~ ?yvD ?Dy
?
1?
ì
T# ?¨=
~ ?yvD ?Dy
G
y
x
o
∫
+
1
L
QdyPdx
5? wLs
∫
+
L
QdyPdx
G
= D
^?í1
B a wLsD
^?í1￥?l
∫
+
2
L
QdyPdx
1
L
2
L
B
A
?T u× G
=μ
=
?5D
^?μ1
￥ ?i
H wL?
=V
^ a ? BAGLL
21
?
?
.s1
,
= ?iB> wL￥Hq
^ u×
D
^?í1￥ 1
= wLs u×
D
QdyPdxD
L
∫
+
~ ?yvD ?Dy
= a wLsD
^?í1￥Hq
! 7 u× G
^B?? ?Y×
f
),(),,( yxQyxP G
= μB¨ ??
ê?
5 wLs
∫
+
L
QdyPdx G
=D
^?í1
G
= ?i> wL￥ wLs1
, ￥
1Hq
^
x
Q
y
P
=
G
=? ?
? ?
~ ?yvD ?Dy
￡
ü s?
^?? wL
= ? |BHμ_;á> GLG,
,,
= ??
??￥ u×
[> wL? ?Y￥ GDL
∫
+
L
QdyPdxQ
dxdy
y
P
x
Q
D
)(
±=
∫∫
,
? ? ? G
y
P
x
Q
=
0=+∴
∫
L
QdyPdx
.
=D
^?í1 ' wLs GQdyPdx
L
∫
+
A1?
,
=D
^?í1 X? wLs GQdyPdx
L
∫
+
5?s G
= ?i> wL￥ wLs1
,,
~ ?yvD ?Dy
¨Q￡E
0
,MGG
y
P
x
Q
=à
μB?5
=?? ?
!
=
,0|)(
0
≠
M
y
P
x
Q
P
.0|)(
0
>=
η
M
y
P
x
Q
?^
!
1??yNA Vs?B?[
= ?? ??
0
,,MG
y
P
x
Q
:,
μ
P¤ ??@l￥??× KK
,
2
η
≥
y
P
x
Q
,,￥
^￥?_H? wL
^
! KKL σ
=+∴
∫
L
QdyPdx
dxdy
y
P
x
Q
K
)(
∫∫
σ
η
2
≥,0>
±
~ ?yvD ?Dy
(1) 7 u× G
^B?? ?Y×,
(2)f
),(),,( yxQyxP G
= μB¨ ?
?
ê?
,
Hq ?B? V
μ1? ?￥
a
ü
x
y
o
)4,2(
A
B
∫ ∫
+=
oA AB
∫
++
)4,2(
)0,0(
2
)()( dyyxedxxe
yy
~ ?yvD ?Dy
? a=íf
￥ ?±s p
! 7 u× G
^B?? ?Y×
f
),(),,( yxQyxP G
= μB¨ ??
ê?
5 dyyxQdxyxP ),(),( + G
=1
B
f
),( yxu ￥ ?±s￥ 1Hq
^?
T
x
Q
y
P
=
G
=? ?
? ?
~ ?yvD ?Dy
￡
ü
A1?
P¤L
!i "
Bf
),,( yxu
,),(),( dyyxQdxyxPdu +=
),( ),,( yxQ
y
u
yxP
x
u
=
=
5Aμ
x
Q
xy
u
y
P
yx
u
=
=
22
,V7
y1 P aQ μB¨ ??
ê?
,),( OM?￥=¨
ê?
??5 yxu
.:
x
Q
y
P
=
#μ
~ ?yvD ?Dy
s?
|G
x
Q
y
P
=? ? ?T
=
=D
^?í1 5 GQdyPdx
L
∫
+
∫
+=
),(
),(
00
),(
yx
yx
QdyPdxyxu
!
x
y
o
),( yxB
A
),(
00
yx
.
.
),( yxxB?+
′
G
x
yxuyxxu
x
u
x
+
=
→?
),(),(
lim
0
),(),( yxuyxxu?+?
∫∫
+
=
),(
),(
),(
),(
0000
yxx
yx
yx
yx
∫∫
+
+=
),(
),(
),(
),(
00
yx
yx
yxx
yx
∫
),(
),(
00
yx
yx
∫
+
=
),(
),(
yxx
yx
∫
′
+=
BB
QdyPdx
.
∫
+
=
xx
x
dxyxP
),(
~ ?yvD ?Dy
),(),( yxuyxxu+
∫
+
=
xx
x
dxyxP ),(
xyxxP+= ),( θ 10 ≤≤θ
x
yxuyxxu
x
u
x
+
=
→?
),(),(
lim
0
x
xyxxP
x
+
=
→?
),(
lim
0
θ
),(lim
0
yxxP
x
+=
→?
θ
).,( yxP=
)),(( ??y1 yxP
)),(( ??y1 yxP
).,(,yxQ
y
u
=
] ? V￡
,),(),( ??y1 yxz 2yxP ￥B¨
ê? ??
[ ),( yxu
|yxu V±5f
),(
.),(),(,dyyxQdxyxPdu += O
~ ?yvD ?Dy
x
Q
y
P
≡
?
∫
+
),(
),(
11
00
yxB
yxA
QdyPdx5
dyyxQdxyxP
y
y
x
x
),(),(
1
0
1
0
10
∫∫
+=
),(
01
yxC?
),(
11
yxB?
x
y
o
),(
00
yxA?
dxyxPdyyxQ
x
x
y
y
),(),(
1
0
1
0
10
∫∫
+=
~ ?yvD ?Dy
è 9
∫
+++
L
dyyxdxxyx )()2(
422
, ?
L1?? )0,0(O ?? )1,1(B ￥ wL
2
sin
x
y
π
=,
xxyx
yy
P
2)2(
2
=+
=
xyx
xx
Q
2)(
42
=+
=
3
x
Q
y
P
=
esD
^?í1
.
15
23
=
x
y
o
)1,1(
A
B
e
T
∫ ∫
+
OA AB
∫ ∫
++=
1
0
1
0
42
)1( dyydxx
~ ?yvD ?Dy
è
! wLs
∫
+
L
dyxydxxy )(
2
D
^?í
1, μ ??￥?
, O 0)0( =?,
9
∫
+
)1,1(
)0,0(
2
)( dyxydxxy,
sD
^?í1
x
Q
y
P
=
3
,2)(
2
xyxy
yy
P
=
=
),()]([ xyxy
xx
Q
′
=?
=
,),(
2
xyyxP =
),(),( xyyxQ?=
~ ?yvD ?Dy
? 0)0( =? ? 0=c
2
)( xx =,
#
∫
+
)1,1(
)0,0(
2
)( dyxydxxy
? xyxy 2)( =?
′
cxx +=
2
)(
∫∫
+=
1
0
1
0
0 ydydx
.
2
1
=
x
y
o
)1,1(B
A
~ ?yvD ?Dy
è
).,(
)2()2(
yxu|
xoydyyxdxyx
pB??"￥f
Bf
￥ ?±s
ü
=
^ ￡ +++
3 yxQyxP +=+= 2,2
2,2 =
=
y
P
x
Q
{xoy
y
P
x
Q
=? ?
=
∴
.
)2()2(
Bf
￥ ?±s
ü
=
^ xoydyyxdxyx +++∴
∫
+=
),(
)0,0(
),(
yx
QdyPdxyxu
!
x
y
o
),( yxB
A
∫∫
+=
oA AB
∫
=
x
xdx
0
∫
++
y
dyyx
0
)2(
.
2
1
2
2
1
22
yxyx ++=
~ ?yvD ?Dy
è
).,(,
)2()2(
2222
yxu
xoydyyxyxdxyxyx
pB??"￥f
±s
ü
=
^
Bf
￥ ?
￡+?+
3
2222
2,2 yxyxQyxyxP=?+=
y
P
yx
x
Q
=?=
22 {xoy
y
P
x
Q
=? ?
=
∴
.
)2()2(
2222
Bf
￥ ?±s
ü
=
^ xoydyyxyxdxyxyx+?+∴
∫
+=
),(
)0,0(
),(
yx
QdyPdxyxu
!
x
y
o
),( yxB
A
∫∫
+=
oA AB
∫
=
x
dxx
0
2
∫
+
y
dyyxyx
0
22
)2(
.
3
1
3
1
3223
yxyyxx+=
~ ?yvD ?Dy
è
dyeyfdxyeyf
x
L
x
]3)([]3)([?
′
+?
∫
9
.5
)1,4()3,2()(
??￥
1 OD°L
?i
^?
￥??1V? ?? ?
AB|
BA| -yf
′
3
x
y
o
B
A
.
.
L
3)(,3)(?
′
=?=
xx
eyfQyeyfP
3)(,)(?
′
=
′
=
xx
eyf
y
P
eyf
x
Q
=+
∫
+
QdydxP
BAL
∫∫
D
dxdy
y
P
x
Q
)(
D
∫∫
=
D
dxdy3
.15?=
14:,5,→+?= xxyBA
L+BA1
¨
HòZ_
~ ?yvD ?Dy
14:,5,→+?= xxyBA
QdydxP
BA
+
∫
∫
+?
′
++?=
1
4
)]3)5((153)5([( dxexfxexf
xx
∫
++?
′
+?=
1
4
]123)5()5([ dxxexfexf
xx
1
4
2
|]12
2
3
)5([ xxexf
x
++?=,
2
33
)1()4(
4
+?= efef
=+
∫
QdydxP
L
∫∫
+
BALBA
.
2
63
)1()4(
4
+?= efef
=+
∫
+
QdydxP
BAL
.15?
L+BA1
I
HòZ_
=+
∫
QdydxP
L
∫∫
+
BALBA
=+
∫
+
QdydxP
BAL
.15
.
2
3
)1()4(
4
+?= efef
~ ?yvD ?Dy
1 al2
D
^?í1￥
1??N
5
H
q
? ?Y 7 u× D
),(),,( yxQyxP μ
??￥B¨
ê?
5[/
1?
5?N
∫
+
L
QdyPdxD D
^?í1
= )1(
∫
=+
C
DCQdyPdx > wL,0)2(
QdyPdxduyxUD +=
P
=i ),()3(
x
Q
y
P
D
=
,)4(
=
?
N
5
~ ?yvD ?Dy
°z A b5
a
!> u× D?s
;á￥ wL L??
f
),(,),( yxQyxP D
μB¨ ??
ê?
5μ
=
∫∫
D
dxdy
y
P
x
Q
)( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a
! D 1
ü
￥B?? ?Y×
f
),(,),( yxQyxP D
=μB¨ ??
ê?
5
∫
+
L
QdyPdx D
=D
^?í1￥ 1Hq
^
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ D
=))? ?
a
! D1?s
;á￥ wL L
??￥> u×
1
? ),( yxP # ),( yxQ D
μB¨ ??
ê
?
O 1=
x
Q
1?=
y
P
5 =+
∫
L
QdyPdx @ @ @
5
~ ?yvD ?Dy
?z 9
∫
++?
L
dyyxdxxxy )()2(
22
? L
^?
t
L
2
xy = xy =
2
??￥ u×￥?_H? wL
i￡ì
T￥? ??
~z ?¨ wLs
p??L taytax
33
sin,cos ==
??￥m?￥
1 a￡
ü wLs
∫
+?
)4,3(
)2,1(
2232
)36()6( dyxyyxdxyxy ?? xoy
=D
^?í1
i9
s′
? a ?¨ì
T
9
/
wLs
a
∫
+
L
dyyxdxyx )sin()(
22
? L
^ ??
2
2 xxy?=
??
??
￥B
~ ?yvD ?Dy
a p wLs
∫
+=
AMB
dyyxdxyxI
22
1
)()(
∫
+=
ANB
dyyxdxyxI
22
2
)()( ￥μ ? AMB
^Ve? )1,1(A
)6,2(B O ?à<°? x
à￥
tL
￥
AMB
^ ?¤ BA,￥L
B a9
∫
+
L
yx
ydxxdy
22
? L1?üVe?￥;á> w
L |
I
H?Z_
t a￡ yxxdxxyyx
2322
8()83( +++ dyye
y
)12+ ?
? xoy
ü
=
^
Bf
),( yxu ￥ ?±s
i p?
"B? ),( yxu
~ ?yvD ?Dy
? a
k ?? λ
P¤ dyr
y
x
dxr
y
x
λλ
2
2
^
?f
),( yxu ￥ ?±s
?
22
yxr +=
i p
),( yxu
? a
! ?
ü
0>x
=μ ? )(
3
jyix
r
k
F +?=? ?
?
? k1è
22
yxr += ￡
ü N ???
? ?
T￥?D
|￥
^?í1
~ ?yvD ?Dy
5s?
B a a
∫
+
L
dyQPdx a
x
Q
y
p
=
a
= a
30
1
? a
2
8
3
aπ
1 a
? a a 2sin
4
1
6
7
+? a
B a a?
?L ￥ D u× ?ce?
H
a?
?L ￥ D u× ce?
? ? O L e?
B ?
H
π2
a?
?L ￥ D u× ce?
? O Lne?
?
H
πn2
~ ?yvD ?Dy
t a )(124),(
223 yy
eyeyxyxyxu?++=
? a
y
r
yxu =?= ),(,1λ
?
1?
ì
T# ?¨=
~ ?yvD ?Dy
G
y
x
o
∫
+
1
L
QdyPdx
5? wLs
∫
+
L
QdyPdx
G
= D
^?í1
B a wLsD
^?í1￥?l
∫
+
2
L
QdyPdx
1
L
2
L
B
A
?T u× G
=μ
=
?5D
^?μ1
￥ ?i
H wL?
=V
^ a ? BAGLL
21
?
?
.s1
,
= ?iB> wL￥Hq
^ u×
D
^?í1￥ 1
= wLs u×
D
QdyPdxD
L
∫
+
~ ?yvD ?Dy
= a wLsD
^?í1￥Hq
! 7 u× G
^B?? ?Y×
f
),(),,( yxQyxP G
= μB¨ ??
ê?
5 wLs
∫
+
L
QdyPdx G
=D
^?í1
G
= ?i> wL￥ wLs1
, ￥
1Hq
^
x
Q
y
P
=
G
=? ?
? ?
~ ?yvD ?Dy
￡
ü s?
^?? wL
= ? |BHμ_;á> GLG,
,,
= ??
??￥ u×
[> wL? ?Y￥ GDL
∫
+
L
QdyPdxQ
dxdy
y
P
x
Q
D
)(
±=
∫∫
,
? ? ? G
y
P
x
Q
=
0=+∴
∫
L
QdyPdx
.
=D
^?í1 ' wLs GQdyPdx
L
∫
+
A1?
,
=D
^?í1 X? wLs GQdyPdx
L
∫
+
5?s G
= ?i> wL￥ wLs1
,,
~ ?yvD ?Dy
¨Q￡E
0
,MGG
y
P
x
Q
=à
μB?5
=?? ?
!
=
,0|)(
0
≠
M
y
P
x
Q
P
.0|)(
0
>=
η
M
y
P
x
Q
?^
!
1??yNA Vs?B?[
= ?? ??
0
,,MG
y
P
x
Q
:,
μ
P¤ ??@l￥??× KK
,
2
η
≥
y
P
x
Q
,,￥
^￥?_H? wL
^
! KKL σ
=+∴
∫
L
QdyPdx
dxdy
y
P
x
Q
K
)(
∫∫
σ
η
2
≥,0>
±
~ ?yvD ?Dy
(1) 7 u× G
^B?? ?Y×,
(2)f
),(),,( yxQyxP G
= μB¨ ?
?
ê?
,
Hq ?B? V
μ1? ?￥
a
ü
x
y
o
)4,2(
A
B
∫ ∫
+=
oA AB
∫
++
)4,2(
)0,0(
2
)()( dyyxedxxe
yy
~ ?yvD ?Dy
? a=íf
￥ ?±s p
! 7 u× G
^B?? ?Y×
f
),(),,( yxQyxP G
= μB¨ ??
ê?
5 dyyxQdxyxP ),(),( + G
=1
B
f
),( yxu ￥ ?±s￥ 1Hq
^?
T
x
Q
y
P
=
G
=? ?
? ?
~ ?yvD ?Dy
￡
ü
A1?
P¤L
!i "
Bf
),,( yxu
,),(),( dyyxQdxyxPdu +=
),( ),,( yxQ
y
u
yxP
x
u
=
=
5Aμ
x
Q
xy
u
y
P
yx
u
=
=
22
,V7
y1 P aQ μB¨ ??
ê?
,),( OM?￥=¨
ê?
??5 yxu
.:
x
Q
y
P
=
#μ
~ ?yvD ?Dy
s?
|G
x
Q
y
P
=? ? ?T
=
=D
^?í1 5 GQdyPdx
L
∫
+
∫
+=
),(
),(
00
),(
yx
yx
QdyPdxyxu
!
x
y
o
),( yxB
A
),(
00
yx
.
.
),( yxxB?+
′
G
x
yxuyxxu
x
u
x
+
=
→?
),(),(
lim
0
),(),( yxuyxxu?+?
∫∫
+
=
),(
),(
),(
),(
0000
yxx
yx
yx
yx
∫∫
+
+=
),(
),(
),(
),(
00
yx
yx
yxx
yx
∫
),(
),(
00
yx
yx
∫
+
=
),(
),(
yxx
yx
∫
′
+=
BB
QdyPdx
.
∫
+
=
xx
x
dxyxP
),(
~ ?yvD ?Dy
),(),( yxuyxxu+
∫
+
=
xx
x
dxyxP ),(
xyxxP+= ),( θ 10 ≤≤θ
x
yxuyxxu
x
u
x
+
=
→?
),(),(
lim
0
x
xyxxP
x
+
=
→?
),(
lim
0
θ
),(lim
0
yxxP
x
+=
→?
θ
).,( yxP=
)),(( ??y1 yxP
)),(( ??y1 yxP
).,(,yxQ
y
u
=
] ? V￡
,),(),( ??y1 yxz 2yxP ￥B¨
ê? ??
[ ),( yxu
|yxu V±5f
),(
.),(),(,dyyxQdxyxPdu += O
~ ?yvD ?Dy
x
Q
y
P
≡
?
∫
+
),(
),(
11
00
yxB
yxA
QdyPdx5
dyyxQdxyxP
y
y
x
x
),(),(
1
0
1
0
10
∫∫
+=
),(
01
yxC?
),(
11
yxB?
x
y
o
),(
00
yxA?
dxyxPdyyxQ
x
x
y
y
),(),(
1
0
1
0
10
∫∫
+=
~ ?yvD ?Dy
è 9
∫
+++
L
dyyxdxxyx )()2(
422
, ?
L1?? )0,0(O ?? )1,1(B ￥ wL
2
sin
x
y
π
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xxyx
yy
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2
=+
=
xyx
xx
Q
2)(
42
=+
=
3
x
Q
y
P
=
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^?í1
.
15
23
=
x
y
o
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A
B
e
T
∫ ∫
+
OA AB
∫ ∫
++=
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0
1
0
42
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~ ?yvD ?Dy
è
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∫
+
L
dyxydxxy )(
2
D
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9
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2
)( dyxydxxy,
sD
^?í1
x
Q
y
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2
xyxy
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=
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#
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2
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′
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)(
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0
1
0
0 ydydx
.
2
1
=
x
y
o
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A
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è
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Bf
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ü
=
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3 yxQyxP +=+= 2,2
2,2 =
=
y
P
x
Q
{xoy
y
P
x
Q
=? ?
=
∴
.
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Bf
￥ ?±s
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∫
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yx
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!
x
y
o
),( yxB
A
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.
2
1
2
2
1
22
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xoydyyxyxdxyxyx
pB??"￥f
±s
ü
=
^
Bf
￥ ?
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3
2222
2,2 yxyxQyxyxP=?+=
y
P
yx
x
Q
=?=
22 {xoy
y
P
x
Q
=? ?
=
∴
.
)2()2(
2222
Bf
￥ ?±s
ü
=
^ xoydyyxyxdxyxyx+?+∴
∫
+=
),(
)0,0(
),(
yx
QdyPdxyxu
!
x
y
o
),( yxB
A
∫∫
+=
oA AB
∫
=
x
dxx
0
2
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+
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dyyxyx
0
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.
3
1
3
1
3223
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x
L
x
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xx
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y
P
eyf
x
Q
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∫
+
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BAL
∫∫
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dxdy
y
P
x
Q
)(
D
∫∫
=
D
dxdy3
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14:,5,→+?= xxyBA
L+BA1
¨
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14:,5,→+?= xxyBA
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BA
+
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xx
∫
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1
4
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xx
1
4
2
|]12
2
3
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x
++?=,
2
33
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4
+?= efef
=+
∫
QdydxP
L
∫∫
+
BALBA
.
2
63
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4
+?= efef
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+
QdydxP
BAL
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L+BA1
I
HòZ_
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L
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+
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+
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BAL
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.
2
3
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4
+?= efef
~ ?yvD ?Dy
1 al2
D
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1??N
5
H
q
? ?Y 7 u× D
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5[/
1?
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∫
+
L
QdyPdxD D
^?í1
= )1(
∫
=+
C
DCQdyPdx > wL,0)2(
QdyPdxduyxUD +=
P
=i ),()3(
x
Q
y
P
D
=
,)4(
=
?
N
5
~ ?yvD ?Dy
°z A b5
a
!> u× D?s
;á￥ wL L??
f
),(,),( yxQyxP D
μB¨ ??
ê?
5μ
=
∫∫
D
dxdy
y
P
x
Q
)( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a
! D 1
ü
￥B?? ?Y×
f
),(,),( yxQyxP D
=μB¨ ??
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5
∫
+
L
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=D
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^
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a
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O 1=
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Q
1?=
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P
5 =+
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L
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5
~ ?yvD ?Dy
?z 9
∫
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L
dyyxdxxxy )()2(
22
? L
^?
t
L
2
xy = xy =
2
??￥ u×￥?_H? wL
i￡ì
T￥? ??
~z ?¨ wLs
p??L taytax
33
sin,cos ==
??￥m?￥
1 a￡
ü wLs
∫
+?
)4,3(
)2,1(
2232
)36()6( dyxyyxdxyxy ?? xoy
=D
^?í1
i9
s′
? a ?¨ì
T
9
/
wLs
a
∫
+
L
dyyxdxyx )sin()(
22
? L
^ ??
2
2 xxy?=
??
??
￥B
~ ?yvD ?Dy
a p wLs
∫
+=
AMB
dyyxdxyxI
22
1
)()(
∫
+=
ANB
dyyxdxyxI
22
2
)()( ￥μ ? AMB
^Ve? )1,1(A
)6,2(B O ?à<°? x
à￥
tL
￥
AMB
^ ?¤ BA,￥L
B a9
∫
+
L
yx
ydxxdy
22
? L1?üVe?￥;á> w
L |
I
H?Z_
t a￡ yxxdxxyyx
2322
8()83( +++ dyye
y
)12+ ?
? xoy
ü
=
^
Bf
),( yxu ￥ ?±s
i p?
"B? ),( yxu
~ ?yvD ?Dy
? a
k ?? λ
P¤ dyr
y
x
dxr
y
x
λλ
2
2
^
?f
),( yxu ￥ ?±s
?
22
yxr +=
i p
),( yxu
? a
! ?
ü
0>x
=μ ? )(
3
jyix
r
k
F +?=? ?
?
? k1è
22
yxr += ￡
ü N ???
? ?
T￥?D
|￥
^?í1
~ ?yvD ?Dy
5s?
B a a
∫
+
L
dyQPdx a
x
Q
y
p
=
a
= a
30
1
? a
2
8
3
aπ
1 a
? a a 2sin
4
1
6
7
+? a
B a a?
?L ￥ D u× ?ce?
H
a?
?L ￥ D u× ce?
? ? O L e?
B ?
H
π2
a?
?L ￥ D u× ce?
? O Lne?
?
H
πn2
~ ?yvD ?Dy
t a )(124),(
223 yy
eyeyxyxyxu?++=
? a
y
r
yxu =?= ),(,1λ
