Gauss' Theorem

(i) R=(x,y,z)
∇. F
=k[∂/∂x(x/r^3)+∂/∂x(x/r^3)+∂/∂x(x/r^3)]
=k{[r^3-x3r^2(x/r)]/r^6+[r^3-y3r^2(y/r)]/r^6+[r^3-z3r^2(z/r)]/r^6}
=k{[r^3-3x^2r]/r^6+[r^3-3y^2r]/r^6+[r^3-3z^2r]/r^6}
=k{[3r^3-3r^3]/r^6}
=0
(ii)
Since n=(x/r,y/r,z/r), F ∙ n=(x^2+y^2+z^2)/r^4
∫∫ F ∙ ndS
=∫∫(x^2+y^2+z^2)/r^4 dS
=r^(-2) ∫∫ dS
=4 pi [The boundary is the sphere of radius r center at (0,0,0)]

∫∫ F ∙ ndS