  We propose a new concept for determining the interior magnetic field vector
components in neutron electric dipole moment experiments. If a closed
three-dimensional boundary surface surrounding the fiducial volume of an
experiment can be defined such that its interior encloses no currents or
sources of magnetization, each of the interior vector field components and the
magnetic scalar potential will satisfy a Laplace equation. Therefore, if either
the vector field components or the normal derivative of the scalar potential
can be measured on the surface of this boundary, thus defining a Dirichlet or
Neumann boundary-value problem, respectively, the interior vector field
components or the scalar potential (and, thus, the field components via the
gradient of the potential) can be uniquely determined via solution of the
Laplace equation. We discuss the applicability of this technique to the
determination of the interior magnetic field components during the operating
phase of neutron electric dipole moment experiments when it is not, in general,
feasible to perform direct in situ measurements of the interior field
components. We also study the specifications that a vector field probe must
satisfy in order to determine the interior vector field components to a certain
precision. The technique we propose here may also be applicable to experiments
requiring monitoring of the vector magnetic field components within some closed
boundary surface, such as searches for neutron-antineutron oscillations along a
flight path or measurements in storage rings of the muon anomalous magnetic
moment $g-2$ and the proton electric dipole moment.
