 對於大距離看似對數極 雙極 coth ( u + i v ) {\displaystyle \coth(u+iv)} 圓, 圓 對於大距離看似點偶極 u + i v {\displaystyle {\sqrt {u+iv}}} 雙曲線, 雙曲線 u = x 2 + 2 y 2 , y = v x 2 {\displaystyle u=x^{2}+2y^{2},\ y=vx^{2}} 橢圓, 拋物線 關閉 直角單極對數極橢圓-拋物線拋物線點偶極sqrt(u+iv)橢圓雙極反對數極 Remove ads 除了直角坐標系之外，下表列出其他常見的正交坐標系[3]，為了簡明性在坐標列中使用了區間符號。 更多資訊 , ... 曲線坐標 (q1, q2, q3) 從直角坐標(x, y, z)轉換 縮放因子 球極坐標系 ( r , θ , ϕ ) ∈ [ 0 , ∞ ) × [ 0 , π ] × [ 0 , 2 π ) {\displaystyle (r,\theta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi )} x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ {\displaystyle {\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}} h 1 = 1 h 2 = r h 3 = r sin θ {\displaystyle {\begin{aligned}h_{1}&=1\\h_{2}&=r\\h_{3}&=r\sin \theta \end{aligned}}} 圓柱坐標系 ( ρ , ϕ , z ) ∈ [ 0 , ∞ ) × [ 0 , 2 π ) × ( − ∞ , ∞ ) {\displaystyle (\rho ,\phi ,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty )} x = ρ cos ϕ y = ρ sin ϕ z = z {\displaystyle {\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}} h 1 = h 3 = 1 h 2 = ρ {\displaystyle {\begin{aligned}h_{1}&=h_{3}=1\\h_{2}&=\rho \end{aligned}}} 拋物柱面坐標系 ( u , v , z ) ∈ ( − ∞ , ∞ ) × [ 0 , ∞ ) × ( − ∞ , ∞ ) {\displaystyle (u,v,z)\in (-\infty ,\infty )\times [0,\infty )\times (-\infty ,\infty )} x = 1 2 ( u 2 − v 2 ) y = u v z = z {\displaystyle {\begin{aligned}x&={\frac {1}{2}}(u^{2}-v^{2})\\y&=uv\\z&=z\end{aligned}}} h 1 = h 2 = u 2 + v 2 h 3 = 1 {\displaystyle {\begin{aligned}h_{1}&=h_{2}={\sqrt {u^{2}+v^{2}}}\\h_{3}&=1\end{aligned}}} 拋物線坐標系 ( u , v , ϕ ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) × [ 0 , 2 π ) {\displaystyle (u,v,\phi )\in [0,\infty )\times [0,\infty )\times [0,2\pi )} x = u v cos ϕ y = u v sin ϕ z = 1 2 ( u 2 − v 2 ) {\displaystyle {\begin{aligned}x&=uv\cos \phi \\y&=uv\sin \phi \\z&={\frac {1}{2}}(u^{2}-v^{2})\end{aligned}}} h 1 = h 2 = u 2 + v 2 h 3 = u v {\displaystyle {\begin{aligned}h_{1}&=h_{2}={\sqrt {u^{2}+v^{2}}}\\h_{3}&=uv\end{aligned}}} 橢圓柱坐標系 ( u , v , z ) ∈ [ 0 , ∞ ) × [ 0 , 2 π ) × ( − ∞ , ∞ ) {\displaystyle (u,v,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty )} x = a cosh u cos v y = a sinh u sin v z = z {\displaystyle {\begin{aligned}x&=a\cosh u\cos v\\y&=a\sinh u\sin v\\z&=z\end{aligned}}} h 1 = h 2 = a sinh 2 u + sin 2 v h 3 = 1 {\displaystyle {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}u+\sin ^{2}v}}\\h_{3}&=1\end{aligned}}} 長球面坐標系 ( ξ , η , ϕ ) ∈ [ 0 , ∞ ) × [ 0 , π ] × [ 0 , 2 π ) {\displaystyle (\xi ,\eta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi )} x = a sinh ξ sin η cos ϕ y = a sinh ξ sin η sin ϕ z = a cosh ξ cos η {\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}}} h 1 = h 2 = a sinh 2 ξ + sin 2 η h 3 = a sinh ξ sin η {\displaystyle {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}\xi +\sin ^{2}\eta }}\\h_{3}&=a\sinh \xi \sin \eta \end{aligned}}} 扁球面坐標系 ( ξ , η , ϕ ) ∈ [ 0 , ∞ ) × [ − π 2 , π 2 ] × [ 0 , 2 π ) {\displaystyle (\xi ,\eta ,\phi )\in [0,\infty )\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]\times [0,2\pi )} x = a cosh ξ cos η cos ϕ y = a cosh ξ cos η sin ϕ z = a sinh ξ sin η {\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}}} h 1 = h 2 = a sinh 2 ξ + sin 2 η h 3 = a cosh ξ cos η {\displaystyle {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}\xi +\sin ^{2}\eta }}\\h_{3}&=a\cosh \xi \cos \eta \end{aligned}}} 雙極圓柱坐標系 ( u , v , z ) ∈ [ 0 , 2 π ) × ( − ∞ , ∞ ) × ( − ∞ , ∞ ) {\displaystyle (u,v,z)\in [0,2\pi )\times (-\infty ,\infty )\times (-\infty ,\infty )} x = a sinh v cosh v − cos u y = a sin u cosh v − cos u z = z {\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}}} h 1 = h 2 = a cosh v − cos u h 3 = 1 {\displaystyle {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&=1\end{aligned}}} 圓環坐標系 ( u , v , ϕ ) ∈ ( − π , π ] × [ 0 , ∞ ) × [ 0 , 2 π ) {\displaystyle (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi )} x = a sinh v cos ϕ cosh v − cos u y = a sinh v sin ϕ cosh v − cos u z = a sin u cosh v − cos u {\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}}} h 1 = h 2 = a cosh v − cos u h 3 = a sinh v cosh v − cos u {\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}}} 雙球坐標系 ( u , v , ϕ ) ∈ ( − π , π ] × [ 0 , ∞ ) × [ 0 , 2 π ) {\displaystyle (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi )} x = a sin u cos ϕ cosh v − cos u y = a sin u sin ϕ cosh v − cos u z = a sinh v cosh v − cos u {\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}}} h 1 = h 2 = a cosh v − cos u h 3 = a sin u cosh v − cos u {\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}}} 圓錐坐標系 ( λ , μ , ν ) ν 2 < b 2 < μ 2 < a 2 λ ∈ [ 0 , ∞ ) {\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\nu ^{2}<b^{2}<\mu ^{2}<a^{2}\\&\lambda \in [0,\infty )\end{aligned}}} x = λ μ ν a b y = λ a ( μ 2 − a 2 ) ( ν 2 − a 2 ) a 2 − b 2 z = λ b ( μ 2 − b 2 ) ( ν 2 − b 2 ) a 2 − b 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}}} h 1 = 1 h 2 2 = λ 2 ( μ 2 − ν 2 ) ( μ 2 − a 2 ) ( b 2 − μ 2 ) h 3 2 = λ 2 ( μ 2 − ν 2 ) ( ν 2 − a 2 ) ( ν 2 − b 2 ) {\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}}} 拋物面坐標系 ( λ , μ , ν ) λ < b 2 < μ < a 2 < ν {\displaystyle {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\lambda <b^{2}<\mu <a^{2}<\nu \end{aligned}}} x 2 q i − a 2 + y 2 q i − b 2 = 2 z + q i {\displaystyle {\frac {x^{2}}{q_{i}-a^{2}}}+{\frac {y^{2}}{q_{i}-b^{2}}}=2z+q_{i}} 其中 ( q 1 , q 2 , q 3 ) = ( λ , μ , ν ) {\displaystyle (q_{1},q_{2},q_{3})=(\lambda ,\mu ,\nu )} h i = 1 2 ( q j − q i ) ( q k − q i ) ( a 2 − q i ) ( b 2 − 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})}}}} 橢球坐標系 ( λ , μ , ν ) λ < c 2 < b 2 < a 2 , c 2 < μ < b 2 < a 2 , c 2 < b 2 < ν < a 2 , {\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}}} x 2 a 2 − q i + y 2 b 2 − q i + z 2 c 2 − q i = 1 {\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 ( q 1 , q 2 , q 3 ) = ( λ , μ , ν ) {\displaystyle (q_{1},q_{2},q_{3})=(\lambda ,\mu ,\nu )} h i = 1 2 ( q j − q i ) ( q k − q i ) ( a 2 − q i ) ( b 2 − q i ) ( c 2 − 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})(c^{2}-q_{i})}}}} 關閉 梯度導引 一個函數 ϕ {\displaystyle \phi } 的梯度朝某個方向 n ^ {\displaystyle {\hat {\mathbf {n} }}} 的分量，等於方向導數 d ϕ d s {\displaystyle {\frac {d\phi }{ds}}} 朝 n ^ {\displaystyle {\hat {\mathbf {n} }}} 方向的值： ∇ Φ ⋅ n ^ = d ϕ d s {\displaystyle \nabla \Phi \cdot {\hat {\mathbf {n} }}={\frac {d\phi }{ds}}} ； 其中， d s {\displaystyle ds} 是朝 n ^ {\displaystyle {\hat {\mathbf {n} }}} 方向的無窮小位移。 假若，這 n ^ {\displaystyle {\hat {\mathbf {n} }}} 與正交坐標軸 e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} 同方向。那麼， d s = h i d q i {\displaystyle ds=h_{i}dq_{i}} 。所以，函數 ϕ {\displaystyle \phi } 的梯度朝 e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} 的分量是 ∂ ϕ h i ∂ q i {\displaystyle {\frac {\partial \phi }{h_{i}\partial q_{i}}}} ；也就是說， ∇ Φ = e ^ 1 1 h 1 ∂ Φ ∂ q 1 + e ^ 2 1 h 2 ∂ Φ ∂ q 2 + e ^ 3 1 h 3 ∂ Φ ∂ q 3 {\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 ∇ ⋅ F = ∇ ⋅ ( e ^ 1 F 1 + e ^ 2 F 2 + e ^ 3 F 3 ) {\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， ∇ ⋅ ( e ^ 1 F 1 ) = ∇ ⋅ [ ( e ^ 1 h 2 h 3 ) ( h 2 h 3 F 1 ) ] {\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 ∇ ⋅ ( A ϕ ) = ϕ ∇ ⋅ A + A ⋅ ( ∇ ϕ ) {\displaystyle \nabla \cdot (\mathbf {A} \phi )=\phi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot (\nabla \phi )} 與 ∇ ⋅ ( ∇ ϕ 1 × ∇ ϕ 2 ) = 0 {\displaystyle \nabla \cdot (\nabla \phi _{1}\times \nabla \phi _{2})=0} ，可以得到 ∇ ⋅ ( e ^ 1 F 1 ) = ( h 2 h 3 F 1 ) ∇ ⋅ ( e ^ 1 h 2 h 3 ) + ( e ^ 1 h 2 h 3 ) ⋅ ∇ ( h 2 h 3 F 1 ) = ( h 2 h 3 F 1 ) ∇ ⋅ [ ( ∇ q 2 ) × ∇ ( q 3 ) ] + ( e ^ 1 h 2 h 3 ) ⋅ ∇ ( h 2 h 3 F 1 ) = ( e ^ 1 h 2 h 3 ) ⋅ ∇ ( h 2 h 3 F 1 ) = 1 h 1 h 2 h 3 ∂ ∂ q 1 ( F 1 h 2 h 3 ) {\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， ∇ ⋅ F = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ( F 1 h 2 h 3 ) + ∂ ∂ q 2 ( F 2 h 3 h 1 ) + ∂ ∂ q 3 ( F 3 h 1 h 2 ) ] {\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 ∇ × F = ∇ × ( e ^ 1 F 1 + e ^ 2 F 2 + e ^ 3 F 3 ) {\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， ∇ × ( e ^ 1 F 1 ) = ∇ × [ ( e ^ 1 h 1 ) ( h 1 F 1 ) ] {\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 ∇ × ( A ϕ ) = ϕ ∇ × A − A × ( ∇ ϕ ) {\displaystyle \nabla \times (\mathbf {A} \phi )=\phi \nabla \times \mathbf {A} -\mathbf {A} \times (\nabla \phi )} ， ∇ × ( e ^ 1 F 1 ) = ( h 1 F 1 ) ∇ × ( e ^ 1 h 1 ) − ( e ^ 1 h 1 ) × ∇ ( h 1 F 1 ) = ( h 1 F 1 ) ∇ × ( ∇ q 1 ) − ( e ^ 1 h 1 ) × ( e ^ 2 h 2 ∂ ∂ q 2 ( h 1 F 1 ) + e ^ 3 h 3 ∂ ∂ q 3 ( h 1 F 1 ) ) {\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 ∇ × ( ∇ ϕ ) = 0 {\displaystyle \nabla \times (\nabla \phi )=0} ， ∇ × ( e ^ 1 F 1 ) = e ^ 2 h 1 h 3 ∂ ∂ q 3 ( h 1 F 1 ) − e ^ 3 h 1 h 2 ∂ ∂ q 2 ( h 1 F 1 ) {\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， ∇ × F = e 1 h 2 h 3 [ ∂ ∂ q 2 ( h 3 F 3 ) − ∂ ∂ q 3 ( h 2 F 2 ) ] + e 2 h 3 h 1 [ ∂ ∂ q 3 ( h 1 F 1 ) − ∂ ∂ q 1 ( h 3 F 3 ) ] + e 3 h 1 h 2 [ ∂ ∂ q 1 ( h 2 F 2 ) − ∂ ∂ 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 ∇ 2 Φ = 1 h 1 h 2 h 3 [ ∂ ∂ q 1 ( h 2 h 3 h 1 ∂ Φ ∂ q 1 ) + ∂ ∂ q 2 ( h 3 h 1 h 2 ∂ Φ ∂ q 2 ) + ∂ ∂ q 3 ( h 1 h 2 h 3 ∂ Φ ∂ q 3 ) ] {\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]} 。 [1]Eric W. Weisstein. Orthogonal Coordinate System. MathWorld. [10 July 2008]. （原始內容存檔於2014-11-12）. [2]Morse and Feshbach 1953，Volume 1, pp. 494-523, 655-666. harvnb模板錯誤: 無指向目標: CITEREFMorse_and_Feshbach1953 (幫助) [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 坐標系 曲線坐標系 斜交坐標系（度規張量有非對角項目） 在圓柱和球坐標系中的del 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。Wikiwand in your browser!Seamless Wikipedia browsing. 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...![{\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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cfdd22e4ec259f3abc462a5b63329c245b33f4a8)
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