Suppose constant value,I is defined as, \begin{align} I=\int_{circle}f(r,\theta)\ dl \end{align} where equation of the circle is defined as $r=\cos(\theta-\frac{\pi}{4})$ (see plot below). Now in an article author claims that I can be written in this way also, \begin{align} I=\int_{0}^{\pi}d\theta \int_{0}^{\infty}dr \ f(r,\theta) \ \delta[r-\cos(\theta-\frac{\pi}{4})] \end{align} $\ \ $Here, line integration is replaced by surface integration by introducing delta factor.
$\ \ $But, in plane polar coordinate surface element factor is $\ \ ``r\ dr\ d\theta" \ \ $. Where is that $\ \ ``r"\ \ $ factor? Is that some how taken care inside delta factor?
