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=(μO/4π) ∫v{J/r}
where ∫v denotes integration over the volume V and μO 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 Aμ=(ρ,A)T, and the Maxwell tensor can then be written as Fμν=dμAν-dνAμ. Remembering that FijijkBk, you can recover the equation that Blush Response has provided above:
    εijlFij = εijlεijkBk = (δjjδkljkδjl)Bk = 2Bi
    εijlFij = εijl(djAl-dlAj) = 2εijldjAl
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 dμFνρ+ dρFμν+ dνFρμ=0. Quite remarkable really.

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