📂수리물리

곡선 좌표계에서 벡터 함수의 컬(회전)

정리

곡선 좌표계에서 벡터 함수 $\mathbf{F}=\mathbf{F}(q_{1},q_{2},q_{3})=F_{1}\hat{\mathbf{q}}_{1}+F_{2}\hat{\mathbf{q}}_{2}+F_{3}\hat{\mathbf{q}}_{3}$의 컬은 다음과 같다.

$$\begin{align*} \nabla \times \mathbf{F} &= \frac{\hat{\mathbf{q}}_{1}}{h_{2}h_{3}}\left( \dfrac{\partial (F_{3}h_{3})}{\partial q_{2}} - \dfrac{\partial (F_{2}h_{2})}{\partial q_{3}} \right) + \frac{\hat{\mathbf{q}}_{2}}{h_{1}h_{3}}\left( \dfrac{\partial (F_{1}h_{1})}{\partial q_{3}} - \dfrac{\partial (F_{3}h_{3})}{\partial q_{1}} \right) \\ &\quad+ \frac{\hat{\mathbf{q}}_{3}}{h_{1}h_{2}}\left( \dfrac{\partial (F_{2}h_{2})}{\partial q_{1}} - \dfrac{\partial (F_{1}h_{1})}{\partial q_{2}} \right) \\ &= \frac{1}{h_{1}h_{2}h_{3}} \begin{vmatrix} h_{1}\hat{\mathbf{q}}_{1} & h_{2}\hat{\mathbf{q}}_{2} & h_{3}\hat{\mathbf{q}}_{3} \\[0.5em] \dfrac{\partial }{\partial q_{1}} & \dfrac{\partial }{\partial q_{2}} & \dfrac{\partial }{\partial q_{3}} \\[1em] F_{1}h_{1} & F_{2}h_{2} & F_{3}h_{3} \end{vmatrix} \end{align*}$$

$h_{i}$는 스케일 팩터이다.

공식

• 직교 좌표계:

  $$h_{1}=h_{2}=h_{3}=1$$

  $$\nabla \times \mathbf{F} = \left( \dfrac{\partial F_{z}}{\partial y}-\dfrac{\partial F_{y}}{\partial z} \right)\hat{\mathbf{x}}+ \left( \dfrac{\partial F_{x}}{\partial z}-\dfrac{\partial F_{z}}{\partial x} \right)\hat{\mathbf{y}}+ \left( \dfrac{\partial F_{y}}{\partial x}-\dfrac{\partial F_{x}}{\partial y} \right)\hat{\mathbf{z}}$$

• 원통 좌표계:

  $$h_{1}=1,\quad h_{2}=\rho,\quad h_{3}=1$$

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

• 구 좌표계:

  $$h_{1}=1,\quad h_{2}=r\quad, h_{3}=r\sin\theta$$

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

유도

방법1 ¹

곡선좌표계를 $(q_{1}, q_{2}, q_{3})$라고 하자.

$$\mathbf{F} = F_{1}\hat{\mathbf{q}}_{1} + F_{2}\hat{\mathbf{q}}_{2} + F_{3}\hat{\mathbf{q}}_{3}$$

컬은 선형성을 가지므로,

$$\begin{equation} \nabla \times \mathbf{F} = \nabla \times \left( F_{1}\hat{\mathbf{q}}_{1} + F_{2}\hat{\mathbf{q}}_{2} + F_{3}\hat{\mathbf{q}}_{3} \right) = \sum_{i=1}^{3} \nabla \times (F_{i}\hat{\mathbf{q}}_{i}) \end{equation}$$

곡선 좌표계에서 그래디언트

$$\nabla f= \frac{1}{h_{1}}\frac{ \partial f }{ \partial q_{1} } \hat{\mathbf{q}}_{1} + \frac{1}{h_{2}}\frac{ \partial f }{ \partial q _{2}}\hat{\mathbf{q}}_{2}+\frac{1}{h_{3}}\frac{ \partial f }{ \partial q_{3} } \hat{\mathbf{q}}_{3}=\sum \limits _{i=1} ^{3}\frac{1}{h_{i}}\frac{ \partial f}{ \partial q_{i}}\hat{\mathbf{q}}_{i}$$

그래디언트 공식에 의해서 다음을 얻는다.

$$\nabla q_{i} = \sum \limits _{m=1} ^{3}\frac{1}{h_{m}}\frac{ \partial q_{i}}{ \partial q_{m}}\hat{\mathbf{q}}_{m} = \frac{1}{h_{i}}\hat{\mathbf{q}}_{i}$$

$$\begin{equation} \implies h_{i}\nabla q_{i} = \hat{\mathbf{q}}_{i} \end{equation}$$

델 연산자가 포함된 곱셈공식

$$\nabla \times (f \mathbf{A}) = (\nabla f) \times \mathbf{A} + f (\nabla \times \mathbf{A})$$

$(2)$를 $(1)$에 대입하고, 위의 곱셈공식을 적용하면 다음을 얻는다.

$$\begin{align*} \nabla \times (F_{i}\hat{\mathbf{q}}_{i}) &= \nabla \times (F_{i}h_{i}\nabla q_{i}) \\ &= [\nabla (F_{i}h_{i})] \times \nabla q_{i} + F_{i}h_{i} \nabla \times (\nabla q_{i}) \\ &= [\nabla (F_{i}h_{i})] \times \nabla q_{i} \end{align*}$$

마지막 등호는 그래디언트의 컬은 $\mathbf{0}$이기 때문에 성립한다. 식에 다시 $(2)$를 대입하고, 그래디언트를 풀어보면

$$\begin{align*} [\nabla (F_{i}h_{i})] \times \nabla q_{i} &= \left[ \sum_{m=1}^{3} \dfrac{1}{h_{m}}\dfrac{\partial (F_{i}h_{i})}{\partial q_{m}} \hat{\mathbf{q}}_{m} \right] \times \frac{1}{h_{i}}\hat{\mathbf{q}}_{i} \\ &= \dfrac{1}{h_{j}}\dfrac{\partial (F_{i}h_{i})}{\partial q_{j}} \hat{\mathbf{q}}_{j} \times \frac{1}{h_{i}}\hat{\mathbf{q}}_{i} + \dfrac{1}{h_{k}}\dfrac{\partial (F_{i}h_{i})}{\partial q_{k}} \hat{\mathbf{q}}_{k} \times \frac{1}{h_{i}}\hat{\mathbf{q}}_{i} \\ &= \dfrac{1}{h_{i}h_{k}}\dfrac{\partial (F_{i}h_{i})}{\partial q_{k}} \hat{\mathbf{q}}_{j} - \dfrac{1}{h_{i}h_{j}}\dfrac{\partial (F_{i}h_{i})}{\partial q_{j}} \hat{\mathbf{q}}_{k} \end{align*}$$

여기서 인덱스를 $\hat{\mathbf{q}}_{i} \times \hat{\mathbf{q}}_{j} = \hat{\mathbf{q}}_{k}$라 두었다. 이를 $(1)$에 대입하면,

$$\begin{align*} \nabla \times \mathbf{F} &= \left( \dfrac{1}{h_{1}h_{3}}\dfrac{\partial (F_{1}h_{1})}{\partial q_{3}} \hat{\mathbf{q}}_{2} - \dfrac{1}{h_{1}h_{2}}\dfrac{\partial (F_{1}h_{1})}{\partial q_{2}} \hat{\mathbf{q}}_{3} \right) + \left( \dfrac{1}{h_{2}h_{1}}\dfrac{\partial (F_{2}h_{2})}{\partial q_{1}} \hat{\mathbf{q}}_{3} - \dfrac{1}{h_{2}h_{3}}\dfrac{\partial (F_{2}h_{2})}{\partial q_{3}} \hat{\mathbf{q}}_{1} \right) \\ &\quad+ \left( \dfrac{1}{h_{3}h_{2}}\dfrac{\partial (F_{3}h_{3})}{\partial q_{2}} \hat{\mathbf{q}}_{1} - \dfrac{1}{h_{3}h_{1}}\dfrac{\partial (F_{3}h_{3})}{\partial q_{1}} \hat{\mathbf{q}}_{2} \right) \\ &= \frac{h_{1}\hat{\mathbf{q}}_{1}}{h_{1}h_{2}h_{3}}\left( \dfrac{\partial (F_{3}h_{3})}{\partial q_{2}} - \dfrac{\partial (F_{2}h_{2})}{\partial q_{3}} \right) + \frac{h_{2}\hat{\mathbf{q}}_{2}}{h_{1}h_{2}h_{3}}\left( \dfrac{\partial (F_{1}h_{1})}{\partial q_{3}} - \dfrac{\partial (F_{3}h_{3})}{\partial q_{1}} \right) \\ &\quad+ \frac{h_{3}\hat{\mathbf{q}}_{3}}{h_{1}h_{2}h_{3}}\left( \dfrac{\partial (F_{2}h_{2})}{\partial q_{1}} - \dfrac{\partial (F_{1}h_{1})}{\partial q_{2}} \right) \\ &= \frac{1}{h_{1}h_{2}h_{3}} \begin{vmatrix} h_{1}\hat{\mathbf{q}}_{1} & h_{2}\hat{\mathbf{q}}_{2} & h_{3}\hat{\mathbf{q}}_{3} \\[0.5em] \dfrac{\partial }{\partial q_{1}} & \dfrac{\partial }{\partial q_{2}} & \dfrac{\partial }{\partial q_{3}} \\[1em] F_{1}h_{1} & F_{2}h_{2} & F_{3}h_{3} \end{vmatrix} \end{align*}$$

■

방법2 ²

$\nabla \times \mathbf{F}$의 첫번째 성분을 구하기 위해 $q_{1}$ 좌표가 상수인 곡면에서의 회전폐곡면을 생각하자.

이때 $\odot$은 평면을 뚫고 나오는 방향을 의미한다. 그러면 $d\boldsymbol{\sigma} = h_{2}h_{3} dq_{2}dq_{3}\hat{\mathbf{q}}_{1}$이므로,

$$\int \nabla \times \mathbf{F} \cdot d\boldsymbol{\sigma} \approx (\nabla \times \mathbf{F})h_{2}h_{3}dq_{2}dq_{3} \cdot \hat{\mathbf{q}}_{1}$$

스토크스 정리

$$\int_{\mathcal{S}} (\nabla \times \mathbf{v} )\cdot d\mathbf{a} = \oint_{\mathcal{P}} \mathbf{v} \cdot d\mathbf{l}$$

스토크스 정리에 의해 다음을 얻는다.

$$(\nabla \times \mathbf{F})h_{2}h_{3}dq_{2}dq_{3} \cdot \hat{\mathbf{q}}_{1} = \oint \mathbf{F} \cdot d \mathbf{r}$$

따라서 $\nabla \times \mathbf{F}$의 첫번째 성분은 다음을 계산하여 얻을 수 있다.

$$(\nabla \times \mathbf{F})_{1} = (\nabla \times \mathbf{F}) \cdot \hat{\mathbf{q}}_{1} = \frac{1}{h_{2}h_{3}dq_{2}dq_{3}}\oint \mathbf{F} \cdot d \mathbf{r}$$

폐곡선 적분을 $\textcircled{1}$~$\textcircled{4}$로 나누어 생각하자.

$$\oint \mathbf{F} \cdot d \mathbf{r} = \int_{\textcircled{1}} \mathbf{F} \cdot d \mathbf{r} + \int_{\textcircled{2}} \mathbf{F} \cdot d \mathbf{r} + \int_{\textcircled{3}} \mathbf{F} \cdot d \mathbf{r} + \int_{\textcircled{4}} \mathbf{F} \cdot d \mathbf{r}$$

경로 $\textcircled{1}$에 대한 적분을 구해보면,

$$\int_{\textcircled{1}} \mathbf{F} \cdot d \mathbf{r} \approx \mathbf{F}(q_{2},q_{3}) \cdot h_{2}(q_{2},q_{3}) dq_{2}\hat{\mathbf{q}}_{2} = F_{2}h_{2}dq_{2}$$

경로 $\textcircled{2}$는,

$$\begin{align*} \int_{\textcircled{2}} \mathbf{F} \cdot d \mathbf{r} &\approx \mathbf{F}(q_{2}+dq_{2},q_{3}) \cdot h_{3}(q_{2}+dq_{2},q_{3}) dq_{3}\hat{\mathbf{q}}_{3} \\ &= F_{3}(q_{2}+dq_{2},q_{3})h_{3}(q_{2}+dq_{2},q_{3})dq_{3} \\ &\approx \left[ F_{3}h_{3} + \dfrac{\partial (F_{3}h_{3})}{\partial q_{2}}dq_{2} \right]dq_{3} \end{align*}$$

마지막 줄에선 테일러 근사를 사용하였다.

테일러 정리

$$f(x+dx) \approx f(x) + f^{\prime}(x)dx$$

마찬가지로 나머지 적분을 계산하면,

$$\begin{align*} \int_{\textcircled{3}} \mathbf{F} \cdot d \mathbf{r} &= -\int_{-\textcircled{3}} \mathbf{F} \cdot d \mathbf{r} \\ &\approx -\mathbf{F}(q_{2},q_{3}+dq_{3}) \cdot h_{2}(q_{2},q_{3}+dq_{3}) dq_{2}\hat{\mathbf{q}}_{2} \\ &= -F_{2}(q_{2},q_{3}+dq_{3})h_{2}(q_{2},q_{3}+dq_{3})dq_{2} \\ &\approx -\left[ F_{2}h_{2} + \dfrac{\partial (F_{2}h_{2})}{\partial q_{3}}dq_{3} \right]dq_{2} \end{align*}$$

$$\int_{\textcircled{4}} \mathbf{F} \cdot d \mathbf{r} = -\int_{-\textcircled{4}} \mathbf{F} \cdot d \mathbf{r} \approx \mathbf{F}(q_{2},q_{3}) \cdot h_{3}(q_{2},q_{3}) dq_{3}\hat{\mathbf{q}}_{3} = F_{3}h_{3}dq_{3}$$

따라서,

$$\begin{align*} (\nabla \times \mathbf{F})_{1} &= \dfrac{1}{h_{2}h_{3}dq_{2}dq_{3}}\oint \mathbf{F} \cdot d \mathbf{r} \\ &= \dfrac{1}{h_{2}h_{3}dq_{2}dq_{3}} \left[ \dfrac{\partial (F_{3}h_{3})}{\partial q_{2}}dq_{2}dq_{3} - \dfrac{\partial (F_{2}h_{2})}{\partial q_{3}}dq_{3} dq_{2} \right]\\ &= \dfrac{1}{h_{2}h_{3}} \left[ \dfrac{\partial (F_{3}h_{3})}{\partial q_{2}} - \dfrac{\partial (F_{2}h_{2})}{\partial q_{3}} \right]\\ \end{align*}$$

두번째, 세번째 성분도 같은 방법으로 얻는다.

■

같이보기

• 곡선좌표계에서 그래디언트, 다이버전스, 컬, 라플라시안 공식
• 그래디언트(기울기)
• 다이버전스(발산)
• 컬(회전)
• 라플라시안