Here [[xi].sub.0] = ([[tau].sub.0],[[psi].sub.0]) is a fixed source point, [xi] = (r, [psi]) is integration variable and [q.sub.1]([xi]) = [partial derivative][u.sub.1]/[partial derivative]r is normal derivative
of the potential function on [[psi].sub.1].
If we further demand the normal derivative
to be constant on S, then we arrive at probably the most common boundary condition used in developing a Neumann-Green's function :
The current flowing into the medium is expressed in terms of the normal derivative
of the potential, taken from the side of the medium.
Here [DELTA] is the Laplacian in [R.sup.m], [ohm] [subset] [R.sup.m] is a bounded domain with smooth boundary [partial derivative][ohm] and [partial derivative]/[partial derivative]n is the outward normal derivative
to [partial derivative][ohm].
Here the outward normal derivative
is [mathematical expression not reproducible]; then
where R(P,Q) is the distance between field point P (x, y, z, t) and boundary point Q ([x.sub.0], [y.sub.0], z), [partial derivative]/[partial derivative][n.sub.Q] is normal derivative
to S at point Q.
n is the unit outward normal field along the boundary [GAMMA], D is the Levi Civita connection on M and [partial derivative]y/[partial derivative]n = [<Dy, n>.sub.g] is the normal derivative
The normal derivative
[partial derivative]u/[partial derivative]n on S is calculated in an analogous way and, thus, the residual [PSI] of the boundary condition on S is updated.
Note that [partial derivative][OMEGA] is part of the interface because the boundary condition for the normal derivative
is only enforced weakly through the penalty term in (1.3).
The approximation of the normal derivative
which represents the flux is the crucial point.
At the outlet, the uniform axial velocity was used to ensure continuity, and tangential velocities were computed from a zero normal derivative
The local potential V(M[member of][OMEGA]) can be written using Green's theorem  in terms of V(P) and of the normal derivative
[partial derivative]V(P)/[partial derivative]n with P being any point on the boundary S (with no overhangs) of [OMEGA]: