how do i derive different equation from cross product of a curl

Asked Sep 20 '23 at 17:23

Active Sep 20 '23 at 19:35

Viewed 33 times

$\vec{a} \times (\nabla \times \vec{a}) = \frac{1}{2}\nabla{a^2}-(\vec{a} \cdot \nabla)\vec{a}$

How do I show this is true? I attempted to use vector triple product rule however it yielded $\vec{a} \times (\nabla \times \vec{a}) = \nabla{a^2}-(\vec{a} \cdot \nabla)\vec{a}$

I even tried to open up this formula to a component form. Yet i got no sensible answer. Maybe I am missing something. Can someone help clarify this problem for me?

edited Sep 20 '23 at 19:35

asked Sep 20 '23 at 17:23

Teemu Mäki-Maunus

1

In $\nabla a^2$, does $a^2$ mean the norm squared of $a$? – peter a g Sep 20 '23 at 17:31
The identity should be $$a\times(\nabla\times a) = \frac12 \nabla a^2 - (a\cdot\nabla)a.$$ – Nicholas Todoroff Sep 20 '23 at 17:37
Also https://math.stackexchange.com/questions/4212674/show-bfa-times-nabla-times-a-frac12-nablaa-cdot-a-a-cdot-nabla – Nicholas Todoroff Sep 20 '23 at 17:38

how do i derive different equation from cross product of a curl

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