  Three-dimensional field distributions from realistic beamline elements can be
obtained only by measurement or by numerical solution of a boundary-value
problem. In numerical charged-particle map generation, fields along a reference
trajectory are differentiated multiple times. Any attempt to differentiate
directly such field data multiple times is soon dominated by "noise" due to
finite meshing and/or measurement errors. This problem can be overcome by the
use of field data on a surface outside of the reference trajectory to
reconstruct the fields along and around the reference trajectory. The integral
kernels for Laplace's equation that provide interior fields in terms of
boundary data or boundary sources are smoothing: interior fields will be
analytic even if the boundary data or source distributions fail to be
differentiable or are even discontinuous. In our approach, we employ all three
components of the field on the surface to find a superposition of single-layer
and double-layer surface source distributions that can be used together with
simple, surface-shape-independent kernels for computing vector potentials and
their multiple derivatives (required for a Hamiltonian map integration) at
interior points. These distributions and kernels are found by the aid of
Helmholtz's theorem (or equivalently, by Green's theorem). A novel application
of the Dirac-monopole vector potential is used to find a kernel for the
single-layer distribution. These methods are the basis for map-generating
modules that can be added to existing numerical electromagnetic field-solving
codes and would produce transfer maps to any order for arbitrary static
charged-particle beamline elements.
