Einstein-Notation: rot(rot(A))

  • #1

Homework Statement


Write [tex]\nabla \times (\nabla \times \vec A)[/tex] in Einstein-Notation, whereas [tex]\vec A[/tex] is the vector potential of the magnetic field.



Homework Equations


[tex](\vec a \times \vec b)=\varepsilon_{ijk} a_j b_k[/tex]




The Attempt at a Solution


[tex]\nabla \times (\nabla \times \vec A)=\varepsilon_{ijk} \partial_j(\varepsilon_{lmn}\partial_m A_n)_k[/tex]

What to do with [tex](\varepsilon_{lmn}\partial_m A_n)_k[/tex] though?

Thank you for your help!
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
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Homework Equations


[tex](\vec a \times \vec b)=\varepsilon_{ijk} a_j b_k[/tex]
On the right hand side, there is summation over j and k, but i is a free index which only occurs once. What you really meant to write, then, is

[tex](\vec a \times \vec b)_i=\varepsilon_{ijk} a_j b_k[/tex]
 
  • #3
1,024
32
In index notation it reads
εijkjklmlum)
after it you can use an identity
εkijεklmilδjmimδjl
(I hope this is not complete solution.if it is then delete this)
 

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