Let f ( S ) be a real 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 second 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 .