复旦大学：数学分析（习题全角、教学教案）：ex7-4





5


?

p/ wL ??￥m? ?
 ò y x = 1
y x=
x = 2
ó
yx 2 4=+()1yx 2 41= ?()
? y x=

yx x=+sin 2 x = 0
x =π
?

y x =e y x = ? e x =1
?

yx=|ln | y = 0 x = 01.
x =10
× =?L
xtt ytt t =? =? ≤≤ ? ? ? 2 2 02 2 23 , , ? ??L
xa t ya t t = = ? ? ? ≤≤ cos , sin , 3 3 02π (8) - ü￡ ?L ra= θ, θ θ π= =02,
(9)  ? ?L ra= e, θ θ θ π= =02,
(10) 
L ra b= +cosθ
ba

≥>0 (11)
rr = 3cosθ =+1cosθ ( 33 π θ π ≤≤? )
(12) ?gL
θ2cos 22 ar = (13) 1= o>L ra= cos 2θb (14) Descartes =?L xy ax 33 3 y+ =
(15) xyaxy 44222 + = +(). 3
1
 ? 2ln 2 3 )ln 2 1 () 1 ( 1 22 2 1 ?=?=?= ∫ xxdx x xA b
2
 ? 3 16 ) 2 2(2)1 4 () 4 1(2 2 0 2 2 0 22 =?= ? ? ? ? ? ? ? ? ???= ∫∫ dy y dy yy A b
3
 ? 2 )2cos1( 2 1 sin 00 2 π ππ =?== ∫∫ dxxxdxA b 231
4
 ? 2 1 )( 1 0 ?+=?= ∫ ? e edxeeA xx b
5
 ? 1.0 1 1 10 1 1.0 10 1 10 1.0 )1(ln)1(lnlnlnln ???=?== ∫∫∫ xxxxxdxxdxdxxA 10 81 10ln 10 99 ?= b
6
 ? 15 8 )32(2)22)(2( 2 0 432 2 0 32 =+?=??= ∫∫ dttttdttttA b
7
 ? ∫∫ ?=?= 2 0 642 2 0 23 )cos(cos12cossin3cos4 ππ dtttatdttataA 22 8 3 96 15 16 3 12 aa πππ = ? ? ? ? ? ? ?= b
8
 ? 23 2 0 22 3 4 2 1 adaA πθθ π == ∫ b
9
 ? == ∫ π θ θ 2 0 22 2 1 deaA 4 1 ( ) 24 1 ae ? π b
10
 ? ∫∫ ++=+= ππ θθθθθ 2 0 222 2 0 2 )cos2cos( 2 1 )cos( 2 1 dbabadbaA =+ + = ∫ πθ θ π 2 2 0 2 2 2cos1 2 bd a 22 2 1 ba ππ + b
11
 ? ∫∫ ?? ?+=+?= 3 3 3 3 22 )cos22cos43( 2 1 ])cos1()cos3[( 2 1 π π π π θθθθθθ ddA π= b
12
 ? 2 4 0 2 2cos 2 1 4 adaA =?= ∫ π θθ b
13
 ? 2 4 0 2 4 0 22 2 1 )4cos1(22cos 2 1 8 adadaA πθθθθ ππ =+=?= ∫∫ b
14
3B
 7
5txy = 3 1 3 t at x + =
 3 2 1 3 t at y + =
 b +∞→0:t ? ^ ? =A ∫ ∞+ ′ ? ? ? ? ? ? ++ 0 33 2 1 3 1 3 dt t at t at = ∫ ∞+ + ? 0 33 23 2 )1( )21( 9 dt t tt a
 7
5 3 tu = 232 =A 2 0 32 2 0 3 2 2 3 )1( 3 )1( 2 3 )1( )21( 3 adu uu adu u u a = ? ? ? ? ? ? ? ? + ? + = + ? ∫∫ ∞+∞+ b 3=
 | θθ sin,cos ryrx == }? xy ax 33 3 y+ = ?
¤? θθ θθ 33 cossin cossin3 + = a r
 ] 2 ,0[ π θ ∈
 ? ^ ? ∫∫ + = + = 2 0 23 22 2 0 233 222 tan )1(tan tan 2 9 )cos(sin cossin 2 9 ππ θ θ θ θ θθ θθ d a d a A 2 0 2 3 2 2 3 1tan 1 2 3 a a = + ??= π θ b
15
| θθ sin,cos ryrx == }? xyaxy 44222 + = +()?
¤? θθ 44 2 2 cossin + = a r
 ? ^ ? ∫∫ + + = + ?= 2 0 4 2 2 2 0 44 2 tan 1tan 1tan 2 cossin2 1 4 ππ θ θ θ θ θθ dad a A
 7 θtan=t
5 ∫∫ ∞+ ? ? ∞+ +? ? = + + = 0 21 1 2 0 4 2 2 2)( )( 2 1 1 2 tt ttd adt t t aA 2 0 1 2 2 2 arctan2 a tt a π= ? = ∞+ ? b ? p??tL DV??￥? ??￥m? ?￥Kl′b ya 2 4= x 3 ê|?? 1? , à1à
y ?USb 5? )0,(a x θθ sin,cos ryarx =+= }??tL￥Z? ya 2 4 x= ?
 V¤?tL￥ USZ?1 θcos1 2 ? = a r b !V??￥?￥?1 α
5 ?D?tL ??￥ ?1 ∫ + ? = πα α θ θ α d a A 2 2 )cos1( 4 2 1 )( b 233 ? α α αα α 4 2 22 2 sin cos8 )cos1( 1 )cos1( 1 2)(' a aA ?= ? ? ? ? ? ? ? ? ? ? + =
 7 0)(' =αA
¤? 2 π α = b??? 2 π α < H
 0)(' <αA
? 2 π α > H
 0)(' >αA
 ?[ )(αA  2 π α = |?l′
9ü ^Kl′ ) 2 ( π A
 ) 2 ( π A 2 2 3 2 22 2 3 2 2 2 2 3 10 2 cot) 2 cot1( )cos1( 1 2 adada =+?= ? = ∫∫ π π π π θθ θ θ b
à p/ wL￥?é
 ò
04yx= 32/ ≤≤x
ó x yy =? 2 42 ln
1≤ ≤y e
?
yx= ln cos 0 2 ≤≤<xa π
? ??L
xa t ya t t = = ? ? ? ≤≤ cos , sin , 3 3 02π ? ?￥v 7L
xa tt t ya tt t t =+ =? ? ? ? ≤≤ (cos sin ), (sin cos ), 02π × ?#L
ra=?(cos1 θ) 02≤ ≤θ π
? - ü￡ ?L ra= θ, 02≤ ≤θ π
ù 3 sin 3 θ ar =
 πθ 30 ≤≤ b 3
1
 =+= ∫ 4 0 4 9 1 dxxL 27 81080 ? b
2
 2 12 1 11 11 1( ) ( ) 42 ee e Lyydyydy ?? 1 4 + =+? = + = ∫∫ b
3
 2 00 1tan sec ln(tan sec) aa Lxdxxd=+ = = + ∫∫ ab
4
 2 0 4 3 sin cos 6L attdx π == ∫ ab
5
? () cos, () sinx tat tytat′′==t
 V¤ 234 2 2 0 2Latd π aπ== ∫ b
6
 22 22 00 2sin 8 2 L rrd a d a ππ θ θθ′=+= = ∫∫ b
7
 22 2 00 1Lrrda d ππ θ θθ′=+= += ∫∫ ( ) 22 412ln 2 41 ππππ ++++ a a b
8
 33 22 2 00 3 sin 32 L rrd a d a ππ θ θ θπ′=+= = ∫∫ b á è }L￥?B 
ps? ￥é1 13￥?￥USb 3 ! ?p? ??￥? ?1 α
5 ) 2 cos1(4sin)cos1( 0 2222 1 α α ?=+?= ∫ adttataL
 ) 2 cos1(4sin)cos1( 2 2222 2 α π α +=+?= ∫ adttataL
 ?
¤ 12 3LL = 2 1 2 cos = α
' πα 3 2 =
 ?[??￥US 1 ) 2 3 ,) 2 3 3 2 (( a a?π b ? p/ +?8￥8
 ò ??? ?
 ? ^é?à1 a?à1 ￥??
/? ^ é?à1 a b Aa?à1 B￥??


ú1
AaBb>>, h ó ?o8 x a y b z c 2 2 2 2 2 2 1++=
? °?? ? ?xya 22 += 2 x za 22 2 + = ??￥+?8
? o ? ?°?? ?xyza 222 ++= 2 xxya 22 + = ??￥+?8b 3
1
 0 ()()(22 6 h Aa Bb h VaxbxdxABabAba hh )B π π ?? =+ + = +++ ∫ b
2
 abcdz c z abV c c ππ 3 4 )1( 2 2 =?= ∫ ? b
3
¨ü?? ü ?￥ü ???? ?8￥?B?K￥?s
? ?1?Z?
? ^ yz0 3 0 22 3 16 )(8 adxxaV a =?= ∫ b
4
¨ü?? ü ?￥ü ???? ?8
5? ?1 yz0 235 ∫ ? ?? ??= 2 2 222 2)( xax xax dyyxaxA xa x xaaxaxax + ?+??= arcsin)(22 2222 b ? ) 3 1 (arcsinarcsin)( 32 00 22 xxad xa x dx xa x xa aa ? + = + ? ∫∫ ∫ + ?? + ?= aa dx xax a xxa xa x xxa 0 32 0 32 )(2 ) 3 1 (arcsin) 3 1 ( 3 ) 45 32 3 ( a?= π
 # 3 00 22 15 4 )( adxxaxadxaxaxax aa =?=?? ∫∫
 ¤? 3 ) 9 8 3 2 ( aV ?= π b ? ￡ ü[/è8￥8 T
 ò ! ^ ??f ?
 ?fx()≥ 0 )(0,0 xfybxa ≤≤≤≤≤ ?V U￥ u ×? àèB? ??￥è8￥81 y ∫ = b a dxxxfV )(2π
ó US/
 ? 00≤ ≤ ≤ ≤ ≤ ≤α θ β π θ,rr() ?V U￥u×? àèB? ??￥è8￥81 ∫ = β α θθθ π drV sin)( 3 2 3 b ￡
1
T u W ￥?s],[ ba bxxxxaP n =<<<<=" 210 : , 51?lu× {)(0,),( 1 xfyxxxyx ii ≤≤≤≤ ? } ? yàè ?¤￥8μ )()( 2 1 2 iiii xfxxV ? ?≈? π iii xxfx ?≈ )(2π b ! )(max 1 i ni x?= ≤≤ λ
 7 0→λ
üμ b ∫ = b a dxxxfV )(2π 236 (2) 3B
 ! θθ cos)(rx = , θθ sin)(ry = , αα cos)(ra = , ββ cos)(rb = , 5 ∫ = a b dxyV 2 π ααπ 22 sin)( 3 1 ar? ββπ 22 sin)( 3 1 br+ ∫ = a b dxy 2 π ∫ + b a xyd )( 3 1 2 π ∫ ?= α β θθθθπ drrr )sincos'(sin 22 () ∫ ?++ β α θθθθθθπ drrrr 332322 sincossin2cossin'3 3 1 ∫ = β α θθθ π dr sin)( 3 2 3 b 3=
 n5
 ? ar ≤≤≤≤≤ 0,0 πβθ ?V U￥
?u×?àèB ? ??￥è8￥81 3 cos 2222 )cos1( 3 2 )(cossin 3 adxxaaaV a a β π πββ π β ?=?+= ∫ b ?aT ],[ βα ￥?s
 βθθθθα =<<<<= n " 210
 I3? )(0, 1 θθθθ rr ii ≤≤≤≤ ? ?V U￥lwH
?u×?àèB? ?? ￥è8￥8
 ?lu× Ví ? AT
?
 ? ^?l v￥8 ?í ??? 33 1 22 ()(1cos) ()(1cos ) iii ii Vr r π π θ θθ ? ?≈ ? ? ? θ iii r θθθ π ??≈ sin)( 3 2 3
 V7 ∑ = ?≈ n i iii rV 1 3 sin)( 3 2 θθθ π b 7 0)(max 1 →?= ≤≤ i ni θλ
üμ Vr= ∫ 2 3 3 π θθ α β ()sindθb ? p/ wL?·?àèB? ???￥è8￥8

1
 x a y b 2 2 2 2 1+=
? xà

2


yx= sin y = 0 0 ≤ ≤x π
 ? ? xà
 ? ? à
y 237
3
 ??L
? xa t ya t t = = ? ? ? ≤≤ cos , sin , 3 3 0 π xà



è }L
 xat t ya t t =? =? ? ? ? ∈ (sin), (cos), [, ] 1 02π y = 0
 ? ? à
 ? ?°Ly ya= 2

5


xyba 222 +? =() 0 < ≤ab

? xà

6
 ?#L ra= ?(cos1 θ)
?à

7
  ? ?L ra= e, θ 0 ≤ ≤θ π
?à

8

?()(xy axy 222222 += ?)xàb 3
1
 222 2 2 3 4 )( abdxxa a b V a a ππ =?= ∫ ? b
2
? 2 0 2 2 1 sin ππ π == ∫ xdxV
? b 2 0 2sin2 ππ π == ∫ xdxxV
3
 ∫∫ == ? π ππ 0 2732 cossin3 tdttadxyV a a 3 2 0 973 105 32 )sin(sin6 adttta ππ π =?= ∫ b
4
?

32 2 0 23 6)cos1)(sin(2)(2 adttttadxxxfV b a πππ π =??== ∫∫ ? b
32 2 0 23 7)cos1()]cos1(2[)2( adttataaaV πππ π =????= ∫
5
 ( ) badxxabdxxabxabV a a a a 2222222222 24)()( πππ =?=????+= ∫∫ ?? b

6
?? 5


¤









3 0 33 3 8 sin)cos1( 3 2 adaV πθθθ π π =?= ∫ b

7
 33 0 33 )1( 15 sin 3 2 aedeaV +== ∫ π π θ π θθ π b

8
 ∫ = 4 0 2 3 3 sin)2(cos 3 4 π θθθ π daV 7 θcos=t
5
238 ∫ ?= 1 2 1 2 3 23 )12( 3 4 dttaV π b ? ∫ ? 1 2 1 2 3 2 )12( dtt ∫ ???= 1 2 1 2 1 22 1 2 3 2 )12(6)12( 2 1 dttttt 31?= ∫ ? 1 2 1 2 3 2 )12( dtt ∫ ?? 1 2 1 2 1 2 )12(3 dtt
 ¤? ∫ ? 1 2 1 2 3 2 )12( dtt 4 1 = ∫ ?? 1 2 1 2 1 2 )12( 4 3 dtt 8 1 )21ln( 16 23 122ln122 28 3 4 1 2 1 1 22 ?+=? ? ? ? ? ? ?+???= tttt
 ?[ ∫ ?= 1 2 1 2 3 23 )12( 3 4 dttaV π  3 3 2 )12ln(2 4 a ? ? ? ? ? ? ?+ π b
? |?tL yxxa=?() ],0[ ax∈ ? xac∈[,]￥? sY? xàè B?a
 ?¤?è8￥8M?
p cD a￥1"b 3
 ∫∫ ?=? c a a dxaxxdxaxx 22 0 22 )()( ππ sa?e
¤? 0615102 54325 =?+? caccaa b ?

: V ^wL()ξ y x x = +1 2  x ∈[, ]0 ξ ￥? ? xàèB? ??? ￥è8￥8
pè ? a P¤ ?@ Va V() lim ()= →+∞ 1 2 ξ ξ b
3


?
239 )1(2)1( )( 2 2 0 22 a a dx x x aV a + = + = ∫ π π

V? 2 )(lim π ξ ξ = +∞→ V
? ^¤? 2 1 1 2 2 = +a a
3¤
1=a b

|? ? x a y b 2 2 2 2 1+=? xàèB???B?è?o8
  
xà Z_¨??1 r
r b<
￥GhuB?,?￥? d
 :/￥8 1e ??o88￥B?
p r￥′b 3 ??
x a y b 2 2 2 2 1+=? xàèB? ???￥è?o8￥81 2 0 2 22 1 3 4 2 abdxxa a b V a ππ = ? ? ? ? ? ? ?= ∫ b é/?s￥81 ? ? ? ? ? ? ? ? ??=?+??= ∫ ? 2 3 222 2 2 2222 2 )( 3 4 )1(22 22 rb b a abdx a x brb b a rV a rb b a π ππ b ?
3¤ 21 2VV = 2 2 1 3 ?= br b  !°L x

D?tL ???￥m?￥ ?1
 O ? ìD°L ???m?￥ ?1 b ay = 10 << a 2 xy = 1 S 1=x 2 S
1

?? ￥′
 P¤a 21 SS + r?Kl
ipKl′

2

?Kl′ ??￥ü ?m?? àèB? ?¤è8￥ 8 b x 3
1
 3 1 2 1 3 1 )()( 3 1 2 0 2 21 +?=?+?=+ ∫∫ aadxaxxdxxaxSS a a b : 3 1 2 1 3 1 )( 3 +?= aaaf
5 aafaaf 2)(, 2 1 )( 2 =′′?=′ b 7
¤ ? 0)( =′ af 2 1 =a
O 02) 2 1 ( >=′′f b ?[ 21 SS +  2 1 =a )|?Kl′
 {} 12 11 1 min ( ) (1 ) 3 22 SS f+= =?b
2
è88 πππ ) 5 1 3 1 15 4 (])([])[( 25 1 24 0 42 +?=?+?= ∫∫ aadxaxxdxxaxV a a b | 2 1 =a }?
ü¤? 240 V 21 30 π + = b 12. !f ? >uW  ??
  7uW v? ,
 i ?@ )(xf ]1,0[ )1,0( 2 2 3 )()( x a xfxfx +=′
1è ?
 b a éB?
L !wL D°L)(xfy = 1=x ? 0=y ??￥m? ￥ ? 1 2b S
1

pf ?
)(xf
2
? 1?′ H
 m ? ? àèB? ?¤è8￥8Kl
$
a S x 3
1
? 2 2 3 )()( x a xfxfx +=′
 V¤ 2 3)( a x xf = ′ ? ? ? ? ? ?
 ?[ cx a x xf += 2 3)(
 ' cxx a xf += 2 2 3 )( b  2 2 3 )()( x a xfxfx +=′ H1? s
μ x 2 )()( 1 0 1 0 a dxxfxxdf += ∫∫
 ?N V¤ 2 4)1( a f +=
V7 ac ?= 4
? ^ xax a xf )4( 2 3 )( 2 ?+= b
2

=V 1 22 0 ( ) ( 10 160) 30 fxdx a a π π =++ ∫ b 7
¤0=′V 5?=a
O? H 0 15 >=′′ π V
 ?[ 5?=a Hè8￥8 |?Kl′b

p/ èw ?￥ ?
 ò
pxy 2 2 = 0 ≤≤x a
? xà
ó

?yx= sin 0 ≤≤x π xà
? x a y b 2 2 2 2 1+=
? xà
? ??L
? xa t ya t t = = ? ? ? ≤≤ cos , sin , 3 3 0 π xà
? ?#L ra=?(cos1 )θ
?à
× ?gL
 θ2cos 22 ar = (i)?à
 (ii) ? L θ π = 2 b 241 3
1
 ? ∫∫ +=+= aa dxpxpdx x p pxA 00 22 2 122 ππ = ? ? ? ? ? ? ? ? ?+ 2 3 2 3 )2( 3 2 ppa pπ b
2
 ? ∫ += π π 0 2 cos1sin2 dxxxA ( ) 0 22 cos1ln(coscos1cos π π xxxx ++++?= = )12ln(222 ++ ππ b
3
 ? ∫∫ ??= ? ? ?+?= ? aa a dxxbaa a b dx xa x a b xa a b A 0 2224 2 2 22 22 )( 4 )(12 ππ = ? ? ? ? ? ? ? ? ? > ? ? + = < ?+ ? + ba a ba ba ba b baab ba a abb ab ba b 22 22 2 2 22 22 2 2 arcsin 2 2 4 ln 2 2 π π π π π b
4
 ? 2 2 0 42 2 0 3 5 12 sinsin12cossin3sin4 attdatdttataA πππ ππ ==?= ∫∫ b
5
 ? ∫∫ ?=′+= ππ θ θ θθπθθπ 0 2 0 22 2 sinsin)cos1(4sin2 dadrrrA 2 0 42 5 32 2 cos 2 sin16 ada πθ θθ π π == ∫ b
6

i
 ? 2 4 0 2 4 0 22 )224(sin4sin4 adadrrrA πθθπθθπ ππ ?==′+= ∫∫

ii
 ? ∫ ′+= 4 0 22 cos4 π θθπ drrrA 2 4 0 2 22cos4 ada πθθπ π == ∫ b 14
 !wL 1?= xy
V e?TML
 p??wLa ?TML# à ???￥ü ?m?? àèB? ?¤è8￥V ?b x x 3 ? 12 1 ? =′ x y
 V !wL 1?= xy V ? ￥MLZ?1 ), 00 yx
)( 2 1 0 0 0 xx y yy ?=?
 7NMLVe?
?N V¤ 1,2 00 == yx
? ^MLZ?1 xy 2 1 = b 242 è8￥V ? ∫∫ ? +?++= 2 1 2 0 )14 1 112 4 1 1 2 2 dx x xdx x A
ππ )1511( 6 34 2 5 2 1 2 0 ?=?+= ∫∫ π ππ dxxxdx b 
￡ ü ? bWwL
xxt yyt zzt tTT = = = ? ? ? ? ? ∈ (), (), (), [, ] 12
<°g?? Oxyü ? ???￥? ?￥ ? T1 S = ′ + ′ ∫ zt xt yt dt T T ( ) [ ( )] [ ( )] 22 1 2
 ? úL ! ′xt()

′yt() ′zt()  ??
O b ],[ 21 TT 0)( ≥tz ￡ T ￥?s
],[ 21 TT 22101 TttttT n =<<<<="
 ! bWwL??
luW ￥l? ],[ 1 ii tt ? Oxyü ?￥g?￥é1 i s?
5
1 22 2 2 ['()]['()] ['()]['()] i i t ii t sxtytdtxyξξ ? ?= + = + ? ∫ t

? ],[ 1 iii tt ? ∈ξ b? ^? l? <°g?? Oxyü ? ???￥? ? ￥ ?1 22 () ()[()] [()] iiii i i Sz sz x y tξξξξ′′?≈ ?= + ? i b 7 1 max( ) i in tλ ≤≤ =?
ü¤? 22 0 1 lim ()[()] [()] n ii ii t? i Szxy λ ξξ ξ → = ′′=+ ∑ 2 1 22 () [ ()] [ ()] T T zt x t y t dt′′=+ ∫ b 
p/ wL·??￥w q?w q??b



?

4=xy )2,2(

 )cos1(),sin( tayttax ?=?=


0>a 2/π=t ?￥?b
3

 32 8 , 4 x y x y =′′?=′
? ^

243 22, 4 2 ])(1[ 2 3 2 == ′+ ′′ = R y y K b


 2 4 2 sin 1 cot , csc (1 cos ) 2 4 2 dy a t t d y t dx a t dx a == =? ?
? ^
aR a y y K 22, 4 2 ])(1[ 2 3 2 == ′+ ′′ = b

p/ wL￥w q?w q??b



?tL



pxy 2 2 = 0>p


?wL 1 2 2 2 2 =? b y a x




??L 3 2 3 2 3 2 ayx =+



0>a


?￥v 7L )cos(sin),sin(cos tttaytttax ?=+=

b
0>a
3

 2 2 1 , dx y d x dy p dy p ==
? ^

2 3 2 3 22 2 2 3 2 )2()(])(1[ 1 xp p yp p p y p K + = + = + =
 p xp R 2 3 )2( + = b


 7
5

tbytax tan,sec ==

t a b tt tt a b dx yd t a b tt t a b dx dy 3 222 22 cot sectan csccot ,csc sectan sec ?= ? ?===

? ^

2 3 4222 4 2 3 222 3 2 3 2 3 2 ])[()csc( cot ])csc1[ cot axba ba tba tab t a b t a b K ?+ = + = + =







ba axba R 4 2 3 4222 ])[( ?+ = b


 7
5
taytax 33 sin,cos == 244

22 22 3sin cos 1 1 tan , 3 cos sin 3 sin cos dy a t t d y t dx a t t dx a t t ==?=? ? 4

? ^

4 3 3 2 2 11 11 13sincos 3sincos 3 (1 tan ) at t K att axy t === +
 3 3R axy= b


 22 23 (cos cos sin ) sec 1 tan , ( sin sin cos ) cos cos dy a t t t t d y t t dx a t t t t dx at t at t ?+ === ?++ =

? ^

at t tat K 1 )tan1( cos 1 2 3 2 3 = + =
 atR = b

pw L ? )￥w q?Z?b
xy ln= )0,1(
3

2 1 , 1 x y x y ?=′′=′
 ?[wL? )￥w q1

)0,1( , 4 2 ) 1 1( 1 2 3 2 2 = + = x x K
w q??1
22=R b
??wL xy ln= ? )￥ML| q1
)0,1( 1=′y
 ? [ELZ? 1
1+?= xy
 ! 1w q?￥??
5),( ba 1+?= ab b
?
3¤
8)0()1( 22 =?+? ba 2,3 ?== ba
 ?[w q?￥Z?1


8)2()3( 22 =++? yx b

!w L￥USZ?1 )(θrr =
 ])2,0[(],[ πβαθ ?∈
O )(θr =¨ V ?b￡ ü ?? ),( θr )￥w q1
2/322 22 )( 2 rr rrrr K ′+ ′′?′+ = . ￡ !wL￥? ?Z?1
5w q1 ? ? ? = = )( )( tyy txx 245 2 3 22 ]))(())([( )()()()( tytx txtytxty K ′+′ ′′′?′′′ = b ? ()cos, ()sinxr yrθ θθ==θ
 V¤ cos sin , sin cosxr r yr rθ θθ′′ ′′=? =+θ
 cos 2 sin cos , sin 2 cos sinxr r r yr r rθ θθ θ θ′′ ′′ ′ ′′ ′′ ′=?? =+ ?θ
 ? ^ 2222 )()( rryx ′+=′+′

 rrrrxyxy ′′?′+=′′′?′′′ 22 2 ?[ 2/322 22 )( 2 rr rrrr K ′+ ′′?′+ = b 246