Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 401

is the tensorial character of a symbol with three subscripts; is it a third order tensor?
Almost. Tensors are identified on the basis of their tensor transformation law.
Recall the tensor transformations laws (A.75) and (A.76) for a vector, (A.86) for
a second order tensor and (A.87) for a tensor of order n . An equation that contains a
transformation law for the alternator is obtained from (A.107) by replacing A by the
orthogonal transformation Q given by (A.64) and changing the indices as follows:
m

! a

, n

! b

, p

! g

, thus

e abg Det Q ¼ e ijk Q i a Q j a Q k g : (A.112)

This is an unusual transformation law because the determinant of an orthogonal
transformation Q is either +1 or 1. The expected transformation law, on the basis

of the tensor transformation laws (A.75) and (A.76) for a vector, (A.86) for a second
order tensor and (A.87) for a tensor of order n , is that Det Q

+1
occurs when the transformation is between coordinate systems of the same hand-
edness (right handed to right handed or left handed to left handed). Recall that a
right (left) hand coordinate system or orthonormal basis is one that obeys the right
(left) hand rule, that is to say if the curl of your fingers in your right (left) hand fist is
in the direction of rotation from the first ordered positive base vector into the second
ordered positive base vector, your extended thumb will point in the third ordered
positive base vector direction. Det Q

¼ +1. Det Q ¼

1 occurs when the transformation is
between coordinate systems of the opposite handedness (left to right or right to
left). Since handedness does not play a role in the transformation law for even order
tensors, this dependence on the sign of Det Q and therefore the relative handedness
of the coordinate systems for the alternator transformation law, is unexpected.

¼

The Cross Product of Vectors

In the indicial notation the vector cross product a b is written in terms of an

alternator as

a b ¼ e ijk a i b j e k ; (A.114)

a result that may be verified by expanding it to show that it coincides with (A.113).
If c

¼ a b denotes the result of the vector cross product, then from (A.114),

c ¼ e ijk a i b j e k ; ð c k ¼ e ijk a i b j Þ: (A.115)

Example A.8.3

Prove that a b ¼ b a .

Solution: In the formula (A.114) let i ! j and j ! i , thus