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

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

• Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. Feb 7 at 14:57
• What have you tried? Have you attempted a brute force approach by carrying out the derivatives? Feb 7 at 15:05
• Possible duplicate: math.stackexchange.com/a/1108604/583883 Feb 7 at 15:06
• You can do it with Levi-Civita symbol May 2 at 17:39

$$\def\n{\nabla}$$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 \\ }