Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 364

Det A B ¼ Det Ae mnp B m1 B n2 B p3 ¼ Det A Det B : Æ

Consider now the question of the tensorial character of the alternator. Vectors
were shown to be characterized by symbols with one subscript, (second order)
tensors were shown to characterized by symbols with two subscripts; what 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 b 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 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 , is that Det Q ¼ +1. Det Q

¼ +1 occurs when the transformation is between coordinate systems of the same

handedness (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 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.
The title to this section mentioned both the alternator and the vector cross
product. How are they connected? If you recall the definition of the vector cross
product a b in terms of a determinant, the connection between the two is made,

e 1 e 2 e 3

a b ¼ a 1 a 2 a 3 ¼ð a 2 b 3 b 2 a 3 Þ e 1 þð a 3 b 1 b 3 a 1 Þ e 2 þð a 1 b 2 b 1 a 2 Þ e 3 :

b 1 b 2 b 3

(A.113)

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)