 Acceleration in cylindrical coordinates

Differentiate the position vector $\mathbf{r}=r{\mathbf{e}}_r+z{\mathbf{e}}_z$ using ${\dot{\mathbf{e}}}_r=\dot{\theta }{\mathbf{e}}_{\theta }$, ${\dot{\mathbf{e}}}_{\theta }=-\dot{\theta }{\mathbf{e}}_r$, and fixed ${\mathbf{e}}_z$. The first derivative is the velocity $\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta }+\dot{z}{\mathbf{e}}_z$; differentiating once more gives the displayed acceleration. In particular, $r$ times the azimuthal acceleration is $d(r^2\dot{\theta })/dt$.