Find an anti-derivative that contains delta functions

Question

Please help me find this anti-derivative: $$\int{r^2\delta(x)\delta(y)\delta(z)dr}$$, with $\delta$ being the delta function and $r=\sqrt{x^2+y^2+z^2}$

Thank you in advance.

Answer 1

In transforming to spherical coordinates, we have

$$\delta (\vec r)=\delta(x)\delta(y)\delta(z)=\frac{\delta(r)\delta(\theta)\delta(\phi)}{r^2\sin(\theta)}\tag 1$$

Applying $(1)$ reveals that

$$\begin{align} \int_0^\infty r^2\delta(x)\delta(y)\delta(z)\,dr&=\int_0^\infty r^2\delta(\vec r)\,dr\\\\ &=\int_0^\infty r^2\frac{\delta(r)\delta(\theta)\delta(\phi)}{r^2\sin(\theta)}\,dr\\\\ &=\frac{\delta(\theta)\delta(\phi)}{\sin(\theta)} \end{align}$$

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Formally, we can write

$$\int r^2\delta(x)\delta(y)\delta(z)\,dr=\int r^2\frac{\delta(r)\delta(\theta)\delta(\phi)}{r^2\sin(\theta)}\,dr=\frac{\delta(\theta)\delta(\phi)}{\sin(\theta)}\,H(r)+C$$

where $H(r)$ is the Heaviside Function.