Let F ( S ) be a second order tensor valued function of the second order tensor S . Then the derivative of F ( S ) with respect to S (or at S ) in the direction T is the fourth order tensor defined as ∂ ∂ F ∂ ∂ S : T = D F ( S ) [ T ] = [ d d α α F ( S + α α T ) ] α α = 0 for all second order tensors T .