As by now you should fully understand from working with the Poisson
equation, one very general way to solve inhomogeneous partial
differential equations (PDEs) is to build a Green's
function^{11.1} and write the solution as an
integral equation.

Let's very quickly review the general concept (for a further discussion
don't forget WIYF
,MWIYF). Suppose is a general (second order)
linear partial differential operator on e.g.
and one wishes to
solve the inhomogeneous equation:

(11.26) |

If one can find a solution
to the associated
differential equation for a *point* source function^{11.2}:

(11.27) |

(11.28) |

(11.29) |

This solution can easily be verified:

(11.30) | |||

(11.31) | |||

(11.32) | |||

(11.33) | |||

(11.34) | |||

(11.35) |

It seems, therefore, that we should *thoroughly understand* the ways
of building Green's functions in general for various important PDEs.
I'm uncertain of how much of this to do within these notes, however.
This isn't really ``Electrodynamics'', it is mathematical physics, one
of the fundamental toolsets you need to do Electrodynamics, quantum
mechanics, classical mechanics, and more. So check out Arfken, Wyld,
WIYF
, MWIYFand we'll content ourselves with a very quick review of
the principle ones we need: