Velocity in polar coordinates ($\mathbf{v}=\dot{r}\hat{\mathbf{r}}+r\dot{\theta }\hat{\mathbf{\theta }}$)

For a differentiable path in plane polar coordinates with $r>0$, the moving unit vectors are $\hat{\mathbf{r}}=(\cos \theta ,\sin \theta )$ and $\hat{\mathbf{\theta }}=(-\sin \theta ,\cos \theta )$. Differentiating the position $\mathbf{r}=r\hat{\mathbf{r}}$ and using $\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathbf{\theta }}$ proves the displayed velocity. Its signed radial and transverse vector components are $\dot{r}$ and $r\dot{\theta }$. A second differentiation gives the acceleration in polar coordinates. The complex representatives of the two moving unit vectors are $e^{i\theta }$ and $i{e}^{i\theta }$.