How Does Green's Function Simplify Understanding Electromagnetic Equations?

[AJ Bentley]

I have a serious blind-spot with mathematics (but I keep trying)
Can someone help me with this.

I have a relation $$A = \mu_{0}/4\pi\int J/r \ dVol$$

Which (apparently!) can be written $$\nabla^{2} A = - \mu_{0} J$$

I know that $$\nabla^{2} A = 1/r \ \delta^{2} ( r A ) / \delta r^{2}$$
which is the polar form of the Laplace operator

but I can't see how to use it to make the step.
The text refers to it as a Green's function but the explanations of that phrase look like a it would take me days to figure out what that means.
Is there a simple explanation?

 

[fresh_42]

Have a look at
https://en.wikipedia.org/wiki/Poisson's_equation#Electrostatics
or a bit more elaborated (but you'll have to use Google translate or something)
https://de.wikipedia.org/wiki/Greensche_Funktion#Bestimmung_des_statischen_elektrischen_Feldes