ナブラの掛算
スカラー場の勾配
\begin{align}
&スカラー場 \phi(x, y, x) に \nabla を作用させると\\
&\nabla\phi = grad \phi = \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right) \phi = \left( e_x \frac{\partial\phi}{\partial x} + e_y \frac{\partial\phi}{\partial y} + e_z \frac{\partial\phi}{\partial z} \right)
\end{align}
ベクトル場の発散
\begin{align}
&ベクトル場 \boldsymbol{A}(x, y, z) = (A_xe_x+A_ye_y + A_ze_z) と \nabla の内積をとると\\
&\nabla\cdot\boldsymbol{A} = div \boldsymbol{A} = \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right) \cdot (A_xe_x+A_ye_y + A_ze_z)\\
& = \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\right)\\
\end{align}
ベクトル場の回転
\begin{align}
&ベクトル場 \boldsymbol{A}(x, y, z) = (A_xe_x+A_ye_y + A_ze_z) と \nabla の外積をとると\\
&\nabla\times\boldsymbol{A} = rot \boldsymbol{A} = \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right) \times (A_xe_x+A_ye_y + A_ze_z)\\
& = \left( e_x(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}) + e_y(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}) + e_z(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}) \right)\\
\end{align}
場の関数のラプラシアン
\begin{align}
&ナブラどうしの内積をとると\\
&\nabla \cdot \nabla = \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right) \cdot \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right)\\
& = \frac{\partial^2}{{\partial x}^2} + \frac{\partial^2}{{\partial y}^2} + \frac{\partial^2}{{\partial z}^2}\\
\end{align}
ベクトル場の勾配
\begin{align}
&ベクトル場 \boldsymbol{A}(x, y, z) = (A_xe_x+A_ye_y + A_ze_z)とのテンソル積をとると\\
&\nabla \otimes \boldsymbol{A} = grad \boldsymbol{A} = \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right) \otimes (A_xe_x+A_ye_y + A_ze_z)\\
& = e_x \otimes e_x \frac{\partial A_x}{\partial x} + e_x \otimes e_y \frac{\partial A_y}{\partial x} + e_x \otimes e_z \frac{\partial A_z}{\partial x}\\
& + e_y \otimes e_x \frac{\partial A_x}{\partial y} + e_y \otimes e_y \frac{\partial A_y}{\partial y} + e_y \otimes e_z \frac{\partial A_z}{\partial y}\\
& + e_z \otimes e_x \frac{\partial A_x}{\partial z} + e_z \otimes e_y \frac{\partial A_y}{\partial z} + e_z \otimes e_z \frac{\partial A_z}{\partial z}\\
&\\
& 転置テンソル\\
& {}^t(\nabla \otimes \boldsymbol{A}) = {}^t(grad \boldsymbol{A})\\
& = e_x \otimes e_x \frac{\partial A_x}{\partial x} + e_x \otimes e_y \frac{\partial A_x}{\partial y} + e_x \otimes e_z \frac{\partial A_x}{\partial z}\\
& + e_y \otimes e_x \frac{\partial A_y}{\partial x} + e_y \otimes e_y \frac{\partial A_y}{\partial y} + e_y \otimes e_z \frac{\partial A_y}{\partial z}\\
& + e_z \otimes e_x \frac{\partial A_z}{\partial x} + e_z \otimes e_y \frac{\partial A_z}{\partial y} + e_z \otimes e_z \frac{\partial A_z}{\partial z}\\
\end{align}
テンソル場の発散
\begin{align}
&& テンソル場 \boldsymbol{G}(x, y, z) &= G_{xx}e_x \otimes e_x + G_{xy}e_x \otimes e_y + G_{xz}e_x \otimes e_z\\
&& &+ G_{yx}e_y \otimes e_x + G_{yy}e_y \otimes e_y + G_{yz}e_y \otimes e_z\\
&& &+ G_{zx}e_z \otimes e_x + G_{zy}e_z \otimes e_y + G_{zz}e_z \otimes e_z\\
&& との内積をとると\\
&& \nabla \cdot \boldsymbol{G} &= div \boldsymbol{G} = \left( e_x \frac{\partial}{\partial x} + e_y \frac{\partial}{\partial y} + e_z \frac{\partial}{\partial z} \right) \cdot
(G_{xx}e_x \otimes e_x + G_{xy}e_x \otimes e_y + G_{xz}e_x \otimes e_z + G_{yx}e_y \otimes e_x + G_{yy}e_y \otimes e_y + G_{yz}e_y \otimes e_z + G_{zx}e_z \otimes e_x + G_{zy}e_z \otimes e_y + G_{zz}e_z \otimes e_z )\\
&& &= e_x\frac{\partial G_{xx}}{\partial x} + e_y\frac{\partial G_{xy}}{\partial x} + e_z\frac{\partial G_{xz}}{\partial x}\\
&& &+ e_x\frac{\partial G_{yx}}{\partial y} + e_y\frac{\partial G_{yy}}{\partial y} + e_z\frac{\partial G_{yz}}{\partial y}\\
&& &+ e_x\frac{\partial G_{zx}}{\partial z} + e_y\frac{\partial G_{zy}}{\partial z} + e_z\frac{\partial G_{zz}}{\partial z}\\
&& &= e_x \left( \frac{\partial G_{xx}}{\partial x} + \frac{\partial G_{yx}}{\partial y} + \frac{\partial G_{zx}}{\partial x} \right)\\
&& &+ e_y \left( \frac{\partial G_{xy}}{\partial x} + \frac{\partial G_{yy}}{\partial y} + \frac{\partial G_{zy}}{\partial x} \right)\\
&& &+ e_z \left( \frac{\partial G_{xz}}{\partial x} + \frac{\partial G_{yz}}{\partial y} + \frac{\partial G_{zz}}{\partial x} \right)\\
\end{align}