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

AI Thread Summary
The discussion centers on deriving the relation F^{ij} = -ε^{ijk} B^k, with a focus on the sign discrepancy encountered by the user. The correct interpretation of the relation may depend on the conventions used, particularly the west-coast convention, which leads to the expression B^i = -ε_{ijk} ∂_j A^k. The user mentions using a metric of diag(1,-1), suggesting that the sign conventions for A^i and A_i do not affect the outcome. A request for a detailed, step-by-step derivation indicates a need for clarity in the application of these conventions. The discussion highlights the importance of consistent notation in tensor calculus.
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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