Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSwhere the elements L ij are given by⎛L = ⎝cos θ sin θ⎞0− sin θ cos θ 0 ⎠ .0 0 1Thus (26.68) and (26.70) agree with our earlier definition in the special case of arigid rotation of Cartesian axes.Following on from (26.68) and (26.70), we proceed in a similar way to definegeneral tensors of higher rank. For example, the contravariant, mixed andcovariant components, respectively, of a second-order tensor must transform asfollows:contravariant components,mixed components,covariant components,T ′ ij = ∂u′ i∂u k ∂u ′ j∂u l T kl ;T ′ ij = ∂u′ i∂u k ∂u l∂u ′ j T k l ;T ′ ij = ∂uk∂u ′ i∂u l∂u ′ j T kl.It is important to remember that these quantities form the components of thesame tensor T but refer to different tensor bases made up from the basis vectorsof the different coordinate systems. For example, in terms of the contravariantcomponents we may writeT = T ij e i ⊗ e j = T ′ ij e ′ i ⊗ e ′ j.We can clearly go on to define tensors of higher order, with arbitrary numbersof covariant (subscript) and contravariant (superscript) indices, by demandingthat their components transform as follows:T ′ ij···klm···n = ∂u′ i∂u ′ j···∂u′ k∂u d ∂u e ∂uf∂u a ∂u b ∂u c ···∂u ′ l∂u′m∂u ′n T ab···c de···f. (26.71)Using the revised summation convention described in section 26.14, the algebraof general tensors is completely analogous to that of the Cartesian tensorsdiscussed earlier. For example, as with Cartesian coordinates, the Kroneckerdelta is a tensor provided it is written as the mixed tensor δj i sinceδ ′i j = ∂u′i ∂u l∂u k ∂u ′j δk l = ∂u′i ∂u k ∂u′i∂u k =∂u′j∂u = ′j δi j,where we have used the chain rule to justify the third equality. This also showsthat δj i is isotropic. As discussed at the end of section 26.15, the δj i can beconsidered as the mixed components of the metric tensor g.962