Biomedical Engineering Reference
In-Depth Information
1.294.
A.7.10 Consider the components of the tensor
T
given in problem A.7.2 to be
relative to a (Latin) coordinate system and denote them by
T
(L)
. Trans-
form these components to a new coordinate system (the Greek) using the
transformation
are 73.227, 2.018. 1.284, 0.602, 0.162, and
2
3
2
q
q
1
4
5
:
1
2
2
p
3
Q
¼
2
2
1
1
p
p
p
2
p
2
0
A.7.11 Show that if a tensor is symmetric (skew-symmetric) in one coordinate
system, then it is symmetric (skew-symmetric) in all coordinate systems.
Specifically show that if
A
(L)
(
A
(L)
)
T
, then
A
(G)
(
A
(G)
)
T
.
¼
¼
A.8 The Alternator and Vector Cross Products
There is a strong emphasis on the indicial notation in this section. It is advised that
the definitions (in Sect. A.3) of free indices and summation indices be reviewed
carefully if one is not altogether comfortable with the indicial notation. It would
also be beneficial to redo some indicial notation problems.
The alternator in three dimensions is a three index numerical symbol that
encodes the permutations that one is taught to use expanding a determinant. Recall
the process of evaluating the determinant of the 3 by 3 matrix
A
,
¼ A
11
A
22
A
33
A
11
A
32
A
23
A
12
A
21
A
33
þ A
12
A
31
A
23
þ A
13
A
21
A
32
A
13
A
31
A
22
:
(A.105)
2
4
3
5
¼
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
Det
A
¼
Det
The permutations that one is taught to use expanding a determinant are
permutations of a set of three objects. The alternator is denoted by
e
ijk
and defined
so that it takes on values +1, 0, or
1 according to the rule:
8
<
:
9
=
;
;
þ
1f
P
is an even permuation
0
123
i
e
ijk
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
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