곡선좌표계에서의 그래디언트, 다이벌전스, 컬, 라플라시안
gradient divergence curl and laplacian in a curved coordinates system
설명
물리학에서 델 연산자 $\nabla$가 포함된 4가지 연산 그래디언트(기울기), 다이벌전스(발산), 컬(회전), 라플라시안은 매우 중요하다. 따라서 3가지 좌표계에 대한 위 연산을 반드시 알아야한다. 물론 이 말은 외워야한다는 뜻이 아니다. 물리학 공부는 이런 식을 외우는 것이 아니기 때문이다. 공부하다 보면 자연스레 외워질 것이므로 일부러 외우려하지 말고 공식을 프린트해서 가지고 다니거나, 이 페이지를 즐겨찾기 해놓고 필요할 때 바로 꺼내보자.
공식
$f$는 스칼라함수, 벡터 함수$\mathbf A$는 $\mathbf A= A_1\mathbf{\hat e_1}+A_2\mathbf{\hat e_2}+A_3\mathbf{\hat e_3}$라고 하자.
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$$ \begin{align*} \nabla f &= \mathbf{\hat e_1}\frac{1}{h_1}\frac{\partial f}{\partial e_1}+ \mathbf{\hat e_2}\frac{1}{h_2}\frac{\partial f}{\partial e_2}+\mathbf{\hat e_3}\frac{1}{h_3}\frac{\partial f}{\partial e_3} \\ &= \sum \limits_{i=1}^3 \mathbf{\hat e_i}\frac{1}{h_i}\frac{\partial f}{\partial e_i} \end{align*} $$
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$$ \nabla \cdot \mathbf A=\frac{1}{h_1h_2h_3} \left[ \frac{\partial}{\partial e_1} (h_2h_3A_1) + \frac{\partial}{\partial e_2} (h_1h_3A_2) + \frac{\partial}{\partial e_3} (h_1h_2A_3) \right] $$
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$$ \nabla \times \mathbf A =\frac{1}{h_1h_2h_3} \begin{vmatrix} h_1 \mathbf{\hat e_1} & h_2 \mathbf{\hat e_2} & h_3 \mathbf{\hat e_3} \\[0.5em] \dfrac{\partial}{\partial e_1} & \dfrac{\partial }{\partial e_2} & \dfrac{\partial}{\partial e_3} \\[1em] h_1A_1 & h_2A_2 & h_3A_3 \end{vmatrix} $$
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$$ \begin{align*} & \nabla \cdot (\nabla f) \\ =&\ \nabla ^2 f \\ =&\ \frac{1}{h_1h_2h_3} \left[ \frac{\partial }{\partial e_1} \left( \frac{h_2h_3}{h_1} \frac{\partial f}{\partial e_1} \right) +\frac{\partial }{\partial e_2} \left( \frac{h_1h_3}{h_2} \frac{\partial f}{\partial e_2} \right) + \frac{\partial }{\partial e_3} \left( \frac{h_1h_2}{h_3} \frac{\partial f}{\partial e_3} \right) \right] \end{align*} $$
이 때 각 좌표계별 단위벡터, 스케일 팩터는 다음과 같다.
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직교 좌표계:
$$ \mathbf{\hat{e_{1}}}=\mathbf{\hat{\mathbf{x}}},\quad\mathbf{\hat{e_{2}}}=\mathbf{\hat{\mathbf{y}}},\quad\mathbf{\hat{e_{3}}}=\mathbf{\hat{\mathbf{z}}},\quad h_{1}=1,\quad h_{2}=1,\quad h_{3}=1 $$
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원통 좌표계:
$$ \mathbf{\hat{e_{1}}}=\boldsymbol{\hat \rho},\quad\mathbf{\hat{e_{2}}}=\boldsymbol{\hat \phi},\quad\mathbf{\hat{e_{3}}}=\mathbf{\hat{\mathbf{z}}},\quad h_{1}=1,\quad h_{2}=\rho,\quad h_{3}=1 $$
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구면 좌표계
$$ \mathbf{\hat{e_{1}}}=\mathbf{\hat r},\quad\mathbf{\hat{e_{2}}}=\boldsymbol{\hat \theta},\quad\mathbf{\hat{e_{3}}}=\boldsymbol{\hat \phi},\quad h_{1}=1,\quad h_{2}=r,\quad h_{3}=r\sin\theta $$
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
$$ \nabla \cdot (\nabla f) = \nabla ^2 f = \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2\frac{\partial f}{\partial r} \right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial^2 \phi} $$