Prove of some vector's differential relations

Question

I want to know how to prove these two relations:

$(V⋅∇)V=\frac12∇(V⋅V)−V×(∇×V)$

$(∇.sv)=(∇s.v)+s(∇.v)$

[These relations are from Bird's Transport Phenomena]

Answer 1

I will use Einstein notation and cartesian coordinates.
For the first one, let's compute $\vec{v} \times (\vec{\nabla} \times \vec{v})$: $$\begin{align} (\vec{v} \times (\vec{\nabla} \times \vec{v}))_i&=\epsilon_{ijk}v_j(\vec{\nabla} \times \vec{v})_k\\ &=\epsilon_{ijk}v_j\epsilon_{kpq}\partial_pv_q\\ &=\epsilon_{ijk}\epsilon_{kpq}v_j\partial_pv_q\\ &=\epsilon_{kij}\epsilon_{kpq}v_j\partial_pv_q\\ &=(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})v_j\partial_pv_q\\ &=v_q\partial_iv_q-v_p\partial_pv_i\\ &=\frac{1}{2}\partial_i(v_qv_q)-v_p\partial_pv_i\\ &=\frac{1}{2}\partial_i(\vec{v} \vec{v})-(\vec{v}\vec{\nabla})v_i \end{align}$$

And if I'm right, your second identity is $\vec{\nabla}(s\vec{v})=(\vec{\nabla}s)\vec{v}+s(\vec{\nabla}\vec{v})$: $$\begin{align} \vec{\nabla}(s\vec{v})&=\partial_i(sv_i)\\ &=(\partial_i s)v_i+s(\partial_iv_i)\\ &=(\vec{\nabla}s)_iv_i+s(\vec{\nabla}\vec{v})\\ &=(\vec{\nabla}s)\vec{v}+s(\vec{\nabla}\vec{v}) \end{align}$$