Vector identity proof using index notation

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1. Nov 22, 2014

darthvishous

1. The problem statement, all variables and given/known data
I am trying to prove
$$\vec{\nabla}(\vec{a}.\vec{b}) = (\vec{a}.\vec{\nabla})\vec{b} + (\vec{b}.\vec{\nabla})\vec{a} + \vec{b}\times\vec{\nabla}\times\vec{a} + \vec{a}\times\vec{\nabla}\times\vec{b}.$$ I can go from RHS to LHS by writng $$\vec{b}\times\vec{\nabla}\times\vec{a}$$ as $$\epsilon_{ijk}b_k\epsilon_{klm}\partial_la_m=b_j\partial_ia_j-a_j\partial_jv_i$$, but I am unable do it the other way. Any help is appreciated.
2. Relevant equations

3. The attempt at a solution

2. Nov 22, 2014

Fredrik

Staff Emeritus
This is the homework forum, so you have to show your attempt up to the point where you're stuck.

If you can do it one way, you can do it the other. Just write down all those things you found to be equal to each other in the reverse order.

There are some typos (I think) in the small piece of your work that you included in your post. The $b_k$ on the left should be $b_i$, right? That last term on the right looks weird? What's $v?$ Should the $a$ be a $b$?