梯度、散度、旋度和拉普拉斯算子
fengxiaot Lv4
  2021-07-03 16:26:14 2021-07-03 16:26  

几何上，一个正交坐标系定义为一组正交坐标 qiq^i，其等值曲面都以直角相交。正交坐标的度规张量无非对角项。常见的正交曲线坐标系包括笛卡尔坐标系、柱坐标系和球坐标系。本文给出这些坐标系中梯度、散度、旋度和拉普拉斯算子的形式，以供快速查阅。

正交曲线坐标系

标度因子 HH

dr=∂r∂q1dq1+∂r∂q2dq2+∂r∂q3dq3\mathrm{d} \boldsymbol{r}=\frac{\partial \boldsymbol{r}}{\partial q_{1}} \mathrm{d} q_{1}+\frac{\partial \boldsymbol{r}}{\partial q_{2}} \mathrm{d} q_{2}+\frac{\partial \boldsymbol{r}}{\partial q_{3}} \mathrm{d} q_{3}

{∣∂r∂q1∣=(∂x∂q1)2+(∂y∂q1)2+(∂z∂q1)2=H1∣∂r∂q2∣=(∂x∂q2)2+(∂y∂q2)2+(∂z∂q2)2=H2∣∂r∂q1∣=(∂x∂q3)2+(∂y∂q3)2+(∂z∂q3)2=H3\left\{\begin{array}{l} \left|\dfrac{\partial \boldsymbol{r}}{\partial q_{1}}\right|=\sqrt{\left(\dfrac{\partial x}{\partial q_{1}}\right)^{2}+\left(\dfrac{\partial y}{\partial q_{1}}\right)^{2}+\left(\dfrac{\partial z}{\partial q_{1}}\right)^{2}}=H_{1} \\ \left|\dfrac{\partial \boldsymbol{r}}{\partial q_{2}}\right|=\sqrt{\left(\dfrac{\partial x}{\partial q_{2}}\right)^{2}+\left(\dfrac{\partial y}{\partial q_{2}}\right)^{2}+\left(\dfrac{\partial z}{\partial q_{2}}\right)^{2}}=H_{2} \\ \left|\dfrac{\partial \boldsymbol{r}}{\partial q_{1}}\right|=\sqrt{\left(\dfrac{\partial x}{\partial q_{3}}\right)^{2}+\left(\dfrac{\partial y}{\partial q_{3}}\right)^{2}+\left(\dfrac{\partial z}{\partial q_{3}}\right)^{2}}=H_{3} \end{array}\right.

弧长

∣dr∣=ds=H12dq12+H22dq22+H32dq32|\mathrm{d} \boldsymbol{r}|=\mathrm{d} s=\sqrt{H_{1}^{2} \mathrm{d} q_{1}^{2}+H_{2}^{2} \mathrm{d} q_{2}^{2}+H_{3}^{2} \mathrm{d} q_{3}^{2}}

面积元

{dσ1=H2H3dq2dq3dσ2=H1H3dq1dq3dσ3=H2H1dq2dq1\left\{\begin{array}{l} \mathrm{d} \sigma_{1}=H_{2} H_{3} \mathrm{d} q_{2} \mathrm{d} q_{3} \\ \mathrm{d} \sigma_{2}=H_{1} H_{3} \mathrm{d} q_{1} \mathrm{d} q_{3} \\ \mathrm{d} \sigma_{3}=H_{2} H_{1} \mathrm{d} q_{2} \mathrm{d} q_{1} \end{array}\right.

体积元

dV=ds1ds2ds3=H1H2H3dq1dq2dq3\mathrm{d} V=\mathrm{d} s_{1} \mathrm{d} s_{2} \mathrm{d} s_{3}=H_{1} H_{2} H_{3} \mathrm{d} q_{1} \mathrm{d} q_{2} \mathrm{d} q_{3}

梯度

∇f=1H1∂f∂q1e1+1H2∂f∂q2e2+1H3∂f∂q3e3\nabla f=\frac{1}{H_{1}} \frac{\partial f}{\partial q_{1}} \boldsymbol{e}_{1}+\frac{1}{H_{2}} \frac{\partial f}{\partial q_{2}} \boldsymbol{e}_{2}+\frac{1}{H_{3}} \frac{\partial f}{\partial q_{3}} \boldsymbol{e}_{3}

散度

∇⋅A=1H1H2H3(∂(A1H2H3)∂q1+∂(A2H1H3)∂q2+∂(A3H2H1)∂q3)\nabla \cdot \boldsymbol{A}=\frac{1}{H_{1} H_{2} H_{3}}\left(\frac{\partial\left(A_{1} H_{2} H_{3}\right)}{\partial q_{1}}+\frac{\partial\left(A_{2} H_{1} H_{3}\right)}{\partial q_{2}}+\frac{\partial\left(A_{3} H_{2} H_{1}\right)}{\partial q_{3}}\right)

旋度

∇×A=1H1H2H3∣H1e1H2e2H3e3∂∂q1∂∂q2∂∂q3H1A1H2A2H3A3∣\nabla \times \boldsymbol{A}=\frac{1}{H_{1} H_{2} H_{3}}\begin{vmatrix} H_{1} \boldsymbol{e}_{1} & H_{2} \boldsymbol{e}_{2} & H_{3} \boldsymbol{e}_{3} \\ \frac{\partial}{\partial q_{1}} & \frac{\partial}{\partial q_{2}} & \frac{\partial}{\partial q_{3}} \\ H_{1} A_{1} & H_{2} A_{2} & H_{3} A_{3} \end{vmatrix}

拉普拉斯算子

Δf=1H1H2H3[∂∂q1(H2H3H1∂f∂q1)+∂∂q2(H1H3H2∂f∂q2)+∂∂q3(H2H1H3∂f∂q3)]\Delta f=\frac{1}{H_{1} H_{2} H_{3}}\left[\frac{\partial}{\partial q_{1}}\left(\frac{H_{2} H_{3}}{H_{1}} \frac{\partial f}{\partial q_{1}}\right)+\frac{\partial}{\partial q_{2}}\left(\frac{H_{1} H_{3}}{H_{2}} \frac{\partial f}{\partial q_{2}}\right)+\frac{\partial}{\partial q_{3}}\left(\frac{H_{2} H_{1}}{H_{3}} \frac{\partial f}{\partial q_{3}}\right)\right]


直角坐标系

梯度

∇f(x,y,z)=∂f∂xi+∂f∂yj+∂f∂zk\nabla f(x, y, z)=\frac{\partial f}{\partial x} \boldsymbol{i}+\frac{\partial f}{\partial y} \boldsymbol{j}+\frac{\partial f}{\partial z} \boldsymbol{k}

散度

∇⋅A=∂Ax∂x+∂Ay∂y+∂Az∂z\nabla \cdot \boldsymbol{A}=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z}

旋度

∇×A=∣ijk∂∂x∂∂y∂∂zAxAyAz∣\nabla \times \boldsymbol{A}=\begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_{x} & A_{y} & A_{z} \end{vmatrix}

拉普拉斯算子

Δf=∂2f∂x2+∂2f∂y2+∂2f∂z2\Delta f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}


柱坐标系

梯度

∇f(ρ,φ,z)=∂f∂ρeρ+1ρ∂f∂φeφ+∂f∂zez\nabla f(\rho, \varphi, z)=\frac{\partial f}{\partial \rho} \boldsymbol{e}_{\rho}+\frac{1}{\rho} \frac{\partial f}{\partial \varphi} \boldsymbol{e}_{\varphi}+\frac{\partial f}{\partial z} \boldsymbol{e}_{z}

散度

∇⋅A=1ρ∂∂ρ(ρAρ)+1ρ∂Aφ∂φ+∂Az∂z\nabla \cdot \boldsymbol{A}=\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho A_{\rho}\right)+\frac{1}{\rho} \frac{\partial A_{\varphi}}{\partial \varphi}+\frac{\partial A_{z}}{\partial z}

旋度

∇×A=∣1ρeρeφ1ρez∂∂ρ∂∂φ∂∂zAρρAφAz∣\nabla \times \boldsymbol{A}=\begin{vmatrix} \frac{1}{\rho} \boldsymbol{e}_{\rho} & \boldsymbol{e}_{\varphi} & \frac{1}{\rho} \boldsymbol{e}_{z} \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \varphi} & \frac{\partial}{\partial z} \\ A_{\rho} & \rho A_{\varphi} & A_{z} \end{vmatrix}

拉普拉斯算子

Δf=1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂φ2+∂2f∂z2\Delta f=\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial f}{\partial \rho}\right)+\frac{1}{\rho^{2}} \frac{\partial^{2} f}{\partial \varphi^{2}}+\frac{\partial^{2} f}{\partial z^{2}}


球坐标系

梯度

∇f(r,θ,ϕ)=∂f∂rer+1r∂f∂θeθ+1rsin⁡θ∂f∂ϕeϕ\nabla f(r, \theta, \phi)=\frac{\partial f}{\partial r} \boldsymbol{e}_{r}+\frac{1}{r} \frac{\partial f}{\partial \theta} \boldsymbol{e}_{\theta}+\frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \boldsymbol{e}_{\phi}

散度

∇⋅A=1r2∂∂r(r2Ar)+1rsin⁡θ∂∂θ(sin⁡θAθ)+1rsin⁡θ∂Aϕ∂ϕ\nabla \cdot \boldsymbol{A}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} A_{r}\right)+\frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta A_{\theta}\right)+\frac{1}{r \sin \theta} \frac{\partial A_{\phi}}{\partial \phi}

旋度

∇×A=1r2sin⁡θ∣erreθrsin⁡θeϕ∂∂r∂∂θ∂∂ϕArrAθrsin⁡θAϕ∣\nabla \times \boldsymbol{A}=\frac{1}{r^{2} \sin \theta}\begin{vmatrix} \boldsymbol{e}_{r} & r \boldsymbol{e}_{\theta} & r \sin \theta \boldsymbol{e}_{\phi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\ A_{r} & r A_{\theta} & r \sin \theta A_{\phi} \end{vmatrix}

拉普拉斯算子

Δf=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂ϕ2\Delta f=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} f}{\partial \phi^{2}}