Dirac delta function of non-linear multivariable arguments How does one compute a dirac delta function with a multivariable argument? For example, compute:
$$
\int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\,
\delta\left(x^{2} + y^{2} - 4\right)
\delta\left(\left[x - 1\right]^{2} + y^{2} -4\right){\rm f}\left(x,y\right).
$$
If we constrain the two delta functions we'll get two intersecting circles, and it seems reasonable to state that we evaluate $f(x,y)$ at the intersecting points, but I feel like there should be some extra identities.
Since for single variable 
$$\delta(f(x))=\sum_i \frac{\delta(x-x_i)}{|f'(x)|},$$ is there a multivariable generalization to be aware of?

I worked with similar objects during my Master's project, and we had to derive a formula for that...couldn't find it anywhere. The formula can be found in the links below. 
See section 3.7
or 
see section A.4.7. 
I've called it a $"\bf Sweet ~Dirac-\delta~ formula"$ or "A lemma on twofold Dirac delta functions".  
