I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral:
$$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$
where $G(z,t)$ is a function which 'pushes' the original function $I(z)$ into $S(x,t)$ at some later time. I've tried using some Dirac delta identities but have not had much success. Any help would be very much appreciated. Thank you.
You'd think of $\int f(x) \delta(x) \; \mathrm{d} x = f(0)$ as picking just the value of $f$ where $\delta$'s argument is zero. I.e., in this case the result is $I(z)$ wherever $G(z, t) = x$.