微分演算子の極座標表示

2次元の極座標表示の場合

デカルト座標と極座標の関係は、
$$\begin{array}{rcl}x& =& r\cos \theta \\ y& =& r\sin \theta \end{array}$$
であり、逆から見ると

$$\begin{array}{rcl}{r}^2& =& x^2+y^2\\ \tan \theta & =& \frac{y}{x}\end{array}$$
となる。それぞれの式の両辺を$x$で偏微分すると、
$$\begin{array}{rcl}\frac{\partial r^2}{\partial r}\frac{\partial r}{\partial x}& =& 2r\frac{\partial r}{\partial x}=2x \mathrm{よ}\mathrm{り} \frac{\partial r}{\partial x}=\frac{x}{r}=\cos \theta \\ \frac{\partial \tan \theta }{\partial \theta }\frac{\partial \theta }{\partial x}& =& \frac{1}{{\cos }^2\theta }\frac{\partial \theta }{\partial x}=-\frac{y}{x^2} \mathrm{よ}\mathrm{り} \frac{\partial \theta }{\partial x}=-\frac{y}{x^2}{\cos }^2\theta =-\frac{\sin \theta }{r}\end{array}$$
となる。同様に$y$で偏微分すると
$$\begin{array}{rcl}\frac{\partial r^2}{\partial r}\frac{\partial r}{\partial y}& =& 2r\frac{\partial r}{\partial y}=2y \mathrm{よ}\mathrm{り} \frac{\partial r}{\partial y}=\frac{y}{r}=\sin \theta \\ \frac{\partial \tan \theta }{\partial \theta }\frac{\partial \theta }{\partial y}& =& \frac{1}{{\cos }^2\theta }\frac{\partial \theta }{\partial y}=\frac{1}{x} \mathrm{よ}\mathrm{り} \frac{\partial \theta }{\partial y}=\frac{1}{x}{\cos }^2\theta =\frac{\cos \theta }{r}\end{array}$$
となる。これらの式を
$$\begin{array}{rcl}\frac{\partial }{\partial x}& =& \frac{\partial r}{\partial x}\frac{\partial }{\partial r}+\frac{\partial \theta }{\partial x}\frac{\partial }{\partial \theta }\\ \frac{\partial }{\partial y}& =& \frac{\partial r}{\partial y}\frac{\partial }{\partial r}+\frac{\partial \theta }{\partial y}\frac{\partial }{\partial \theta }\end{array}$$
に代入すれば、
$$\boxed{{\displaystyle \begin{array}{rcl}\frac{\partial }{\partial x}& =& \cos \theta \frac{\partial }{\partial r}-\frac{1}{r}\sin \theta \frac{\partial }{\partial \theta }\\ \frac{\partial }{\partial y}& =& \sin \theta \frac{\partial }{\partial r}+\frac{1}{r}\cos \theta \frac{\partial }{\partial \theta }\end{array}}}$$
となる。

3次元の極座標表示の場合

デカルト座標と極座標の関係は、
$$\begin{array}{rcl}x& =& r\sin \theta \cos \varphi \\ y& =& r\sin \theta \sin \varphi \\ z& =& r\cos \theta \end{array}$$
であり、逆から見ると
$$\begin{array}{rcl}{r}^2& =& x^2+y^2+z^2\\ \tan \varphi & =& \frac{y}{x}\\ {\tan }^2\theta & =& \frac{x^2+y^2}{z^2}\end{array}$$
となる。それぞれの式の両辺を$x$で偏微分すると、
$$\begin{array}{rcl}\frac{\partial r^2}{\partial r}\frac{\partial r}{\partial x}& =& 2r\frac{\partial r}{\partial x}=2x \mathrm{よ}\mathrm{り} \frac{\partial r}{\partial x}=\frac{x}{r}=\sin \theta \cos \varphi \\ \frac{\partial \tan \varphi }{\partial \varphi }\frac{\partial \varphi }{\partial x}& =& \frac{1}{{\cos }^2\varphi }\frac{\partial \varphi }{\partial x}=-\frac{y}{x^2} \mathrm{よ}\mathrm{り} \frac{\partial \varphi }{\partial x}=-\frac{y}{x^2}{\cos }^2\varphi =-\frac{\sin \varphi }{r\sin \theta }\\ \frac{\partial {\tan }^2\theta }{\partial \theta }\frac{\partial \theta }{\partial x}& =& \frac{2\tan \theta }{{\cos }^2\theta }\frac{\partial \theta }{\partial x}=\frac{2x}{z^2} \mathrm{よ}\mathrm{り} \frac{\partial \theta }{\partial x}=\frac{x}{z^2}\frac{{\cos }^2\theta }{\tan \theta }=\frac{1}{r}\cos \theta \cos \varphi \end{array}$$
となる。同様に$y$で偏微分すると
$$\begin{array}{rcl}\frac{\partial r^2}{\partial r}\frac{\partial r}{\partial y}& =& 2r\frac{\partial r}{\partial y}=2y \mathrm{よ}\mathrm{り} \frac{\partial r}{\partial y}=\frac{y}{r}=\sin \theta \sin \varphi \\ \frac{\partial \tan \varphi }{\partial \varphi }\frac{\partial \varphi }{\partial y}& =& \frac{1}{{\cos }^2\varphi }\frac{\partial \varphi }{\partial y}=-\frac{1}{x} \mathrm{よ}\mathrm{り} \frac{\partial \varphi }{\partial y}=\frac{1}{x}{\cos }^2\varphi =\frac{\cos \varphi }{r\sin \theta }\\ \frac{\partial {\tan }^2\theta }{\partial \theta }\frac{\partial \theta }{\partial y}& =& \frac{2\tan \theta }{{\cos }^2\theta }\frac{\partial \theta }{\partial y}=\frac{2y}{z^2} \mathrm{よ}\mathrm{り} \frac{\partial \theta }{\partial y}=\frac{y}{z^2}\frac{{\cos }^2\theta }{\tan \theta }=\frac{1}{r}\cos \theta \sin \varphi \end{array}$$
となり、$z$で偏微分すると
$$\begin{array}{rcl}\frac{\partial r^2}{\partial r}\frac{\partial r}{\partial z}& =& 2r\frac{\partial r}{\partial z}=2z \mathrm{よ}\mathrm{り} \frac{\partial r}{\partial z}=\frac{z}{r}=\cos \theta \\ \frac{\partial \tan \varphi }{\partial \varphi }\frac{\partial \varphi }{\partial z}& =& \frac{1}{{\cos }^2\varphi }\frac{\partial \varphi }{\partial z}=0 \mathrm{よ}\mathrm{り} \frac{\partial \varphi }{\partial z}=0\\ \frac{\partial {\tan }^2\theta }{\partial \theta }\frac{\partial \theta }{\partial z}& =& \frac{2\tan \theta }{{\cos }^2\theta }\frac{\partial \theta }{\partial z}=-2\frac{x^2+y^2}{z^3} \mathrm{よ}\mathrm{り} \frac{\partial \theta }{\partial z}=-\frac{x^2+y^2}{z^3}\frac{{\cos }^2\theta }{\tan \theta }=-\frac{1}{r}\sin \theta \end{array}$$
となる。これらの式を
$$\begin{array}{rcl}\frac{\partial }{\partial x}& =& \frac{\partial r}{\partial x}\frac{\partial }{\partial r}+\frac{\partial \theta }{\partial x}\frac{\partial }{\partial \theta }+\frac{\partial \varphi }{\partial x}\frac{\partial }{\partial \varphi }\\ \frac{\partial }{\partial y}& =& \frac{\partial r}{\partial y}\frac{\partial }{\partial r}+\frac{\partial \theta }{\partial y}\frac{\partial }{\partial \theta }+\frac{\partial \varphi }{\partial y}\frac{\partial }{\partial \varphi }\\ \frac{\partial }{\partial z}& =& \frac{\partial r}{\partial z}\frac{\partial }{\partial r}+\frac{\partial \theta }{\partial z}\frac{\partial }{\partial \theta }+\frac{\partial \varphi }{\partial z}\frac{\partial }{\partial \varphi }\end{array}$$
に代入すれば、
$$\boxed{{\displaystyle \begin{array}{rcl}\frac{\partial }{\partial x}& =& \sin \theta \cos \varphi \frac{\partial }{\partial r}+\frac{1}{r}\cos \theta \cos \varphi \frac{\partial }{\partial \theta }-\frac{1}{r}\frac{\sin \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }\\ \frac{\partial }{\partial y}& =& \sin \theta \sin \varphi \frac{\partial }{\partial r}+\frac{1}{r}\cos \theta \sin \varphi \frac{\partial }{\partial \theta }+\frac{1}{r}\frac{\cos \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }\\ \frac{\partial }{\partial z}& =& \cos \theta \frac{\partial }{\partial r}-\frac{1}{r}\sin \theta \frac{\partial }{\partial \theta }\end{array}}}$$
となる。${\mathbf{\nabla }}^2$については、上記の式を計算すれば求まるのだが、とても計算が面倒くさい。ほとんどの教科書がそうしているように途中式は省略して、結果のみ書くと、
$$\boxed{{\displaystyle {\mathbf{\nabla }}^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial }{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}}}$$
となる。