Biomedical Engineering Reference
In-Depth Information
On the right-hand side of this expression for
r
ð
p
q
Þ
,
e
ijk
is now changed to
e
ikj
and the first of (A.111) is then employed,
r
ð
p
q
Þ¼ðd
im
d
kn
d
in
d
km
Þ
r
i
p
m
q
n
e
k
;
then summing over
k
and
I
,
r
ð
p
q
Þ¼
r
i
p
k
q
i
e
k
r
i
p
i
q
k
e
k
¼ð
r
q
Þ
p
ð
r
p
Þ
q
:
Æ
Problems
A.3.1 Simplify the following expression by using the Einstein summation index
convention for a range of three:
0
¼
r
1
w
1
þ
r
2
w
2
þ
r
3
w
3
;
c ¼ð
u
1
v
1
þ
u
2
v
2
þ
u
3
v
3
Þð
u
1
v
1
þ
u
2
v
2
þ
u
3
v
3
Þ;
A
11
x
1
þ
A
22
x
2
þ
A
33
x
3
þ
f ¼
A
12
x
1
x
2
þ
A
21
x
1
x
2
þ
A
13
x
1
x
3
þ
A
31
x
1
x
3
þ
A
23
x
3
x
2
þ
A
32
x
3
x
2
:
Prove that
@x
i
A.3.3
@x
j
¼ d
ij
.
B
)
T
B
T
A
T
.
A.3.9
Show that (
A
¼
If
F
is a square matrix and
a
is an n-tuple, show that
a
T
F
T
A.5.8
¼
F
a
.
A
T
, then
A.8.2
Show that if
A
is a skew-symmetric 3 by 3 matrix,
A
¼
Det
A
¼
0.
A.8.3 Evaluate Det(
a
b
).
Det
A
T
.
Show Det
A
¼
A.8.4
A.12.9 If
v
x
and
a
is a constant vector, using the indicial notation,
evaluate the div
v
and the curl
v
.
¼
a
A.14 Exact Differentials
In one dimension a differential d
q
¼
f
(
x
)d
x
is always exact and, in two dimensions,
in order that a differential d
q
be an exact differential in a simply-connected
2D region
R
of the
x
1
,
x
2
plane, it is necessary and sufficient that between
a
1
and
a
2
there exists the relation
¼ a
d
x
@
a
1
x
2
¼
@
a
2
x
1
:
(A.192)
@
@
(Note that the notation often used for this result is d
q
¼
M
(
x
,
y
)d
x
þ
N
(
x
,
y
)d
y
leading to the condition
@
M
@y
¼
@
N
). Continuing now with the considerations
@x
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