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Tpqr. . = XSpXtqXUr> · -Tstu. . 7. A scalar is a zeroth order tensor and a vector is afirstorder tensor. 8. In η dimensions, a Mi order tensor has n components. In η dimensions, a scalar, a vector and a second order tensor may be represented respectively as a 1 χ 1 matrix, an « χ 1 matrix, and a n « x « matrix. The transformation laws of vectors and second order tensors may be written in the following forms k Vector Suffix notation Matrix notation A\ = X^Aj and A = X^Aj [X] {A } and {Α} = [λ] {Α'} % {A'}= τ Second order tensor Suffix notation T' = Matrix notation [Γ] st XSpXtqTPq = [A] [T] and T = [X] and [Τ] pq T XSpXtqT'st = Τ [Χ] [Τ'] [A].

By taking a unit force in the x' direction the reader will readily 2 2 2 2 verify that / 2 = I and /21 = 2 2 )~ \ - ^ 2 U )° 2^ j 22 s i n 2 0 - It may be noted in passing that f[ = / 2 1 in accordance with the reciprocal theorem. The steps 1, 2 and 3 given above enable one to give a physical significance to the tensor transformation as follows. (a) Starting with a unit load in the x{ direction, this is repre" 1 sented as the column vector 0 referred to X19X2 axes. e. the components cos0 and sin0 in the x± and x directions are given by cos0 1 Γ cos0 — sin0 " 1 ~ [λ]~ 0 ^ sin0 J ~~ [ ^ ^ 2 s n c o s (c) Step 2 above converts known force components into displacements in the x± and x directions.

A χ A = 0 because this cross product Ε 58 CARTESIAN TENSORS I N E N G I N E E R I N G SCIENCE is e qrApAq\ the terms of which vanish in identical pairs (see a similar example below). In particular P r ii χ h = 0, h X i = 0, i Χ i = 0. e. is perpendicular to the plane defined by A and B. To see that this is so it is only necessary to examine the scalar product of C with A and of C with B. The scalar product C . A is C r A r = ερατΑρΒςΑτ. 15) The right-hand side of eqn. 15) gives zero identically, since the terms vanish in identical pairs.

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