How Does Green's Function Simplify Understanding Electromagnetic Equations?

AJ Bentley
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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?
 
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