Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSanother, without reference to any coordinate system) and consider the matrixcontaining its components as a representation of the tensor with respect to aparticular coordinate system. Moreover, the matrix T =[T ij ], containing thecomponents of a second-order tensor, behaves in the same way under orthogonaltransformations T ′ = LTL T as a linear operator.However, not all linear operators are second-order tensors. More specifically,the two subscripts in a second-order tensor must refer to the same coordinatesystem. In particular, this means that any linear operator that transforms a vectorinto a vector in a different vector space cannot be a second-order tensor. Thus,although the elements L ij of the transformation matrix are written with twosubscripts, they cannot be the components of a tensor since the two subscriptseach refer to a different coordinate system.As examples of sets of quantities that are readily shown to be second-ordertensors we consider the following.(i) The outer product of two vectors. Letu i and v i , i =1, 2, 3, be the componentsof two vectors u and v, and consider the set of quantities T ij defined byT ij = u i v j . (26.20)The set T ij are called the components of the the outer product of u and v. Underrotations the components T ij becomeT ij ′ = u ′ iv j ′ = L ik u k L jl v l = L ik L jl u k v l = L ik L jl T kl , (26.21)which shows that they do transform as the components of a second-order tensor.Use has been made in (26.21) of the fact that u i and v i are the components offirst-order tensors.The outer product of two vectors is often denoted, without reference to anycoordinate system, asT = u ⊗ v. (26.22)(This is not to be confused with the vector product of two vectors, which is itselfa vector and is discussed in chapter 7.) The expression (26.22) gives the basis towhich the components T ij of the second-order tensor refer: since u = u i e i andv = v i e i , we may write the tensor T asT = u i e i ⊗ v j e j = u i v j e i ⊗ e j = T ij e i ⊗ e j . (26.23)Moreover, as for the case of first-order tensors (see equation (26.10)) we notethat the quantities T ij ′ are the components of the same tensor T, but referred toa different coordinate system, i.e.T = T ij e i ⊗ e j = T ije ′ ′ i ⊗ e ′ j.These concepts can be extended to higher-order tensors.936