Let

ginC^(\\\\infty )(R^(3))

. If

F

is any distribution acting on test functions in the class\

C_(c)^(\\\\infty )(R^(3))

, we can define the product of

g

with

F

as a distribution; for any

\\\\phi in

\

C^(\\\\infty )(R^(3)),(:gF,\\\\phi :)=(:F,g\\\\phi :)

.\ (a) Why was this a reasonable definition?\ (b) Prove that

|x|^(2)\\\\Delta \\\\delta _(0)=6\\\\delta _(0)

in the sense of distributions where

\\\\xi nR^(3)

and

\\\\Delta

is\ the 3D Laplacian.

Let gC(R3). If F is any distribution acting on test functions in the class Cc(R3), we can define the product of g with F as a distribution; for any C(R3),gF,=F,g. (a) Why was this a reasonable definition? (b) Prove that x20=60 in the sense of distributions where xR3 and is the 3D Laplacian