I am given the following path and asked to find acceleration and velocity in cylindrical coordinates $$x(t) = \cos wt \,\,\, \,\,\, y(t) = \sin wt \,\,\, \,\,\, z(t) = t$$
I called $\underline{r}(t) = \cos (wt) \underline{i}+\sin( wt)\underline{j}+t\underline{k} $ the position vector and I differentiated it obtaining the following: $$\dot{\underline{r}}(t) = -w\sin(wt)\underline{i} + w\cos(wt)\underline{j}+\underline{k}$$ Then I obtained the expressions of the Cartesian coordinates in terms of the cylindrical ones, $\underline{i} = -\sin(wt)\underline{e}_\theta+\cos(wt)\underline{e}_r$ and $\underline{j} = \cos(wt)\underline{e}_\theta+\sin(wt)\underline{e}_r$ and $\underline{k} = \underline{e}_z$ , and substituted back into obtaining $$\dot{\underline{r}}(t) = w\underline{e}_\theta +\underline{e}_z$$ then by differentiating wrt $t$ and letting $\dot{\underline{e}}_\theta = -\dot{\theta}\underline{e}_r$ I get $$\ddot{\underline{r}}(t) = -w\dot{\theta\underline{e}}_r+\dot{\underline{e}}_z$$
Is this correct and does $\dot{\underline{e}}_z$ vanishes? Cause technically $\underline{e}_z = \underline{k}$ and it is a fixed vector, hence the derivative should be zero