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

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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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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 [tex]diag(1,-1)[/tex] and the expression you gave [tex]B^i = - \epsilon_{ijk} \partial_j A^k[/tex] contains also [tex]A^i = - A_i[/tex], so I think it does not make any difference. Could you do it for me in complete and explicit steps?