By subtracting expressions in both systems (eq. 6 and eq. 12) then using eq. 9, it is straightforward to show, that both representations for the density gradient are equivalent. The difference between both vanishes, as expected: representation are observables characterizing well defined atoms and their links in a closed system. Direct relations between these physical derivatives and local derivatives represent an alternative to condensation of the local derivatives to atoms by their integration over an arbitrary chosen part of space. By this method, bonded atoms -the focus of all chemistry, may be endowed with an accurate physical measure of their changes in a reaction.