- #1

- 2

- 0

## Homework Statement

Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity.

## Homework Equations

## The Attempt at a Solution

This integral actually came up in a homework problem for my E&M class and I'm confused about how to handle it. I understand that the delta function will make the volume integral come out to one, but when I express that integral as a triple integral over spherical coordinates, $$\int_0^{2\pi}\int_0^\pi\int_0^\infty \delta^3(\vec r)~ r^2 sin(\theta)~ dr~ d\theta~ d\phi$$ The integrating factor apparently makes the integral go to zero, since ##\delta^3(\vec r)~r^2=0##.

In my textbook, the author uses the delta function to say that ##\delta^3(\vec r)~r=0## and I can understand that since ##\delta^3(\vec r)~r = \delta(x)\delta(y)\delta(z)~\sqrt{x^2+y^2+z^2} = 0##. But then, given that, I'm at a loss as to what went wrong when I set up the integral.