Biomedical Engineering Reference
In-Depth Information
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
@
@
x 1 e 1 þ @
x 2 e 2 þ @
x 3 e 3 :
(A.171)
@
@
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