📂수리물리

원통 좌표계에서의 델 연산자

공식

원통 좌표계에서의 델 연산자는 다음과 같다.

$$ \nabla = \dfrac{\partial}{\partial \rho} \widehat{\boldsymbol{\rho}} + \dfrac{1}{\rho}\dfrac{\partial}{\partial \phi} \widehat{\boldsymbol{\phi}} + \dfrac{\partial}{\partial z} \widehat{\mathbf{z}} $$

설명

델 연산자는 벡터가 아니지만 편의상 위와 같이 표기하였다.

• 그래디언트:

  $$ \nabla f = \frac{\partial f}{\partial \rho}\boldsymbol{\hat \rho} + \frac{1}{\rho}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}} $$

• 다이벌전스:

  $$ \begin{align*} \nabla \cdot \mathbf{F} &= \frac{1}{\rho} \left( \frac{\partial (\rho F_{\rho})}{\partial \rho} + \frac{\partial (F_{\phi})}{\partial \phi} + \frac{\partial (\rho F_{z})}{\partial z} \right) \\ &= \frac{1}{\rho} \frac{\partial (\rho F_{\rho})}{\partial \rho} + \frac{1}{\rho}\frac{\partial F_{\phi}}{\partial \phi} + \frac{\partial F_{z}}{\partial z} \end{align*} $$

• 컬:

  $$ \nabla \times \mathbf{F} = \left(\frac{1}{\rho}\dfrac{\partial F_{z}}{\partial \phi} - \dfrac{\partial F_{\phi}}{\partial z} \right)\boldsymbol{\hat \rho} + \left(\dfrac{\partial F_{\rho}}{\partial z} - \dfrac{\partial F_{z}}{\partial \rho} \right)\boldsymbol{\hat \phi} + \frac{1}{\rho}\left(\dfrac{\partial (\rho F_{\phi})}{\partial \rho} - \dfrac{\partial F_{\rho}}{\partial \phi} \right)\mathbf{\hat{\mathbf{z}}} $$

• 라플라시안:

  $$ \nabla ^{2}f= \frac{1}{\rho} \frac{ \partial }{ \partial \rho }\left( \rho\frac{ \partial f}{ \partial \rho} \right) + \frac{1}{\rho^{2}} \frac{\partial ^{2} f}{\partial \phi^{2} }+ \frac{\partial ^{2} f}{\partial z^{2} } $$