Biomedical Engineering Reference
In-Depth Information
was before the operation. For example the trace of the gradient of a vector function is
a scalar called the divergence of the vector, tr[
D
r
]
¼ D
r
¼
div
r
,
¼
@
r
1
x
1
þ
@
x
2
þ
@
r
2
r
3
r
r
¼
div
r
x
3
:
(A.178)
@
@
@
The divergence operation is similar to the scalar product of two vectors in that
the effect of the operation is to reduce the order of the quantity by two from the sum
of the ranks of the combined quantities before the operation. The curl operation is
the gradient operator cross product with a vector function
r
(
x
1
,
x
2
,
x
3
,
t
), thus
e
1
e
2
e
3
r
r
¼
curl
r
¼
@
@x
1
@
@x
2
@
@x
3
:
(A.179)
r
1
r
2
r
3
2
)
A three-dimensional double gradient tensor defined by
O
¼rr
(tr
O
¼r
and its six-dimensional vector counterpart
O
(
O
U
2
) are often conve-
¼
tr
O
¼r
nient notations to employ. The components of
O
are
T
p
p
p
2
2
2
2
2
2
2
2
@
x
1
;
@
x
2
;
@
@
@
@
O ¼
x
3
;
x
3
;
x
3
;
;
(A.180)
@
@
@
@
x
2
@
@
x
1
@
@
x
1
@
x
2
and the operation of
O
on a six-dimensional vector representation of a second order
tensor in three dimensions,
O
T
¼
tr
O
T
, is given by
2
T
11
@
2
T
22
@
2
T
33
@
2
T
23
2
T
13
2
T
12
¼
@
x
1
þ
@
x
2
þ
@
2
@
2
@
2
@
O
T
x
3
þ
x
3
þ
x
3
þ
x
2
:
(A.181)
@
x
2
@
@
x
1
@
@
x
1
@
The divergence of a second order tensor
T
is defined in a similar fashion to the
divergence of a vector; it is a vector given by
e
1
þ
e
2
Þ¼
@
T
11
@
x
1
þ
@
T
12
@
x
2
þ
@
T
13
@
@
T
21
@
x
1
þ
@
T
22
@
x
2
þ
@
T
23
@
r
T
ð
x
1
;
x
2
;
x
3
;
t
x
3
x
3
e
3
:
@
T
31
@
x
1
þ
@
T
32
@
x
2
þ
@
T
33
@
þ
x
3
(A.182)
The divergence theorem (also called Gauss' theorem, Green's theorem or
Ostrogradsky's theorem, depending on the nationality) relates a volume integral
to a surface integral over the volume. The divergence of a vector field
r
(
x
1
,
x
2
,
x
3
,
t
)
integrated over a volume of space is equal to the integral of the projection of the
field
r
(
x
1
,
x
2
,
x
3
,
t
) on the normal to the boundary of the region, evaluated on the
boundary of the region, and integrated over the entire boundary
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