What is the divergence of this specific vector field?

Question

I'm trying to find the divergence of a specific vector field and I just want to make sure I did it right. The field in question is given in cylindrical coordinates $(r, \theta, z)$, depends purely on $r$, and $z$ and is always in the $\theta$ direction. Am I right in thinking that it's divergence free?

Answer 1

Yes. In cylindrical coordinates, using a normalized basis, the divergence is $$\frac{1}{r}\frac{\partial}{\partial r}\left(ru_r\right)+\frac{1}{r}\frac{\partial u_\theta}{\partial \theta}+\frac{\partial u_z}{\partial z}$$ So if $\boldsymbol u=(0,u_\theta,0)$ depends only on $r,z$ then $\nabla\cdot\boldsymbol u=0$.