Biomedical Engineering Reference
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T
¼
T
ij
e
i
e
j
¼
T
ab
e
a
e
b
:
(A.80)
The formulas (A.78) and (A.80) have similar content in that the vectors
r
and
t
and the tensor
T
are quantities independent of base or coordinate system while the
components of
r
,
t
, and
T
may be expressed relative to any basis. In the formulas
(A.78) and (A.80),
r
,
t
, and
T
are expressed as components relative to two different
bases. The vectors are expressed as components relative to the bases
e
i
and
e
a
, while
the tensor
T
is expressed relative to the bases
e
i
e
j
and
e
a
e
b
. The tensor bases
e
i
e
j
and
e
a
e
b
are constructed from the vector bases
e
i
and
e
a
.
Example A.7.1
[1,0,0]
T
,
e
2
¼
[0, 1, 0]
T
,and
If the base vectors
e
1
,
e
2
,and
e
3
are expressed as
e
1
¼
[0,0,1]
T
, then it follows from (A.77) that
v
e
3
¼
¼
v
i
e
i
andwe can express
v
in the form
2
3
2
3
2
3
1
0
0
0
1
0
0
0
1
4
5
þ
4
5
þ
4
5
:
v
¼
v
1
v
2
v
3
(A.81)
Create a similar representation for
T
given by (A.80) for
n
¼
3.
Solution: The representation for
T
given by (A.80),
T
¼
T
ij
e
i
e
j
, involves the
base vectors
e
1
e
1
,
e
1
e
2
etc. These “base vectors” are expressed as matrices of
tensor components by
2
3
2
3
100
000
000
010
000
000
4
5;
4
5;
e
1
e
1
¼
e
1
e
2
¼
2
3
000
100
000
4
5;
e
2
e
1
¼
etc
:
(A.82)
The representation for
T
,
T
¼
T
ij
e
i
e
j
, then can bewritten in analogy to (A.81) as
2
3
2
3
2
3
2
3
100
000
000
000
100
000
010
000
000
001
000
000
4
5
þ
4
5
þ
4
5
þ
4
5
T
¼
T
11
T
21
T
12
T
13
2
3
2
3
2
3
000
000
100
000
010
000
000
001
000
4
5
þ
4
5
þ
4
5
þ
T
31
T
22
T
23
2
3
2
3
000
000
010
000
000
001
4
5
þ
4
5
:
þ
T
32
T
33
Æ
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