Green second identity

Green’s second identity is the divergence theorem applied to phi grad psi minus psi grad phi.

The whole-space form of Green's second identity requires the displayed boundary limit, together with the existence of the relevant volume integrals. Pointwise decay of the fields does not by itself control their normal derivatives or the growing sphere area. In three dimensions, sufficient power bounds are $u=O(r^{-a})$, $\nabla u=O(r^{-a-1})$, $v=O(r^{-b})$ and $\nabla v=O(r^{-b-1})$, with $a+b>1$: the surface integral is then $O(R^{1-a-b})$.

For a smooth counterexample to pointwise decay alone, take $u=(1+r^2{)}^{-1/2}$ and $v=\cos r(1+r^2{)}^{-1/2}$. Both are smooth radial functions on all of ${\mathbb{R}}^3$ and tend to zero. Nevertheless $u{v}^{\prime }-v{u}^{\prime }=-\sin r/(1+r^2)$, giving surface integral $-4\pi R^2\sin R/(1+R^2)$, which has no zero limit. Screened Green functions with exponential decay avoid this particular problem.