If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that G (x, x ′) vanishes when either x or x′ is on the bounding surface. Thus only one of the two terms in the surface integral remains. If the problem is to solve a Neumann boundary value problem, it might seem logical to choose Green's function so that its normal derivative vanishes on the bounding surface. However, application of Gauss's theorem to the differential equation defining the Green's function yields ∫ ∫ S ∇ ∇ ′ G ( x , x ′ ) ⋅ ⋅ d σ σ ^ ^ ′ = ∫ ∫ V ∇ ∇ ′ 2 G ( x , x ′ ) d 3 x ′ = ∫ ∫ V δ δ ( x − − x ′ ) d 3 x ′ = 1 , meaning the normal derivative of G (x, x ′) cannot vanish on the surface, because it must integrate to 1 on the surface.