rotを計算する

\(x,y,z\)について偏微分可能な関数\(f\)について

\begin{align}
\mathrm{rot} f &=
\begin{vmatrix}
\boldsymbol{i} &\boldsymbol{j} &\boldsymbol{k} \\
f_i & f_j & f_k \\
\frac{\partial}{\partial x} &\frac{\partial}{\partial y} &\frac{\partial}{\partial z}
\end{vmatrix}\\[1.5ex]
&=\left(\dfrac{\partial f_k}{\partial y}-\dfrac{\partial f_j}{\partial z} \right) \boldsymbol{i} + \left ( \dfrac{\partial f_i}{\partial z}-\dfrac{\partial f_k}{\partial x} \right ) \boldsymbol{j} + \left ( \dfrac{\partial f_j}{\partial x}-\dfrac{\partial f_i}{\partial y} \right) \boldsymbol{k}
\end{align}

を\(f\)の回転(rotation)という。

\(f=xy \boldsymbol{i} +yz \boldsymbol{j} +xz \boldsymbol{k} \)の時のrotは

\begin{align}
\mathrm{rot} f = -x \boldsymbol{i} -y\boldsymbol{j} – z \boldsymbol{k}
\end{align}