~ ?yvD ?Dy
? ??
ì
T# ?¨B
~ ?yvD ?Dy
B a u× ?Y?￥s ?
! D1
ü
u×
?T D
= ?B> wL
??￥?s?
? D
5? D1
ü
? ?Y u
×
?5?1ˉ ?Y u×
ˉ ?Y u×? ?Y u×
D
D
~ ?yvD ?Dy
! bW u× G
?T G
= ?B> w
??
￥ u× ?
? G
5? G
^ bW=?? ?Y×
?T G
= ?B> wL9 V[fB
? ?
?
G￥ w
5? G1 bWB?? ?Y u×
G
G
G
B?? ?Y
=?? ?Y
B?? ?Y
=?? ?Y
B?? ?Y
=?? ?Y
~ ?yvD ?Dy
!> u× D?s
;á￥ wL L?
?
f
),(),( yxQyxP # D
μB¨ ?
?
ê?
5μ
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
? L
^ D￥ |?_￥H? wL
T
?S ì
T
= aì
T
? ?
~ ?yvD ?Dy
??D?
21
LLL F?D?
21
LLL
H? wL L￥?_,?43?H??<
H, u
× D9
e￥ PH,
2
L
D
1
L
2
L
1
L
D
~ ?yvD ?Dy
}),()(),{(
21
bxaxyxyxD ≤≤≤≤=
￡
ü
? u× D;
^?X ?
?
^?Y ?,'
ü??
USà￥°L Là
??
?,
}),()(),{(
21
dycyxyyxD ≤≤≤≤= ψψ
y
x
o a
b
D
c
d
)(
1
xy?=
)(
2
xy?=
A
B
C
E
)(
2
yx ψ=
)(
1
yx ψ=
~ ?yvD ?Dy
dx
x
Q
dydxdy
x
Q
y
y
d
c
D
∫∫∫∫
=
)(
)(
2
1
ψ
ψ
∫∫
=
d
c
d
c
dyyyQdyyyQ )),(()),((
12
ψψ
∫∫
=
CAECBE
dyyxQdyyxQ ),(),(
∫∫
+=
EACCBE
dyyxQdyyxQ ),(),(
∫
=
L
dyyxQ ),(
] ? V￡
∫∫∫
=
L
D
dxyxPdxdy
y
P
),(
y
x
o
d
)(
2
yx ψ=
D
c
C
E
)(
1
yx ψ=
~ ?yvD ?Dy
? u× D??
;
á￥> wL?? ?m
￡
ü
L
1
L
2
L3
L
D
1
D
2
D3
D
TMF¤
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
| Ds? ??;
^?X ??
^
Y ?￥ u×
1
D
2
D
3
D
∫∫∫∫
++
=
321
)()(
DDDD
dxdy
y
P
x
Q
dxdy
y
P
x
Q
~ ?yvD ?Dy
∫∫∫∫∫∫
+
+
321
)()()(
DDD
dxdy
y
P
x
Q
dxdy
y
P
x
Q
dxdy
y
P
x
Q
∫∫∫
+++++=
321
LLL
QdyPdxQdyPdxQdyPdx
∫
+=
L
QdyPdx
1
D
2
D
3
D
L
1
L
2
L3
L
),(
32,1
?
a1?Z_ DLLL
~ ?yvD ?Dy
G
D
3
L
2
L
F
C
E
1
L
A
B
￡
ü
? u×???BH> w
L
?? @F°L
" #
$ &
5 D￥H? wL? " #
2
L
# "
" ' $
$ &
3
L
& $# $ ( "?
? (2)?
∫∫
D
dxdy
y
P
x
Q
)(
∫∫∫∫∫
++++=
CEAFCBALAB
2
{
∫∫∫
+?+++
CGAECL
QdyPdx )(}
3
~ ?yvD ?Dy
∫
+=
L
QdyPdx
∫∫∫
+++=
231
))((
LLL
QdyPdx
),(
32,1
?
a1?Z_ DLLL
ì
T￥
Lé
y ?
ü
> wL￥ wLsD=×
s-W￥ ó",
,)(
∫∫∫
=+
D
L
dxdy
y
P
x
Q
QdyPdx
1 a ? L
^ D￥?_H? wL
2 aP aQ D
B¨
ê?
??
?i
,3
?¨ì
T a
∫ ∫
LL
QdyPdx
~ ?yvD ?Dy
￥?_H? wL
ü
> u×1 DxoyL
{DA ￥
1 u×
=?
∫
ydxxdy
L
∫∫
=
D
dxdy2
dxdy
y
P
x
Q
D
)(
∫∫
A2=
=A ydxxdy
L
∫
2
1
∫
=
L
xdy
.
∫
=
L
ydx
∫
+=
L
xdyydx 43
∫
+=
L
xdyydx 32
~ ?yvD ?Dy
? ae??¨
e wLs
è 9
dyxydxyx
L
)635()42(
∫
+++?
{L H?1??￥ ???￥?_[ )2,3(),0,3(),0,0(:
3
x
y
o
)2,3(
635,42?+=+?= xyQyxP
,3=
x
Q
,1?=
y
P
D
dyxydxyx
L
)635()42(
∫
+++?
dxdy
y
P
x
Q
D
)(
=
∫∫
dxdy
D
∫∫
= 4
.12=
~ ?yvD ?Dy
x
y
o
L
è 9
∫
AB
xdy, ? w
L AB
^??1 r￥?
?B`K?s,
3 ? ?￡ù wL L,
A
B
D
BOABOAL ++=
(
?¨ì
T,xQP ==,0 μ
∫∫∫
=?
L
D
xdydxdy,
∫∫∫
++=
BOABOA
xdyxdyxdy
,0,0 ==
∫∫
BOOA
xdyxdy??
.
4
1
2
rdxdyxdy
D
AB
π?=?=∴
∫∫∫
~ ?yvD ?Dy
è 9
∫
+
L
dyyxdxyx )sin()(
22
.)1,1()0,0(2:
2
￥B
??
?xxyL?=
3
x
y
o
)1,1(A
D
#
BOABLl ++=
yxQyxP
22
sin,=?=
1,1?=
=
y
P
x
Q
∫
+
l
dyyxdxyx )sin()(
22
dxdy
y
P
x
Q
D
)(
=
∫∫
,0=
=+
∫
AB
dyyxdxyx )sin()(
22
dyy)sin1(
0
1
2
∫
+?
4
2sin
2
3
=
=+
∫
BO
dyyxdxyx )sin()(
22
∫
0
1
2
dxx
3
1
=
e
T
∫∫∫
BOABl
)
3
1
()
4
2sin
2
3
(0=
.
6
7
4
2sin
=
~ ?yvD ?Dy
è 9
∫∫
D
y
dxdye
2
, ? D
^
[ )1,0(),1,1(),0,0( BAO 1??
￥ ???> u×,
3
7
2
,0
y
xeQP
==,
e=×s
x
y
o
A
B
1
1
D
5
2
y
e
y
P
x
Q
=
,
~ ?yvD ?Dy
?¨ì
T
μ
∫∫∫
++
=
BOABOA
y
D
y
dyxedxdye
22
∫∫
==
1
0
22
dxxedyxe
x
OA
y
).1(
2
1
1?
= e
~ ?yvD ?Dy
è 9
∫
+
L
yx
ydxxdy
22
? L1BHí×?
s
;á O?üVe?￥ ??> wL
L￥Z
_1
I
H?Z_
5? 0
22
≠+ yx
H
μ
y
P
yx
xy
x
Q
=
+
=
222
22
)(
.
,L
??￥> u×1 D,3
7
2222
,
yx
x
Q
yx
y
P
+
=
+
=,
~ ?yvD ?Dy
L
? D?)0,0(
H
(2) ? D∈)0,0(
H,
1
D
r
l
x
y
o
L
D
?ì
T?
∫
=
+
L
yx
ydxxdy
0
22
Tê? D
=??
222
,ryxl =+,
:
1
D? L l
??,
?¨ì
T,¤
y
x
o
~ ?yvD ?Dy
∫∫
+
=
+
lL
yx
ydxxdy
yx
ydxxdy
2222
x
y
o
r
1
D
l
L
0
2222
=
+
+
∫∫
lL
yx
ydxxdy
yx
ydxxdy
( ? l￥Z_
|
I
H?Z_ )
.2π=
(?iì
T￥Hq )
θ
θθ
d
r
rr
2
2222
sincos +
∫
π
=
2
0
~ ?yvD ?Dy
ì
T,
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
|,,xQyP =?= ¤
∫∫∫
=
L
D
ydxxdydxdy2
> u× D￥
∫
=
L
ydxxdyA
2
1
.
|,,0 xQP == ¤
∫
=
L
xdyA
|,0,=?= QyP ¤
∫
=
L
ydxA
9
ü
~ ?yvD ?Dy
wL AMO?f
],0[,axxaxy ∈?= V
U,
è 9
tL )0()(
2
>=+ aaxyx D xà
??￥
3
ONA1°L 0=y,
∫
=∴
L
ydxxdyA
2
1
∫∫
+?=
AMOONA
ydxxdyydxxdy
2
1
2
1
)0,(aA
N
M
~ ?yvD ?Dy
∫
=
AMO
ydxxdy
2
1
dxxaxdx
ax
a
x
a
)()1
2
(
2
1
0
=
∫
.
6
1
4
2
0
adxx
a
a
==
∫
)0,(aA
N
M
~ ?yvD ?Dy
1 al2
?Y u×￥à
Q ;
=×sD wLs￥1"
ì
T￥?¨
——ì
T
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
~ ?yvD ?Dy
? u× ?m1
ˉ ?Y×
k
í
ì
T? wLs? L
￥Z_ b
∫∫∫
+=
LD
QdyPdxdxdy
y
P
x
Q
o
x
y
A
B
C
D
E F
G
± I5
~ ?yvD ?Dy
± I53s
o
x
y
A
B
C
D
E F
G
?
?sF?
L
? H?
= H?
BCDAB
EGFE
? ??
ì
T# ?¨B
~ ?yvD ?Dy
B a u× ?Y?￥s ?
! D1
ü
u×
?T D
= ?B> wL
??￥?s?
? D
5? D1
ü
? ?Y u
×
?5?1ˉ ?Y u×
ˉ ?Y u×? ?Y u×
D
D
~ ?yvD ?Dy
! bW u× G
?T G
= ?B> w
??
￥ u× ?
? G
5? G
^ bW=?? ?Y×
?T G
= ?B> wL9 V[fB
? ?
?
G￥ w
5? G1 bWB?? ?Y u×
G
G
G
B?? ?Y
=?? ?Y
B?? ?Y
=?? ?Y
B?? ?Y
=?? ?Y
~ ?yvD ?Dy
!> u× D?s
;á￥ wL L?
?
f
),(),( yxQyxP # D
μB¨ ?
?
ê?
5μ
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
? L
^ D￥ |?_￥H? wL
T
?S ì
T
= aì
T
? ?
~ ?yvD ?Dy
??D?
21
LLL F?D?
21
LLL
H? wL L￥?_,?43?H??<
H, u
× D9
e￥ PH,
2
L
D
1
L
2
L
1
L
D
~ ?yvD ?Dy
}),()(),{(
21
bxaxyxyxD ≤≤≤≤=
￡
ü
? u× D;
^?X ?
?
^?Y ?,'
ü??
USà￥°L Là
??
?,
}),()(),{(
21
dycyxyyxD ≤≤≤≤= ψψ
y
x
o a
b
D
c
d
)(
1
xy?=
)(
2
xy?=
A
B
C
E
)(
2
yx ψ=
)(
1
yx ψ=
~ ?yvD ?Dy
dx
x
Q
dydxdy
x
Q
y
y
d
c
D
∫∫∫∫
=
)(
)(
2
1
ψ
ψ
∫∫
=
d
c
d
c
dyyyQdyyyQ )),(()),((
12
ψψ
∫∫
=
CAECBE
dyyxQdyyxQ ),(),(
∫∫
+=
EACCBE
dyyxQdyyxQ ),(),(
∫
=
L
dyyxQ ),(
] ? V￡
∫∫∫
=
L
D
dxyxPdxdy
y
P
),(
y
x
o
d
)(
2
yx ψ=
D
c
C
E
)(
1
yx ψ=
~ ?yvD ?Dy
? u× D??
;
á￥> wL?? ?m
￡
ü
L
1
L
2
L3
L
D
1
D
2
D3
D
TMF¤
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
| Ds? ??;
^?X ??
^
Y ?￥ u×
1
D
2
D
3
D
∫∫∫∫
++
=
321
)()(
DDDD
dxdy
y
P
x
Q
dxdy
y
P
x
Q
~ ?yvD ?Dy
∫∫∫∫∫∫
+
+
321
)()()(
DDD
dxdy
y
P
x
Q
dxdy
y
P
x
Q
dxdy
y
P
x
Q
∫∫∫
+++++=
321
LLL
QdyPdxQdyPdxQdyPdx
∫
+=
L
QdyPdx
1
D
2
D
3
D
L
1
L
2
L3
L
),(
32,1
?
a1?Z_ DLLL
~ ?yvD ?Dy
G
D
3
L
2
L
F
C
E
1
L
A
B
￡
ü
? u×???BH> w
L
?? @F°L
" #
$ &
5 D￥H? wL? " #
2
L
# "
" ' $
$ &
3
L
& $# $ ( "?
? (2)?
∫∫
D
dxdy
y
P
x
Q
)(
∫∫∫∫∫
++++=
CEAFCBALAB
2
{
∫∫∫
+?+++
CGAECL
QdyPdx )(}
3
~ ?yvD ?Dy
∫
+=
L
QdyPdx
∫∫∫
+++=
231
))((
LLL
QdyPdx
),(
32,1
?
a1?Z_ DLLL
ì
T￥
Lé
y ?
ü
> wL￥ wLsD=×
s-W￥ ó",
,)(
∫∫∫
=+
D
L
dxdy
y
P
x
Q
QdyPdx
1 a ? L
^ D￥?_H? wL
2 aP aQ D
B¨
ê?
??
?i
,3
?¨ì
T a
∫ ∫
LL
QdyPdx
~ ?yvD ?Dy
￥?_H? wL
ü
> u×1 DxoyL
{DA ￥
1 u×
=?
∫
ydxxdy
L
∫∫
=
D
dxdy2
dxdy
y
P
x
Q
D
)(
∫∫
A2=
=A ydxxdy
L
∫
2
1
∫
=
L
xdy
.
∫
=
L
ydx
∫
+=
L
xdyydx 43
∫
+=
L
xdyydx 32
~ ?yvD ?Dy
? ae??¨
e wLs
è 9
dyxydxyx
L
)635()42(
∫
+++?
{L H?1??￥ ???￥?_[ )2,3(),0,3(),0,0(:
3
x
y
o
)2,3(
635,42?+=+?= xyQyxP
,3=
x
Q
,1?=
y
P
D
dyxydxyx
L
)635()42(
∫
+++?
dxdy
y
P
x
Q
D
)(
=
∫∫
dxdy
D
∫∫
= 4
.12=
~ ?yvD ?Dy
x
y
o
L
è 9
∫
AB
xdy, ? w
L AB
^??1 r￥?
?B`K?s,
3 ? ?￡ù wL L,
A
B
D
BOABOAL ++=
(
?¨ì
T,xQP ==,0 μ
∫∫∫
=?
L
D
xdydxdy,
∫∫∫
++=
BOABOA
xdyxdyxdy
,0,0 ==
∫∫
BOOA
xdyxdy??
.
4
1
2
rdxdyxdy
D
AB
π?=?=∴
∫∫∫
~ ?yvD ?Dy
è 9
∫
+
L
dyyxdxyx )sin()(
22
.)1,1()0,0(2:
2
￥B
??
?xxyL?=
3
x
y
o
)1,1(A
D
#
BOABLl ++=
yxQyxP
22
sin,=?=
1,1?=
=
y
P
x
Q
∫
+
l
dyyxdxyx )sin()(
22
dxdy
y
P
x
Q
D
)(
=
∫∫
,0=
=+
∫
AB
dyyxdxyx )sin()(
22
dyy)sin1(
0
1
2
∫
+?
4
2sin
2
3
=
=+
∫
BO
dyyxdxyx )sin()(
22
∫
0
1
2
dxx
3
1
=
e
T
∫∫∫
BOABl
)
3
1
()
4
2sin
2
3
(0=
.
6
7
4
2sin
=
~ ?yvD ?Dy
è 9
∫∫
D
y
dxdye
2
, ? D
^
[ )1,0(),1,1(),0,0( BAO 1??
￥ ???> u×,
3
7
2
,0
y
xeQP
==,
e=×s
x
y
o
A
B
1
1
D
5
2
y
e
y
P
x
Q
=
,
~ ?yvD ?Dy
?¨ì
T
μ
∫∫∫
++
=
BOABOA
y
D
y
dyxedxdye
22
∫∫
==
1
0
22
dxxedyxe
x
OA
y
).1(
2
1
1?
= e
~ ?yvD ?Dy
è 9
∫
+
L
yx
ydxxdy
22
? L1BHí×?
s
;á O?üVe?￥ ??> wL
L￥Z
_1
I
H?Z_
5? 0
22
≠+ yx
H
μ
y
P
yx
xy
x
Q
=
+
=
222
22
)(
.
,L
??￥> u×1 D,3
7
2222
,
yx
x
Q
yx
y
P
+
=
+
=,
~ ?yvD ?Dy
L
? D?)0,0(
H
(2) ? D∈)0,0(
H,
1
D
r
l
x
y
o
L
D
?ì
T?
∫
=
+
L
yx
ydxxdy
0
22
Tê? D
=??
222
,ryxl =+,
:
1
D? L l
??,
?¨ì
T,¤
y
x
o
~ ?yvD ?Dy
∫∫
+
=
+
lL
yx
ydxxdy
yx
ydxxdy
2222
x
y
o
r
1
D
l
L
0
2222
=
+
+
∫∫
lL
yx
ydxxdy
yx
ydxxdy
( ? l￥Z_
|
I
H?Z_ )
.2π=
(?iì
T￥Hq )
θ
θθ
d
r
rr
2
2222
sincos +
∫
π
=
2
0
~ ?yvD ?Dy
ì
T,
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
|,,xQyP =?= ¤
∫∫∫
=
L
D
ydxxdydxdy2
> u× D￥
∫
=
L
ydxxdyA
2
1
.
|,,0 xQP == ¤
∫
=
L
xdyA
|,0,=?= QyP ¤
∫
=
L
ydxA
9
ü
~ ?yvD ?Dy
wL AMO?f
],0[,axxaxy ∈?= V
U,
è 9
tL )0()(
2
>=+ aaxyx D xà
??￥
3
ONA1°L 0=y,
∫
=∴
L
ydxxdyA
2
1
∫∫
+?=
AMOONA
ydxxdyydxxdy
2
1
2
1
)0,(aA
N
M
~ ?yvD ?Dy
∫
=
AMO
ydxxdy
2
1
dxxaxdx
ax
a
x
a
)()1
2
(
2
1
0
=
∫
.
6
1
4
2
0
adxx
a
a
==
∫
)0,(aA
N
M
~ ?yvD ?Dy
1 al2
?Y u×￥à
Q ;
=×sD wLs￥1"
ì
T￥?¨
——ì
T
∫∫∫
+=
L
D
QdyPdxdxdy
y
P
x
Q
)(
~ ?yvD ?Dy
? u× ?m1
ˉ ?Y×
k
í
ì
T? wLs? L
￥Z_ b
∫∫∫
+=
LD
QdyPdxdxdy
y
P
x
Q
o
x
y
A
B
C
D
E F
G
± I5
~ ?yvD ?Dy
± I53s
o
x
y
A
B
C
D
E F
G
?
?sF?
L
? H?
= H?
BCDAB
EGFE
