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

_{μ}A

_{ν}-d

_{ν}A

_{μ}. Remembering that F

_{ij}=ε

_{ijk}B

_{k}, you can recover the equation that

Blush Response has provided above:

ε_{ijl}F_{ij} = ε_{ijl}ε_{ijk}B_{k} = (δ_{jj}δ_{kl}-δ_{jk}δ_{jl})B_{k} = 2B_{i}

ε_{ijl}F_{ij} = ε_{ijl}(d_{j}A_{l}-d_{l}A_{j}) = 2ε_{ijl}d_{j}A_{l}

ie.

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

_{μ}F

_{νρ}+
d

_{ρ}F

_{μν}+
d

_{ν}F

_{ρμ}=0. Quite remarkable really.