Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 400

d im d in d ik

e ijk e mnk ¼ d jm d jn d jk ;

d km d kn 3

one finds that

e ijk e mnk ¼ 3 d im d jn d im d jk d kn 3 d in d jm þ d in d km d jk þ d ik d jm d kn d ik d km d jn :

Carrying out the indicated summation over the index k in the expression above,

e ijk e mnk ¼ 3 d im d jn d im d jn 3 d in d jm þ d in d jm þ d in d jm d im d jn :

This is the desired result, the first of (A.111).

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Example A.8.2

Prove that Det( A B ) ¼ Det A Det B .

Solution: Replacing A in (A.107) by C and selecting the values of mnp to be 123,
then (A.107) becomes

Det C ¼ e ijk C i1 C j2 C k3 ¼ e ijk C 1i C 2j C 3k :

Now C is replaced by the product A B using

C i1 ¼ A im B m1 ; C j2 ¼ A jn B n2 ; C k3 ¼ A kp B p3 ;

thus

Det A B ¼ e ijk A im B m1 A jn B n2 A kp B p3 ; or Det A B ¼ð e ijk A im A jn A kp Þ B m1 B n2 B p3 ;

where the order of the terms in the second sum has been rearranged from the first.
Comparison of the first four rearranged terms from the second sum with the right-
hand side of (A.107) shows that the first four terms in the sum on the right may be
replaced by e mnp Det A ; thus applying the first equation of this solution again with C
replaced by B , the desired result is obtained:

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

Æ

The Tensorial Character of the Alternator

Vectors were shown to be characterized by symbols with one subscript, (second
rank) tensors were shown to be characterized by symbols with two subscripts; what