Definition of the variation of velocity gradient in Abaqus

I am confused by the definition of the variation of the velocity gradient in the Abaqus Theory Manual (4.6.1).

Normally δL = δ(grad(v)) = grad(δv); where δ denotes the variational operator.  But in the manual its given as δL = grad(δu); where u is the displacement field.

Could someone explain how the definition in the Abaqus theory Manual came about? I have attached a screen capture of the definition given in the manual.