Electrostatic reciprocity identity (${\int }_V({\varphi }_1{\rho }_2-{\varphi }_2{\rho }_1)dV={\epsilon }_0{\int }_{\partial V}({\varphi }_2\nabla {\varphi }_1-{\varphi }_1\nabla {\varphi }_2)\cdot d\mathbf{S}$)

For two electrostatic potentials satisfying the Poisson equation $\Delta {\varphi }_i=-{\rho }_i/{\epsilon }_0$, Green's second identity yields the displayed reciprocity formula. The product rule for divergence applied to ${\varphi }_1\nabla {\varphi }_2-{\varphi }_2\nabla {\varphi }_1$ cancels the cross-gradient terms. The normal points outward from the integration region; for an exterior region this means inward toward the excluded object on its inner boundary. Boundary terms may be removed only under suitable decay or boundary conditions.