A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an axial vector, w , as follows W i j = − − ϵ ϵ i j k w k ; w i = − − 1 2 ϵ ϵ i j k W j k where ϵ ϵ i j k is the permutation symbol. In matrix form W _ _ _ _ = [ 0 − − w 3 w 2 w 3 0 − − w 1 − − w 2 w 1 0 ] ; w _ _ = [ w 1 w 2 w 3 ] The axial vector is also called the infinitesimal rotation vector. The rotation vector is related to the displacement gradient by the relation w = 1 2 ∇ ∇ × × u In index notation w i = 1 2 ϵ ϵ i j k u k , j If ‖ ‖ W ‖ ‖ ≪ ≪ 1 and ε ε = 0 then the material undergoes an approximate rigid body rotation of magnitude | w | around the vector w .