Curl of Curl Identity Proof

MoogPoo

Hi, does anyone know how to prove the identity that

curl(curl ) A = grad(divA) - grad(grad A)

I know how to do it in cartesians or in given coordinates but how do I prove it generally. I googled it and stuff but can't find anything, can anyone point me in the right direction please?? Thanks.

ZorbaTehZ

Ignore the del operator and focus on proving [latex]A \times (B\times C) = B(A\cdot C) - C(A\cdot [/latex] keeping in mind what does and does not commute. It's usually called the vector triple product if you want to look it up.

MoogPoo

Well I get the vector triple product with vector product and scalar product, but how does that all relate to curl, div and grad?
Sorry I'm still fairy new to those operations.

ZorbaTehZ

Because from definition, you have:

[latex]\textrm{grad} \ F=\nabla F[/latex]
[latex]\textrm{div} \ F=\nabla \cdot F[/latex]
[latex]\textrm{curl} \ F=\nabla \times F[/latex]