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

[Zohaib_aarfi]

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

 

[haushofer]

What is correct probably depends on your conventions, which you don't give.

 

[vanhees71]

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
$$\partial_j=\frac{\partial}{\partial x^j}.$$

 

[Zohaib_aarfi]

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?