Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSA useful application of (26.30) is in obtaining alternative expressions for vectorquantities that arise from the vector product of a vector product.◮Obtain an alternative expression for ∇×(∇×v).As shown in the previous example, ∇×(∇×v) can be expressed in tensor form as∂ 2 v m[∇×(∇×v)] i = ɛ ijk ɛ klm∂x j ∂x l=(δ il δ jm − δ im δ jl ) ∂2 v m∂x j ∂x l= ∂ ( ) ∂vj−∂2 v i∂x i ∂x j ∂x j ∂x j=[∇(∇ · v)] i −∇ 2 v i ,where in the second line we have used the identity (26.30). This result has alreadybeen mentioned in chapter 10 and the reader is referred there for a discussion of itsapplicability. ◭By examining the various possibilities, it is straightforward to verify that, moregenerally,∣ δ ip δ iq δ ir ∣∣∣∣∣ɛ ijk ɛ pqr =δ jp δ jq δ jr(26.34)∣ δ kp δ kq δ krand it is easily seen that (26.30) is a special case of this result. From (26.34) wecan derive alternative forms of (26.30), for example,ɛ ijk ɛ ilm = δ jl δ km − δ jm δ kl . (26.35)The pattern of subscripts in these identities is most easily remembered by notingthat the subscripts on the first δ on the RHS are those that immediately follow(cyclically, if necessary) the common subscript, here i, ineachɛ-term on the LHS;the remaining combinations of j,k,l,m as subscripts in the other δ-terms on theRHS can then be filled in automatically.Contracting (26.35) by setting j = l (say) we obtain, since δ kk = 3 when usingthe summation convention,ɛ ijk ɛ ijm =3δ km − δ km =2δ km ,and by contracting once more, setting k = m, we further find thatɛ ijk ɛ ijk =6. (26.36)26.9 Isotropic tensorsIt will have been noticed that, unlike most of the tensors discussed (except forscalars), δ ij and ɛ ijk have the property that all their components have valuesthat are the same whatever rotation of axes is made, i.e. the component values944