Define angular velocity ($\mathrm{\omega }$). Derive V= r$\mathrm{\omega }$?

$\text{ Angular velocity }\left(\overrightarrow{\mathrm{\omega }}\right)$
It is defined as rate of change of angular displacement
$\overrightarrow{\mathrm{\omega }}=\frac{\mathrm{d\theta }}{\mathrm{dt}}$
Relation between linear velocty and angular velocity

Let a particle p be displaced by ds in time interval dt on the circumference of circle of radius r through an angle $\mathrm{d\theta }$
$\text{ angular veoocity }\mathrm{\omega }=\frac{\mathrm{d\theta }}{\mathrm{dt}}$
$\text{ but Linear velocity }\mathrm{V}=\frac{\mathrm{ds}}{\mathrm{dt}}$
$\text{ we know that angle }=\frac{\mathrm{arc}}{\text{ radius }}$
$\mathrm{d\theta }=\frac{\mathrm{ds}}{\mathrm{r}};\mathrm{ds}=\mathrm{rd\theta }\Rightarrow \mathrm{V}=\frac{\mathrm{rd\theta }}{\mathrm{dt}}\Rightarrow \mathrm{V}=\mathrm{r\omega }$