What is the actual meaning of the Green identities:
Is there a picture/geometric interpretation of these, as well as intuition going beyond the usual integration-by-parts meaning associated to these identites? For instance, why do they really arise in the Green function for Poisson's equation.
There should be a nice way to interpret these geometrically in terms of differential forms, where you derive the second one by subtracting
$$d(\psi \wedge * d \phi ) = d \psi \wedge * d \phi + \psi d * d \phi = d \phi \wedge * d \psi + \psi \nabla ^2 \phi * 1$$
from
$$d(\phi \wedge * d \psi ) = d \phi \wedge * d \psi + \phi d * d \psi = d \phi \wedge * d \psi + \phi \nabla ^2 \psi * 1$$
to get
$$d(\phi \wedge * d \psi) - d( \phi \wedge * d \psi ) = d(\phi * d \psi - \phi * d \psi ) = (\phi \nabla ^2 \psi - \psi \nabla ^2 \phi)* 1$$
which, by Stokes theorem, gives
$$\int _{V} (\phi \nabla ^2 \psi - \psi \nabla ^2 \phi)* 1 = \int _{V} d(\phi * d \psi - \phi * d \psi ) = \int _{\partial V} (\phi * d \psi - \phi * d \psi ) $$
We know Stokes theorem has a pictorial interpretation:
as does the Hodge star, so what about the Green identities? There should be pictorial meaning to every step of those calculations...