物理で使う数式集(∇演算子)

概要

　物理を勉強する中で出てきた数式の導出を備忘録としてまとめていきます。この記事では主に∇演算子に関する公式について扱い適宜追加していく予定です。

　以下の公式ではスカラー関数を$A$、ベクトル関数を$\boldsymbol B=(B_{x}, B_{y}, B_{z})$とします。

1.$\bigtriangleup=\mathrm{div}\,\mathrm{grad}$

　まず左辺の$\bigtriangleup$ですがラプラシアンと読み、式(1)で定義されます。

$$\bigtriangleup=(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})=\nabla^2 \tag{1}$$

　これを式変形していくと単純ですが上記の公式になります。

$$\begin{align}\bigtriangleup&=(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}) \\ &=\mathrm{div}(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}) \\ &=\mathrm{div}\,\mathrm{grad} \tag{2} \end{align}$$

2.$\mathrm{rot}\,\mathrm{rot}\boldsymbol{B}=\mathrm{grad}\,\mathrm{div}\boldsymbol{B}-\mathrm{div}\,\mathrm{grad}\boldsymbol{B}$

　左辺右側の$\mathrm{rot}\boldsymbol B$から計算を行い式変形をしていきます。3成分について計算していますが、同じ計算なので$x$成分のみ計算してもよいと思います。

$$\begin{align}\mathrm{rot}\,\mathrm{rot}\boldsymbol B&=\mathrm{rot}\left( \begin{array}{ccc} \frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z}\\ \frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\\ \frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y}\\ \end{array} \right) \\ &=\left( \begin{array}{ccc} \frac{\partial}{\partial y}(\frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y})-\frac{\partial}{\partial z}(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x})\\ \frac{\partial}{\partial z}(\frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z})-\frac{\partial}{\partial x}(\frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y})\\ \frac{\partial}{\partial x}(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x})-\frac{\partial}{\partial y}(\frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z})\\ \end{array} \right) \\ &=\left( \begin{array}{ccc} \frac{\partial}{\partial x}(\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z})-(\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})B_x\\ \frac{\partial}{\partial y}(\frac{\partial B_x}{\partial x}+\frac{\partial B_z}{\partial z})-(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial z^2})B_y\\ \frac{\partial}{\partial z}(\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y})-(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})B_y\\ \end{array} \right) \\ &=\left( \begin{array}{ccc} \frac{\partial}{\partial x}(\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z})-(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})B_x\\ \frac{\partial}{\partial y}(\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z})-(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})B_y\\ \frac{\partial}{\partial z}(\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z})-(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})B_z\\ \end{array} \right) \\ &=\left( \begin{array}{ccc} \frac{\partial}{\partial x}(\mathrm{div}\boldsymbol B)\\ \frac{\partial}{\partial y}(\mathrm{div}\boldsymbol B)\\ \frac{\partial}{\partial z}(\mathrm{div}\boldsymbol B)\\ \end{array} \right)-\mathrm{div}\,\mathrm{grad}\boldsymbol B \\ &=\mathrm{grad}\,\mathrm{div}\boldsymbol B-\mathrm{div}\,\mathrm{grad}\boldsymbol B \\ \end{align}$$

参考文献
1.電磁気学、砂川重信