직교 원통 구면 좌표계에 대한 기울기 발산 회전 라플라스 연산

del operator in several coordinates

설명

직교좌표계, 원통좌표계, 구면좌표계에서의 그래디언트, 다이벌전스, 컬, 라플라시안을 한데 모아 정리했다. 이렇게 정리한 이유는 보고 암기하라는 것이 아니라 갑자기 필요한데 생각이 안나거나 할 때 검색해서 보고 참고하라고 정리한 것이다. 굳이 외우려하지 않아도 어차피 공부 열심히하다보면 외워진다. $f$는 스칼라 함수, $\mathbf A$는 벡터 함수이다.

직교 좌표계

그래디언트

$$ \nabla f = \frac{\partial f}{\partial x}\mathbf{\hat{\mathbf{x}} }+ \frac{\partial f}{\partial y}\mathbf{\hat{\mathbf{y}}} + \frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}} $$

다이벌전스

$$ \nabla \cdot \mathbf A=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z} $$

컬

$$ \begin{align*} \nabla \times \mathbf A&=\left(\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z} \right) \mathbf{\hat{\mathbf{x}}}+\left(\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{\partial x} \right) \mathbf{\hat{\mathbf{y}}}+\left(\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{\partial y} \right) \mathbf{\hat{\mathbf{z}}} \\ &= \begin{vmatrix} \mathbf{\hat{\mathbf{x}}} & \mathbf{\hat{\mathbf{y}}} & \mathbf{\hat{\mathbf{z}}} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial}{\partial z} \\ A_{x} & A_{y} & A_{z} \end{vmatrix} \end{align*} $$

라플라시안

$$ \begin{align*} \nabla \cdot (\nabla f) = \nabla ^2 f &= \left( \frac{\partial}{\partial x}\mathbf{\hat{\mathbf{x}}}+\frac{\partial}{\partial y}\mathbf{\hat{\mathbf{y}}}+\frac{\partial}{\partial z}\mathbf{\hat{\mathbf{z}}} \right) \cdot \left( \frac{\partial f}{\partial x}\mathbf{\hat{\mathbf{x}}}+\frac{\partial f}{\partial y}\mathbf{\hat{\mathbf{y}}}+\frac{\partial f}{\partial z}\mathbf{\hat{\mathbf{z}}} \right) \\ &= \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2} \end{align*} $$

원통 좌표계

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

$$ \nabla \cdot \mathbf A=\frac{1}{\rho}\frac{\partial (\rho A_\rho)}{\partial \rho}+\frac{1}{\rho}\frac{\partial A_\phi}{\partial \phi}+\frac{\partial A_{z}}{\partial z} $$

컬

$$ \begin{align*} \nabla \times \mathbf A&=\left[\frac{1}{\rho}\frac{\partial A_{z}}{\partial \phi}-\frac{\partial A_\phi}{\partial z} \right] \boldsymbol{\hat \rho}+\left[\frac{\partial A_\rho}{\partial z}-\frac{\partial A_{z}}{\partial \rho} \right] \boldsymbol{\hat \phi}+\frac{1}{\rho}\left[\frac{\partial (\rho A_\phi)}{\partial \rho}-\frac{\partial A_\rho}{\partial \phi} \right] \mathrm{\hat{\mathbf{z}}} \\ &= \frac{1}{\rho}\begin{vmatrix} \boldsymbol{\hat \rho} & \rho\boldsymbol{ \hat \phi} & \mathbf{\hat{\mathbf{z}}} \\ \dfrac{\partial}{\partial \rho} & \dfrac{\partial }{\partial \phi} & \dfrac{\partial}{\partial z} \\ A_\rho & \rho A_\phi & A_{z} \end{vmatrix} \end{align*} $$

라플라시안

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

구 좌표계

그래디언트

$$ \nabla f = \frac{\partial f}{\partial r} \mathbf{\hat{\mathbf{r}}} + \frac{1}{r}\frac{\partial f}{\partial \theta} \boldsymbol{\hat{\boldsymbol{\theta}}} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi} $$

다이벌전스

$$ \nabla \cdot \mathbf A=\frac{1}{r^2}\frac{\partial (r^2 A_r)}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial (\sin\theta A_\theta)}{\partial \theta}+\frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi} $$

컬

$$ \begin{align*} \nabla \times \mathbf A &=\frac{1}{r\sin\theta} \left[\frac{\partial (\sin\theta A_\phi)}{\partial \theta}-\frac{\partial A_\theta}{\partial \phi} \right]\mathbf{\hat{\mathbf{r}}}+\frac{1}{r}\left[\frac{1}{\sin\theta} \frac{\partial A_r}{\partial \phi}-\frac{\partial (rA_\phi)}{\partial r} \right] \boldsymbol{\hat{\boldsymbol{\theta}}} \\ & \quad+ \frac{1}{r} \left[\frac{\partial (rA_\theta)}{\partial r}-\frac{\partial A_r}{\partial \theta} \right]\boldsymbol{\hat \phi} \\ &= \frac{1}{r^2\sin\theta}\begin{vmatrix} \mathbf{\hat{\mathbf{r}}} & r\boldsymbol{\hat{\boldsymbol{\theta}}} & r\sin\theta\boldsymbol{\hat \phi} \\ \dfrac{\partial}{\partial r} & \dfrac{\partial }{\partial \theta} & \dfrac{\partial}{\partial \phi} \\ A_r & r A_\theta & r\sin\theta A_\phi \end{vmatrix} \end{align*} $$

라플라시안

$$ \nabla \cdot (\nabla f) = \nabla ^2 f = \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2\frac{\partial f}{\partial r} \right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial^2 \phi} $$