PX275 - C1f - example of curl II

consider a cylinder rotating about the $z$ -axis, so the angular velocity: $\vec{ω} = ω_{z} \hat{k}$
looking down the $z$ -axis, the linear velocity, $\vec{v}$ , of a point on the cylinder will be perpendicular to its position vector, and direction of $\vec{v}$ will be $- y \hat{i} + x \hat{j}$
the angular velocity:

| ω | = \frac{| \vec{v} |}{r} ⟹ | \vec{v} | = | ω | r

\vec{v} = - ω y \hat{i} + ω x \hat{j}

\vec{\nabla} \times \vec{v} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ - ω_{z} y & ω_{z} x & 0 \end{matrix} | = 2 ω_{z} \hat{k}