In an irrotational region of flow, we can write the velocity vector as the gradient of the scalar velocity potential function, $\vec{V}=\vec{\nabla} \phi$. The components of $\vec{V}$ in cylindrical coordinates, $(r, \theta, z)$ and $\left(u_r, u_\theta, u_z\right)$, are
$$
\begin{aligned}
u_r & =\frac{\partial_\phi}{\partial r} \\
u_\theta & =\frac{1}{r} \frac{\partial_\phi}{\partial \theta} \\
u_z & =\frac{\partial_\phi}{\partial z}
\end{aligned}
$$
From Chap. 9, we also write the components of the vorticity vector in cylindrical coordinates as $\zeta_r=\frac{1}{r} \frac{\partial u_z}{\partial \theta}-\frac{\partial u_\theta}{\partial z}$, $\zeta_\theta=\frac{\partial u_r}{\partial z}-\frac{\partial u_z}{\partial r}$, and $\zeta_z=\frac{1}{r} \frac{\partial}{\partial r}\left(r u_\theta\right)-\frac{1}{r} \frac{\partial u_r}{\partial \theta}$. Substitute the velocity components into the vorticity components to show that all three components of the vorticity vector are indeed zero in an irrotational region of flow.