vector potential (idea)

In electrodynamics this vector potential A taken together with the electrical scalar potential ρ give the four-vector A_μ=(ρ,A)^T, and the Maxwell tensor can then be written as F_μν=d_μA_ν-d_νA_μ. Remembering that F_ij=ε_ijkB_k, you can recover the equation that Blush Response has provided above:

ε_ijlF_ij = ε_ijlε_ijkB_k = (δ_jjδ_kl-δ_jkδ_jl)B_k = 2B_i
ε_ijlF_ij = ε_ijl(d_jA_l-d_lA_j) = 2ε_ijld_jA_l

ie.

B_i=ε_ijld_jA_l

Which is just B=curl A.
The fact that the Maxwell tensor can be written in terms of the vector potential implies half of Maxwell's equations - when expressed in the form d_μF_νρ+ d_ρF_μν+ d_νF_ρμ=0. Quite remarkable really.