원통좌표계의 벡터 연산 (Vector Operation of Cylindrical Coordinates)

대부분의 수학/역학 서적에서 원통좌표계를 많이 다루지 않으나 실제로 원은 이상적인 형상일 경우가 많으므로 산업체에서는 유용한 개념이다. 본 글에서는 원통좌표계에서 기본적인 벡터 연산자의 유도 과정을 소개한다.

원통좌표계 위치 벡터

원통좌표계의 단위벡터는 좌표계의 함수이다. 단위벡터의 좌표계 미분을 복습하면 다음과 같다.

${\displaystyle \frac{\partial \hat{r}}{\partial \theta }}=\hat{\theta },{\displaystyle \frac{\partial \hat{\theta }}{\partial \theta }}=-\hat{r},{\displaystyle \frac{\partial \hat{r}}{\partial r}}={\displaystyle \frac{\partial \hat{r}}{\partial z}}={\displaystyle \frac{\partial \hat{\theta }}{\partial r}}={\displaystyle \frac{\partial \hat{\theta }}{\partial z}}={\displaystyle \frac{\partial \hat{z}}{\partial r}}={\displaystyle \frac{\partial \hat{z}}{\partial \theta }}={\displaystyle \frac{\partial \hat{z}}{\partial z}}=0$

경로 증분 (Path increment)

원통좌표계로 표현된 경로 증분 $d\mathbf{p}$는 앞으로 많이 쓰일 것이다. 단위벡터의 좌표계 미분과 합성함수의 편미분법을 적용하면 다음과 같이 유도된다.

$\begin{array}{rl}d\mathbf{p}& =d(r\hat{r}+z\hat{z})=\hat{r}dr+rd\hat{r}+\hat{z}dz+zd\hat{z}\\ & =\hat{r}dr+r(\frac{\partial \hat{r}}{\partial r}dr+\frac{\partial \hat{r}}{\partial \theta }d\theta +\frac{\partial \hat{r}}{\partial z}dz)+\hat{z}dz+z(\frac{\partial \hat{z}}{\partial r}dr+\frac{\partial \hat{z}}{\partial \theta }d\theta +\frac{\partial \hat{z}}{\partial z}dz)\\ & =\hat{r}dr+\hat{\theta }rd\theta +\hat{z}dz\end{array}$

단위벡터의 시간미분 (Time derivative of the unit vectors)

또한 원통좌표계로 표현된 단위벡터의 시간미분도 앞으로 많이 활용될 것이다.

$\begin{array}{rl}\frac{d\hat{r}}{dt}& =\frac{\partial \hat{r}}{\partial r}\dot{r}+\frac{\partial \hat{r}}{\partial \theta }\dot{\theta }+\frac{\partial \hat{r}}{\partial z}\dot{z}=\hat{\theta }\dot{\theta }\\ \frac{d\hat{\theta }}{dt}& =\frac{\partial \hat{\theta }}{\partial r}\dot{r}+\frac{\partial \hat{\theta }}{\partial \theta }\dot{\theta }+\frac{\partial \hat{\theta }}{\partial z}\dot{z}=-\hat{r}\dot{\theta }\\ \frac{d\hat{z}}{dt}& =\frac{\partial \hat{z}}{\partial r}\dot{r}+\frac{\partial \hat{z}}{\partial \theta }\dot{\theta }+\frac{\partial \hat{z}}{\partial z}\dot{z}=0\end{array}$

속도와 가속도 (Velocity and Acceleration)

원통좌표계에서 한점의 속도와 가속도는 단위벡터 시간미분으로 표현될 수 있다.

$\begin{array}{rl}\mathbf{v}& =\dot{\mathbf{p}}=\frac{d\hat{r}}{dt}r+\hat{r}\dot{r}+\frac{d\hat{z}}{dt}z+\hat{z}\dot{z}=\hat{r}\dot{r}+\hat{\theta }r\dot{\theta }+\hat{z}\dot{z}\\ \mathbf{a}& =\dot{\mathbf{v}}=\frac{d\hat{r}}{dt}\dot{r}+\hat{r}\ddot{r}+\frac{d\hat{\theta }}{dt}r\dot{\theta }+\hat{\theta }\dot{r}\dot{\theta }+\hat{\theta }r\ddot{\theta }+\frac{d\hat{z}}{dt}\dot{z}+\hat{z}\ddot{z}\\ & =\hat{\theta }\dot{\theta }\dot{r}+\hat{r}\ddot{r}-\hat{r}r{\dot{\theta }}^2+\hat{\theta }\dot{r}\dot{\theta }+\hat{\theta }r\ddot{\theta }+\hat{z}\ddot{z}\\ & =\hat{r}(\ddot{r}-r{\dot{\theta }}^2)+\hat{\theta }(r\ddot{\theta }+2\dot{r}\dot{\theta })+\hat{z}\ddot{z}\end{array}$

구배 연산자 (The del operator from the definition of the gradient)

어느 스칼라 장(場, field) f가 원통좌표계 r, θ, z의 함수라고 하자. 한 점에 dp 만큼의 미소변위가 있을 때 스칼라 함수 f도 df 만큼의 미소변화가 생긴다. 그 변화는 함수 f의 편미분으로 다음과 같이 결정된다.

$df={\displaystyle \frac{\partial f}{\partial r}}dr+{\displaystyle \frac{\partial f}{\partial \theta }}d\theta +{\displaystyle \frac{\partial f}{\partial z}}dz$

한편으로는 구배의 정의에 의해서 다음식을 얻는다.

$df=\mathrm{\nabla }f\cdot d\mathbf{p}$

위의 두 식은 같아야 하므로

${\displaystyle \frac{\partial f}{\partial r}}dr+{\displaystyle \frac{\partial f}{\partial \theta }}d\theta +{\displaystyle \frac{\partial f}{\partial z}}dz=\mathrm{\nabla }f\cdot d\mathbf{p}=(\mathrm{\nabla }f{)}_{r}dr+(\mathrm{\nabla }f{)}_{\theta }rd\theta +(\mathrm{\nabla }f{)}_{z}dz$

위의 식은 임의의 dr, dθ 및 dz에 대하여 성립되어야 한다. 따라서,

$(\mathrm{\nabla }f{)}_r={\displaystyle \frac{\partial f}{\partial r}},(\mathrm{\nabla }f{)}_{\theta }={\displaystyle \frac{\partial f}{r\partial \theta }},(\mathrm{\nabla }f{)}_z={\displaystyle \frac{\partial f}{\partial z}}$

위의 식은 구배의 각 성분을 나타내므로

$\mathrm{\nabla }=\hat{r}{\displaystyle \frac{\partial }{\partial r}}+\hat{\theta }{\displaystyle \frac{\partial }{r\partial \theta }}+\hat{z}{\displaystyle \frac{\partial }{\partial z}}$

발산 (Divergence)

발산 $\mathrm{\nabla }\cdot \mathbf{F}$는 또 다시 원통좌표계의 단위벡터가 좌표계의 함수라는 것을 고려하여 유도된다.

$\begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{F}& =(\hat{r}\frac{\partial }{\partial r}+\hat{\theta }\frac{\partial }{r\partial \theta }+\hat{z}\frac{\partial }{\partial z})\cdot (F_r\hat{r}+F_{\theta }\hat{\theta }+F_z\hat{z})\\ & =(\hat{r}\frac{\partial }{\partial r}+\hat{\theta }\frac{\partial }{r\partial \theta }+\hat{z}\frac{\partial }{\partial z})\cdot \mathbf{F}\\ & =\hat{r}\cdot \frac{\partial \mathbf{F}}{\partial r}+\hat{\theta }\cdot \frac{\partial \mathbf{F}}{r\partial \theta }+\hat{z}\cdot \frac{\partial \mathbf{F}}{\partial z}\\ & =\hat{r}\cdot (\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}+F_r\frac{\partial \hat{r}}{\partial r}+F_{\theta }\frac{\partial \hat{\theta }}{\partial r}+F_z\frac{\partial \hat{z}}{\partial r})\\ & +\frac{\hat{\theta }}{r}\cdot (\frac{\partial F_r}{\partial \theta }\hat{r}+\frac{\partial F_{\theta }}{\partial \theta }\hat{\theta }+\frac{\partial F_z}{\partial \theta }\hat{z}+F_r\frac{\partial \hat{r}}{\partial \theta }+F_{\theta }\frac{\partial \hat{\theta }}{\partial \theta }+F_z\frac{\partial \hat{z}}{\partial \theta })\\ & +\hat{z}\cdot (\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_{\theta }}{\partial z}\hat{\theta }+\frac{\partial F_z}{\partial z}\hat{z}+F_r\frac{\partial \hat{r}}{\partial z}+F_{\theta }\frac{\partial \hat{\theta }}{\partial z}+F_z\frac{\partial \hat{z}}{\partial z})\end{array}$

단위벡터의 좌표계 미분을 적용하면

$\begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{F}& =\hat{r}\cdot (\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}+0+0+0)\\ & +\frac{\hat{\theta }}{r}\cdot (\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}+F_r\hat{\theta }-F_{\theta }\hat{r})\\ & +\hat{z}\cdot (\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}+0+0+0)\\ & =\frac{\partial F_r}{\partial r}+\frac{F_r}{r}+\frac{\partial F_{\theta }}{r\partial \theta }+\frac{\partial F_z}{\partial z}\\ & =\frac{\partial }{r\partial r}\left(r{F}_r\right)+\frac{\partial F_{\theta }}{r\partial \theta }+\frac{\partial F_z}{\partial z}\end{array}$

컬 (Curl)

컬 $\mathrm{\nabla }\times \mathbf{F}$ 또한 원통좌표계의 단위벡터는 좌표계의 함수라는 것을 고려하여 유도된다.

$\begin{array}{rl}\mathrm{\nabla }\times \mathbf{F}& =(\hat{r}\frac{\partial }{\partial r}+\hat{\theta }\frac{\partial }{r\partial \theta }+\hat{z}\frac{\partial }{\partial z})\times (F_r\hat{r}+F_{\theta }\hat{\theta }+F_z\hat{z})\\ & =(\hat{r}\frac{\partial }{\partial r}+\hat{\theta }\frac{\partial }{r\partial \theta }+\hat{z}\frac{\partial }{\partial z})\times \mathbf{F}\\ & =\hat{r}\times \frac{\partial \mathbf{F}}{\partial r}+\hat{\theta }\times \frac{\partial \mathbf{F}}{r\partial \theta }+\hat{z}\times \frac{\partial \mathbf{F}}{\partial z}\\ & =\hat{r}\times (\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}+F_r\frac{\partial \hat{r}}{\partial r}+F_{\theta }\frac{\partial \hat{\theta }}{\partial r}+F_z\frac{\partial \hat{z}}{\partial r})\\ & +\frac{\hat{\theta }}{r}\times (\frac{\partial F_r}{\partial \theta }\hat{r}+\frac{\partial F_{\theta }}{\partial \theta }\hat{\theta }+\frac{\partial F_z}{\partial \theta }\hat{z}+F_r\frac{\partial \hat{r}}{\partial \theta }+F_{\theta }\frac{\partial \hat{\theta }}{\partial \theta }+F_z\frac{\partial \hat{z}}{\partial \theta })\\ & +\hat{z}\times (\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_{\theta }}{\partial z}\hat{\theta }+\frac{\partial F_z}{\partial z}\hat{z}+F_r\frac{\partial \hat{r}}{\partial z}+F_{\theta }\frac{\partial \hat{\theta }}{\partial z}+F_z\frac{\partial \hat{z}}{\partial z})\end{array}$

단위벡터의 좌표계 미분을 적용하면

$\begin{array}{rl}\mathrm{\nabla }\times \mathbf{F}& =\hat{r}\times (\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}+0+0+0)\\ & +\frac{\hat{\theta }}{r}\times (\frac{\partial F_r}{\partial \theta }\hat{r}+\frac{\partial F_{\theta }}{\partial \theta }\hat{\theta }+\frac{\partial F_z}{\partial \theta }\hat{z}+F_r\hat{\theta }-F_{\theta }\hat{r})\\ & +\hat{z}\times (\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_{\theta }}{\partial z}\hat{\theta }+\frac{\partial F_z}{\partial z}\hat{z}+0+0+0)\\ & =(\frac{\partial F_{\theta }}{\partial r}\hat{z}-\frac{\partial F_z}{\partial r}\hat{\theta })+(-\frac{\partial F_r}{r\partial \theta }\hat{z}+\frac{\partial F_z}{r\partial \theta }\hat{r}+\frac{F_{\theta }}{r}\hat{z})+(\frac{\partial F_r}{\partial z}\hat{\theta }-\frac{\partial F_{\theta }}{\partial z}\hat{r})\\ & =\hat{r}(\frac{\partial F_z}{r\partial \theta }-\frac{F_{\theta }}{\partial z})+\hat{\theta }(\frac{\partial F_r}{\partial z}-\frac{\partial F_z}{\partial r})+\hat{z}(\frac{\partial }{r\partial r}\left(r{F}_{\theta }\right)-\frac{\partial F_r}{r\partial r})\end{array}$ 

라플라시안 (Laplacian)

라플라시안은 다음과 같이 정의에 의해 유도될 수 있는 스칼라 연산자이다.

$\begin{array}{rl}{\mathrm{\nabla }}^{2}f& =\mathrm{\nabla }\cdot (\mathrm{\nabla }f)=(\hat{r}\frac{\partial }{\partial r}+\hat{\theta }\frac{\partial }{r\partial \theta }+\hat{z}\frac{\partial }{\partial z})\cdot (\hat{r}\frac{\partial f}{\partial r}+\hat{\theta }\frac{\partial f}{r\partial \theta }+\hat{z}\frac{\partial f}{\partial z})\\ & =\hat{r}\cdot \frac{\partial }{\partial r}(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta }\frac{\partial f}{r\partial \theta }+\hat{z}\frac{\partial f}{\partial z})\\ & +\hat{\theta }\cdot \frac{\partial }{r\partial \theta }(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta }\frac{\partial f}{r\partial \theta }+\hat{z}\frac{\partial f}{\partial z})\\ & +\hat{z}\cdot \frac{\partial }{\partial z}(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta }\frac{\partial f}{r\partial \theta }+\hat{z}\frac{\partial f}{\partial z})\end{array}$

적(積)의 미분법과 단위벡터의 좌표계 미분을 적용하면

$\begin{array}{rl}{\mathrm{\nabla }}^{2}f& =\hat{r}\cdot (\hat{r}\frac{{\partial }^{2}f}{\partial r^2}-\hat{\theta }\frac{\partial f}{r^2\partial \theta }+\hat{\theta }\frac{{\partial }^{2}f}{r\partial \theta \partial r}+\hat{z}\frac{{\partial }^{2}f}{\partial z\partial r})\\ & +\frac{\hat{\theta }}{r}\cdot (\hat{\theta }\frac{\partial f}{\partial r}+\hat{r}\frac{{\partial }^{2}f}{\partial r\partial \theta }-\hat{r}\frac{\partial f}{r\partial \theta }+\hat{\theta }\frac{{\partial }^{2}f}{r\partial {\theta }^2}+\hat{z}\frac{{\partial }^{2}f}{\partial z\partial \theta })\\ & +\hat{z}\cdot (\hat{r}\frac{{\partial }^{2}f}{\partial r\partial z}+\hat{\theta }\frac{{\partial }^{2}f}{r\partial \theta \partial z}+\hat{z}\frac{{\partial }^{2}f}{\partial z^2})\\ & =\frac{{\partial }^{2}f}{\partial r^2}+\frac{\partial f}{r\partial r}+\frac{{\partial }^{2}f}{r^2\partial {\theta }^2}+\frac{{\partial }^{2}f}{\partial z^2}\\ & =\frac{\partial }{r\partial r}\left(r\frac{\partial f}{\partial r}\right)+\frac{{\partial }^{2}f}{r^2\partial {\theta }^2}+\frac{{\partial }^{2}f}{\partial z^2}\end{array}$

따라서, 라플라시안 연산자는 다음과 같이 쓸 수 있다.

${\mathrm{\nabla }}^2={\displaystyle \frac{\partial }{r\partial r}}\left(r{\displaystyle \frac{\partial }{\partial r}}\right)+{\displaystyle \frac{{\partial }^2}{r^2\partial {\theta }^2}}+{\displaystyle \frac{{\partial }^2}{\partial z^2}}$

라플라시안 연산자를 벡터 F에 적용하면

$${\mathrm{\nabla }}^2\mathbf{F}=\frac{\partial }{r\partial r}\left(r\frac{\partial \mathbf{F}}{\partial r}\right)+\frac{{\partial }^2\mathbf{F}}{r^2\partial {\theta }^2}+\frac{{\partial }^2\mathbf{F}}{\partial z^2}$$

각 항별로 벡터 F의 성분으로 편미분하면 다음과 같다.

$$\begin{array}{rl}\frac{\partial \mathbf{F}}{\partial r}& =\frac{\partial }{\partial r}(F_r\hat{r}+F_{\theta }\hat{\theta }+F_z\hat{z})=\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z}\\ \frac{\partial \mathbf{F}}{\partial \theta }& =\frac{\partial }{\partial \theta }(F_r\hat{r}+F_{\theta }\hat{\theta }+F_z\hat{z})=(\frac{\partial F_r}{\partial \theta }-F_{\theta })\hat{r}+(\frac{\partial F_{\theta }}{\partial \theta }+F_r)\hat{\theta }+\frac{\partial F_z}{\partial \theta }\hat{z}\\ \frac{\partial \mathbf{F}}{\partial z}& =\frac{\partial }{\partial z}(F_r\hat{r}+F_{\theta }\hat{\theta }+F_z\hat{z})=\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_{\theta }}{\partial z}\hat{\theta }+\frac{\partial F_z}{\partial z}\hat{z}\\ \frac{{\partial }^2\mathbf{F}}{\partial r^2}& =\frac{\partial }{\partial r}(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_{\theta }}{\partial r}\hat{\theta }+\frac{\partial F_z}{\partial r}\hat{z})=\frac{{\partial }^2{F}_r}{\partial r^2}\hat{r}+\frac{{\partial }^2{F}_{\theta }}{\partial r^2}\hat{\theta }+\frac{{\partial }^2{F}_z}{\partial r^2}\hat{z}\\ \frac{{\partial }^2\mathbf{F}}{\partial {\theta }^2}& =\frac{\partial }{\partial \theta }\{(\frac{\partial F_r}{\partial \theta }-F_{\theta })\hat{r}+(\frac{\partial F_{\theta }}{\partial \theta }+F_r)\hat{\theta }+\frac{\partial F_z}{\partial \theta }\hat{z}\}\\ & =(\frac{{\partial }^2{F}_r}{\partial {\theta }^2}-2\frac{\partial F_{\theta }}{\partial \theta }-F_r)\hat{r}+(2\frac{\partial F_r}{\partial \theta }-F_{\theta }+\frac{{\partial }^2{F}_{\theta }}{\partial {\theta }^2})\hat{\theta }+\frac{{\partial }^2{F}_z}{\partial {\theta }^2}\hat{z}\\ \frac{{\partial }^2\mathbf{F}}{\partial z^2}& =\frac{\partial }{\partial z}(\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_{\theta }}{\partial z}\hat{\theta }+\frac{\partial F_z}{\partial z}\hat{z})=\frac{{\partial }^2{F}_r}{\partial z^2}\hat{r}+\frac{{\partial }^2{F}_{\theta }}{\partial z^2}\hat{\theta }+\frac{{\partial }^2{F}_z}{\partial z^2}\hat{z}\end{array}$$

위의 결과를 앞의 각 항에 대입하면 벡터 F의 라플라시안을 얻는다.

$$\begin{array}{rl}{\mathrm{\nabla }}^2\mathbf{F}& =\{\frac{\partial }{r\partial r}\left(r\frac{\partial F_r}{\partial r}\right)+\frac{{\partial }^2{F}_r}{r^2\partial {\theta }^2}+\frac{{\partial }^2{F}_r}{\partial z^2}-\frac{F_r}{r^2}-\frac{2\partial F_{\theta }}{r^2\partial \theta }\}\hat{r}\\ & +\{\frac{\partial }{r\partial r}\left(r\frac{\partial F_{\theta }}{\partial r}\right)+\frac{{\partial }^2{F}_{\theta }}{r^2\partial {\theta }^2}+\frac{{\partial }^2{F}_{\theta }}{\partial z^2}-\frac{F_{\theta }}{r^2}+\frac{2\partial F_r}{r^2\partial \theta }\}\hat{\theta }\\ & +\{\frac{\partial }{r\partial r}\left(r\frac{\partial F_z}{\partial r}\right)+\frac{{\partial }^2{F}_z}{r^2\partial {\theta }^2}+\frac{{\partial }^2{F}_z}{\partial z^2}\}\hat{z}\end{array}$$