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.

\(\dfrac{\partial\hat{r}}{\partial\theta}=\hat{\theta},\,\dfrac{\partial\hat{\theta}}{\partial\theta}=-\hat{r},\,\dfrac{\partial\hat{r}}{\partial r}=\dfrac{\partial\hat{r}}{\partial z}=\dfrac{\partial\hat{\theta}}{\partial r}=\dfrac{\partial\hat{\theta}}{\partial z}=\dfrac{\partial\hat{z}}{\partial r}=\dfrac{\partial\hat{z}}{\partial\theta}=\dfrac{\partial\hat{z}}{\partial z}=0\)

경로 증분 (Path increment)

원통좌표계로 표현된 경로 증분 \(d\bf p\)는 앞으로 많이 쓰일 것이다. 단위벡터의 좌표계 미분과 합성함수의 편미분법을 적용하면 다음과 같이 유도된다.

\(\begin{split}d{\bf p}&=d(r\hat{r}+z\hat{z})=\hat{r}dr+rd\hat{r}+\hat{z}dz+zd\hat{z}\\&=\hat{r}dr+r\left(\frac{\partial\hat{r}}{\partial r}dr+\frac{\partial\hat{r}}{\partial\theta}d\theta+\frac{\partial\hat{r}}{\partial z}dz\right)+\hat{z}dz+z\left(\frac{\partial\hat{z}}{\partial r}dr+\frac{\partial\hat{z}}{\partial\theta}d\theta+\frac{\partial\hat{z}}{\partial z}dz\right)\\&=\hat{r}dr+\hat{\theta}rd\theta+\hat{z}dz\end{split}\)

단위벡터의 시간미분 (Time derivative of the unit vectors)

또한 원통좌표계로 표현된 단위벡터의 시간미분도 앞으로 많이 활용될 것이다.

\(\begin{split}\frac{d\hat{r}}{dt}&=\frac{\partial\hat{r}}{\partial r}\dot{r}+\frac{\partial\hat{r}}{\partial\theta}\dot{\theta}+\frac{\partial\hat{r}}{\partial z}\dot{z}=\hat{\theta}\dot{\theta}\\\frac{d\hat{\theta}}{dt}&=\frac{\partial\hat{\theta}}{\partial r}\dot{r}+\frac{\partial\hat{\theta}}{\partial\theta}\dot{\theta}+\frac{\partial\hat{\theta}}{\partial z}\dot{z}=-\hat{r}\dot{\theta}\\\frac{d\hat{z}}{dt}&=\frac{\partial\hat{z}}{\partial r}\dot{r}+\frac{\partial\hat{z}}{\partial\theta}\dot{\theta}+\frac{\partial\hat{z}}{\partial z}\dot{z}=0\end{split}\)

속도와 가속도 (Velocity and Acceleration)

원통좌표계에서 한점의 속도와 가속도는 단위벡터 시간미분으로 표현될 수 있다.

\(\begin{split}{\bf v}&=\dot{\bf p}=\frac{d\hat{r}}{dt}r+\hat{r}\dot{r}+\frac{d\hat{z}}{dt}z+\hat{z}\dot{z}=\hat{r}\dot{r}+\hat{\theta}r\dot{\theta}+\hat{z}\dot{z}\\{\bf a}&=\dot{\bf v}=\frac{d\hat{r}}{dt}\dot{r}+\hat{r}\ddot{r}+\frac{d\hat{\theta}}{dt}r\dot{\theta}+\hat{\theta}\dot{r}\dot{\theta}+\hat{\theta}r\ddot{\theta}+\frac{d\hat{z}}{dt}\dot{z}+\hat{z}\ddot{z}\\&=\hat{\theta}\dot{\theta}\dot{r}+\hat{r}\ddot{r}-\hat{r}r\dot{\theta}^2+\hat{\theta}\dot{r}\dot{\theta}+\hat{\theta}r\ddot{\theta}+\hat{z}\ddot{z}\\&=\hat{r}\left(\ddot{r}-r\dot{\theta}^2\right)+\hat{\theta}\left(r\ddot{\theta}+2\dot{r}\dot{\theta}\right)+\hat{z}\ddot{z}\end{split}\)

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.

\(df=\dfrac{\partial f}{\partial r}dr+\dfrac{\partial f}{\partial\theta}d\theta+\dfrac{\partial f}{\partial z}dz\)

한편으로는 구배의 정의에 의해서 다음식을 얻는다.

\(df=\nabla f\cdot d{\bf p}\)

위의 두 식은 같아야 하므로

\(\dfrac{\partial f}{\partial r}dr+\dfrac{\partial f}{\partial\theta}d\theta+\dfrac{\partial f}{\partial z}dz=\nabla f\cdot d{\bf p}=(\nabla f)_rdr+(\nabla f)_\theta rd\theta+(\nabla f)_zdz\)

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

\((\nabla f)_r=\dfrac{\partial f}{\partial r},\quad(\nabla f)_\theta=\dfrac{\partial f}{r\partial\theta},\quad(\nabla f)_z=\dfrac{\partial f}{\partial z}\)

위의 식은 구배의 각 성분을 나타내므로

\(\nabla=\hat{r}\dfrac{\partial}{\partial r}+\hat{\theta}\dfrac{\partial}{r\partial\theta}+\hat{z}\dfrac{\partial}{\partial z}\)

발산 (Divergence)

발산 \(\nabla\cdot{\bf F}\)는 또 다시 원통좌표계의 단위벡터가 좌표계의 함수라는 것을 고려하여 유도된다.

\(\begin{split}\nabla\cdot{\bf F}&=\left(\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{\partial}{r\partial\theta}+\hat{z}\frac{\partial}{\partial z}\right)\cdot\left(F_r\hat{r}+F_\theta\hat{\theta}+F_z\hat{z}\right)\\&=\left(\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{\partial}{r\partial\theta}+\hat{z}\frac{\partial}{\partial z}\right)\cdot{\bf F}\\&=\hat{r}\cdot\frac{\partial{\bf F}}{\partial r}+\hat{\theta}\cdot\frac{\partial{\bf F}}{r\partial\theta}+\hat{z}\cdot\frac{\partial{\bf F}}{\partial z}\\&=\hat{r}\cdot\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}+F_r\frac{\partial\hat{r}}{\partial r}+F_\theta\frac{\partial\hat{\theta}}{\partial r}+F_z\frac{\partial\hat{z}}{\partial r}\right)\\&+\frac{\hat{\theta}}{r}\cdot\left(\frac{\partial F_r}{\partial\theta}\hat{r}+\frac{\partial F_\theta}{\partial\theta}\hat{\theta}+\frac{\partial F_z}{\partial\theta}\hat{z}+F_r\frac{\partial\hat{r}}{\partial\theta}+F_\theta\frac{\partial\hat{\theta}}{\partial\theta}+F_z\frac{\partial\hat{z}}{\partial\theta}\right)\\&+\hat{z}\cdot\left(\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_\theta}{\partial z}\hat{\theta}+\frac{\partial F_z}{\partial z}\hat{z}+F_r\frac{\partial\hat{r}}{\partial z}+F_\theta\frac{\partial\hat{\theta}}{\partial z}+F_z\frac{\partial\hat{z}}{\partial z}\right)\end{split}\)

단위벡터의 좌표계 미분을 적용하면

\(\begin{split}\nabla\cdot{\bf F}&=\hat{r}\cdot\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}+0+0+0\right)\\&+\frac{\hat{\theta}}{r}\cdot\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}+F_r\hat{\theta}-F_\theta\hat{r}\right)\\&+\hat{z}\cdot\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}+0+0+0\right)\\&=\frac{\partial F_r}{\partial r}+\frac{F_r}{r}+\frac{\partial F_\theta}{r\partial\theta}+\frac{\partial F_z}{\partial z}\\&=\frac{\partial}{r\partial r}\left(rF_r\right)+\frac{\partial F_\theta}{r\partial\theta}+\frac{\partial F_z}{\partial z}\end{split}\)

컬 (Curl)

컬 \(\nabla\times{\bf F}\) 또한 원통좌표계의 단위벡터는 좌표계의 함수라는 것을 고려하여 유도된다.

\(\begin{split}\nabla\times{\bf F}&=\left(\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{\partial}{r\partial\theta}+\hat{z}\frac{\partial}{\partial z}\right)\times\left(F_r\hat{r}+F_\theta\hat{\theta}+F_z\hat{z}\right)\\&=\left(\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{\partial}{r\partial\theta}+\hat{z}\frac{\partial}{\partial z}\right)\times{\bf F}\\&=\hat{r}\times\frac{\partial{\bf F}}{\partial r}+\hat{\theta}\times\frac{\partial{\bf F}}{r\partial\theta}+\hat{z}\times\frac{\partial{\bf F}}{\partial z}\\&=\hat{r}\times\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}+F_r\frac{\partial\hat{r}}{\partial r}+F_\theta\frac{\partial\hat{\theta}}{\partial r}+F_z\frac{\partial\hat{z}}{\partial r}\right)\\&+\frac{\hat{\theta}}{r}\times\left(\frac{\partial F_r}{\partial\theta}\hat{r}+\frac{\partial F_\theta}{\partial\theta}\hat{\theta}+\frac{\partial F_z}{\partial\theta}\hat{z}+F_r\frac{\partial\hat{r}}{\partial\theta}+F_\theta\frac{\partial\hat{\theta}}{\partial\theta}+F_z\frac{\partial\hat{z}}{\partial\theta}\right)\\&+\hat{z}\times\left(\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_\theta}{\partial z}\hat{\theta}+\frac{\partial F_z}{\partial z}\hat{z}+F_r\frac{\partial\hat{r}}{\partial z}+F_\theta\frac{\partial\hat{\theta}}{\partial z}+F_z\frac{\partial\hat{z}}{\partial z}\right)\end{split}\)

단위벡터의 좌표계 미분을 적용하면

\(\begin{split}\nabla\times{\bf F}&=\hat{r}\times\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}+0+0+0\right)\\&+\frac{\hat{\theta}}{r}\times\left(\frac{\partial F_r}{\partial\theta}\hat{r}+\frac{\partial F_\theta}{\partial\theta}\hat{\theta}+\frac{\partial F_z}{\partial\theta}\hat{z}+F_r\hat{\theta}-F_\theta\hat{r}\right)\\&+\hat{z}\times\left(\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_\theta}{\partial z}\hat{\theta}+\frac{\partial F_z}{\partial z}\hat{z}+0+0+0\right)\\&=\left(\frac{\partial F_\theta}{\partial r}\hat{z}-\frac{\partial F_z}{\partial r}\hat{\theta}\right)+\left(-\frac{\partial F_r}{r\partial\theta}\hat{z}+\frac{\partial F_z}{r\partial\theta}\hat{r}+\frac{F_\theta}{r}\hat{z}\right)+\left(\frac{\partial F_r}{\partial z}\hat{\theta}-\frac{\partial F_\theta}{\partial z}\hat{r}\right)\\&=\hat{r}\left(\frac{\partial F_z}{r\partial\theta}-\frac{F_\theta}{\partial z}\right)+\hat{\theta}\left(\frac{\partial F_r}{\partial z}-\frac{\partial F_z}{\partial r}\right)+\hat{z}\left(\frac{\partial}{r\partial r}\left(rF_\theta\right)-\frac{\partial F_r}{r\partial r}\right)\end{split}\) 

라플라시안 (Laplacian)

라플라시안은 다음과 같이 정의에 의해 유도될 수 있는 스칼라 연산자이다.

\(\begin{align*}\nabla^2f&=\nabla\cdot(\nabla f)=\left(\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{\partial}{r\partial\theta}+\hat{z}\frac{\partial}{\partial z}\right)\cdot\left(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta}\frac{\partial f}{r\partial\theta}+\hat{z}\frac{\partial f}{\partial z}\right)\\&=\hat{r}\cdot\frac{\partial}{\partial r}\left(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta}\frac{\partial f}{r\partial\theta}+\hat{z}\frac{\partial f}{\partial z}\right)\\&+\hat{\theta}\cdot\frac{\partial}{r\partial\theta}\left(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta}\frac{\partial f}{r\partial\theta}+\hat{z}\frac{\partial f}{\partial z}\right)\\&+\hat{z}\cdot\frac{\partial}{\partial z}\left(\hat{r}\frac{\partial f}{\partial r}+\hat{\theta}\frac{\partial f}{r\partial\theta}+\hat{z}\frac{\partial f}{\partial z}\right)\end{align*}\)

적(積)의 미분법과 단위벡터의 좌표계 미분을 적용하면

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

따라서, 라플라시안 연산자는 다음과 같이 쓸 수 있다.

\(\nabla^2=\dfrac{\partial}{r\partial r}\left(r\dfrac{\partial}{\partial r}\right)+\dfrac{\partial^2}{r^2\partial\theta^2}+\dfrac{\partial^2}{\partial z^2}\)

라플라시안 연산자를 벡터 F에 적용하면

\[\nabla^2{\bf F}=\frac{\partial}{r\partial r}\left(r\frac{\partial{\bf F}}{\partial r}\right)+\frac{\partial^2{\bf F}}{r^2\partial\theta^2}+\frac{\partial^2{\bf F}}{\partial z^2}\]

각 항별로 벡터 F의 성분으로 편미분하면 다음과 같다.

\[\begin{align}\frac{\partial{\bf 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{\bf F}}{\partial\theta}&=\frac{\partial}{\partial\theta}(F_r\hat{r}+F_\theta\hat{\theta}+F_z\hat{z})=\left(\frac{\partial F_r}{\partial\theta}-F_\theta\right)\hat{r}+\left(\frac{\partial F_\theta}{\partial\theta}+F_r\right)\hat{\theta}+\frac{\partial F_z}{\partial\theta}\hat{z}\\\frac{\partial{\bf 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{\bf F}}{\partial r^2}&=\frac{\partial}{\partial r}\left(\frac{\partial F_r}{\partial r}\hat{r}+\frac{\partial F_\theta}{\partial r}\hat{\theta}+\frac{\partial F_z}{\partial r}\hat{z}\right)=\frac{\partial^2F_r}{\partial r^2}\hat{r}+\frac{\partial^2F_\theta}{\partial r^2}\hat{\theta}+\frac{\partial^2F_z}{\partial r^2}\hat{z}\\\frac{\partial^2{\bf F}}{\partial\theta^2}&=\frac{\partial}{\partial\theta}\left\{\left(\frac{\partial F_r}{\partial\theta}-F_\theta\right)\hat{r}+\left(\frac{\partial F_\theta}{\partial\theta}+F_r\right)\hat{\theta}+\frac{\partial F_z}{\partial\theta}\hat{z}\right\}\\&=\left(\frac{\partial^2F_r}{\partial\theta^2}-2\frac{\partial F_\theta}{\partial\theta}-F_r\right)\hat{r}+\left(2\frac{\partial F_r}{\partial\theta}-F_\theta+\frac{\partial^2F_\theta}{\partial\theta^2}\right)\hat{\theta}+\frac{\partial^2F_z}{\partial\theta^2}\hat{z}\\\frac{\partial^2{\bf F}}{\partial z^2}&=\frac{\partial}{\partial z}\left(\frac{\partial F_r}{\partial z}\hat{r}+\frac{\partial F_\theta}{\partial z}\hat{\theta}+\frac{\partial F_z}{\partial z}\hat{z}\right)=\frac{\partial^2F_r}{\partial z^2}\hat{r}+\frac{\partial^2F_\theta}{\partial z^2}\hat{\theta}+\frac{\partial^2F_z}{\partial z^2}\hat{z}\end{align}\]

위의 결과를 앞의 각 항에 대입하면 벡터 F의 라플라시안을 얻는다.

\[\begin{split}\nabla^2{\bf F}&=\left\{\frac{\partial}{r\partial r}\left(r\frac{\partial F_r}{\partial r}\right)+\frac{\partial^2F_r}{r^2\partial\theta^2}+\frac{\partial^2F_r}{\partial z^2}-\frac{F_r}{r^2}-\frac{2\partial F_\theta}{r^2\partial\theta}\right\}\hat{r}\\&+\left\{\frac{\partial}{r\partial r}\left(r\frac{\partial F_\theta}{\partial r}\right)+\frac{\partial^2F_\theta}{r^2\partial\theta^2}+\frac{\partial^2F_\theta}{\partial z^2}-\frac{F_\theta}{r^2}+\frac{2\partial F_r}{r^2\partial\theta}\right\}\hat{\theta}\\&+\left\{\frac{\partial}{r\partial r}\left(r\frac{\partial F_z}{\partial r}\right)+\frac{\partial^2F_z}{r^2\partial\theta^2}+\frac{\partial^2F_z}{\partial z^2}\right\}\hat{z}\end{split}\]