vector potential

The divergence of the curl of any vector is zero. The divergence of a magnetic field B is zero. (See Maxwell's equations). Thus,

B=curl A

can always be set. A is known as the vector potential.

From the Biot-Savart law an expression for A may be derived. For a current density distribution J within a volume V, the vector potential at distance r is given by-

A=(μ_O/4π) ∫_v{J/r}

where ∫_v denotes integration over the volume V and μ_O is the vacuum permeability.

The vector potential is not uniquely defined (i.e. for any magnetic field configuration there are an infinite number of valid vector potentials). It is analagous to the scalar potential involving the electric field.

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.