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How can I proof this statement? As $a$ and $b$ are vector fields, $$∇ \times (a \times b)=a(\nabla\cdot b)+(b\cdot \nabla)a-b(\nabla\cdot a)-(a\cdot ∇)b$$

And $$ (a \cdot \nabla) a = \nabla ((a^2)/2) - a \times ( \nabla \times a)$$

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    What about writing down *explicitly* all those expressions...? – 2017-01-25
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    As DonAntonio says, just give some general expressions for the two vector fields and then take their cross products, for example $\vec{a} = (a_1,a_2,a_3)$. When you've take in try rearraging the terms so that you get the right hand side. To succeed, expand the right hand side as well and try matching the two – 2017-01-25
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    you have written on the right hand side the formula for the **curl** of the cross product, not the gradient. Please edit your question – 2017-01-25
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    use suffices, maybe? – 2017-01-25

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