It seems the Dirac delta function is invariant under any invertible coordinate transfromation. Using the scaling and composition rule for the Dirac delta function, I show this on the following website:
I'm trying to understand what this means. Is this unique to the Dirac delta because of its scaling and composition properties? Or is every integral diffeomorphism invariant? You can always change coordinates to solve an integral, right? But that integral does not always have the same from. You may introduce new factors in the integrand, etc. But with the Dirac delta, it seems you get that same integral, only replacing the old variable with new one, getting the exact same form. What does that mean? If the Dirac delta function is required for other reasons, does this mean that the dirac delta is specifying the existence of an equivalence class of diffeomorphisms? Thanks.