The calculation of the magnetic field from a current distribution: Application to finite element techniques

In finite element calculations of the magnetic field associated with a distributed current density
\overrightarrow{J}
it is useful to cast the law of Biot and Savart in a form where the integration over
\overrightarrow{J}
need include only those elements of the mesh containing either a current source or a conductivity boundary, i.e. those elements where
\nabla \times \overrightarrow{J}
is non-zero.