How to prove that for every two scalar fields $u(x,y,z)$ and $v(x,y,z)$ the identity above holds true?
I guess it says the Laplacian of the dot product of two scalar fields equals the Laplacian of each field times the other field plus the gradient of each field times the gradient of the other field?
We have: $(uv)_x=u_xv+uv_x,$ hence
$$(uv)_{xx}=u_{xx}v+2u_xv_x+v_{xx}u.$$
Now compute $(uv)_{yy}$ and $(uv)_{zz}$ and add.