logo

曲線座標系における勾配、発散、および回転 📂数理物理学

曲線座標系における勾配、発散、および回転

公式

球座標系でのデル演算子は以下の通りだ。

∇=∂∂rr^+1r∂∂θθ^+1rsin⁡θ∂∂ϕϕ^ \nabla = \dfrac{\partial}{\partial r} \widehat{\mathbf{r}} + \dfrac{1}{r}\dfrac{\partial}{\partial \theta} \widehat{\boldsymbol{\theta}} + \dfrac{1}{r\sin\theta}\dfrac{\partial}{\partial \phi} \widehat{\boldsymbol{\phi}}

説明

デル演算子はベクトルではないが、便宜上上記のように表記された。

  • グラディエント:

    ∇f=∂f∂rr^+1r∂f∂θθ^+1rsin⁡θ∂f∂ϕϕ^ \nabla f= \frac{\partial f}{\partial r} \mathbf{\hat r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \boldsymbol{\hat \theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi}

  • ダイバージェンス:

    ∇⋅F=1r2sin⁡θ(∂(r2sin⁡θFr)∂r+∂(rsin⁡θFθ)∂θ+∂(rFϕ)∂ϕ)=1r2∂(r2Fr)∂r+1rsin⁡θ∂(sin⁡θFθ)∂θ+1rsin⁡θ∂Fϕ∂ϕ \begin{align*} \nabla \cdot \mathbf{F} &= \frac{1}{r^{2}\sin\theta}\left( \frac{\partial (r^{2}\sin\theta F_{r})}{\partial r}+\frac{\partial (r\sin\theta F_{\theta})}{\partial \theta}+\frac{\partial (rF_{\phi})}{\partial \phi} \right) \\ &= \frac{1}{r^{2}}\frac{\partial (r^{2} F_{r})}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial (\sin\theta F_{\theta})}{\partial \theta}+\frac{1}{r\sin\theta}\frac{\partial F_{\phi}}{\partial \phi} \end{align*}

  • カール:

    ∇×F=1rsin⁡θ(∂(Fϕsin⁡θ)∂θ−∂Fθ∂ϕ)r^+1r(1sin⁡θ∂Fr∂ϕ−∂(rFϕ)∂r)θ^+1r(∂(rFθ)∂r−∂Fr∂θ)ϕ^ \nabla \times \mathbf{F} = \frac{1}{r\sin\theta}\left(\dfrac{\partial (F_{\phi} \sin\theta)}{\partial \theta} - \dfrac{\partial F_{\theta}}{\partial \phi} \right)\mathbf{\hat r} + \frac{1}{r}\left(\frac{1}{\sin\theta}\dfrac{\partial F_{r}}{\partial \phi} - \dfrac{\partial (r F_{\phi})}{\partial r} \right)\boldsymbol{\hat \theta} + \frac{1}{r}\left(\dfrac{\partial (r F_{\theta})}{\partial r} - \dfrac{\partial F_{r}}{\partial \theta} \right)\boldsymbol{\hat \phi}

  • ラプラシアン:

    ∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂ϕ2 \nabla ^{2}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} }