本稿では、
\begin{eqnarray}
x &=& r \cos \theta \\
y &=& r \sin \theta \\
z &=& z \tag{1}
\end{eqnarray}で表される3次元の円柱座標系$(r, \theta, z)$の偏微分について述べます。
極座標 - 数式で独楽する
円柱座標系による直交座標系の偏微分
式(1)より、容易に
\begin{eqnarray}
&& \frac{\partial x}{\partial r} = \cos \theta, & \quad \frac{\partial x}{\partial \theta} = -r \sin \theta , & \quad \frac{\partial x}{\partial z} = 0 \\
&& \frac{\partial y}{\partial r} = \sin \theta, & \quad \frac{\partial y}{\partial \theta} = r \cos \theta , & \quad \frac{\partial y}{\partial z} = 0 \\
&& \frac{\partial z}{\partial r} = 0, & \quad \frac{\partial z}{\partial \theta} = 0, & \quad \frac{\partial z}{\partial z} = 1 \tag{2}
\end{eqnarray}を得ることができます。
極座標系による偏微分を直交座標系による偏微分で表す
式(2)を用いて、極座標系による偏微分を直交座標系による偏微分で表すと、次のようになります。
\begin{eqnarray}
\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} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial r} = \cos \theta \, \frac{\partial u}{\partial x} + \sin \theta \, \frac{\partial u}{\partial y}\\
\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} = -r\sin \theta \, \frac{\partial u}{\partial x} + r\cos \theta \, \frac{\partial u}{\partial y} \\
\frac{\partial u}{\partial z} &=& \frac{\partial u}{\partial x} \frac{\partial x}{\partial z} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial z} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial z} = \frac{\partial u}{\partial z} \tag{3}
\end{eqnarray}
行列
\begin{equation}
R = \left( \begin{array}{ccc} \cos \theta & \sin \theta & 0 \\
-\sin \theta & \cos \theta & 0 \\
0 & 0 & 1 \end{array} \right)
\end{equation}を用いて表すと、
\begin{equation}
\left( \begin{array}{c} \frac{\partial u}{\partial r} \\ \frac{1}{r} \frac{\partial u}{\partial \theta} \\ \frac{\partial u}{\partial z} \end{array} \right)
= R \left( \begin{array}{c} \frac{\partial u}{\partial x} \\ \frac{\partial u}{\partial y} \\ \frac{\partial u}{\partial z} \end{array} \right)
\end{equation}となります。
直交座標系による極座標系の偏微分
式(1)より
\begin{eqnarray}
r &=& \sqrt{x^2+y^2} \\
\tan \theta &=& \frac{y}{x}
\end{eqnarray}となります。
偏微分が煩雑になりますが、計算していきます。
\begin{eqnarray}
\frac{\partial r}{\partial x} &=& \frac{x}{\sqrt{x^2+y^2}} &=\frac{x}{r} &= \cos \theta \\
\frac{\partial r}{\partial y} &=& \frac{y}{\sqrt{x^2+y^2}} &=\frac{y}{x} &= \sin \theta \\
\frac{\partial r}{\partial z} && &&= 0 \\
\frac{\partial \theta}{\partial x} &=& \cos^2 \theta \cdot \left( -\frac{y}{x^2} \right) &= \frac{\cos^2 \theta \cdot r\sin \theta}{r^2 \cos^2 \theta} &= -\frac{\sin \theta}{r} \\
\frac{\partial \theta}{\partial y} &=& \cos^2 \theta \cdot \frac{1}{x} &= \frac{\cos^2 \theta}{r \cos \theta} &= \frac{\cos \theta}{r} \\
\frac{\partial \theta}{\partial z} && &&= 0 \\
\frac{\partial z}{\partial x} && &&= 0 \\
\frac{\partial z}{\partial y} && &&= 0 \\
\frac{\partial z}{\partial z} && &&= 1 \tag{4}
\end{eqnarray}を得ます。
直交座標系による偏微分を極座標系による偏微分で表す
式(4)を用いて、直交座標系による偏微分を極座標系による偏微分で表すと、次のようになります。
\begin{eqnarray}
\frac{\partial u}{\partial x} &=& \frac{\partial u}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial u}{\partial \theta} \frac{\partial \theta}{\partial x} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial x} = \cos \theta \, \frac{\partial u}{\partial r} -\frac{\sin \theta}{r} \, \frac{\partial u}{\partial \theta}\\
\frac{\partial u}{\partial y} &=& \frac{\partial u}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial u}{\partial \theta} \frac{\partial \theta}{\partial y} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial y} = \sin \theta \, \frac{\partial u}{\partial r} + \frac{\cos \theta}{r} \, \frac{\partial u}{\partial \theta} \\
\frac{\partial u}{\partial z} &=& \frac{\partial u}{\partial r} \frac{\partial r}{\partial z} + \frac{\partial u}{\partial \theta} \frac{\partial \theta}{\partial z} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial z} = \frac{\partial u}{\partial z} \tag{5}
\end{eqnarray}
こちらを行列で表すと
\begin{equation}
\left( \begin{array}{c} \frac{\partial u}{\partial x} \\ \frac{\partial u}{\partial y} \\ \frac{\partial u}{\partial z} \end{array} \right)
= R^{-1} \left( \begin{array}{c} \frac{\partial u}{\partial r} \\ \frac{1}{r} \frac{\partial u}{\partial \theta} \\ \frac{\partial u}{\partial z}\end{array} \right)
\end{equation}となります。
