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**1. Homework Statement**

Prove the following identity:

[tex] \vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A}) [/tex]

**2. Homework Equations**

Kronecker's delta, levi-civita tensor

**3. The Attempt at a Solution**

My solution consisted of simply solving the RHS by decomposition. I wrote [tex] \vec{A} = A_{1}A_{\hat{x}} + A_{2}A_{\hat{y}} + A_{3}A_{\hat{z}} [/tex] likewise with B. I manually solved for each term in RHS and I did get the correct result which was LHS. My problem is that there is probably a more efficient way of doing this, perhaps by incorporating Einstein notation and solve LHS immediately without having to look at RHS.

How should I do this?

I tried [tex] \vec{\nabla}(\vec{A} \cdot \vec{B}) = \partial_{x_{i}}A_{i}B_{i} = \frac{\partial A_{i}}{\partial_{x_{i}}}B_{i} + A_{i} \frac{\partial B_{i}}{\partial_{x_{i}}}[/tex] but then I got stuck. How to continiue? I'm not sure how all those [tex] \times [/tex] appear using einstein notation.