However, in many practical cases we may assume that the field lines
of the magnetic induction
do not enter or leave
a bounded domain
representing the device
(e.g. the machine).
The latter condition, called flux-surface condition, requires that
the normal component vanishes on the boundary
which is assumed to be sufficiently smooth, i.e.
and is an immediate consequence of
Therefore, we will demand that the tangential components of vanish on , i.e.
If (12) holds on a smooth surface, then we get
if and denote two perpendicular tangential unit vectors. Thus, in view of (5), we should formulate a Neumann condition
8), i.e. we multiply with a test function v and obtain
The second integral will be zero if , i.e. . The latter will be satisfied if is taken from an appropriate test space related to , see Subsection 3.1.