TITLE: Dirac delta function of non-linear multivariable arguments
QUESTION [16 upvotes]: 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?

REPLY [5 votes]: $\newcommand{\+}{^{\dagger}}%
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\begin{align}
&\color{#0000ff}{\large%
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm d}x\,{\rm d}y\,
\delta\pars{x^{2} + y^{2} - 4}
\delta\pars{\bracks{x - 1}^{2} + y^{2} -4}\fermi\pars{x,y}}
\\[3mm]&=
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm d}x\,{\rm d}y\,
\delta\pars{x^{2} + y^{2} - 4}
\delta\pars{\bracks{x^{2} - 2x + 1} + y^{2} -4}\fermi\pars{x,y}
\\[3mm]&=
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm d}x\,{\rm d}y\,
\delta\pars{x^{2} + y^{2} - 4}
\delta\pars{2x - 1}\fermi\pars{x,y}
\\[3mm]&=
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm d}x\,{\rm d}y\,
\delta\pars{x^{2} + y^{2} - 4}\,
{\delta\pars{x - 1/2} \over 2}\,\fermi\pars{x,y}
\\[3mm]&=
\half\int_{-\infty}^{\infty}{\rm d}y\,
\delta\pars{\bracks{\half}^{2} + y^{2} - 4}\,\fermi\pars{\half,y}
\\[3mm]&=
\half\int_{-\infty}^{\infty}{\rm d}y\,\bracks{%
{\delta\pars{y + \root{15}/2} \over \root{15}}
+ {\delta\pars{y - \root{15}/2} \over \root{15}}}\fermi\pars{\half,y}
\\[3mm]&=\color{#0000ff}{\large%
{\root{15} \over 30}\bracks{\fermi\pars{\half,-\,{\root{15} \over 2}} + \fermi\pars{\half,{\root{15} \over 2}}}}
\end{align}