The quantity 
is the local area dA as a function of the
`tilt' 
of the surface z(x,y).
For small slopes, the lowest order terms for additional energy due to the tilt goes like: 
.
If motion of the surface requires no long-range diffusion, as in grain and antiphase boundaries or
domain walls, then gradient flow in the 
inner product is motion by curvature.
The small slope approximation to gradient flow appears in the table above and the exact expression
appears below.
Motion of an interface by surface diffusion has the proximity effect: gradient flow in the
inner products gives the Mullins surface diffusion equation[10]
(the approximate expression appears in Table I).