  The geometric phase techniques for swimming in viscous flows express the net
displacement of a swimmer as a path integral of a field in configuration space.
This representation can be transformed into an area integral for simple
swimmers using Stokes theorem. Since this transformation applies for any loop,
the integrand of this area integral can be used to help design these swimmers.
However, the extension of this Stokes theorem technique to more complicated
swimmers is hampered by problems with variables that do not commute and by how
to visualise and understand the higher dimensional spaces. In this paper, we
develop a treatment for each of these problems, thereby allowing the
displacement of general swimmers in any environment to be designed and
understood similarly to simple swimmers. The net displacement arising from
non-commuting variables is tackled by embedding the integral into a higher
dimensional space, which can then be visualised through a suitability
constructed surface. These methods are developed for general swimmers and
demonstrated on {three} benchmark examples: Purcell's two-hinged swimmer, an
axisymmetric squirmer in free space {and an axisymmetric squirmer approaching a
free interface}. We show in particular that for swimmers with more than two
modes of deformation, there exists an infinite set of strokes that generate
each net displacement. Hence, in the absence of additional restrictions,
general microscopic swimmers do not have a single stroke that maximises their
displacement.
