Consider the map $\varphi:S^1\times \mathbb (0, \infty)\to \mathbb R^2$ which takes $(p, t)$ to $tp$. (Here $S^1$ is thought of as a subset of $\mathbb R^2$. Thus writing $tp$ means scaling $p$ by $t$.)
Question. Equipping $S^1\times (0, \infty)$ with the product of the respective Lebesgue measures, I want to calculate what is the push-forward of this measure under $\varphi$.
This is a basic question about measures but the only way I could think of doing this is by computing the pull-back of the volume form on $\mathbb R^2$ under $\varphi$, which is doable by a simple derivatives calculation.