Biomedical Engineering Reference
In-Depth Information
This result must hold for all constant vectors
c
, and the divergence theorem for
the second order tensor, (A.184), follows.
Stokes theorem relates line integrals to surface integrals,
ð
A
r
þ
v
d
A
¼
v
d
x
;
(A.188)
@A
specifically, it relates the surface integral of the curl of a vector field
v
over a surface
A
in Euclidean three-space to the line integral of the vector field over its boundary,
@
A
must have positive orientation, such
that d
x
points counterclockwise when the surface normal to
A
. The closed curve of the line integral on
@
A
points toward the
viewer, following the right-hand rule. Note that if
v
is the gradient of a scalar
function
@
f
,
v
¼rf
, then (A.188) reduces to
þ
0
¼
v
d
x
;
(A.189)
@A
for all closed paths since
0, that is to say that the curl of the gradient is
zero. It follows that when the vector field
v
satisfies the condition all closed paths, it
may be represented as the gradient of a potential,
v
rrf ¼
¼rf
. From (A.179) one can
see that
r
v
¼
0 implies the three conditions
@
v
1
x
2
¼
@
v
2
x
1
;
@
v
1
x
3
¼
@
v
3
x
1
;
@
v
3
x
2
¼
@
v
2
or
@
v
i
x
j
¼
@
v
j
x
i
;
(A.190)
@
@
@
@
@
@
x
3
@
@
which are the conditions that insure that
v
d
x
be an exact differential. Note that
these there conditions are equivalent to the requirement that the tensor
r
v
be
symmetric,
T
r
v
¼ðr
v
Þ
:
(A.191)
We return to this topic in the next section where exact differentials are
considered.
Problems
(
x
1
)
2
(
x
2
)
2
(
x
3
)
3
and evaluate the
A.12.1 Calculate the gradient of the function
f
¼
gradient at the point (1, 2, 3).
A.12.2 Calculate the gradient of a vector function
r
(
x
1
,
x
2
,
x
3
)
[(
x
1
)
2
(
x
2
)
2
,
¼
x
1
x
2
,(
x
3
)
2
].
A.12.3 Calculate the divergence of a vector function
r
(
x
1
,
x
2
,
x
3
)
[(
x
1
)
2
(
x
2
)
2
,
¼
x
1
x
2
,(
x
3
)
2
].
A.12.4 Calculate the curl of a vector function
r
(
x
1
,
x
2
,
x
3
)
¼
[(
x
1
)
2
(
x
2
)
2
,
x
1
x
2
,
(
x
3
)
2
].
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