Show that $\vec\nabla\times(\vec\nabla\times\vec A )=- \nabla^ 2\vec A + \vec\nabla (\vec\nabla\cdot\vec A )$

Question

I just started learning this and I don't understand much so how can I prove this? $$\vec\nabla\times(\vec\nabla\times\vec A )=-\vec\nabla^2\vec A +\vec\nabla (\vec\nabla\cdot\vec A )$$

Answer 1

$\def\n{\nabla}$The vector aspect can be addressed using the triple product (aka bac-cab) rule $a\times(b\times c) = b\,(a\cdot c)- c\,(a\cdot b)$, but the derivative aspect requires keeping the $\n$ operator on the LHS of each expression
$\eqalign{\n\times(\n\times c) &= \n\big(\n\cdot c\big)- \big(\n\cdot \n\big)\,c \\ &= \n\big(\n\cdot c\big)- \n^2c \\ }$