Biomedical Engineering Reference
In-Depth Information
coordinates called physical components of the tensors may be constructed if the
curvilinear coordinate system is orthogonal. Below, the standard formulas used in
this text for cylindrical coordinates are recorded. In the case when cylindrical
coordinates are employed, the vectors and tensors introduced are all considered as
functions of the coordinate positions
r
,
,
and z
in place of the Cartesian
coordinates
x
1
,
x
2
,and
x
3
. In the case of a vector or tensor this dependence is
written
v
(
r
,
y
,
z
,
t
), which means that each element of the vector
v
or
the tensor
T
is a function of
r
,
y
,
z
,
t
)or
T
(
r
,
y
y
,
z
,and
t
. The gradient operator is denoted by
D
and defined, in three dimensions, by
r¼
@
@
1
r
@
@y
e
y
þ
@
@
r
e
r
þ
z
e
z
;
(A.196)
where
e
r
,
e
y
, and
e
z
are the unit base vectors in the cylindrical coordinate system.
The gradient of a scalar function
f
(
x
1
,
x
2
,
x
3
,
t
) is a vector given by
¼
@
f
r
@
f
@y
e
y
þ
@
f
1
r
f
r
e
r
þ
z
e
z
:
(A.197)
@
@
The gradient of a vector function
v
(
r
,
y
,
z
,
t
) is given by
2
3
@
v
r
@
1
r
@
v
r
@y
@
v
r
@
4
5
r
z
@
v
y
@
1
r
@
v
y
@y
@
v
y
@
T
½
r
v
¼
:
(A.198)
r
z
@
v
z
@
1
r
@
v
z
@y
@
v
z
@
r
z
The formula for the divergence of the vector
v
in cylindrical coordinates is
obtained by taking the trace of (A.198), tr[
D
v
]
¼ D
v
¼
div
v
. The curl is the
gradient operator cross product with a vector function
v
(
r
,
y
,
z
,
t
), thus
2
3
e
r
e
y
e
z
4
5
:
@
@
@
@y
@
@
r
v
¼
curl
v
¼
(A.199)
r
z
v
r
rv
y
v
z
The form of the double gradient three-dimensional second order tensor defined
by
O
O
ð O
U
2
) and its six-dimensional vector counterpart
¼rr
(tr
O
¼r
2
¼
tr
O
¼r
Þ
have the following cylindrical coordinate representations:
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