  The present paper is intended to provide the basis for the study of weakly
differentiable functions on rectifiable varifolds with locally bounded first
variation. The concept proposed here is defined by means of integration by
parts identities for certain compositions with smooth functions. In this class
the idea of zero boundary values is realised using the relative perimeter of
superlevel sets. Results include a variety of Sobolev Poincar\'e type
embeddings, embeddings into spaces of continuous and sometimes H\"older
continuous functions, pointwise differentiability results both of approximate
and integral type as well as coarea formulae.
  As prerequisite for this study decomposition properties of such varifolds and
a relative isoperimetric inequality are established. Both involve a concept of
distributional boundary of a set introduced for this purpose.
  As applications the finiteness of the geodesic distance associated to
varifolds with suitable summability of the mean curvature and a
characterisation of curvature varifolds are obtained.
