Suppose the problem is to solve for φ (x) inside the region. Then the integral ∫ ∫ V φ φ ( x ′ ) δ δ ( x − − x ′ ) d 3 x ′ reduces to simply φ (x) due to the defining property of the Dirac delta function and we have φ φ ( x ) = − − ∫ ∫ V G ( x , x ′ ) ρ ρ ( x ′ ) d 3 x ′ + ∫ ∫ S [ φ φ ( x ′ ) ∇ ∇ ′ G ( x , x ′ ) − − G ( x , x ′ ) ∇ ∇ ′ φ φ ( x ′ ) ] ⋅ ⋅ d σ σ ^ ^ ′ .