Background: The closed path "P" can be defined at any initial time anywhere in the fluid. Following the fluid motion of the individual fluid parcels on "P", "P" becomes a material property of the parcel group on "P". Circulation can be defined around path "P" and for non divergent and homogeneous flows its conserved.
Now, it implicitly means that the parcels will have always the same parcels as neighbors, since if new parcels come in the place between two parcels on path "P" so that after a while there would be not closed path "P" possible anymore tracking the same initial group, "P" is not anymore a material property of the initial parcel group.
It is kind of weird, because the evolution of the distance between two water parcels is also a material property given as the dot product of the vector distance between them and the gradient of velocity tensor. (both informations are on the same book)
If this is true, how can the distance between parcels change if they are always neighbors and infinitesimally small. And if they slide on each other making possible that parcels occupy the position of other parcels without creating holes in the continuum mesh how can the circulation theorem as a material property be true?