How Do You Prove the Vector Identity ∇ x ∇φ = 0 Using Index Notation?

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SUMMARY

The vector identity ∇ x ∇φ = 0 can be proven using index notation by applying the Levi-Civita symbol and the properties of partial derivatives. The discussion highlights the expression E[jrt]E[tpq]∂[r]∂[p]φ[q], which simplifies to the difference of mixed partial derivatives, leading to the conclusion that the identity holds true. The key step involves recognizing that the mixed partial derivatives are equal, thus confirming that the curl of the gradient is indeed zero.

PREREQUISITES
  • Understanding of vector calculus, specifically vector identities.
  • Familiarity with index notation and the Levi-Civita symbol.
  • Knowledge of partial derivatives and their properties.
  • Basic skills in mathematical proofs and manipulation of equations.
NEXT STEPS
  • Study the properties of the Levi-Civita symbol in tensor calculus.
  • Learn about the implications of the equality of mixed partial derivatives.
  • Explore additional vector identities and their proofs in index notation.
  • Review advanced topics in vector calculus, such as Stokes' theorem and its applications.
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Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector calculus and its applications in theoretical contexts.

flame_m13
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Homework Statement


Prove vector identity \nabla x\nabla\phi = 0 using index notation.

Homework Equations



\nabla x A = Ejrt\partialrAt

The Attempt at a Solution


I'm treating this as \nablax A, where A = del \Phi = Etpq\partialp\Phiq

putting A back in the equation:

= EjrtEtpq\partialr\partialp\Phiq

=EtrjEtpq\partialr\partialp\Phiq

=(\partialjp\partialrq - \partialjq\partialrp)\partialr\partialp\Phiq

It's here that I'm stuck. I know it should be simple after this, and somehow I need to get zero, but I'm completely stuck. Maybe I messed up somewhere?
 
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Hi flame_m13! :smile:

(have a del: ∇ and a curly d: ∂ :wink:)
flame_m13 said:
… where A = del \Phi = Etpq\partialp\Phiq


Nooo … (∇φ)p = ∂pφ :wink:
 
thanks. i have skill at making things unnecessarily complicated. :)
 

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