Q:

Greens's formula with Hodge-*

Why is it the Green's formula?
$$0=\int_{\partial N'} u \ast dv-v \ast du$$
where $\ast$ is the Hodge-$\ast$, and $u,v$ are Green functions, and $N'=N \backslash D_{1} \cup D_{2}$, and $D_{1}, D_{2}$ are disks, $N$ is sub-Riemann surface with piecewise $C^{2}$ boundary and such that $\bar{N}$ is compact.

A:

Let $\phi, \psi$ be scalar valued(or, if you prefer, '$0$-forms'). Then 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
$$d(\phi \wedge * d \psi ) = d \phi \wedge * d \psi + \phi d * d \psi = d \phi \wedge * d \psi + \phi \nabla ^2 \psi * 1$$
reduces to
$$d(\phi \wedge * d \psi) - d( \psi \wedge * d \phi ) = d(\phi * d \psi - \psi  * d \phi ) = (\phi \nabla ^2 \psi - \psi \nabla ^2 \phi)* 1$$
which, by Stokes theorem, gives
$$\int _{V} (\phi \nabla ^2 \psi - \psi \nabla ^2 \phi)* 1 = \int _{V} d(\phi * d \psi - \phi  * d \psi ) = \int _{\partial V} (\phi * d \psi - \phi  * d \psi ) $$

