Reynolds transport theorem

The Reynolds transport theorem (after Osborne Reynolds ) makes a connection between the Eulerian and Lagrangian approach of a time-dependent volume control. It is used to derive the basic conservation laws of continuum mechanics.

It is a general field size depends on both the location and the time. If, for example, the density for a, we obtain a formulation for the conservation of mass:

Is the volume of the enclosing surface ( the last term is to be understood as a dot product of the velocity vector and normal vector of the surface). A clear derivation is obtained by transformation to the time-independent initial volume with the introduction of time-dependent Jacobian.