The mean curvature of surface of revolution$$(x,y,z)=\big(r(\gamma) \cos(\theta),r(\gamma)\sin(\theta),z(\gamma)\big)$$ parameterized by $(\gamma,\theta)$ where $$\dot{r}^2(\gamma)+\dot{z}^2(\gamma)=1$$
the mean curvature $2 C=-\nabla.n$, where $n$ is the unit normal $$n=\big(\dot{z}(\gamma)\cos(\theta),-\sin(\theta)\dot{z}(\gamma),\dot{r}(\gamma)\big),$$ the $\nabla=\partial/\partial x\hat{x}+\partial/\partial y\hat{y}+\partial/\partial z\hat{z}$.
How to change the $\nabla$ into the parameterized form $\gamma$, $\theta$, where $\gamma $ is the arc length of the curve?
There are other ways to find the principal curvatures (and hence the mean curvature) of a surface of revolution, but you can find the answer to your question about $\nabla$ in cylindrical coordinates here.