  Advective transport of scalar quantities through surfaces is of fundamental
importance in many scientific applications. From the Eulerian perspective of
the surface it can be quantified by the well-known integral of the flux
density. The recent development of highly accurate semi-Lagrangian methods for
solving scalar conservation laws and of Lagrangian approaches to coherent
structures in turbulent (geophysical) fluid flows necessitate a new approach to
transport from the (Lagrangian) material perspective. We present a Lagrangian
framework for calculating transport of conserved quantities through a given
surface in $n$-dimensional, fully aperiodic, volume-preserving flows. Our
approach does not involve any dynamical assumptions on the surface or its
boundary.
