Direct Proof of Div, Grad, and Curl Operator Identities

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The discussion revolves around proving identities related to the divergence, gradient, and curl operators. The user successfully demonstrated that the triple cross product equals 2a using Einstein notation and confirmed that 2∇(a·r) equals 2a, aligning with the left-hand side of the equation. However, they express a desire for a more elegant or direct proof method. Other participants agree that while the current approach is valid, it feels like there should be a simpler shortcut available. The conversation highlights the search for more efficient proof techniques in vector calculus.
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Homework Statement



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Homework Equations





The Attempt at a Solution



For part (c), I showed that the tripple cross product = 2a using einstein notation. Then, I showed that 2∇(a.r) = 2a which is the same as LHS. I don't think this is as elegant as it can get..

How do I prove it directly? I've looked up some operator identities:
d5b8119de472d7a6d75c1672e93ccd34.png
 
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Your method is fine. I would have done the same thing. I know what you mean though. It feels like there should be some kind of shortcut method.
 

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