The continuum definition of surface free energy is the amount of reversible work  performed to create new area  of surface, expressed as:
In this definition the number of atoms at the surface is proportional to the area. Gibbs was the first to define another surface quantity, different from the surface free energy , that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface. In a continuum approach one can define a surface stress tensor  that relates the work associated with the variation in , the total excess free energy of the surface due to a strain tensor 
In general there is no change in area for shear, which means that for the second term on the right  and , using the Kronecker delta. Cancelling the area then gives
called the Shuttleworth equation. 
An alternative approach is an atomistic one, which defines all quantities in terms of the number of atoms, not continuum measures such as areas. This is related to the ideal of using Gibb's equimolar quantities rather than continuum numbers such as area, that is keeping the number of surface atoms constant. In this case the surface stress is defined as the derivative of the surface energy with strain, that is (deliberately using a different symbol)
This second definition is more convenient in many cases. A conventional liquid cannot sustain strains, so in the continuum definition the surface stress and surface energies are the same, whereas in the atomistic approach the surface stress is zero for a liquid. So long as care is taken the choice of the two does not matter, although this has been a little contentious in the literature.