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[Community Question] Calculus: I need to prove a few vector identities using Cartesion Tensor Notation, and I can't figure out how!

One of our user asked:

I have been all over the internet, but I just can't make sense of this stuff. I have done my best to learn from my textbook and different websites, but this is confusing for me. I haven't taken any calculus in years, and I'm jumping in headfirst. If anyone can help me understand how to prove these using Cartesian Tensor Notation, I would really appreciate it!

First identity: ∇ x (∇ x a) = ∇(∇ . a) - (∇^2)a

Second identity: ∇ . (a b) = a . ∇ b + b(∇ . a)

Third identity: ∇ . (f δ) = ∇ f

Fourth identity: δ : ∇ a = ∇ . a

Thanks everyone


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