Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORS◮Using (26.40), show that A ij = ɛ ijk p k .By contracting both sides of (26.40) with ɛ ijk , we findɛ ijk p k = 1 ɛ 2 ijkɛ klm A lm .Using the identity (26.30) then givesɛ ijk p k = 1 (δ 2 ilδ jm − δ im δ jl )A lm= 1 (A 2 ij − A ji )= 1 (A 2 ij + A ij )=A ij ,whereinthelastlineweusethefactthatA ij = −A ji . ◭By a simple extension, we may associate a dual pseudoscalar s with everytotally antisymmetric third-rank tensor A ijk , i.e. one that is antisymmetric withrespect to the interchange of every possible pair of subscripts; s is given bys = 1 3! ɛ ijkA ijk . (26.41)Since A ijk is a totally antisymmetric three-subscript quantity, we expect it toequal some multiple of ɛ ijk (since this is the only such quantity). In fact A ijk = sɛ ijk ,as can be proved by substituting this expression into (26.41) and using (26.36).26.12 Physical applications of tensorsIn this section some physical applications of tensors will be given. First-ordertensors are familiar as vectors and so we will concentrate on second-order tensors,starting with an example taken from mechanics.Consider a collection of rigidly connected point particles of which the αth,which has mass m (α) and is positioned at r (α) with respect to an origin O, istypical. Suppose that the rigid assembly is rotating about an axis through O withangular velocity ω.The angular momentum J about O of the assembly is given byJ = ∑ (r (α) × p (α)) .αBut p (α) = m (α) ṙ (α) and ṙ (α) = ω × r (α) , for any α, and so in subscript form thecomponents of J are given byJ i = ∑ αm (α) ɛ ijk x (α)jẋ (α)k= ∑ α= ∑ α= ∑ αm (α) ɛ ijk x (α)jɛ klm ω l x m(α)m (α) (δ il δ jm − δ im δ jl )x (α)j x m (α) ω lm (α) [ (r(α) ) 2δil − x (α)ix (α)l]ω l ≡ I il ω l , (26.42)where I il is a symmetric second-order Cartesian tensor (by the quotient rule, see950