物理数学I ベクトル解析/付録

用語集(Wikipediaへのリンク)

• ja:w:ベクトル解析 w:Vector analysis
• ja:w:デル w:Del
• ja:w:勾配 w:Gradient
• ja:w:発散 w:Divergence
• ja:w:回転 w:Curl
• ja:w:ラプラシアン w:Laplace operator

定義式

3次元デカルト座標での定義式。

$\mathrm{\nabla }=\left(\begin{array}{c}\partial /\partial x\\ \partial /\partial y\\ \partial /\partial z\end{array}\right)$

$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\varphi =\mathrm{\nabla }\varphi =\left(\begin{array}{c}\partial \varphi /\partial x\\ \partial \varphi /\partial y\\ \partial \varphi /\partial z\end{array}\right)$

$\mathrm{d}\mathrm{i}\mathrm{v}𝐅=\mathrm{\nabla }\cdot 𝐅=\partial F_x/\partial x+\partial F_y/\partial y+\partial F_z/\partial z$

$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}𝐅=\mathrm{r}\mathrm{o}\mathrm{t}𝐅=\mathrm{\nabla }\times 𝐅=\left(\begin{array}{c}\partial F_z/\partial y-\partial F_y/\partial z\\ \partial F_x/\partial z-\partial F_z/\partial x\\ \partial F_y/\partial x-\partial F_x/\partial y\end{array}\right)$

$\mathrm{\Delta }={\mathrm{\nabla }}^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$

$\mathrm{\Delta }\varphi ={\mathrm{\nabla }}^2\varphi =\mathrm{\nabla }\cdot (\mathrm{\nabla }\varphi )=\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\varphi )$

重要な公式

$\mathrm{div}(a𝐅+b𝐆)=a\mathrm{div}(𝐅)+b\mathrm{div}(𝐆)$

$\mathrm{div}(\varphi 𝐅)=\mathrm{grad}(\varphi )\cdot 𝐅+\varphi \mathrm{div}(𝐅)$

$\mathrm{\nabla }\cdot (\varphi 𝐅)=(\mathrm{\nabla }\varphi )\cdot 𝐅+\varphi (\mathrm{\nabla }\cdot 𝐅)$

$\mathrm{div}(𝐅\times 𝐆)=\mathrm{curl}(𝐅)\cdot 𝐆-𝐅\cdot \mathrm{curl}(𝐆)$

div(curl F)

$\mathrm{div}(\mathrm{curl}𝐅)=\mathrm{div}(\mathrm{\nabla }\times 𝐅)=\mathrm{curl}(\mathrm{\nabla })\cdot 𝐅-\mathrm{\nabla }\cdot \mathrm{curl}(𝐅)$

ここで ${[\mathrm{curl}(\mathrm{\nabla })]}_x=\frac{{\partial }^2}{\partial z\partial y}-\frac{{\partial }^2}{\partial y\partial z}=0$ (演算対象の関数が連続でなめらかな場合) であるので

$\mathrm{div}(\mathrm{curl}𝐅)=-\mathrm{\nabla }\cdot \mathrm{curl}(𝐅)=-\mathrm{div}(\mathrm{curl}𝐅)$

結局 $\mathrm{div}(\mathrm{curl}𝐅)=0$

curl(curl F)

$\mathrm{curl}(\mathrm{curl}(𝐅))=-\mathrm{\Delta }𝐅+\mathrm{grad}(\mathrm{div}𝐅)$

x成分をとって証明する。

${[\mathrm{curl}(\mathrm{curl}(𝐅))]}_x={[\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐅)]}_x=\frac{\partial }{\partial y}{[\mathrm{\nabla }\times 𝐅]}_z-\frac{\partial }{\partial z}{[\mathrm{\nabla }\times 𝐅]}_y$ $=\frac{\partial }{\partial y}(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})-\frac{\partial }{\partial z}(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x})$ $=-(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2})F_x+\frac{\partial }{\partial x}(\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z})$ $=-\mathrm{\Delta }{F}_x+\frac{\partial }{\partial x}\mathrm{div}𝐅={[-\mathrm{\Delta }𝐅+\mathrm{grad}(\mathrm{div}𝐅)]}_x$