∇×(u×v)=v∙(∇u)-u(∇∙v)+u(∇∙v)-v(∇∙u) where u and v are vectors
I tried to do this but it could not expand them
∇×(u×v)=ε_ijk ∂/(∂x_j )(ϵ_klm u_l v_m)
Also, I could not figure out what to do to these other 3 identities:
1 v∙(∇v)=(∇×v)×v+∇(v^2/2) where v is a vector
2 ∇∙(φuu)=φu∙∇u+u[∇∙(φu)] where φ is a scalar and u is a vector
3 ∇∙(T×x)=-x×(∇∙T)+ϵ:T where T is a second order tensor and ϵ is the third order alternating tensor.
Please give me some help. Thanks in advance.