\[\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\]

\[\mathbf{v} = \frac{\partial \mathbf{r}}{\partial t} = \frac{\partial x}{\partial t} \mathbf{i} + \frac{\partial y}{\partial t} \mathbf{j} + \frac{\partial z}{\partial t} \mathbf{k}\]

\[\mathbf{a} = \frac{\partial \mathbf{v}}{\partial t}= \frac{\partial^2 \mathbf{r}}{\partial t^2} = \frac{\partial^2 x}{\partial t^2} \mathbf{i} + \frac{\partial^2 y}{\partial t^2} \mathbf{j} + \frac{\partial^2 z}{\partial t^2} \mathbf{k}\]

\[\mathbf{i} = cos \theta \mathbf{e_r} - sin \theta \mathbf{e_{\theta}}\]

\[\mathbf{j} = sin \theta \mathbf{e_r} + cos \theta \mathbf{e_{\theta}}\]

\[\mathbf{e_z} = \mathbf{k}\]

\[x= r cos \theta , \; y= r sin \theta , z=z\]

\[\begin{equation} \begin{aligned} \mathbf{r} &= x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \\ &= r cos \theta (cos \theta \mathbf{e_r} - sin \theta \mathbf{e_{\theta}}) + r sin \theta ( sin \theta \mathbf{e_r} + cos \theta \mathbf{e_{\theta}}) + z \mathbf{e_z} \\ &- = r \mathbf{e_{\theta} } + z \mathbf{e_z}\end{aligned} \end{equation}\]

\[\begin{equation} \begin{aligned} \mathbf{v} &= \frac{d x}{d t} \mathbf{i} + \frac{d y}{d t} \mathbf{j} + \frac{d z}{d t} \mathbf{k} \\ &= \frac{d}{dt}( r cos \theta) \mathbf{i} + \frac{d}{dt}( r sin \theta) \mathbf{j} + \frac{dz}{dt} \mathbf{e_z} \\ &= \frac{dr}{dt} cos \theta \mathbf{i} - r sin \theta \frac{d \theta}{dt} \mathbf{i} + \frac{dr}{dt} sin \theta \mathbf{j} + r cos \theta \frac{d \theta}{dt} \mathbf{j} + \frac{dz}{dt} \mathbf{e_z} \\ &= \frac{dr}{dt}(cos \theta \mathbf{i} + sin \theta \mathbf{j}) + \frac{d \theta}{dt} (- r sin \theta) \mathbf{i} + r cos \theta \mathbf{j}) + \frac{dz}{dt} \mathbf{e_z} \\ &= \frac{dr}{dt}\mathbf{e_r} + r \frac{d \theta}{dt} \mathbf{e_{\theta}} + \frac{dz}{dt} \mathbf{e_z} \end{aligned} \end{equation}\]

\[\begin{equation} \begin{aligned} \mathbf{a} &= \frac{d \mathbf{v}}{dt} \\ &= \frac{d}{dt} (\frac{dr}{dt}(cos \theta \mathbf{i} + sin \theta \mathbf{j}) + \frac{d \theta}{dt} (- r sin \theta) \mathbf{i} + r cos \theta \mathbf{j}) + \frac{dz}{dt} \mathbf{e_z}) \\ &= \frac{d^2 r}{dt^2}((cos \theta \mathbf{i} + sin \theta \mathbf{j}) + \frac{dr}{dt} \frac{d \theta}{dt} (-sin \theta \mathbf{i} + cos \theta \mathbf{j}) \\ &+ \frac{d^2 \theta}{dt^2} (- r sin \theta \mathbf{i} + r cos \theta \mathbf{j}) + (\frac{d \theta}{dt})^2 (- r cos \theta \mathbf{i} -r sin \theta \mathbf{j}) \\ &= \frac{d^2 r}{dt^2} \mathbf{e_r} + 2 \frac{dr}{dt} \frac{d \theta}{dt} \mathbf{e_{\theta}} + \frac{d^2 \theta}{dt^2} r \mathbf{e_{\theta}} - r(\frac{d \theta}{dt})^2 \mathbf{e_{\theta}} + \frac{d^2 z}{dt^2} \mathbf{e_z} \\ &= (\frac{d^2 r}{dt^2} - r (\frac{d \theta}{dt})^2 ) \mathbf{e_r} + (r \frac{d^2 \theta}{dt^2} +2 \frac{dr}{dt} \frac{d \theta}{dt} ) \mathbf{e_{\theta}} + \frac{d^2z}{dt^2} \mathbf{e_z} \end{aligned} \end{equation}\]