How to Prove the Vector Identity Involving Curl and Dot Product Operations?

[aanabtawi]

Homework Statement

Prove that:
∇×(a∙∇a) = a∙∇(∇×a) + (∇∙a)(∇×a) - (∇×a)∙∇a

Homework Equations

Related to the vorticity transport equation.

The Attempt at a Solution

Brand new to index/tensor notation, any suggestions on where to begin? For example, I am having trouble converting to index notation with differentials inside the curl on the LHS.

 

[tiny-tim]

welcome to pf!

hi aanabtawi! welcome to pf!

(have a curly d: ∂ and try using the X₂ button just above the Reply box )

Prove that:
∇×(a∙∇a) = a∙∇(∇×a) + (∇∙a)(∇×a) - (∇×a)∙∇a
…
Brand new to index/tensor notation, any suggestions on where to begin? For example, I am having trouble converting to index notation with differentials inside the curl on the LHS.

write the LHS out carefully with index notation, then use the product rule for derivatives …

show us how far you get ​