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Cylindrical Coordinates에서의 Del 연산자 📂Mathematical Physics

Cylindrical Coordinates에서의 Del 연산자

Formula

The del operator in the cylindrical coordinate system is as follows.

∇=∂∂ρρ^+1ρ∂∂ϕϕ^+∂∂zz^ \nabla = \dfrac{\partial}{\partial \rho} \widehat{\boldsymbol{\rho}} + \dfrac{1}{\rho}\dfrac{\partial}{\partial \phi} \widehat{\boldsymbol{\phi}} + \dfrac{\partial}{\partial z} \widehat{\mathbf{z}}

Description

The del operator is not a vector, but for convenience, it is represented as above.

  • Gradient:

    ∇f=∂f∂ρρ^+1ρ∂f∂ϕϕ^+∂f∂zz^ \nabla f = \frac{\partial f}{\partial \rho}\boldsymbol{\hat \rho} + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}}

  • Divergence:

    ∇⋅F=1ρ(∂(ρFρ)∂ρ+∂(Fϕ)∂ϕ+∂(ρFz)∂z)=1ρ∂(ρFρ)∂ρ+1ρ∂Fϕ∂ϕ+∂Fz∂z \begin{align*} \nabla \cdot \mathbf{F} &= \frac{1}{\rho} \left( \frac{\partial (\rho F_{\rho})}{\partial \rho} + \frac{\partial (F_{\phi})}{\partial \phi} + \frac{\partial (\rho F_{z})}{\partial z} \right) \\ &= \frac{1}{\rho} \frac{\partial (\rho F_{\rho})}{\partial \rho} + \frac{1}{\rho}\frac{\partial F_{\phi}}{\partial \phi} + \frac{\partial F_{z}}{\partial z} \end{align*}

  • Curl:

    ∇×F=(1ρ∂Fz∂ϕ−∂Fϕ∂z)ρ^+(∂Fρ∂z−∂Fz∂ρ)ϕ^+1ρ(∂(ρFϕ)∂ρ−∂Fρ∂ϕ)z^ \nabla \times \mathbf{F} = \left(\frac{1}{\rho}\dfrac{\partial F_{z}}{\partial \phi} - \dfrac{\partial F_{\phi}}{\partial z} \right)\boldsymbol{\hat \rho} + \left(\dfrac{\partial F_{\rho}}{\partial z} - \dfrac{\partial F_{z}}{\partial \rho} \right)\boldsymbol{\hat \phi} + \frac{1}{\rho}\left(\dfrac{\partial (\rho F_{\phi})}{\partial \rho} - \dfrac{\partial F_{\rho}}{\partial \phi} \right)\mathbf{\hat{\mathbf{z}}}

  • Laplacian:

    ∇2f=1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂ϕ2+∂2f∂z2 \nabla ^{2}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 \phi^{2} }+ \frac{\partial ^{2} f}{\partial z^{2} }