Need help proving vector identities

[dooogle]

Homework Statement

i have to prove that

∇x(FxG)=(G⋅∇)F-(F⋅∇)G+F(∇⋅G)-G(∇⋅F)

where F and G are vector fields with F=F1,F2,F3 and G=G1,G2,G3 ∇=d/dx,d/dy/d/dz

Homework Equations

The Attempt at a Solution

i have tried applying scalar multiplication and the cross product to multiply out the terms
and have got the right hand side as twice the left hand side.

is (G⋅∇)F equal to F(∇⋅G)?

thank you for your time

dooogle

 

[henry_m]

Do you know about the completely antisymmetric or Levi-Civita symbol \epsilon_{ijk}? This is by far the easiest way I know to prove this sort of identity.

is (G⋅∇)F equal to F(∇⋅G)?

What if G is a constant vector and F is not?