Question

Prove that

(v.del)v = 1/2 del v^2 - v×(del×v)

v is a vector

Answer 1

SInce, (https://en.wikipedia.org/wiki/Del)

$${\displaystyle \nabla ({\vec {u}}\cdot {\vec {v}})=({\vec {u}}\cdot \nabla ){\vec {v}}+({\vec {v}}\cdot \nabla ){\vec {u}}+{\vec {u}}\times (\nabla \times {\vec {v}})+{\vec {v}}\times (\nabla \times {\vec {u}})}$$

Now, put $\overrightarrow{u}=\overrightarrow{v}$ ,we get

$${\displaystyle \nabla ({\vec {v}}\cdot {\vec {v}})=({\vec {v}}\cdot \nabla ){\vec {v}}+({\vec {v}}\cdot \nabla ){\vec {v}}+{\vec {v}}\times (\nabla \times {\vec {v}})+{\vec {v}}\times (\nabla \times {\vec {v}})}$$

$$\implies {\displaystyle \nabla ( {v^2})=2({\vec {v}}\cdot \nabla ){\vec {v}}+2{\vec {v}}\times (\nabla \times {\vec {v}})}\\ \implies{ ({\vec {v}}\cdot \nabla ){\vec {v}}= \frac{1}{2}{\displaystyle \nabla ( {v^2})}-{\vec {v}}\times (\nabla \times {\vec {v}})}$$