I'm interested in the behavior of Dirac deltafunctions within multivariate integrals. Here is a simple example to which I do not know the answer:
$$\iint\limits_{[0,1]\times [0,1]} \delta\left(x - y\right)\, dA$$
This integrates the constant function $1$ over the line $y=x$. Is there a prefactor involed though due to the delta? I can imagine this evaluating either to $1$ or to $\sqrt2$.
The information from the relevant Wikipedia page suggests the value $\sqrt2$ while WolframAlpha suggests the value $1$.
How is the behavior of a Dirac deltafunction within a multivariate integral defined? My goal is to create an algorithm to solve a broad class of these problems in general.
In this case, where the coordinates appear linearly in the argument of the delta function, no special treatment is required; you can just evaluate the inner integral as you'd normally evaluate an integral over a delta function, and then evaluate the outer integral:
$$\int_{0}^{1}\int_{0}^{1} \delta\left(x - y\right)\, \mathrm dx\, \mathrm dy=\int_0^11\,\mathrm dy=1\;.$$
In case you got the value $\sqrt2$ from the length of the diagonal, you didn't take into account the factor $1/|\nabla g|=1/\sqrt2$.