在数学里，一个正交坐标系定义为一组正交坐标 q = ( q 1 , q 2 , q 3 , … q n ) {\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 q i {\displaystyle q_{i}} 的等值曲面，即 q i {\displaystyle q_{i}} 为常数的曲线、曲面或超曲面。例如，三维直角坐标 ( x , y , z ) {\displaystyle (x,\ y,\ z)} 是一种正交坐标系，它的 x {\displaystyle x} 为常数， y {\displaystyle y} 为常数， z {\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。 Remove ads 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 H 2 − {\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量法来成功的解析关于这问题的方程式。分离变量法是一种数学技巧，专门用来将一个复杂的 n {\displaystyle n} 维问题变为 n {\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]。 Remove ads 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 d s 2 {\displaystyle ds^{2}} ，可以写为无穷小坐标位移的平方和： 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} 是维数，标度因子 h i {\displaystyle h_{i}} 是度规张量的对角元素 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} )}}} 。 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 ( x , y ) {\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 z = x + i y {\displaystyle z=x+iy} 的任何全纯函数 w = f ( z ) {\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 w = u + i v {\displaystyle w=u+iv} ，则 u {\displaystyle u} 与 v {\displaystyle v} 的等值曲线以直角相交，就如同原本的 x {\displaystyle x} 与 y {\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着其对称轴旋转。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。 Remove ads 在正交坐标系里，内积的公式仍旧不变： A ⋅ B = ∑ i = 1 n A i B i {\displaystyle \mathbf {A} \cdot \mathbf {B} =\sum _{i=1}^{n}A_{i}B_{i}} 。 从前面的距离公式，可以观察出，一个正交坐标 q i {\displaystyle q_{i}} 的无穷小改变 d q i {\displaystyle dq_{i}} ，其相伴的长度是 d s i = h i d q i {\displaystyle ds_{i}=h_{i}dq_{i}} 。因此，一个位移向量的全微分 d r {\displaystyle d\mathbf {r} } 等于 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}} 是垂直于 q i {\displaystyle q_{i}} 等值曲面的单位向量，指向着 q i {\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 F {\displaystyle \mathbf {F} } 沿着周线 C {\displaystyle \mathbb {C} } 的线积分等于 ∫ 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} } 在单位向量 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} } 。 类似地，一个无穷小面积元素是 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} ， 一个无穷小体积元素是 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} } 对于一个曲面 S {\displaystyle \mathbb {S} } 的曲面积分是 ∫ 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球坐标系实例 直角坐标 ( x , y , z ) {\displaystyle (x,\ y,\ z)} 与球坐标 ( r , θ , ϕ ) {\displaystyle (r,\ \theta ,\phi )} 的变换方程式为 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 } 。 直角坐标的全微分是 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，无穷小距离的平方是 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}}} 。 标度因子是 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} } 沿着周线 C {\displaystyle \mathbb {C} } 的线积分等于 ∫ 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} } 对于一个曲面 S {\displaystyle \mathbb {S} } 的曲面积分是 ∫ 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 主条目：向量分析和Nabla算子 更多信息 , ... 算子 正交坐标公式 标量场的梯度 ∇ 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}}}} 向量场的散度 ∇ ⋅ 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]} 向量场的旋度 ∇ × 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}}} 标量场的拉普拉斯算子 ∇ 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]} 关闭 上面表达式可以使用列维-奇维塔符号 ϵ {\displaystyle \epsilon } 的更简洁形式书写，定义 H = h 1 h 2 h 3 {\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: 更多信息 , ... 算子 表达式 标量场的梯度 ∇ ϕ = e ^ k h k ∂ ϕ ∂ q k {\displaystyle \nabla \phi ={\frac {{\hat {\mathbf {e} }}_{k}}{h_{k}}}{\frac {\partial \phi }{\partial q^{k}}}} 向量场的散度 ∇ ⋅ 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)} 向量场（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)} 标量场的拉普拉斯算子 ∇ 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)} 关闭 更多信息 , ... 坐标系 复数变换 x + i y = f ( u + i v ) {\displaystyle x+iy=f(u+iv)} u {\displaystyle u} 和 v {\displaystyle v} 等值线的形状 注释 直角 u + i v {\displaystyle u+iv} 直线, 直线 对数极（英语：Log-polar coordinates） exp ( u + i v ) {\displaystyle \exp(u+iv)} 圆, 直线 若 u = ln r {\displaystyle u=\ln r} 则为极坐标系 抛物线 1 2 ( u + i v ) 2 {\displaystyle {\frac {1}{2}}(u+iv)^{2}} 抛物线, 抛物线 点偶极 ( u + i v ) − 1 {\displaystyle (u+iv)^{-1}} 圆, 圆 椭圆 cosh ( u + i v ) {\displaystyle \cosh(u+iv)} 椭圆, 双曲线 对于大距离看似对数极 双极 coth ( u + i v ) {\displaystyle \coth(u+iv)} 圆, 圆 对于大距离看似点偶极 u + i v {\displaystyle {\sqrt {u+iv}}} 双曲线, 双曲线 u = x 2 + 2 y 2 , y = v x 2 {\displaystyle u=x^{2}+2y^{2},\ y=vx^{2}} 椭圆, 抛物线 关闭 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 Remove ads 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。 更多信息 , ... 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 ( 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}}} 圆柱坐标系 ( 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}}} 抛物柱面坐标系 ( 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}}} 抛物线坐标系 ( 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}}} 椭圆柱坐标系 ( 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}}} 长球面坐标系 ( 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}}} 扁球面坐标系 ( 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}}} 双极圆柱坐标系 ( 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}}} 圆环坐标系 ( 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}}} 双球坐标系 ( 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}}} 圆锥坐标系 ( 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}}} 抛物面坐标系 ( 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})}}}} 椭球坐标系 ( 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})}}}} 关闭 梯度导引 一个函数 ϕ {\displaystyle \phi } 的梯度朝某个方向 n ^ {\displaystyle {\hat {\mathbf {n} }}} 的分量，等于方向导数 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} }}} 方向的无穷小位移。 假若，这 n ^ {\displaystyle {\hat {\mathbf {n} }}} 与正交坐标轴 e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} 同方向。那么， d s = h i d q i {\displaystyle ds=h_{i}dq_{i}} 。所以，函数 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}}}} ；也就是说， ∇ 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}}}} 。 散度导引 ∇ ⋅ 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})} 。 取右手边第一个项目， ∇ ⋅ ( 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]} 。 应用向量恒等式 ∇ ⋅ ( A ϕ ) = ϕ ∇ ⋅ A + A ⋅ ( ∇ ϕ ) {\displaystyle \nabla \cdot (\mathbf {A} \phi )=\phi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot (\nabla \phi )} 与 ∇ ⋅ ( ∇ 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}}} 。 总合所有项目， ∇ ⋅ 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]} 。 旋度导引 ∇ × 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})} 。 取右手边第一个项目， ∇ × ( 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]} 。 应用向量恒等式 ∇ × ( 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}}} 。 应用向量恒等式 ∇ × ( ∇ 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})} 。 总合所有项目， ∇ × 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 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 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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