Math Help - Delta function in 3D

1. Delta function in 3D

Hi,I need to prove the following relation $\nabla^{2} = −4π\delta (R)$where $R=(x-y), x,y \in R^3$ and $\nabla^{2}$ is taken over x_{i} and $\delta(R)$ is the 3D delta function (=1 iff x=y)Thanks!Isatis

2. Re: Delta function in 3D

Hey isatis55.

Can you please define specifically the definition of the 3D delta function given <x,y,z> in R^3 where you map <x,y,z> -> w?

3. Re: Delta function in 3D

Originally Posted by chiro
Hey isatis55.

Can you please define specifically the definition of the 3D delta function given <x,y,z> in R^3 where you map <x,y,z> -> w?
3D delta functional with the usual abuse of notation, for any function on $\mathbb{R}^3$ continuous at $\bf{0}$:

$\delta(f) = \int_{\mathbb{R}^3} f({\bf{x}}) \delta({\bf{x}}) d{\bf{x}}=f({\bf{0})$

(the first and last terms above are the definition, the middle term is a common suggestive abuse of notation.)