How to Prove the Vector Identity Involving Curl and Dot Product Operations?

AI Thread Summary
To prove the vector identity involving curl and dot product operations, the equation ∇×(a∙∇a) must be shown to equal a∙∇(∇×a) + (∇∙a)(∇×a) - (∇×a)∙∇a. A user new to index and tensor notation seeks guidance on converting the left-hand side to index notation, particularly with differentials in the curl. Suggestions include carefully writing out the left-hand side in index notation and applying the product rule for derivatives. Engaging with the community for further assistance is encouraged as the user progresses. Mastery of these concepts is essential for understanding the vorticity transport equation.
aanabtawi
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Homework Statement



Prove that:
∇×(a∙∇a) = a∙∇(∇×a) + (∇∙a)(∇×a) - (∇×a)∙∇a


Homework Equations



Related to the vorticity transport equation.


The Attempt at a Solution



Brand new to index/tensor notation, any suggestions on where to begin? For example, I am having trouble converting to index notation with differentials inside the curl on the LHS.
 
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welcome to pf!

hi aanabtawi! welcome to pf! :smile:

(have a curly d: ∂ and try using the X2 button just above the Reply box :wink:)
aanabtawi said:
Prove that:
∇×(a∙∇a) = a∙∇(∇×a) + (∇∙a)(∇×a) - (∇×a)∙∇a
…
Brand new to index/tensor notation, any suggestions on where to begin? For example, I am having trouble converting to index notation with differentials inside the curl on the LHS.

write the LHS out carefully with index notation, then use the product rule for derivatives …

show us how far you get :smile:​
 

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