How to treat Dirac delta function of two variable? We can treat one variable delta function as 
$$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$   
Then how do we treat two variable delta function, such as $\delta(f(x,y))$?
for example, how calculate $\int\int \delta(x-y)$ ?
I first thought using  $\int f(x)\delta(x-y)dx = f(y)$
$$\int\int \delta(x-y)dxdy = \int\int1 * \delta(x-y)dx dy = \int dy.$$ 
but this is nonsense, since we can also think it as $\int\int \delta(x-y)dxdy = \int dx$

I think something is still wrong. Because according to your answer in the integration limits $(-\infty,+\infty)$ it doesn't matter which is the function inside the $\delta$, the answer is always the same.
