Convert polar velocity components to Cartesian I haven't been able to find an answer to velocity component transformation from polar to Cartesian on here, so I'm hoping that someone might be able to answer this question for me.
I am given a position in Cartesian coordinates $(1,0)$ and a velocity in polar form ($v_r = 0$, $v_\theta = \pi r$) and am looking to solve for my position forward in time IN Cartesian coordinates. How do I go about converting the radial and rotational velocity components into $U$ and $V$ in order to keep everything in the same form?
Thanks!
 A: $\vec v = v_r \hat r + v_\theta \hat \theta$, where $\hat r$ and $\hat \theta$ are the unit radial vector and unit rotational vector at $(r, \theta)$. Letting $\vec p = x\hat i + y \hat j$ be the position vector, and noting that $x = r\cos \theta, y = r\sin \theta$, we have $$\vec p = r\cos \theta \hat i + r\sin \theta\hat j$$. Now $\hat r$ is the direction that $\vec p$ changes when $r$ increases, and $\hat \theta$ is the direction that $\vec p$ changes then $\theta$ increases: 
$$\hat r = \frac{\frac{\partial\vec p}{\partial r}}{\left\|\frac{\partial\vec p}{\partial r}\right\|}\qquad\hat \theta = \frac{\frac{\partial\vec p}{\partial \theta}}{\left\|\frac{\partial\vec p}{\partial \theta}\right\|}$$
So $$\hat r = \cos \theta \hat i + \sin \theta\hat j\\\hat \theta = -\sin \theta \hat i + \cos \theta\hat j$$
Thus $$\begin{align}\vec v &= \pi r\hat \theta\\&=\pi(-r\sin \theta\hat i + r\cos \theta\hat j)\\&=-\pi y\hat i + \pi x \hat j\end{align}$$
Since you've not defined what $U$ and $V$ mean, I can't answer any further than that.
