the delta function
The Laplacian ∇2 operator is shorthand for the divergence of the gradient. From Appendix F, the gradient of a function that depends only on r equals (∂f/∂r)r^. So we have ∇(1/r) = −r^/r2. The volume integral of ∇2(1/r) is therefore (using the divergence theorem)
∫V∇2(r1)dv=∫V∇⋅∇(r1)dv=∫V∇⋅(r2−r^)dv=−∮Sr2r^⋅da.
(12.131)
Since ∇2(1/r) = 0 everywhere away from the origin, any volume containing the origin will yield the same integral. Choosing a sphere of radius R gives
∫V∇2(r1)dv=−∮Sr2r^⋅da=−∫R21da=−R24πR2=−4π,
(12.132)
as desired. A function that is (1) zero everywhere except at one point and (2) infinite enough at that one point so that it has a nonzero integral, is called a delta function (up to numerical factors).