Motivation
One physical system of this type is a charge distribution in electrostatics. In such a system, the electric field is expressed as the negative gradient of the electric potential, and Gauss's law in differential form applies:
Combining these expressions gives
- (Poisson's equation.)
We can find the solution to this equation for an arbitrary charge distribution by temporarily considering the distribution created by a point charge located at :
In this case,
which shows that for will give the response of the system to the point charge . Therefore, from the discussion above, if we can find the Green's function of this operator, we can find to be
for a general charge distribution.
Read more about this topic: Green's Function For The Three-variable Laplace Equation
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