Biomedical Engineering Reference
In-Depth Information
Solution:
2
3
2
3
ð
0
456
8 0 2
12
1
2
1
2
3
2
4
5
;
T
4
5
:
a
b
¼
a
b
ð
a
b
Þ
Þ¼
10
1
210
15
18
and tr{
a
32.
Frequently n-tuples are considered as functions of coordinate positions
x
1
,
x
2
,
x
3
,
and time
t
. In this case the n-tuple is written
r
(
x
1
,
x
2
,
x
3
,
t
) that means that each
element of
r
is a function of
x
1
,
x
2
,
x
3
, and
t
,
b
}
¼
a
b
¼
r
ð
x
1
;
x
2
;
x
3
;
t
Þ¼½
r
1
ð
x
1
;
x
2
;
x
3
;
t
Þ;
r
2
ð
x
1
;
x
2
;
x
3
;
t
Þ; ...;
r
n
ð
x
1
;
x
2
;
x
3
;
t
Þ:
(A.33)
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 n-tuple follows the same rule as the
multiplication of an n-tuple by a scalar (A.9), thus
Again letting the operator
}
}
r
ð
x
1
;
x
2
;
x
3
;
t
Þ¼½}
r
1
ð
x
1
;
x
2
;
x
3
;
t
Þ; }
r
2
ð
x
1
;
x
2
;
x
3
;
t
Þ; ...; }
r
n
ð
x
1
;
x
2
;
x
3
;
t
Þ:
(A.34)
The following distributive laws connect matrix addition and operator operations:
}ðr þ tÞ¼}r þ}t
and
ð}
1
þ}
2
Þ
r
¼}
1
r
þ}
2
r
;
(A.35)
where
}
1
and
}
2
are two different operators.
Problems
A.4.1.
Find the derivative of the n-tuple
r
(
x
1
,
x
2
,
x
3
,
t
)
¼
[
x
1
x
2
x
3
,10
x
1
x
2
,
x
3
]
T
with respect to
x
3
.
cosh
a
A.4.2.
Find the symmetric and skew-symmetric parts of the matrix
r
s
where
r
¼
[1, 2, 3, 4] and
s
¼
[5, 6, 7, 8].
A.5 The Types andLinear Transformations
A system of linear equations
r
1
¼
A
11
t
1
þ
A
12
t
2
þþ
A
1n
t
n
;
r
2
¼
A
21
t
1
þ
A
22
t
2
þþ
A
2n
t
n
;
...
r
n
¼
A
n1
t
1
þ
A
n2
t
2
þ ...þ
A
nn
t
n
;
(A.36)
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