직교 원통 구면 좌표계에 대한 기울기 발산 회전 라플라스 연산

직교 원통 구면 좌표계에 대한 기울기 발산 회전 라플라스 연산

설명

직교좌표계, 원통좌표계, 구면좌표계에서의 그래디언트, 다이벌전스, , 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

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

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

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

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

원통 좌표계

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

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

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

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

구 좌표계

-그래디언트:

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

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

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

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

댓글