Biomedical Engineering Reference
In-Depth Information
Solution: The products
A
B
and
B
A
are given by
2
3
2
3
84
90
96
138
171
204
4
5
;
4
5
:
A
B
¼
201
216
231
B
A
¼
174
216
258
318
342
366
210
261
312
A
.
The
double dot
notation between the two second order tensors is an extension of
the single dot notation between the matrices,
A
Observe that
A
B
6¼
B
B
, which indicates that one index
from
A
and one index from
B
are to be summed over; the double dot notation
between the matrices,
A
B
, indicates that both indices of
A
are to be summed with
different indices from
B
, thus
:
A
:
B
¼
A
ik
B
ik
:
This colon notation stands for the same operation as the trace of the product,
A
:
B ¼
tr(
AB
). Although tr(
AB
) and
A
:
B
mean the same thing,
A
:
B
involves
fewer characters and it will be the notation of choice. Note that
A
:
B
A
T
:
B
T
and
¼
A
T
:
B
A
T
:
B
in general.
In the considerations of mechanics, matrices are often functions of coordinate
positions
x
1
,
x
2
,
x
3
, and time
t
. In this case the matrix is written
A
(
x
1
,
x
2
,
x
3
,
t
) which
means that each element of
A
is a function of
x
1
,
x
2
,
x
3
, and
t
,
A
:
B
T
but that
A
:
B
¼
6¼
2
4
3
5
:
A
11
ð
x
1
;
x
2
;
x
3
;
t
Þ
A
12
ð
x
1
;
x
2
;
x
3
;
t
Þ
...
A
1n
ð
x
1
;
x
2
;
x
3
;
t
Þ
A
21
ð
x
1
;
x
2
;
x
3
;
t
Þ
A
22
ð
x
1
;
x
2
;
x
3
;
t
Þ
...
A
2n
ð
x
1
;
x
2
;
x
3
;
t
Þ
A
ð
x
1
;
x
2
;
x
3
;
t
Þ¼
:
:
...
:
A
n1
ð
x
1
;
x
2
;
x
3
;
t
Þ
:
...
A
nn
ð
x
1
;
x
2
;
x
3
;
t
Þ
(A.27)
stand for a total derivative, or a partial derivative with respect
to
x
1
,
x
2
,
x
3
,or
t
, or a definite or indefinite (single or multiple) integral; then the
operation of the operator on the matrix follows the same rule as the multiplication
of a matrix by a scalar (A.9), thus
Let the operator
}
2
4
3
5
:
}
A
11
ð
x
1
;
x
2
;
x
3
;
t
Þ}
A
12
ð
x
1
;
x
2
;
x
3
;
t
Þ ::: }
A
1n
ð
x
1
;
x
2
;
x
3
;
t
Þ
}
A
21
ð
x
1
;
x
2
;
x
3
;
t
Þ}
A
22
ð
x
1
;
x
2
;
x
3
;
t
Þ ::: }
A
2n
ð
x
1
;
x
2
;
x
3
;
t
Þ
}
A
ð
x
1
;
x
2
;
x
3
;
t
Þ¼
:
:
:::
:
}
A
n1
ð
x
1
;
x
2
;
x
3
;
t
Þ
:
: : }
A
nn
ð
x
1
;
x
2
;
x
3
;
t
Þ
(A.28)
}ð
A
þ
B
Þ¼}
B
þ}
A
and
ð}
1
þ}
2
Þ
A
¼}
1
A
þ}
2
A
;
(A.29)
where
}
1
and
}
2
are two different operators.
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