I'm reading Dynamical Processes on Complex Networks (link), which makes frequent use of dirac delta integrals to examine evolving networks. I'm trying to get a good sense of how to evalute them and when they are appropriate to use. I am currently unable to crack the following integral in Equation 3.31:
$$ P(k, t) = \int_0^t \delta[k-k_s(t)] ds = -\bigg(\frac{\partial k_s(t)}{\delta s}\bigg)^{-1}\bigg|_{s=s(k,t)} $$
Where $k_s(t)$ is $0$ for $s<t$ and monotonically increasing otherwise, and $s(k,t)$ is the solution of the implicit equation $k = k_s(t)$.
The book just states this result and moves on and I'd love to understand what is going on here. I should say, I'm not so interested in the results so much as the machinery of the dirac delta function and how one can use such tricks/witchcraft in general.
By substitution and composition,$$P(k,\,t)=\int_{k_0(t)}^{k_s(t)}\delta[k-k_s(t)]\frac{\partial s}{\partial k}dk=-\left(\frac{\partial s}{\partial k}\right)_{k=k_s(t)},$$where the overall $-$ sign is due to the order of the integral's limits. But of course, we can rewrite this as$$-\left(\frac{\partial k_s(t)}{\partial s}\right)^{-1}_{s=s(k,\,t)},$$as required.