# 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?

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Hints:

1. There exists an $n$-dimensional generalization $$\tag{1} \delta^n({\bf f}({\bf x})) ~=~\sum_{{\bf x}_{(0)}}^{{\bf f}({\bf x}_{(0)})=0}\frac{1}{|\det\frac{\partial {\bf f}({\bf x})}{\partial {\bf x}} |}\delta^n({\bf x}-{\bf x}_{(0)})$$ of the substitution formula for the Dirac delta distribution under pertinent assumptions, such as e.g., that the function ${\bf f}:\Omega \subseteq \mathbb{R}^n \to \mathbb{R}^n$ has isolated zeros. Here the sum on the rhs. of eq. (1) extends to all the zeros ${\bf x}_{(0)}$ of the function ${\bf f}$.

2. Example: The function $$\tag{2} {\bf f}(x,y)~=~(x^2+y^2-4,(x-1)^2+y^2-4)$$ with Jacobian determinant $$\tag{3}\det\frac{\partial {\bf f}(x,y)}{\partial (x,y)} ~=~4y,$$ has two zeros $$\tag{4} (x,y)~=~(\frac{1}{2},\pm \frac{\sqrt{15}}{2}),$$ leading to $$\tag{5} \delta^2({\bf f}(x,y)) ~\stackrel{(1)}{=}~\frac{1}{2\sqrt{15}}\delta(x-\frac{1}{2})\sum_{\pm} \delta(y\mp\frac{\sqrt{15}}{2}).$$

• I cannot comment so I will put my question to Qmechanic here as an answer. @Qmechanic, can you give a reference where I can find the formula you quoted? Thanks. – Poor Soul Dec 29 '13 at 21:53
• I second @PoorSoul's request. Is there a reference where your Eq. (1) is proved? – becko Jan 13 '16 at 2:12
• @Qmechanic what does $|\text{det} \frac{d\mathbf{f}}{d\mathbf{x}}|$ mean when f is a multivariate scalar function? What is the equivalent relation to Eq. (1) in this case ? – Mencia Feb 11 '18 at 18:50
• The formula (1) only makes sense if the domain and codomain of ${\bf f}$ have same dimension. – Qmechanic Feb 11 '18 at 19:23
• However, see e.g. this Math.SE post. – Qmechanic Feb 11 '18 at 19:31


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

I've called it a $"\bf Sweet ~Dirac-\delta~ formula"$ or "A lemma on twofold Dirac delta functions".

• I haven't proved it, but I'm sure the "sweet Dirac-$\delta$ formula" can be generalized to higher dimensions. Have you tried it? – becko Jan 13 '16 at 1:44
• Ah, just found the generalization in Qmechanics answer. – becko Jan 13 '16 at 2:13

There is indeed a multivariable generalization of the identity you mentioned. See this Wikipedia article, where it says "As in the one-variable case, it is possible to define the composition of $δ$..."