Correct relation is F^{ij} = - epsilon^{ijk} B^k.

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SUMMARY

The correct relation for the electromagnetic field tensor is given by F^{ij} = -\epsilon^{ijk} B^k, which is derived using the west-coast convention where the derivative operator is defined as \partial^l = -\partial_l. The expression for B^i is B^i = -\epsilon_{ijk} \partial_j A^k, where A^i = -A_i. The discussion emphasizes the importance of consistent conventions in tensor calculus, particularly when dealing with metrics such as diag(1,-1).

PREREQUISITES
  • Understanding of tensor calculus and electromagnetic field theory
  • Familiarity with the west-coast and east-coast conventions in physics
  • Knowledge of partial derivatives and their notation
  • Experience with metric signatures, specifically diag(1,-1)
NEXT STEPS
  • Study the implications of different metric signatures in tensor calculus
  • Learn about the derivation of the electromagnetic field tensor F^{ij}
  • Explore the differences between west-coast and east-coast conventions in physics
  • Investigate the role of the Levi-Civita symbol \epsilon^{ijk} in vector calculus
USEFUL FOR

This discussion is beneficial for physicists, particularly those specializing in electromagnetism, as well as students and researchers working with tensor analysis and differential geometry.

Zohaib_aarfi
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When I tried to derive this relation I got the wrong sign. Please check the pic and tell me my mistakes.
 

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What is correct probably depends on your conventions, which you don't give.
 
If you use the west-coast convention you have ##\partial^l=-\partial_l##, and thus
$$B^i=-\epsilon_{ijk} \partial_j A^k.$$
Note that
$$\partial_j=\frac{\partial}{\partial x^j}.$$
 
Thank you for your response. I am using metric diag(1,-1) and the expression you gave B^i = - \epsilon_{ijk} \partial_j A^k contains also A^i = - A_i, so I think it does not make any difference. Could you do it for me in complete and explicit steps?
 

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