Mathematical Methods for Physics and Engineering - Matematica.NET

26.5 SECOND- AND HIGHER-ORDER CARTESIAN TENSORS(ii) The gradient of a vector. Suppose v i represents the components of a vector;let us consider the quantities generated by forming the derivatives of each v i ,i =1, 2, 3, with respect to each x j , j =1, 2, 3, i.e.T ij = ∂v i.∂x jThese nine quantities form the components of a second-order tensor, as can beseen from the fact thatT ′ij = ∂v′ i∂x ′ j= ∂(L ikv k ) ∂x l ∂v k∂x l ∂x ′ = L ik L jl = L ik L jl T kl .j∂x lIn coordinate-free language the tensor T may be written as T = ∇v and hencegives meaning to the concept of the gradient of a vector, a quantity that was notdiscussed in the chapter on vector calculus (chapter 10).A test of whether any given set of quantities forms the components of a secondordertensor can always be made by direct substitution of the x ′ i in terms of thex i , followed by comparison with the right-hand side of (26.16). This procedure isextremely laborious, however, and it is almost always better to try to recognisethe set as being expressible in one of the forms just considered, or to makealternative tests based on the quotient law of section 26.7 below.◮Show that the T ij given by(x2T =[T ij ]= 2−x 1 x 2−x 1 x 2 x 2 1are the components of a second-order tensor.)(26.24)Again we consider a rotation θ about the e 3 -axis. Carrying out the direct evaluation firstwe obtain, using (26.7),T 11 ′ = x ′ 2 2 = s 2 x 2 1 − 2scx 1 x 2 + c 2 x 2 2,T 12 ′ = −x ′ 1x ′ 2 = scx 2 1 +(s 2 − c 2 )x 1 x 2 − scx 2 2,T 21 ′ = −x ′ 1x ′ 2 = scx 2 1 +(s 2 − c 2 )x 1 x 2 − scx 2 2,T 22 ′ = x ′ 1 2 = c 2 x 2 1 +2scx 1 x 2 + s 2 x 2 2.Now, evaluating the right-hand side of (26.16),T 11 ′ = ccx 2 2 + cs(−x 1 x 2 )+sc(−x 1 x 2 )+ssx 2 1,T 12 ′ = c(−s)x 2 2 + cc(−x 1 x 2 )+s(−s)(−x 1 x 2 )+scx 2 1,T 21 ′ =(−s)cx 2 2 +(−s)s(−x 1 x 2 )+cc(−x 1 x 2 )+csx 2 1,T 22 ′ =(−s)(−s)x 2 2 +(−s)c(−x 1 x 2 )+c(−s)(−x 1 x 2 )+ccx 2 1.After reorganisation, the corresponding expressions are seen to be the same, showing, asrequired, that the T ij are the components of a second-order tensor.The same result could be inferred much more easily, however, by noting that the T ijare in fact the components of the outer product of the vector (x 2 , −x 1 ) with itself. That(x 2 , −x 1 ) is indeed a vector was established by (26.12) and (26.13). ◭937