Biomedical Engineering Reference
In-Depth Information
(
)
(
)
Let us consider a function
F
x
that contains another function
f
x
. One example
2
(
)
=
(
)
+
(
)
(
)
(
)
is
F
x
f
x
2
x
where
F
x
consists of
f
x
and
x
. The integral of
F
x
,
ʲ
I
[
f
(
x
)
]=
F
(
x
)
dx
,
(C.69)
ʱ
is a functional of
f
(
x
)
. Let us define a small change in
f
(
x
)
as
ʴ
f
(
x
)
, and a change
in
I
[
f
(
x
)
]
due to
ʴ
f
(
x
)
as
ʴ
I
[
f
(
x
)
]
. For simplicity,
ʴ
I
[
f
(
x
)
]
is denoted
ʴ
I
in the
following arguments. To derive the relationship between
ʴ
I
and
ʴ
f
(
x
)
, the region
[
ʱ, ʲ
]
is divided into small regions
x
. A contribution from the
j
th small region onto
ʴ
I
,
ʴ
I
j
, is equal to
ʴ
I
j
=
A
j
ʴ
f
(
x
j
)
x
.
(C.70)
Here we assume that
ʴ
I
j
is proportional to
ʴ
f
(
x
j
)
and
A
j
is a proportional constant.
Then,
ʴ
I
is derived as a sum of contributions from all small regions such that
ʴ
I
=
A
j
ʴ
f
(
x
j
)
x
.
(C.71)
j
Accordingly, when
x
ₒ
0, the relationship between
ʴ
f
(
x
)
and
ʴ
I
becomes
ʲ
ʴ
I
=
A
(
x
)ʴ
f
(
x
)
dx
.
(C.72)
ʱ
In Eq. (
C.72
),
A
(
x
)
is defined as the derivative of the functional
I
[
f
(
x
)
]
with respect
to
f
f
.
Let us compute this derivative of a functional. When a functional is given in
Eq. (
C.68
), i.e.,
F
(
x
)
, and it is denoted
ʴ
I
/ʴ
(
x
)
=
f
(
x
)
, the amount of change
ʴ
I
due to the change from
f
(
x
)
to
f
(
x
)
+
ʴ
f
(
x
)
is expressed as
ʲ
ʲ
ʲ
f
)
dx
ʴ
I
=
(
x
)
+
ʴ
f
(
x
−
f
(
x
)
dx
=
ʴ
f
(
x
)
dx
.
(C.73)
ʱ
ʱ
ʱ
Therefore, we obtain
A
(
x
)
=
ʴ
I
/ʴ
f
=
1. In the general case where
F
(
x
)
consists of
f
(
x
)
, the relationship
ʲ
ʲ
ʲ
ʴ
I
=
[
F
(
x
)
+
ʴ
F
(
x
)
]
dx
−
F
(
x
)
dx
=
ʴ
F
(
x
)
dx
(C.74)
ʱ
ʱ
ʱ
holds. Using the relationship
)
=
∂
F
∂
ʴ
F
(
x
f
ʴ
f
(
x
),
