Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 362

permutations of 123. The only 1 values of e ijk are e 132 , e 321 , and e 213 . It is easy to

verify that 132, 321, and 213 are odd permutations of 123. The other 21 components
of e ijk are all zero because they are neither even nor odd permutations of 123 due to
the fact that one number (either 1, 2, or 3) occurs more than once in the indices (e.g.,
e 122 ¼

0 since 122 is not a permutation of 123). One mnemonic device for the even
permutations of 123 is to write 123123, then read the first set of three digits 123, the
second set 231, and the third set 312. The odd permutations may be read off 123123
also by reading from right to left rather than from left to right; reading from the right
(but recording them then from the left, as usual) the first set of three digits 321, the
second set 213, and the third set 132.
The alternator may now be employed to shorten the formula (A.105) for
calculating the determinant;

e mnp Det A ¼ e ijk A im A jn A kp ¼ e ijk A mi A nj A pk : (A.107)

This result may be verified by selecting the values of mnp to be 123, 231, 312,
132, 321, or 213, then performing the summation over the three indices i , j , and k
over 1, 2, and 3 as indicated on the right-hand side of (A.107). In each case the
result is the right-hand side of (A.105). It should be noted that (A.107) may be used
to show Det A ¼ Det A T .
The alternator may be used to express the fact that interchanging two rows or
two columns of a determinant changes the sign of the determinant,

A 1m A 1n A 1p
A 2m A 2n A 2p
A 3m A 3n A 3p

A m1 A m2 A m3
A n1 e mnp Det A ¼ ¼ A n2 A n3 : (A.108)

A p1 A p2 A p3

Using the alternator again may combine these two representations:

A im A in A ip
A jm A jn A jp
A km A kn A kp

e ijk e mnp Det A ¼ : (A.109)

In the special case when A ¼ 1 ( A ij ¼ d ij ), an important identity relating the
alternator to the Kronecker delta is obtained:

d im d in d ip

e ijk e mnp ¼ d jm d jn d jp : (A.110)

d km d kn d kp

The following special cases of (A.110) provide three more very useful relations
between the alternator and the Kronecker delta:

e mnk e ijk ¼ d im d jn d jm d in ; e mjk e ijk ¼ 2 d im ; e ijk e ijk ¼ 6 : (A.111)