원통좌표계, 구면좌표계에서의 라플라스 방정식 유도

원통좌표계, 구면좌표계에서의 라플라스 방정식 유도

직각좌표계\((\mathbb{R}^{3})\)에서의 라플라스 방정식은 \(\displaystyle\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}=0\)이다. 

라플라스 방정식을 각각 원통좌표계와 구면좌표계에 대해서 나타내면 

원통좌표계: \(\displaystyle\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}}+\frac{\partial^{2}u}{\partial z^{2}}=0\)

구면좌표계: \(\displaystyle\frac{\partial^{2}u}{\partial\rho^{2}}+\frac{2}{\rho}\frac{\partial u}{\partial\rho}+\frac{\cot\phi}{\rho^{2}}\frac{\partial u}{\partial\phi}+\frac{1}{\rho^{2}}\frac{\partial^{2}u}{\partial\phi^{2}}+\frac{1}{\rho^{2}\sin^{2}\phi}\frac{\partial^{2}u}{\partial\theta^{2}}=0\)

이다. 편미분에서의 연쇄법칙을 이용하여 위 식들을 유도할 수 있다.

원통좌표계에 대한 라플라스 방정식 유도:

\(x=r\cos\theta\), \(y=r\sin\theta\), \(z=z\,(r\geq0,\,0\leq\theta\leq2\pi)\)이고

$$\begin{align*}\frac{\partial u}{\partial r}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}=\cos\theta\frac{\partial u}{\partial x}+\sin\theta\frac{\partial u}{\partial y}\\ \frac{\partial^{2}u}{\partial r^{2}}&=\frac{\partial}{\partial r}\left(\cos\theta\frac{\partial u}{\partial x}+\sin\theta\frac{\partial u}{\partial y}\right)=\cos\theta\frac{\partial}{\partial r}\left(\frac{\partial u}{\partial x}\right)+\sin\theta\frac{\partial}{\partial r}\left(\frac{\partial u}{\partial y}\right)\\ \frac{\partial}{\partial r}\left(\frac{\partial u}{\partial x}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}\right)\frac{\partial x}{\partial r}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\frac{\partial y}{\partial r}=\cos\theta\frac{\partial^{2}u}{\partial x^{2}}+\sin\theta\frac{\partial^{2}u}{\partial x\partial y}\\ \frac{\partial}{\partial r}\left(\frac{\partial u}{\partial y}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right)\frac{\partial x}{\partial r}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)\frac{\partial y}{\partial r}=\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\sin\theta\frac{\partial^{2}u}{\partial y^{2}}\end{align*}$$이므로 \(\displaystyle\frac{\partial^{2}u}{\partial r^{2}}=\cos^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+2\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\sin^{2}\theta\frac{\partial^{2}u}{\partial^{2}y}\)이다. 

같은 방법으로

$$\begin{align*}\frac{\partial u}{\partial\theta}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta}=-r\sin\theta\frac{\partial u}{\partial x}+r\cos\theta\frac{\partial u}{\partial y}\\ \frac{\partial^{2}u}{\partial\theta^{2}}&=\frac{\partial}{\partial\theta}\left(-r\sin\theta\frac{\partial u}{\partial x}+r\cos\theta\frac{\partial u}{\partial y}\right)=-r\cos\theta\frac{\partial u}{\partial x}-r\sin\theta\frac{\partial u}{\partial y}-r\sin\theta\frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial x}\right)+r\cos\theta\frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial y}\right)\\ \frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial x}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}\right)\frac{\partial x}{\partial\theta}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\frac{\partial y}{\partial\theta}=-r\sin\theta\frac{\partial^{2}u}{\partial x^{2}}+r\cos\theta\frac{\partial^{2}u}{\partial x\partial y}\\ \frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial y}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right)\frac{\partial x}{\partial\theta}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)\frac{\partial x}{\partial\theta}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)\frac{\partial y}{\partial\theta}=-r\sin\theta\frac{\partial^{2}u}{\partial x\partial y}+r\cos\theta\frac{\partial^{2}u}{\partial y^{2}}\end{align*}$$이므로$$\begin{align*}\frac{\partial^{2}u}{\partial\theta^{2}}&=-r\cos\theta\frac{\partial u}{\partial x}-r\sin\theta\frac{\partial u}{\partial y}+r^{2}\sin^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}-2r\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+r^{2}\cos^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}\\&=-r\frac{\partial u}{\partial r}+r^{2}\sin^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}-2r^{2}\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+r^{2}\cos^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}\end{align*}$$이고 \(\displaystyle\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}}=\sin^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}-2\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\cos^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}\)이다. 따라서

$$\begin{align*}\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}}+\frac{\partial^{2}u}{\partial z^{2}}&=(\cos^{2}\theta+\sin^{2}\theta)\frac{\partial^{2}u}{\partial x^{2}}+(\sin^{2}\theta+\cos^{2}\theta)\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial^{2}z}\\&=\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial^{2}y}+\frac{\partial^{2}u}{\partial z^{2}}\\&=0\end{align*}$$이다. 

이렇게 원통좌표계에서의 라플라스 방정식을 유도했다.

구면좌표계에 대한 라플라스 방정식 유도:

\(x=\rho\sin\phi\cos\theta\), \(y=\rho\sin\phi\sin\theta\), \(z=\rho\cos\phi\,(\rho\geq0,\,0\leq\phi\leq\pi,\,0\leq\theta\leq2\pi)\)이고

$$\begin{align*}\frac{\partial u}{\partial\rho}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\rho}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\rho}+\frac{\partial u}{\partial z}\frac{\partial z}{\partial\rho}=\sin\phi\cos\theta\frac{\partial u}{\partial x}+\sin\phi\sin\theta\frac{\partial u}{\partial y}+\cos\phi\frac{\partial u}{\partial z}\\ \frac{\partial^{2}u}{\partial\rho^{2}}&=\sin\phi\cos\theta\frac{\partial}{\partial\rho}\left(\frac{\partial u}{\partial x}\right)+\sin\phi\sin\theta\frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial y}\right)+\cos\phi\frac{\partial}{\partial\rho}\left(\frac{\partial u}{\partial z}\right)\\ \frac{\partial}{\partial\rho}\left(\frac{\partial u}{\partial x}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}\right)\frac{\partial x}{\partial\rho}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\frac{\partial y}{\partial\rho}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial x}\right)\frac{\partial z}{\partial\rho}=\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x^{2}}+\sin\phi\sin\theta\frac{\partial^{2}u}{\partial x\partial y}+\cos\phi\frac{\partial^{2}u}{\partial y\partial z}\\ \frac{\partial}{\partial\rho}\left(\frac{\partial u}{\partial y}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right)\frac{\partial x}{\partial\rho}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)\frac{\partial y}{\partial\rho}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial y}\right)\frac{\partial z}{\partial \rho}=\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\sin\phi\sin\theta\frac{\partial^{2}u}{\partial y^{2}}+\cos\phi\frac{\partial^{2}u}{\partial y\partial z}\\ \frac{\partial}{\partial\rho}\left(\frac{\partial u}{\partial z}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial z}\right)\frac{\partial x}{\partial\rho}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial z}\right)\frac{\partial y}{\partial\rho}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial z}\right)\frac{\partial z}{\partial\rho}=\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\sin\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}+\cos\phi\frac{\partial^{2}u}{\partial z^{2}}\end{align*}$$이므로

$$\begin{align*}\frac{\partial^{2}u}{\partial\rho^{2}}&=\sin\phi\cos\theta\left(\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x^{2}}+\sin\phi\sin\theta\frac{\partial^{2}u}{\partial x\partial y}+\cos\phi\frac{\partial^{2}u}{\partial z\partial x}\right)\\&+\sin\phi\sin\theta\left(\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\sin\phi\sin\theta\frac{\partial^{2}u}{\partial y^{2}}+\cos\phi\frac{\partial^{2}u}{\partial y\partial z}\right)\\&+\cos\phi\left(\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial z}+\sin\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}+\cos\phi\frac{\partial^{2}u}{\partial z^{2}}\right)\\&=\sin^{2}\phi\cos^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+\sin^{2}\phi\sin^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}+\cos^{2}\phi\frac{\partial^{2}u}{\partial z^{2}}\\&+2\sin^{2}\phi\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+2\sin\phi\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}+2\sin\phi\cos\phi\cos\theta\frac{\partial^{2}u}{\partial z\partial x}\end{align*}$$이다.

$$\begin{align*}\frac{\partial u}{\partial\phi}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\phi}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\phi}+\frac{\partial u}{\partial z}\frac{\partial z}{\partial\phi}=\rho\cos\phi\cos\theta\frac{\partial u}{\partial x}+\rho\cos\phi\sin\theta\frac{\partial u}{\partial y}-\rho\sin\phi\frac{\partial u}{\partial z}\\ \frac{\partial^{2}u}{\partial\phi^{2}}&=-\rho\sin\phi\cos\theta\frac{\partial u}{\partial x}+\rho\sin\phi\sin\theta\frac{\partial u}{\partial y}-\rho\cos\phi\frac{\partial u}{\partial z}\\&+\rho\cos\phi\cos\theta\frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial x}\right)+\rho\cos\phi\sin\theta\frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial y}\right)-\rho\sin\phi\frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial z}\right)\\ \frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial x}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}\right)\frac{\partial x}{\partial\phi}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\frac{\partial y}{\partial\phi}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial x}\right)\frac{\partial z}{\partial\phi}=\rho\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x^{2}}+\rho\cos\phi\sin\theta\frac{\partial^{2}u}{\partial x\partial y}-\rho\sin\phi\frac{\partial^{2}u}{\partial x\partial z}\\  \frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial y}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial  y}\right)\frac{\partial y}{\partial\phi}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)\frac{\partial y}{\partial\phi}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial y}\right)\frac{\partial z}{\partial\phi}=\rho\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\rho\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y^{2}}-\rho\sin\phi\frac{\partial^{2}u}{\partial y\partial z}\\ \frac{\partial}{\partial\phi}\left(\frac{\partial u}{\partial z}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial z}\right)\frac{\partial x}{\partial\phi}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial z}\right)\frac{\partial y}{\partial\phi}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial z}\right)\frac{\partial z}{\partial\phi}=\rho\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial z}+\rho\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}-\rho\sin\phi\frac{\partial^{2}u}{\partial z^{2}}\end{align*}$$이므로

$$\begin{align*}\frac{\partial^{2}u}{\partial\phi^{2}}&=-\rho\sin\phi\cos\theta\frac{\partial u}{\partial x}-\rho\sin\phi\sin\theta\frac{\partial u}{\partial y}-\rho\cos\phi\frac{\partial u}{\partial z}\\&+\rho\cos\phi\cos\theta\left(\rho\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x^{2}}+\rho\cos\phi\sin\theta\frac{\partial^{2}u}{\partial x\partial y}-\rho\sin\phi\frac{\partial^{2}u}{\partial x\partial z}\right)\\&+\rho\cos\phi\sin\theta\left(\rho\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+\rho\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y^{2}}-\rho\sin\phi\frac{\partial^{2}u}{\partial y\partial z}\right)\\&-\rho\sin\phi\left(\rho\cos\phi\frac{\partial^{2}u}{\partial x\partial z}+\rho\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}-\rho\sin\phi\frac{\partial^{2}u}{\partial z^{2}}\right)\\&=-\left(\rho\sin\phi\cos\theta\frac{\partial u}{\partial x}+\rho\sin\phi\sin\theta\frac{\partial u}{\partial y}+\rho\cos\phi\frac{\partial u}{\partial z}\right)+\rho^{2}\cos^{2}\phi\cos^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+\rho^{2}\cos^{2}\phi\sin^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}+\rho^{2}\sin^{2}\phi\frac{\partial^{2}u}{\partial z^{2}}\\&+2\rho^{2}\cos^{2}\phi\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}-2\rho^{2}\sin\phi\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}-2\rho^{2}\sin\phi\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial z}\end{align*}$$이다.

$$\begin{align*}\frac{\partial u}{\partial\theta}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta}+\frac{\partial u}{\partial z}\frac{\partial z}{\partial\theta}=-\rho\sin\phi\sin\theta\frac{\partial u}{\partial x}+\rho\sin\phi\cos\theta\frac{\partial u}{\partial y}\\ \frac{\partial^{2}u}{\partial\theta^{2}}&=-\rho\sin\phi\cos\theta\frac{\partial u}{\partial x}-\rho\sin\phi\sin\theta\frac{\partial u}{\partial y}-\rho\sin\phi\sin\theta\frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial  x}\right)+\rho\sin\phi\cos\theta\frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial y}\right)\\ \frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial x}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}\right)\frac{\partial x}{\partial\theta}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\frac{\partial y}{\partial\theta}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial x}\right)\frac{\partial z}{\partial\theta}=-\rho\sin\phi\sin\theta\frac{\partial^{2}u}{\partial x^{2}}+\rho\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}\\ \frac{\partial}{\partial\theta}\left(\frac{\partial u}{\partial y}\right)&=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right)\frac{\partial x}{\partial\theta}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}\right)\frac{\partial y}{\partial\theta}+\frac{\partial}{\partial z}\left(\frac{\partial u}{\partial y}\right)\frac{\partial z}{\partial\theta}=-\rho\sin\phi\sin\theta\frac{\partial^{2}u}{\partial x\partial y}+\rho\sin\phi\cos\theta\frac{\partial^{2}u}{\partial y^{2}}\end{align*}$$이므로

$$\begin{align*}\frac{\partial^{2}u}{\partial\theta^{2}}&=-\rho\sin\phi\cos\theta\frac{\partial u}{\partial y}-\rho\sin\phi\cos\theta\frac{\partial u}{\partial y}\\&-\rho\sin\phi\sin\theta\left(-\rho\sin\phi\sin\theta\frac{\partial^{2}u}{\partial x^{2}}+\rho\sin\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial y}\right)+\rho\sin\phi\cos\theta\left(-\rho\sin\phi\sin\theta\frac{\partial^{2}u}{\partial x\partial y}+\rho\sin\phi\cos\theta\frac{\partial^{2}u}{\partial y^{2}}\right)\\&=-\rho\sin\phi\cos\theta\frac{\partial u}{\partial x}-\rho\sin\phi\sin\theta\frac{\partial u}{\partial y}+\rho^{2}\sin^{2}\phi\sin^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+\rho^{2}\sin^{2}\phi\cos^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}-2\rho^{2}\sin^{2}\phi\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}\end{align*}$$이다. 식들을 종합하면

$$\begin{align*}\frac{\partial^{2}u}{\partial\rho^{2}}&=\sin^{2}\phi\cos^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+\sin^{2}\phi\sin^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}+\cos^{2}\phi\frac{\partial^{2}u}{\partial z^{2}}\\&+2\sin^{2}\phi\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}+2\sin\phi\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}+2\sin\phi\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial z}\\ \frac{2}{\rho}\frac{\partial u}{\partial\rho}&=\frac{2\sin\phi\cos\theta}{\rho}\frac{\partial u}{\partial x}+\frac{2\sin\phi\sin\theta}{\rho}\frac{\partial u}{\partial y}+\frac{2\cos\phi}{\rho}\frac{\partial u}{\partial z}\\ \frac{\cot\phi}{\rho^{2}}\frac{\partial u}{\partial\phi^{2}}&=\frac{\cos^{2}\phi\cos\theta}{\rho\sin\phi}\frac{\partial u}{\partial x}+\frac{\cos^{2}\phi\sin\theta}{\rho\sin\phi}\frac{\partial u}{\partial y}-\frac{\cos\phi}{\rho}\frac{\partial u}{\partial z}\\ \frac{1}{\rho^{2}}\frac{\partial^{2}u}{\partial\phi^{2}}&=-\frac{\sin\phi\cos\theta}{\rho}\frac{\partial u}{\partial x}-\frac{\sin\phi\sin\theta}{\rho}\frac{\partial u}{\partial y}-\frac{\cos\phi}{\rho}\frac{\partial u}{\partial z}+\cos^{2}\phi\cos^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+\cos^{2}\phi\sin^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}+\sin^{2}\phi\frac{\partial^{2}u}{\partial z^{2}}\\&+2\cos^{2}\phi\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}-2\sin\phi\cos\phi\sin\theta\frac{\partial^{2}u}{\partial y\partial z}-2\sin\phi\cos\phi\cos\theta\frac{\partial^{2}u}{\partial x\partial z}\\ \frac{1}{\rho^{2}\sin^{2}\phi}\frac{\partial^{2}u}{\partial\theta^{2}}&=-\frac{\cos\theta}{\rho\sin\phi}\frac{\partial u}{\partial x}-\frac{\sin\theta}{\rho\sin\phi}\frac{\partial u}{\partial y}+\sin^{2}\theta\frac{\partial^{2}u}{\partial x^{2}}+\cos^{2}\theta\frac{\partial^{2}u}{\partial y^{2}}-2\sin\theta\cos\theta\frac{\partial^{2}u}{\partial x\partial y}\end{align*}$$이고 이때$$\begin{align*}\frac{2\sin\phi\cos\theta}{\rho}+\frac{\cos^{2}\phi\cos\theta}{\rho\sin\phi}-\frac{\sin\phi\cos\theta}{\rho}-\frac{\cos\theta}{\rho\sin\phi}&=\frac{\sin\phi\cos\theta}{\rho}-\frac{(1-\cos^{2}\phi)\cos\theta}{\rho\sin\phi}=\frac{\sin\phi\cos\theta}{\rho}-\frac{\sin\phi\cos\theta}{\rho}=0\\ \frac{2\sin\phi\sin\theta}{\rho}+\frac{\cos^{2}\phi\sin\theta}{\rho\sin\phi}-\frac{\sin\phi\sin\theta}{\rho}-\frac{\sin\theta}{\rho\sin\phi}&=\frac{\sin\phi\sin\theta}{\rho}-\frac{(1-\cos^{2}\phi)\sin\theta}{\rho\sin\phi}=\frac{\sin\phi\sin\theta}{\rho}-\frac{\sin\phi\sin\theta}{\rho}=0\end{align*}$$이므로 따라서 \(\displaystyle\frac{\partial^{2}u}{\partial\rho^{2}}+\frac{2}{\rho}\frac{\partial u}{\partial\rho}+\frac{\cot\phi}{\rho^{2}}\frac{\partial u}{\partial\phi}+\frac{1}{\rho^{2}}\frac{\partial^{2}u}{\partial\theta^{2}}+\frac{1}{\rho^{2}\sin^{2}\phi}\frac{\partial^{2}u}{\partial\theta^{2}}=0\)이 성립한다.