Vector and tensor identities. Prove the following, using indicial notation.

$($a$)v*gradv=(gradxv)xv+$grad$(\frac{v^2}{2})$

$($b$)gra{d}^{2}v=$grad$(grad*v)-$grad$(gradx)$

$($c$)$:$T=0$ where $$ is the alternating tensor and $T$ is a symmetric second order tensor.

$($d$)$ grad$(vu)=v(grad*u)-u(grad*v)+u.$gradv$-v.$gradu Note: ${}_{\text{i}\text{j}\text{k}}{}_{klm}={}_{il}{}_{jm}-{}_{im}{}_{jl}$