Biomedical Engineering Reference
In-Depth Information
The transition across the second equal sign is a simple rearrangement of terms.
The transition across the second equal sign is based on the condition
Q
k
a
Q
k
b
¼ d
ab
(A.89)
which is an alternate form of (A.68), a form equivalent to
Q
T
1
. The transition
across the fourth equal sign employs the definition of the Kronecker delta and the
summation over
Q
¼
. The result is that the trace of the matrix of second order tensor
components relative to any basis is the same number,
b
A
kk
¼
A
aa
:
(A.90)
The Alternator and the Permutation Symbol
The
alternator
is denoted by
e
ijk
and defined so that it takes on values +1, 0, or
1
according to the rule:
<
=
;
;
þ
1 f
P
is an even permuation
123
i
e
ijk
0
otherwise
P
;
(A.106)
:
j k
1 f
P
is an odd permuation
where
P
is the
permutation symbol
on a set of three objects. The only +1 values of
e
ijk
are
e
123
,
e
231
, and
e
312
. It is easy to verify that 123, 231, and 312 are even
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 and Determinants
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)
Search WWH ::
Custom Search