ナブラ演算子とは

ナブラ演算子：ベクトル解析における微分演算子。$\nabla$ で表す。
たとえば3次元空間におけるナブラ演算子は下式で表される。

\[\nabla = \left( \cfrac{\partial}{\partial x}, \cfrac{\partial}{\partial y}, \cfrac{\partial}{\partial z} \right)\]

ベクトル三重積・スカラー三重積

ラプラシアン

定義

ラプラシアン：ナブラ演算子同士の内積 $\nabla \cdot \nabla$ で定義される演算子。$\nabla^2$ または $\Delta$ で表す（大文字デルタと同じ記号）。

\[\Delta = \nabla^2 := \nabla \cdot \nabla = \cfrac{\partial^2}{\partial x^2} + \cfrac{\partial^2}{\partial y^2} + \cfrac{\partial^2}{\partial z^2}\]

極座標形式

極座標 $(r,\theta,\phi)$ を用いてラプラシアンを表すと、

\[\Delta f = \cfrac{1}{r^2} \cfrac{\partial}{\partial r} \left( r^2 \cfrac{\partial f}{\partial r} \right) + \cfrac{1}{r^2 \sin\theta} \cfrac{\partial}{\partial \theta} \left( \sin\theta \cfrac{\partial f}{\partial \theta} \right) + \cfrac{1}{r^2\sin^2\theta} \cfrac{\partial^2 f}{\partial \phi^2}\]

導出

極座標 $(r,\theta,\phi)$ と直交座標 $(x,y,z)$ の関係は以下の通り。

\[\begin{cases} x = r\sin\theta \cos\phi \\ y = r \sin\theta \sin\phi \\ z = r \cos\theta \end{cases}\] \[\begin{cases} r = \sqrt{x^2+y^2+z^2} \\ \theta = \arctan \cfrac{\sqrt{x^2+y^2}}{z} \\ \phi = \arctan \cfrac{y}{x} \end{cases}\]

ここで、後の計算で使うために必要な偏微分を計算しておく。

\[\begin{eqnarray} \cfrac{\partial r}{\partial x} &=& \cfrac{x}{\sqrt{x^2+y^2+z^2}} = \cfrac{x}{r} = \sin\theta\cos\phi \\ \cfrac{\partial r}{\partial y} &=& \cfrac{y}{\sqrt{x^2+y^2+z^2}} = \cfrac{y}{r} = \sin\theta\sin\phi \\ \cfrac{\partial r}{\partial z} &=& \cfrac{z}{\sqrt{x^2+y^2+z^2}} = \cfrac{z}{r} = \cos\theta \\ \cfrac{\partial \theta}{\partial x} &=& 2x \cdot \cfrac{1}{2z\sqrt{x^2+y^2}} \cdot \cfrac{1}{1+(x^2+y^2)/z^2} = \cfrac{xz}{(x^2+y^2+z^2)\sqrt{x^2+y^2}} \\ &=& \cfrac{r\sin\theta \cos\phi \cdot r\cos\theta}{r^2 \cdot r \sin \theta} = \cfrac{1}{r} \cos\theta \cos\phi \\ \cfrac{\partial \theta}{\partial y} &=& 2y \cdot \cfrac{1}{2z\sqrt{x^2+y^2}} \cdot \cfrac{1}{1+(x^2+y^2)/z^2} = \cfrac{yz}{(x^2+y^2+z^2)^2\sqrt{x^2+y^2}} \\ &=& \cfrac{r\sin\theta \sin\phi \cdot r\cos\theta}{r^2 \cdot r \sin \theta} = \cfrac{1}{r} \cos\theta \sin\phi \\ \cfrac{\partial \theta}{\partial z} &=& - \cfrac{\sqrt{x^2+y^2}}{z^2} \cdot \cfrac{1}{1+(x^2+y^2)/z^2} = - \cfrac{\sqrt{x^2+y^2}}{x^2+y^2+z^2} \\ &=& - \cfrac{r\sin\theta}{r^2} = - \cfrac{1}{r} \sin\theta \\ \cfrac{\partial \phi}{\partial x} &=& - \cfrac{y}{x^2} \cdot \cfrac{1}{1+y^2/x^2} = -\cfrac{y}{x^2+y^2} \\ &=& - \cfrac{r\sin\theta \sin\phi}{r^2 \sin^2\theta} = -\cfrac{\sin\phi}{r\sin\theta} \\ \cfrac{\partial \phi}{\partial y} &=& \cfrac{1}{x} \cdot \cfrac{1}{1+y^2/x^2} = \cfrac{x}{x^2+y^2} \\ &=& \cfrac{r\sin\theta \cos\phi}{r^2 \sin^2\theta} = \cfrac{\cos\phi}{r\sin\theta} \\ \cfrac{\partial \phi}{\partial z} &=& 0 \end{eqnarray}\]

途中、逆三角関数の微分公式

\[\cfrac{d}{du} \arctan u = \cfrac{1}{1+u^2}\]

と、極座標では $0 \le \theta \le \pi$ より $\sin \theta \ge 0$ であることを用いた。

以上を用いて、極座標の関数 $f(r, \theta, \phi)$ に関して、$x,y,z$ による偏微分を行う。
まず $x$ について、

\[\begin{eqnarray} \cfrac{\partial f}{\partial x} &=& \cfrac{\partial r}{\partial x} \cfrac{\partial f}{\partial r} + \cfrac{\partial \theta}{\partial x} \cfrac{\partial f}{\partial \theta} + \cfrac{\partial \phi}{\partial x} \cfrac{\partial f}{\partial \phi} \\ &=& \sin\theta\cos\phi \cfrac{\partial f}{\partial r} + \cfrac{1}{r} \cos\theta \cos\phi \cfrac{\partial f}{\partial \theta} - \cfrac{\sin\phi}{r\sin\theta} \cfrac{\partial f}{\partial \phi} \\ \\ \cfrac{\partial^2 f}{\partial x^2} &=& \sin\theta\cos\phi \cfrac{\partial}{\partial r} \left(\cfrac{\partial f}{\partial x}\right) + \cfrac{1}{r} \cos\theta \cos\phi \cfrac{\partial}{\partial \theta} \left(\cfrac{\partial f}{\partial x}\right) - \cfrac{\sin\phi}{r\sin\theta} \cfrac{\partial}{\partial \phi} \left(\cfrac{\partial f}{\partial x}\right) \\ &=& \sin\theta\cos\phi \left( \sin\theta\cos\phi \cfrac{\partial^2 f}{\partial r^2} - \cfrac{1}{r^2} \cos\theta \cos\phi \cfrac{\partial f}{\partial \theta} + \cfrac{1}{r} \cos\theta \cos\phi \cfrac{\partial^2 f}{\partial \theta \partial r} + \cfrac{\sin\phi}{r^2\sin\theta} \cfrac{\partial f}{\partial \phi} - \cfrac{\sin\phi}{r\sin\theta} \cfrac{\partial^2 f}{\partial \phi \partial r} \right) \\ && + \cfrac{1}{r} \cos\theta \cos\phi \left( \cos\theta\cos\phi \cfrac{\partial f}{\partial r} + \sin\theta\cos\phi \cfrac{\partial^2 f}{\partial r \partial \theta} - \cfrac{1}{r} \sin\theta \cos\phi \cfrac{\partial f}{\partial \theta} + \cfrac{1}{r} \cos\theta \cos\phi \cfrac{\partial^2 f}{\partial \theta^2} + \cfrac{\cos\theta\sin\phi}{r\sin^2\theta} \cfrac{\partial f}{\partial \phi} - \cfrac{\sin\phi}{r\sin\theta} \cfrac{\partial^2 f}{\partial \phi \partial \theta} \right) \\ && - \cfrac{\sin\phi}{r\sin\theta} \left( -\sin\theta\sin\phi \cfrac{\partial f}{\partial r} + \sin\theta\cos\phi \cfrac{\partial^2 f}{\partial r \partial \phi} - \cfrac{1}{r} \cos\theta \sin\phi \cfrac{\partial f}{\partial \theta} + \cfrac{1}{r} \cos\theta \cos\phi \cfrac{\partial^2 f}{\partial \theta \partial \phi} - \cfrac{\cos\phi}{r\sin\theta} \cfrac{\partial f}{\partial \phi} - \cfrac{\sin\phi}{r\sin\theta} \cfrac{\partial^2 f}{\partial \phi^2} \right) \\ &=& \sin^2\theta\cos^2\phi \cfrac{\partial^2 f}{\partial r^2} + \cfrac{\cos^2\theta \cos^2\phi}{r^2} \cfrac{\partial^2 f}{\partial \theta^2} + \cfrac{\sin^2\phi}{r^2\sin^2\theta} \cfrac{\partial^2 f}{\partial \phi^2} \\ &&+ \cfrac{2\sin\theta\cos\theta\cos^2\phi}{r} \cfrac{\partial^2 f}{\partial r \partial \theta} - \cfrac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} \cfrac{\partial^2 f}{\partial \theta \partial \phi} - \cfrac{2\sin\phi\cos\phi}{r} \cfrac{\partial^2 f}{\partial \phi \partial r} \\ &&+ \cfrac{\cos^2\theta\cos^2\phi+\sin^2\phi}{r} \cfrac{\partial f}{\partial r} + \cfrac{\cos\theta\sin^2\phi-2\sin^2\theta\cos\theta\cos^2\phi}{r^2\sin\theta} \cfrac{\partial f}{\partial \theta} + \cfrac{2\sin\phi\cos\phi}{r^2\sin^2\theta} \cfrac{\partial f}{\partial \phi} \end{eqnarray}\]

次に $y$ について、

\[\begin{eqnarray} \cfrac{\partial f}{\partial y} &=& \cfrac{\partial r}{\partial y} \cfrac{\partial f}{\partial r} + \cfrac{\partial \theta}{\partial y} \cfrac{\partial f}{\partial \theta} + \cfrac{\partial \phi}{\partial y} \cfrac{\partial f}{\partial \phi} \\ &=& \sin\theta\sin\phi \cfrac{\partial f}{\partial r} + \cfrac{1}{r} \cos\theta \sin\phi \cfrac{\partial f}{\partial \theta} + \cfrac{\cos\phi}{r\sin\theta} \cfrac{\partial f}{\partial \phi} \\ \\ \cfrac{\partial^2 f}{\partial y^2} &=& \sin\theta\sin\phi \cfrac{\partial}{\partial r} \left(\cfrac{\partial f}{\partial y}\right) + \cfrac{1}{r} \cos\theta \sin\phi \cfrac{\partial}{\partial \theta} \left(\cfrac{\partial f}{\partial y}\right) + \cfrac{\cos\phi}{r\sin\theta} \cfrac{\partial}{\partial \phi} \left(\cfrac{\partial f}{\partial z}\right) \\ &=& \sin\theta\sin\phi \left( \sin\theta\sin\phi \cfrac{\partial^2 f}{\partial r^2} - \cfrac{1}{r^2} \cos\theta \sin\phi \cfrac{\partial f}{\partial \theta} + \cfrac{1}{r} \cos\theta \sin\phi \cfrac{\partial^2 f}{\partial \theta \partial r} - \cfrac{\cos\phi}{r^2\sin\theta} \cfrac{\partial f}{\partial \phi} + \cfrac{\cos\phi}{r\sin\theta} \cfrac{\partial^2 f}{\partial \phi \partial r} \right) + \\ && + \cfrac{1}{r} \cos\theta \sin\phi \left( \cos\theta\sin\phi \cfrac{\partial f}{\partial r} + \sin\theta\sin\phi \cfrac{\partial^2 f}{\partial r \partial \theta} - \cfrac{1}{r} \sin\theta \sin\phi \cfrac{\partial f}{\partial \theta} + \cfrac{1}{r} \cos\theta \sin\phi \cfrac{\partial^2 f}{\partial \theta^2} - \cfrac{\cos\theta\cos\phi}{r\sin^2\theta} \cfrac{\partial f}{\partial \phi} + \cfrac{\cos\phi}{r\sin\theta} \cfrac{\partial^2 f}{\partial \phi \partial \theta} \right) + \\ && + \cfrac{\cos\phi}{r\sin\theta} \left( \sin\theta\cos\phi \cfrac{\partial f}{\partial r} + \sin\theta\sin\phi \cfrac{\partial^2 f}{\partial r \partial \phi} + \cfrac{1}{r} \cos\theta \cos\phi \cfrac{\partial f}{\partial \theta} + \cfrac{1}{r} \cos\theta \sin\phi \cfrac{\partial^2 f}{\partial \theta \partial \phi} - \cfrac{\sin\phi}{r\sin\theta} \cfrac{\partial f}{\partial \phi} + \cfrac{\cos\phi}{r\sin\theta} \cfrac{\partial^2 f}{\partial \phi^2} \right) \\ &=& \sin^2\theta\sin^2\phi \cfrac{\partial^2 f}{\partial r^2} + \cfrac{\cos^2\theta \sin^2\phi}{r^2} \cfrac{\partial^2 f}{\partial \theta^2} + \cfrac{\cos^2\phi}{r^2\sin^2\theta} \cfrac{\partial^2 f}{\partial \phi^2} \\ &&+ \cfrac{2\sin\theta\cos\theta \sin^2\phi}{r} \cfrac{\partial^2 f}{\partial r \partial \theta} + \cfrac{2\cos\theta \sin\phi\cos\phi}{r^2\sin\theta} \cfrac{\partial^2 f}{\partial \theta \partial \phi} + \cfrac{2\sin\phi\cos\phi}{r} \cfrac{\partial^2 f}{\partial \phi \partial r} \\ &&+ \cfrac{\cos^2\theta\sin^2\phi+\cos^2\phi}{r} \cfrac{\partial f}{\partial r} + \cfrac{\cos\theta\cos^2\phi-2\sin^2\theta\cos\theta\sin^2\phi}{r^2\sin\theta} \cfrac{\partial f}{\partial \theta} - \cfrac{2\sin\phi\cos\phi}{r^2\sin^2\theta} \cfrac{\partial f}{\partial \phi} \end{eqnarray}\]

最後に $z$ について、

\[\begin{eqnarray} \cfrac{\partial f}{\partial z} &=& \cfrac{\partial r}{\partial z} \cfrac{\partial f}{\partial r} + \cfrac{\partial \theta}{\partial z} \cfrac{\partial f}{\partial \theta} + \cfrac{\partial \phi}{\partial z} \cfrac{\partial f}{\partial \phi} \\ &=& \cos\theta \cfrac{\partial f}{\partial r} - \cfrac{\sin\theta}{r} \cfrac{\partial f}{\partial \theta} \\ \\ \cfrac{\partial^2 f}{\partial z^2} &=& \cos\theta \cfrac{\partial}{\partial r} \left(\cfrac{\partial f}{\partial z}\right) - \cfrac{\sin\theta}{r} \cfrac{\partial}{\partial \theta} \left(\cfrac{\partial f}{\partial z}\right) \\ &=& \cos\theta \left( \cos\theta \cfrac{\partial^2 f}{\partial r^2} + \cfrac{\sin\theta}{r^2} \cfrac{\partial f}{\partial \theta} - \cfrac{\sin\theta}{r} \cfrac{\partial f}{\partial \theta \partial r} \right) - \cfrac{\sin\theta}{r} \left( -\sin\theta \cfrac{\partial f}{\partial r} + \cos\theta \cfrac{\partial^2 f}{\partial r \partial \theta} - \cfrac{\cos\theta}{r} \cfrac{\partial f}{\partial \theta} - \cfrac{\sin\theta}{r} \cfrac{\partial^2 f}{\partial \theta^2} \right) \\ &=& \cos^2\theta \cfrac{\partial^2 f}{\partial r^2} + \cfrac{\sin^2\theta}{r^2} \cfrac{\partial^2 f}{\partial \theta^2} - \cfrac{2\sin\theta\cos\theta}{r} \cfrac{\partial f}{\partial r \partial \theta} + \cfrac{\sin^2\theta}{r} \cfrac{\partial f}{\partial r} + \cfrac{2\sin\theta\cos\theta}{r^2} \cfrac{\partial f}{\partial \theta} \end{eqnarray}\]

以上により、

\[\begin{eqnarray} \Delta f &=& \cfrac{\partial^2 f}{\partial x^2} + \cfrac{\partial^2 f}{\partial y^2} + \cfrac{\partial^2 f}{\partial z^2} \\ &=& \cfrac{\partial^2 f}{\partial r^2} + \cfrac{1}{r^2} \cfrac{\partial^2 f}{\partial \theta^2} + \cfrac{1}{r^2\sin^2\theta} \cfrac{\partial^2 f}{\partial \phi^2} + \cfrac{2}{r} \cfrac{\partial f}{\partial r} + \cfrac{\cos\theta}{r^2\sin\theta} \cfrac{\partial f}{\partial \theta} \\ &=& \cfrac{1}{r^2} \cfrac{\partial}{\partial r} \left( r^2 \cfrac{\partial f}{\partial r} \right) + \cfrac{1}{r^2 \sin\theta} \cfrac{\partial}{\partial \theta} \left( \sin\theta \cfrac{\partial f}{\partial \theta} \right) + \cfrac{1}{r^2\sin^2\theta} \cfrac{\partial^2 f}{\partial \phi^2} \end{eqnarray}\]

勾配・発散・回転

勾配・発散・回転を参照。