Acceleration in cylindrical coordinates ($\mathbf{a}=(\ddot{r}-r{\dot{\theta }}^2){\mathbf{e}}_r+(r\ddot{\theta }+2\dot{r}\dot{\theta }){\mathbf{e}}_{\theta }+\ddot{z}{\mathbf{e}}_z$)

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 $d(r^2\dot{\theta })/dt$.