By L. G. Jaeger and B. G. Neal (Auth.)
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Extra resources for Cartesian Tensors in Engineering Science
Ij,m,n survive the summation process so this is Q'^ the bracketed components being respectively S' and T' . e. 2) It is seen that the 81 componented quantity β is a fourth order tensor. The extension of the above argument to tensors of order other than two is obvious. Multiplication of anrathorder tensor's components by an nih. order tensor's components gives the components of an (m + /i)th order tensor. 5(a). g. S T Rij = QpqvwRij — L ij, where L is a sixth order tensor. 3 Products Involving the Kronecker Delta.
Orders. In particular it is possible, by contracting often enough to arrive at either a scalar or a vector quantity, according as (m + n) is even or odd. 4 Examples of Contraction (a) If A and Β are vectors, then the nine quantities A B are the components of a second order tensor. Consider however the expression A B ô . With summation over ρ and q, this means the following: v p q q pq (i) Take in turn each element of A B P ΑιΒι A2B1 A3B1 viz. each element of A1B2 A±Bs A2B2 A2B2 A3B2 A3Bz (ii) Multiply each of the A B p Q9 q by its associated element of 3 PQi 0 1 0 viz.
It will be noted that the order in which A and Β are written down is important because if, in eqn. 11) the roles of A and Β are reversed, the associated e changes sign owing to the fact that the subscripts ρ and q change places. r r pqr Hence Α χ Β = - Β χ Α. 12) Important particular examples of eqn. 11) are the cross products of unit vectors ii,i2 and 13. Using eqn. 11), letting A = ii so that A \ = 1, A2 = 0, Az = 0, and Β = i so that Bi = 0, B2 = 1, Bz = 0 one finds ii X 12 = 13. There are five more such equations.
Cartesian Tensors in Engineering Science by L. G. Jaeger and B. G. Neal (Auth.)