Biomedical Engineering Reference
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components of
Q
. Thus, while (A.167) uniquely determines
Q
given
Q
, the
components of
Q
must be considerably restricted in order to determine
Q
given
Q
, and selections of signs must be made.
Problems
T
that corresponds to the three-
A.11.1 Construct the six-dimensional vector
dimensional tensor
T
given by
2
4
3
3
p
p
3
3
5
:
13
3
3
p
1
2
T
¼
7
1
p
3
1
8
A.11.2 Prove that the relationship
T
J
¼
T
:
J
(A.164) is correct by substitution
of components.
A.11.3 Construct the six-dimensional orthogonal transformation
Q
that
corresponds to the three-dimensional orthogonal transformation
Q
where
2
4
3
5
:
cos
c
010
sin
c
0
sin
Q
¼
c
0
cos
c
A.11.4 Construct the six-dimensional orthogonal transformation
Q
that corresponds
to the three-dimensional orthogonal transformation
Q
where
2
4
2
q
q
3
5
:
1
2
1
2
p
3
Q
¼
1
2
1
2
p
p
p
2
p
2
0
A.12 The Gradient Operator and the Divergence Theorem
The vectors and tensors introduced are all considered as functions of coordinate
positions
x
1
,
x
2
,
x
3
, and time
t
. In the case of a vector or tensor this dependence is
written
r
(
x
1
,
x
2
,
x
3
,
t
)or
T
(
x
1
,
x
2
,
x
3
,
t
), which means that each element of the vector
r
or the tensor
T
is a function of
x
1
,
x
2
,
x
3
, and
t
,. The gradient operator is denoted by
D
and defined, in three dimensions, by
r¼
@
@
x
1
e
1
þ
@
x
2
e
2
þ
@
x
3
e
3
:
(A.171)
@
@
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