grad, div, rot
定義？
$$
\nabla=i\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{\partial}{\partial z}\\=\begin{bmatrix}\frac{\partial}{\partial x}\\\frac{\partial}{\partial y}\\\frac{\partial}{\partial z}\end{bmatrix}=\begin{bmatrix}\partial_x\\\partial_y\\\partial_z\end{bmatrix}
$$$$
\mathbf{u}=\begin{bmatrix}u_x&u_y&u_z\end{bmatrix}=\begin{bmatrix}uv&w\end{bmatrix}
$$grad
$$
\rm{grad}\space\mathbf{u}=\nabla\mathbf{u}=\begin{bmatrix}\partial_x\\\partial_y\\\partial_z\end{bmatrix}\begin{bmatrix}u&v&w\end{bmatrix}=\begin{bmatrix}\partial_x u&\partial_x v&\partial_x w\\\partial_y u&\partial_y v&\partial_y w\\\partial_z u&\partial_z v&\partial_z w\end{bmatrix}
$$div
$$
\mathrm{div}\space\mathbf{u}=\nabla\cdot\mathbf{u}=\partial_{x}u+\partial_{y}v+\partial_{z}w
$$rot
$$
\begin{aligned}\mathrm{rot}\space\mathbf{u}=\nabla\times\mathbf{u}=\begin{vmatrix}i&j&k\\\partial_x&\partial_y&\partial_z\\u&v&w\end{vmatrix}=i\begin{vmatrix}\partial_y&\partial_z\\v&w\end{vmatrix}-j\begin{vmatrix}\partial_x&\partial_z\\u&w\end{vmatrix}+k\begin{vmatrix}\partial_x&\partial_y\\u&v\end{vmatrix}\\=(\partial_y w-\partial_z v)-(\partial_x w-\partial_z u)+(\partial_x v-\partial_y u)=\partial_y w-\partial_z v-\partial_x w+\partial_z u+\partial_x v-\partial_y u\end{aligned}
$$