nabla算子

发表于2019-10-06更新于2020-10-30阅读次数

基础

∇=ex⃗∂∂x+ey⃗∂∂y+ez⃗∂∂z\nabla = \vec{e_x}\frac{\partial}{\partial x} + \vec{e_y}\frac{\partial}{\partial y} + \vec{e_z}\frac{\partial}{\partial z}

∇⋅f⃗≡divf⃗\nabla \cdot \vec{f} \equiv div \vec{f}

∇×f⃗≡rotf⃗\nabla \times \vec{f} \equiv rot \vec{f}

∇ϕ≡gradϕ\nabla \phi \equiv grad \phi

积分公式

  • Gauss积分公式

∮Sf⃗⋅dS⃗=∫V∇f⃗dV\oint_S \vec{f}\cdot d\vec{S} = \int_V \nabla \vec{f} dV

  • Stokes定理

∮Lf⃗⋅dl⃗=∫S(∇×f⃗)⋅dS⃗\oint_L \vec{f}\cdot d\vec{l} = \int_S(\nabla \times \vec{f})\cdot d\vec{S}

  • Green公式

∮ψ∇ϕ⋅dS⃗=∫V(ψ∇2ϕ+∇ψ⋅∇ϕ)dV\oint \psi \nabla \phi \cdot d\vec{S} = \int_V(\psi\nabla^2\phi+\nabla\psi\cdot\nabla\phi)dV

∮(ψ∇ϕ−ϕ∇ψ)⋅dS⃗=∫V(ψ∇2ϕ−ϕ∇2ψ)dV\oint(\psi\nabla\phi-\phi\nabla\psi)\cdot d\vec{S}=\int_V(\psi\nabla^2\phi-\phi\nabla^2\psi)dV

  • 亥姆霍兹定理：任意矢量可以分解成无源场和无旋场之和

f⃗=∇ϕ+∇×A⃗\vec{f} = \nabla\phi+\nabla\times\vec{A}

常用公式

∇2ϕ=(∇⋅∇)ψ∇2f⃗=∇(∇⋅f⃗)−∇×(∇×f⃗)∇×∇ϕ=0,∇⋅(∇×f⃗)∇(ϕψ)=ϕ∇ψ+ψ∇ϕ∇(f⃗⋅g⃗)=f⃗×(∇×g⃗)+g⃗×(∇×f⃗)+(f⃗⋅∇)g⃗+(g⃗⋅∇)f⃗∇⋅(ϕf⃗)=ϕ(∇⋅f⃗)+f⃗⋅(∇ϕ)∇⋅(f⃗×g⃗)=g⃗⋅(∇×f⃗)−f⃗⋅(∇×g⃗)∇×(ϕf⃗)=ϕ(∇×f⃗)−f⃗×(∇ϕ)∇×(f⃗×g⃗)=(g⃗⋅∇)f⃗−(f⃗⋅∇)g⃗+f⃗(∇⋅g⃗)−g⃗(∇⋅f⃗).\begin{aligned} &\nabla^2\phi=(\nabla\cdot\nabla)\psi \\ &\nabla^2\vec{f}=\nabla(\nabla\cdot\vec{f})-\nabla\times(\nabla\times\vec{f}) \\ &\nabla\times\nabla\phi=0, \quad \nabla\cdot(\nabla\times\vec{f}) \\ &\nabla(\phi\psi) = \phi\nabla\psi + \psi\nabla\phi \\ &\nabla(\vec{f}\cdot\vec{g})=\vec{f}\times(\nabla\times\vec{g})+\vec{g}\times(\nabla\times\vec{f})+(\vec{f}\cdot\nabla)\vec{g}+(\vec{g}\cdot\nabla)\vec{f} \\ &\nabla\cdot(\phi\vec{f}) = \phi(\nabla\cdot\vec{f})+\vec{f}\cdot(\nabla\phi) \\ &\nabla\cdot(\vec{f}\times\vec{g}) = \vec{g}\cdot(\nabla\times\vec{f})-\vec{f}\cdot(\nabla\times\vec{g}) \\ &\nabla\times(\phi\vec{f})=\phi(\nabla\times\vec{f})-\vec{f}\times(\nabla\phi) \\ &\nabla\times(\vec{f}\times\vec{g}) = (\vec{g}\cdot\nabla)\vec{f}-(\vec{f}\cdot\nabla)\vec{g}+\vec{f}(\nabla\cdot\vec{g})-\vec{g}(\nabla\cdot\vec{f}). \end{aligned}

对一些简单函数的作用

∇⋅r⃗=3,∇r=r⃗r,∇×r⃗=0,∇1r=−r⃗r3,∇f(r)=f(r′)r⃗r,∇21r=−4πδ(r⃗)∇×(f(r)r⃗)=0,∇r⃗=I↔∇(a⃗⋅r⃗)=(a⃗⋅∇)r⃗=a⃗,∇eia⃗⋅r⃗=ia⃗r⃗eia⃗⋅r⃗.\begin{aligned} &\nabla\cdot\vec{r}=3, \quad \nabla r = \frac{\vec{r}}{r}, \quad \nabla\times\vec{r} = 0, \\ &\nabla\frac{1}{r} = -\frac{\vec{r}}{r^3}, \quad \nabla f(r)=f(r')\frac{\vec{r}}{r}, \quad \nabla^2\frac{1}{r} = -4\pi\delta(\vec{r}) \\ &\nabla\times(f(r)\vec{r})=0, \quad \nabla\vec{r}=\overleftrightarrow{I} \\ &\nabla(\vec{a}\cdot\vec{r})=(\vec{a}\cdot\nabla)\vec{r}=\vec{a}, \quad \nabla e^{i\vec{a}\cdot\vec{r}}=i\vec{a}\vec{r}e^{i\vec{a}\cdot\vec{r}}. \end{aligned}

球坐标

∇=er⃗∂∂r+eθ⃗1r∂∂θ+eϕ⃗1rsin⁡θ∂∂ϕ∇ψ=∂ψ∂rer⃗+1r∂ψ∂θeθ⃗+1rsin⁡θ∂ψ∂ϕeϕ⃗∇⋅A⃗=1r2∂∂r(r2Ar)+1rsin⁡θ∂∂θ(sin⁡θAθ)+1rsin⁡θ∂∂ϕ(Aϕ)∇×A⃗=1rsin⁡θ[∂∂θ(sin⁡θAϕ)−∂∂ϕ(Aθ)]er⃗+1r[1sin⁡θ∂∂ϕ(Ar)−∂∂r(rAϕ)]eθ⃗+1r[∂∂r(rAθ)−∂∂θAr]eϕ⃗∇2ψ=1r2∂∂r(r2∂ψ∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂ψ∂θ)+1r2sin⁡2θ∂2ψ∂ϕ2.\begin{aligned} &\nabla = \vec{e_r}\frac{\partial}{\partial r}+\vec{e_\theta}\frac{1}{r}\frac{\partial}{\partial \theta} + \vec{e_\phi}\frac{1}{r\sin \theta}\frac{\partial}{\partial \phi} \\ &\nabla \psi = \frac{\partial \psi}{\partial r}\vec{e_r} + \frac{1}{r}\frac{\partial \psi}{\partial \theta}\vec{e_\theta}+\frac{1}{r\sin \theta}\frac{\partial \psi}{\partial \phi}\vec{e_\phi} \\ &\nabla \cdot \vec{A} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 A_r) + \frac{1}{r\sin \theta}\frac{\partial}{\partial \theta}(\sin\theta A_\theta) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \phi}(A_\phi) \\ &\nabla\times\vec{A} = \frac{1}{r\sin\theta}\left[\frac{\partial}{\partial \theta}(\sin\theta A_\phi)-\frac{\partial}{\partial\phi}(A_\theta)\right]\vec{e_r} + \frac{1}{r}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}(A_r)-\frac{\partial}{\partial r}(rA_\phi)\right]\vec{e_\theta} +\frac{1}{r}\left[\frac{\partial}{\partial r}(rA_\theta)-\frac{\partial}{\partial\theta}A_r\right]\vec{e_\phi} \\ &\nabla^2\psi = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial\psi}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial\psi}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2\psi}{\partial\phi^2}. \end{aligned}

柱坐标

∇=er⃗∂∂r+eθ⃗1r∂∂θ+ez⃗∂∂z∇ψ=∂ψ∂rer⃗+1r∂ψ∂θeθ⃗+∂ψ∂zez⃗∇⋅A⃗=1r∂∂r(rAr)+1r∂∂θAθ+∂∂zAz∇×A⃗=(1r∂Az∂θ−∂Aθ∂z)er⃗+(∂Ar∂z−∂Az∂r)eθ⃗+1r(∂∂r(rAθ)−∂Ar∂θ)ez⃗∇2ψ=1r∂∂r(r∂ψ∂r)+1r2∂2ψ∂θ2+∂2ψ∂z2.\begin{aligned} &\nabla = \vec{e_r}\frac{\partial}{\partial r}+\vec{e_\theta}\frac{1}{r}\frac{\partial}{\partial\theta} + \vec{e_z}\frac{\partial}{\partial z} \\ &\nabla\psi = \frac{\partial\psi}{\partial r}\vec{e_r}+\frac{1}{r}\frac{\partial\psi}{\partial\theta}\vec{e_\theta}+\frac{\partial\psi}{\partial z}\vec{e_z} \\ &\nabla\cdot\vec{A} = \frac{1}{r}\frac{\partial}{\partial r}(rA_r)+\frac{1}{r}\frac{\partial}{\partial\theta}A_\theta+\frac{\partial}{\partial z}A_z \\ &\nabla\times\vec{A} = \left(\frac{1}{r}\frac{\partial A_z}{\partial\theta} - \frac{\partial A_\theta}{\partial z}\right)\vec{e_r} + \left(\frac{\partial A_r}{\partial z} -\frac{\partial A_z}{\partial r}\right)\vec{e_\theta} + \frac{1}{r}\left(\frac{\partial}{\partial r}(rA_\theta)-\frac{\partial A_r}{\partial \theta}\right)\vec{e_z} \\ &\nabla^2\psi = \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial \psi}{\partial r}) + \frac{1}{r^2}\frac{\partial^2\psi}{\partial \theta^2}+\frac{\partial^2 \psi}{\partial z^2}. \end{aligned}

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