/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
1
§ + + + +
" " " " x x x x
1,M-+ – \ o M-+ – \ o M-+ – \ o M-+ – \ o
+ M-+,X) C? n) ¨ G+ o 0 s C? X · G,X?B ¨?-?M-+ – \ o?
1-57 + o 0 s C? X · G
A' v& + 9K q
0
" QAPBQ < 0,X s L? ¨ G
=
QAPBQ
0
0dlEq
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
2
+ !8? k
=
QAP QBP
dlEdlE
G" K` ) C?,X/? L? ¨ /? C? X · G1
2 M-+ M-+ M-+ M-+
– \ oEfl+ ′ ‘+,X V?
1 ˉ ˉ ˉ ˉ + 6 + 6 + 6 + 6
M-+ o 2O!currency1 ¨?5 F,s C? X · G,X-?G￡ ¨ G – \ o ¨ FEfl 6,X V o ¨
B & ￠,X& P /ˇ & Q? ¨? o,s1 b+ P Q & 6,X ￡?
1-58 + o 0 s ¨ + 6 ￡
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
3
1-59 " # "ˇ5 ¨ ",X 6 3 ￡
2O ¨ M-+ ¨ ’ ^A' v+ 9K P /ˇ & Q? ¨ + o 0,X s h ’1 b+ P Q A' v+ 9K+ 6,X ￡
QPPQ
WWW -=
==
Q
P
0PQpQ
dlEqAW
W
P
/?+ 6 P &,X+ 6 ¨ E ^ ·L$E,X+ 6 0 5 & ¨ ·? b b L$0NK¨8 ¨ Y+ 9K {*,X+ 9A¨ ¨
^ ·L$E,X+ 6 0 L?& ¨ G
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
4
￥
=
P
0P
dlEqW
2 ˉ ˉ ˉ ˉ + + ′+ + ′+ + ′+ + ′
A' v+ 9K q
0
U?+ o,s? W
PQ
/q
0
A' v+ 9K · G ¨ M-+,X?B G?
==
Q
P0
PQ
PQ
dlE
q
W
U
/? + ′ ¨ E,·0KE & + L? ¨?Ax q
0
P,X+
￥
￥
-= =
P
P
dlEdlE)P(U
Fw ¨ PQ?& K¨,X+ ′
￥
￥
￥ ￥
- = + =
Q
P P
Q
P Q
dlEdlEdlEdlEdlE
)Q(U)P(UU
PQ
-=
G 5 + · G?
^+ 6,XL?& E ·0KE ¨,h+,XL?& 3E ·0KE? rL K′Nl ¨+ <? ˙,X+
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
5
LL?&,X !5B ¨ &,X+ 6 ‘+,X D Lc- ¨ F?,G￡ ¨?7˙? E& K′,X+ 6 ′?
& K′,X+ ′?
3 ˉ ˉ ˉ ˉ +,X ) !+,X ) !+,X ) !+,X ) !
+ 6,X ) ! 6 G￡,X ) !,¨ * &?6? J ˉ? 5 +,X ) ! + 6 ‘ <? ¨ + 4 2 ￡E?+?B,X(=) G￡ ¨
+ "u + 9K · G?
+ ′ ‘+,X ) ! &?6? g ￥ ¨ SI /? a(M ¨ * 8ˉ [ V a? m V GG￡ 6 a
1C
1J
1V,,
1
1
1 ==?
g ￥
&?6
a(M
" " " " h – 5 h – 5 h – 5 h – 5
1 & + 9K,X+ & + 9K,X+ & + 9K,X+ & + 9K,X+
& + 9K,X+ z
r
2
0
e
r
q
4
1
E
pe
=
+ +,X n? k
r4
q
r
dr
4
q
r
dle
4
q
dlE)r(U
0r
2
0r
2
r
r 0
pepepe
==
= =
￥￥￥
2 + · t s) + · t s) + · t s) + · t s)
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
6
& + 9K2ˇ ¨ +,X · t s) ¨
= = = =
￥￥ ￥
i
i
i
r
i
r r
i
i
)r(UdlEdl)E(dlE)r(U
!8 a>< ¨ & + 9K4,X+ 1 + 9K )(,?+,X? D ‘? D ‘? D ‘? D ‘?
3 & + 9K4,X+ & + 9K4,X+ & + 9K4,X+ & + 9K4,X+
N & + 9K4,X ’2ˇ ¨ q
1
,q
2
,q
3
,…,q
N
,! b r
1
,r
2
,r
3
,…,r
N
¨ ! b r
i
q
i
)( r {*,X+
i
i
0
i
rr
q
4
1
)r(U
-
=
pe
+ + · t s) ¨ N + 9K r {*,X +
-
==
N
i i
i
0
N
i
i
rr
q
4
1
)r(U)r(U
pe
4+ ’,X+?+ ’,X+?+ ’,X+?+ ’,X+
E?4`,X?+ ’ {*,X+? ¨ ^?+ ’ F A A,X& + 9K ¨ ! b r’ & + 9K,X+ G￡ dq ¨ + +,X ·
t s) ¨ E?4`?+ ’ r {*,X +
-
=
'rr4
dq
)r(U
pe
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
7
1-60?+ ’,X+
’?+ ’? M6?+ ’ ‘4?+ ’ ¨ 8? J+ 9K z r? r’ ˉ ¨ s? r’ ˉ ‘ l? r’ ˉ ¨
'dV
'rr
)'r(
4
1
)r(U
'V0
-
=
r
pe
'dS
'rr
)'r(
4
1
)r(U
'S0
-
=
s
pe
'dl
'rr
)'r(
4
1
)r(U
'L0
-
=
l
pe
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
8
l fi l fi l fi l fi
1 1 M61 M61 M61 M6
￠ E?+,X n J 8 ><E a?-? ¨ + 0NK¨ $,X G￡ D ¨ G￡? G￡ * 1 M6 9Efl> 6B5,X · ) ￡E?
+,X1 M6/? 1 M6 ¨ 1 M6 ¨ +,1?
h,X!7B & + 9K2ˇ4?,X1 M6? r4 ˉ,X/ a? E?* <.4 +? Z+ 4,X? 4 Z+ J U $,X1 M6?
1-61 !7B & + 9K,X1 M6
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
9
1-62? !7+ 9K,X1 M6
2 1 M6,X(M?1 M6,X(M?1 M6,X(M?1 M6,X(M?
1 ˉ ˉ ˉ ˉ + 46 1 M6,x + 46 1 M6,x + 46 1 M6,x + 46 1 M6,x
2 ˉ ˉ ˉ ˉ 0NK¨ E&,X+ z h A?,X1 M6 V,¨0NK¨ E&,X+ z h A?,X1 M6 V,¨0NK¨ E&,X+ z h A?,X1 M6 V,¨0NK¨ E&,X+ z h A?,X1 M6 V,¨
*?A "' 9A? A A?4§A?,¨ G+ z? V,¨1 M6 ¨ E+ z " 1 M6,X"' ‘,X?
G￡ ¨? G￡ H? 0? a+ +,X @ a ¨ 1 M6 ! bA?,X?& K¨,+ ′ ¨?7˙ 1 M6,X n
￥* -fl,¨? s4§A 0? + E? 4§A?-? ¨ + 4 ‘1 M6 K¨ !7 x?
3 ˉ ˉ ˉ ˉ + z,X? a 3?* 1 M6,X+c / z 9G￡ z+ z,X? a 3?* 1 M6,X+c / z 9G￡ z+ z,X? a 3?* 1 M6,X+c / z 9G￡ z+ z,X? a 3?* 1 M6,X+c / z 9G￡ z
!8?A,F 1 M6,X+ ′F ¨ Fw ) !!7+ 9K" "'4 ￠ 1 M6/ˇ J,F,X1 M6 ¨ +
0,X s,X? a 3? E? s,X? a + z,F 1 M6K¨C–/,X,/ ·!8 ¨ 1 M6K¨C–C^ a ¨ + C^
1 M6K¨C–,X? a!7? Z1 M6,X+c / z ¨ +,X? a?* 1 M6,X+c / z 9G￡ z?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
10
| | | | " " " " E U h h h h
+ G￡ ¨ ￠+ 9K Au1k+ !currency1Au1k? 8?6 ￠+ "? ¨ E? ’ M2 a,X?
1,+ z+ z+ z+ z
M2 M4E￥,X1 M6 A ¨ B ‘ C ¨ W,X+
UU,U,UU DD +-
) !& + 9K ￠ B /ˇ7˙ C ¨ + o,,X s1 b+ 6,X ￡ ¨ G
UlE DD -=
lEU
l
DD -=
z G
l
U
E
l
D
D
-=
1-63 +,X — D
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
11
¨?,X DU ¨ "?,X ¨ + b Dl,XKS z? ¨ E
l
,X? J?,?
D,¨? b ˇ ) G￡ ¨? n J z ¨ W -?G￡ ¨? a1 bA? G￡ D" J1 M6,X"'4,X
— D ¨ " 1 M6,X"'4 ¨ J* grad ></ z ¨ b,
n
n
U
UgradU
-= =
2,+ z + z + z + z U,X G2ˇ,X G2ˇ,X G2ˇ,X G2ˇ
’ l,"'4? ¨ + A?,X)[? ¨ z G+,X z? ¨ B-?G￡,X n ¨ -?G￡4–,X G￡
-?G￡ D ¨? k + z,X? a
n
U
n
U
EE
n
D
D
D
D
=-==
’ Dn C_ bL ¨
n
U
n
U
limE
0n
==
fi D
D
+ +,X z ‘ a ¨
gradUE -=
GM-+ ˇ ) &,X+ z,X? a D 1 bA?& + z,X? a ¨ + z,X,? ¨ + L!:,X
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
12
Oxy,¨ $ 2ˇ ¨ E?* A? $ 2ˇ,X G￡ 9></ ¨ G
U
e
z
U
e
y
U
e
x
U
eEeEeEE
zyx
zzyyxx
- =
-
-
-=
++=
E?1k0? ¨,¨ $ 2ˇ
zyx
e
z
e
y
e
x?
+
+
=
) $ 2ˇ
jq
jqq
e
sinr
1
e
r
1
e
r
r
+
+
=
-?+,X ¨?" k+ z,X?
3 A|A A|A A|A A|A
1 ˉ + U? x ¨ y ¨ z ˉ G￡ ¨ G￡ ¨ -?G￡ ¨ G￡ ¨ )+ E=- U 6 4 D E
x
E
y
‘
E
z
6 J r ¨ M-+ JM2 ‘ < ˇ a,X-?G￡ ¨ W NO$ C ) C? n) ¨ ·5 E,X G￡ J? ( 0,X? 6 *
G￡ D U 9 ￡ mM-+ ¨ J+ k -?G￡? ˉ ¨ + M-+ – \,X?B n,X?
2 ˉ M-+,X) C? n) ￠ g ￥ n _ —?,X ¨ · g ￥ n _? V ZM-+ o E? (M? ¨ o ¨ J) #
F & L 6 * G￡ D ￡ mM-+,X !? M-+ o ¨ 5U" M-+ o C Z?
7˙ b D,X K ’ 6 a ¨ E?,b o,X K ’ 6 a ¨ GL } bP? n)? + + 9K,B n,X+ D @ a ¨?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
13
Z+ 9KK¨,f 0*,X G?!currency1 _E? Y ¨ G? Z g ￥ n _,X <F… C? ￡ mM-+,X? /
0E,E =·=
e
r
+ ) C? n)? k E ‘ U,X G2ˇ ¨ G E=- U ¨? 9 P? n)
0
2
U
e
r
-=
/?" R / ¨,¨ $ 2ˇ ¨? m
0
2
2
2
2
2
2
z
U
y
U
x
U
e
r
=
+
+
-?+ 9K ¨ E E "A? / ¨ z? k +?
3 ˉ + D,X +,XL? 5 &,XE ‰ G ¨ 5 + L? 5 &,XE ‰ \?,X ˇ a? ¨ + ˇ ) & F 0 +
,XL?&? ^+,XL?&,·0KE · L$?,X+ 9K {*,X+ E / + 9K,X 1? r
2
￡ ¨
·L$E ˇ a?&,X+ ′ L? ¨ ·L$E +,X1? ¨ ·5 ^+,XL?&,E
¨ ’+ 9K ·L$ ¨ ·L$E J? 1 E?/? ¨ <Q’,E?,B n&
0 +,XL?& ¨ H?6 ^ ·L$E 0 +,XL? 5 &? ·L$E? ˉ?
h h h h h* _ h* _ h* _ h* _
[ _ _ _ _ ] " + J U $,X+ +,X?
[ ],+ J U $,X & $ s& O ¨ r? l ¨
)
rr
1
rr
1
(
4
q
)r(U
0 -+
-
-
-
=
pe
+ D,,X4{ D ) ¨ E￥ k
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
14
q
q
cos
r
l
1rrr
cos
r
l
1rrr
+?-
-?-
-
+
1-64 + J U $,X+ +,X
^? a? 9 ¨ J +9 2 P?L NM ¨
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
15
2
0
r4
cosql
)r(U
pe
q
=
5 <% + J U $,X ¨ P &,X+
3
0
r4
rp
)r(U
pe
=
+ A? a ‘+ z +,X G2ˇ a?" k+ z
qq
q
jq
q
pe
q
pe
jqq
eEeE
e
r
sinp
4
1
e
r
cosp2
4
1
e
U
sinr
1
e
U
r
1
e
r
U
UE
rr
3
0
r
3
0
r
+=
+=
-
-
-=- =
=
=
=
0E
r
sinp
4
1
E
r
cosp2
4
1
E
3
0
3
0
r
j
q
q
pe
q
pe
+ J U $,X?KS4 ¨ q=0 ¨
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
16
3
0
r
r
p2
4
1
EE
pe
==
+ J U $,X VM6 ¨ q=p/2 ¨
3
0
r
p
4
1
E
pe
-=
+ J U $,X+ z 3? m
5
0
3
0
r4
r)rp(3
r4
p
E
pepe
+-=
[ _ _ _ _ ] " M6+ 9K z s,X?+ ;X,?EH4,X + + ¨,? X R?
[ ] 0 ¨ EH4 ˇ & Z,X+
-
=
S0
'rr
dS
4
1
)z(U
s
pe
E?G ¨ dS=rdfdr ¨ ‰r-r’‰=(z
2
+r
2
)
1/2
,
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
17
[ ]
[ ]
[ ]
++
-+
=
-+=
+
=
+
=
zzR
2
,zzR
2
zzR
2
zr
rdr
2
zr
rd
dr
4
1
)z(U
22
0
22
0
22
0
R
0
22
0
R
0
2
0
22
0
e
s
e
s
e
s
e
s
js
pe
p
Z>0
Z<0
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
18
1-65?+,?EH4,X+ ‘+ z
¨ + z E
z
e
z
U
UE
-=- =
,
e
zR
z
1
2
e
zR
z
1
2
E
z
22
0
z
22
0
+
+
+
-
=
e
s
e
s
Z>0
Z<0
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
19
’ Rfi? ¨ G k ·L$+ GM6,X+ z
-
=
z
0
z
0
e
2
e
2
E
e
s
e
s
1-66 `,?EH,X+ Z,X G2ˇ
[ _ _ _ _ ]?+ ¨ z r ¨ Y? X R
1
‘ R
2
,X) ˙ ¨ " J+ ‘+?
Z>0
Z<0
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
20
[ ] V / ¨ * P? n) "?+ ¨ a" +?
0NK¨ K¨ ¨ 0 X r,XP? M6 ¨
)R(r,
r
e
)RR(
3
e
r4
Q
E
2
2
r3
1
3
2
0
r
2
0
1
-==
e
r
pe
)Rr(R,e)
r
R
r(
3
E
21r
2
3
1
0
2
￡￡-=
e
r
)R(r,0E
13
￡=
B U? r ˉ ‘ E,X G2ˇ ¨+ /? k U? r ˉ
)R(r,
r
1
)RR(
3r
dlr
)RR(
3
)r(U
2
3
1
3
2
0
3
2
1
2
2
r
0
1
-=
--=
￥
e
r
e
r
)Rr(R ),
2
r
r
R
R
2
3
(
3
dlr)
r
R
1(
3
)R(UdlEdlE)r(U
21
23
12
2
0
r
R
3
3
1
0
2
R r
R
12
2
2
2
￡￡--=
--= - -=
￥
e
r
e
r
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
21
)R(r ),RR(
2
)R(U
dlEdlEdlE)r(U
1
2
1
2
2
0
12
R R
R
r
R
3
2 1
2 1
￡-==
- - -=
￥
e
r
A|A U
3
r,X G2ˇ?-? ¨ ) ˙ Y0N7( 1 ’ ¨ J+ ˙ Y><M6,1?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
22
1-67?+ ) ˙,X+ ‘+
/ u / u / u / u
1? + 9K \ &+ 9K \ &+ 9K \ &+ 9K \ &
b?+ (,X(= ’?+ ’ 7 ’ +?/?+ 9K !7+ 9K ‘ B + 9K? /?+ 9K,y /?+ 9K,? 7 ’ + ¨ + 9K
+ $+ 9K,X H D) ¨ + 9K,XE? (M,+ 9K,XG￡ $ + 9K6 * ¨ 3?6 > #\&A ¨ 6 ￠ (= ’E@/ˇ?,
(= ’ ¨ 2ˇ4? Y ¨ ·A ￥*,X(=) E / ¨ A?2ˇ4? Y+ 9K,X+ G￡,X? D ‘ – !8? ¨ + 9K,?A
G￡?
2? g ￥ n _ g ￥ n _ g ￥ n _ g ￥ n _
g ￥ rP‘ ￥) ¨? M-!6& + 9KK¨,f 0* o? a!7!currency1 b!￡ & + 9K+ G￡ ¨?!currency1 b?& + 9K K¨C–/,X G ¨ J"?&
+ 9K,XE?4?
r
2
12
21
12
e
r
qq
4
1
F
pe
=
g ￥ n _>< + 0* o o? g ￥ n _,X S* 8 ¨ M-+ 9K ‘C–/ 10
7
m- 10
17
m 8 ¨?
3 · t s) · t s) · t s) · t s)
1u ’2ˇ,& + 9K ¨ !￡?& + 9K K¨,X 0* oF? ￠ g ￥ n _ 5 ˇ & + 9K?,X o1 b J,
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
23
& + 9K )( 0* bA?& + 9K,X g ￥ o,X-?G￡ ‘ ¨ E? 4§A /? · t s)?
=
i
i
FF
+ ’ K¨,X 0* o?* · t s) Au1k ¨ V? ’?+ ’ K¨,X 0* o
-
-
=
'
V
''
3
'
'
V0
dVdV)rr(
rr
)r()r(
4
1
F
rr
pe
4? + z+ z+ z+ z
+ 9K K¨,X 0* o M4+ E,X + 9K,X ¨ < ¨,+ ¨ G + 9K 7 ˉ,X < ¨ {* +?% ￥+ ¨ +
Y,X J,+ 9K o 0*? + 9K? +,X 0* o + A?+ 9K,X+ n ¨ J,,X+ · G? M-!6+ 9K {*
,X+ /? M-+ ¨ W?Lc?K¨5
* A' v& + 9K-?0J+ z? + Y ˇ a &,X+ z D 1 b ) !+ G￡,X!7+ 9K A?&?,X 0* o ¨
W A?& + 9K? o,X,? + z 0NK¨ $,X-?G￡ D ¨ E E? x ¨ y ¨ z ˉ ¨ G-?G￡ ¨?+ ’ < ¨,X K (M
n?B,X0NK¨? W K 6 G￡ ¨ /? v? (=B? & + 9K,X+ z
r
2
0
e
r
q
4
1
E
pe
=
+ z $ C +,X · t s)? & + 9K2ˇ ‘?+ ’ {*,X+ z?* · t s) " k ¨ V?+ ’,X+ z
'dV
'rr
)r(
4
1
E
'V
3
'
0
-
r
e
=
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
24
5? P? n) P? n) P? n) P? n)
0NK¨+* + o4 ></? + o4,X 4,X,h&,X 7¨ ¨? W,X D z A?&,X,X? a !7
!currency1? + 4 L,X ¨? ¨ + 4 / +c,X ¨ a?
+? ˇ a K`?M6,X+ E G￡ n b> ¨ K`?M6 YF…,X+ 9K ¨?1 b ¨ K`?M6 Y+ G￡? D ‘L8? e
0
¨
K`?M6?,X+ 9K · G?
=
=
Y
1
0
1
i
i
S
qdSE
e
P? n) >< M-+ $d ¨ V p P? M6 YF…F…+ 9K ￥*? ¨ 0NK¨+ ￥* ¨?UP? M6 Y
F… + 9K D ¨ P? n)? ¨ +?E E A? K`?M6,X+ E G￡ J ·
* P? n)Au1k K · )?/? 6(,X?+ ’,X+ z? P? n) M-+,X 5GU? n) ¨ W ￠ g ￥ n _ —
9 ¨ !currency1 g ￥ n _ ¨ BF!?
6 ) C? n) ) C? n) ) C? n) ) C? n)
M-+,X) G￡></ M-+?" A?K` C? X/ˇ |,X ) !!7+ 9K 0,X s? M-+ · ¨ M-+ 0 s C? X · G ¨ CK&
4 &,X !5B G?M-+ " ˇ )K` ) C? 0 s L
0dlE
L
=
+,XEB 9$d b g ￥ o,X o(M? 5? G?!currency1 _? ) C? n) >< M-+ + o46 K`?4?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
25
7 + + + +
M-+,X) C? n) ¨ G+ o 0 s C? X · G,X?B ¨?-?M-+ – \ o? – \ o,X(M?,D? M-+
¨ ’ ^+ 9K + E? |? ¨ + o 0,X s h ’1 b+ 6,X ￡? + +,X D (M? ¨ W D 1 b ) !& + 9K5B bA?
&,X+ 6 ) !& + 9K ￠ ·L$E 4￡ ˇ aC? X/ˇ A?& + o,,X s? + K,? a ¨? L$,X+ 9K ¨ E,·
L$E +,X 5 &? L?& ˉ?
M-+ ¨?&,X+ ′
=
b
a
ab
dlEU
& + 9K {*,X+
r4
q
)r(U
0
pe
=
+ $ C · t s) ¨ + 9K4 ‘?+ ’ {*,X+?* · t s) Au1k ¨ V?+ ’,X+
'dV
'rr
)'r(
4
1
)r(U
'V0
-
=
r
pe
+ 0NK¨ $,X G￡ D ¨ +,X1 M6/? 1 M6 ¨ 1 M6 ¨ +,1? M-+ ˇ ) &,X+ z
,X? a D 1 bA?& + z,X? a ¨ + z,X,? ¨ + L!:,X ¨
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
26
gradUE -=
-?+ U ¨?E E Au1k J z" k E ¨?-?+ z E ¨ 3?" k+?
1
§ + + + +
" " " " x x x x
1,M-+ – \ o M-+ – \ o M-+ – \ o M-+ – \ o
+ M-+,X) C? n) ¨ G+ o 0 s C? X · G,X?B ¨?-?M-+ – \ o?
1-57 + o 0 s C? X · G
A' v& + 9K q
0
" QAPBQ < 0,X s L? ¨ G
=
QAPBQ
0
0dlEq
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
2
+ !8? k
=
QAP QBP
dlEdlE
G" K` ) C?,X/? L? ¨ /? C? X · G1
2 M-+ M-+ M-+ M-+
– \ oEfl+ ′ ‘+,X V?
1 ˉ ˉ ˉ ˉ + 6 + 6 + 6 + 6
M-+ o 2O!currency1 ¨?5 F,s C? X · G,X-?G￡ ¨ G – \ o ¨ FEfl 6,X V o ¨
B & ￠,X& P /ˇ & Q? ¨? o,s1 b+ P Q & 6,X ￡?
1-58 + o 0 s ¨ + 6 ￡
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
3
1-59 " # "ˇ5 ¨ ",X 6 3 ￡
2O ¨ M-+ ¨ ’ ^A' v+ 9K P /ˇ & Q? ¨ + o 0,X s h ’1 b+ P Q A' v+ 9K+ 6,X ￡
QPPQ
WWW -=
==
Q
P
0PQpQ
dlEqAW
W
P
/?+ 6 P &,X+ 6 ¨ E ^ ·L$E,X+ 6 0 5 & ¨ ·? b b L$0NK¨8 ¨ Y+ 9K {*,X+ 9A¨ ¨
^ ·L$E,X+ 6 0 L?& ¨ G
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
4
￥
=
P
0P
dlEqW
2 ˉ ˉ ˉ ˉ + + ′+ + ′+ + ′+ + ′
A' v+ 9K q
0
U?+ o,s? W
PQ
/q
0
A' v+ 9K · G ¨ M-+,X?B G?
==
Q
P0
PQ
PQ
dlE
q
W
U
/? + ′ ¨ E,·0KE & + L? ¨?Ax q
0
P,X+
￥
￥
-= =
P
P
dlEdlE)P(U
Fw ¨ PQ?& K¨,X+ ′
￥
￥
￥ ￥
- = + =
Q
P P
Q
P Q
dlEdlEdlEdlEdlE
)Q(U)P(UU
PQ
-=
G 5 + · G?
^+ 6,XL?& E ·0KE ¨,h+,XL?& 3E ·0KE? rL K′Nl ¨+ <? ˙,X+
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
5
LL?&,X !5B ¨ &,X+ 6 ‘+,X D Lc- ¨ F?,G￡ ¨?7˙? E& K′,X+ 6 ′?
& K′,X+ ′?
3 ˉ ˉ ˉ ˉ +,X ) !+,X ) !+,X ) !+,X ) !
+ 6,X ) ! 6 G￡,X ) !,¨ * &?6? J ˉ? 5 +,X ) ! + 6 ‘ <? ¨ + 4 2 ￡E?+?B,X(=) G￡ ¨
+ "u + 9K · G?
+ ′ ‘+,X ) ! &?6? g ￥ ¨ SI /? a(M ¨ * 8ˉ [ V a? m V GG￡ 6 a
1C
1J
1V,,
1
1
1 ==?
g ￥
&?6
a(M
" " " " h – 5 h – 5 h – 5 h – 5
1 & + 9K,X+ & + 9K,X+ & + 9K,X+ & + 9K,X+
& + 9K,X+ z
r
2
0
e
r
q
4
1
E
pe
=
+ +,X n? k
r4
q
r
dr
4
q
r
dle
4
q
dlE)r(U
0r
2
0r
2
r
r 0
pepepe
==
= =
￥￥￥
2 + · t s) + · t s) + · t s) + · t s)
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
6
& + 9K2ˇ ¨ +,X · t s) ¨
= = = =
￥￥ ￥
i
i
i
r
i
r r
i
i
)r(UdlEdl)E(dlE)r(U
!8 a>< ¨ & + 9K4,X+ 1 + 9K )(,?+,X? D ‘? D ‘? D ‘? D ‘?
3 & + 9K4,X+ & + 9K4,X+ & + 9K4,X+ & + 9K4,X+
N & + 9K4,X ’2ˇ ¨ q
1
,q
2
,q
3
,…,q
N
,! b r
1
,r
2
,r
3
,…,r
N
¨ ! b r
i
q
i
)( r {*,X+
i
i
0
i
rr
q
4
1
)r(U
-
=
pe
+ + · t s) ¨ N + 9K r {*,X +
-
==
N
i i
i
0
N
i
i
rr
q
4
1
)r(U)r(U
pe
4+ ’,X+?+ ’,X+?+ ’,X+?+ ’,X+
E?4`,X?+ ’ {*,X+? ¨ ^?+ ’ F A A,X& + 9K ¨ ! b r’ & + 9K,X+ G￡ dq ¨ + +,X ·
t s) ¨ E?4`?+ ’ r {*,X +
-
=
'rr4
dq
)r(U
pe
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
7
1-60?+ ’,X+
’?+ ’? M6?+ ’ ‘4?+ ’ ¨ 8? J+ 9K z r? r’ ˉ ¨ s? r’ ˉ ‘ l? r’ ˉ ¨
'dV
'rr
)'r(
4
1
)r(U
'V0
-
=
r
pe
'dS
'rr
)'r(
4
1
)r(U
'S0
-
=
s
pe
'dl
'rr
)'r(
4
1
)r(U
'L0
-
=
l
pe
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
8
l fi l fi l fi l fi
1 1 M61 M61 M61 M6
￠ E?+,X n J 8 ><E a?-? ¨ + 0NK¨ $,X G￡ D ¨ G￡? G￡ * 1 M6 9Efl> 6B5,X · ) ￡E?
+,X1 M6/? 1 M6 ¨ 1 M6 ¨ +,1?
h,X!7B & + 9K2ˇ4?,X1 M6? r4 ˉ,X/ a? E?* <.4 +? Z+ 4,X? 4 Z+ J U $,X1 M6?
1-61 !7B & + 9K,X1 M6
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
9
1-62? !7+ 9K,X1 M6
2 1 M6,X(M?1 M6,X(M?1 M6,X(M?1 M6,X(M?
1 ˉ ˉ ˉ ˉ + 46 1 M6,x + 46 1 M6,x + 46 1 M6,x + 46 1 M6,x
2 ˉ ˉ ˉ ˉ 0NK¨ E&,X+ z h A?,X1 M6 V,¨0NK¨ E&,X+ z h A?,X1 M6 V,¨0NK¨ E&,X+ z h A?,X1 M6 V,¨0NK¨ E&,X+ z h A?,X1 M6 V,¨
*?A "' 9A? A A?4§A?,¨ G+ z? V,¨1 M6 ¨ E+ z " 1 M6,X"' ‘,X?
G￡ ¨? G￡ H? 0? a+ +,X @ a ¨ 1 M6 ! bA?,X?& K¨,+ ′ ¨?7˙ 1 M6,X n
￥* -fl,¨? s4§A 0? + E? 4§A?-? ¨ + 4 ‘1 M6 K¨ !7 x?
3 ˉ ˉ ˉ ˉ + z,X? a 3?* 1 M6,X+c / z 9G￡ z+ z,X? a 3?* 1 M6,X+c / z 9G￡ z+ z,X? a 3?* 1 M6,X+c / z 9G￡ z+ z,X? a 3?* 1 M6,X+c / z 9G￡ z
!8?A,F 1 M6,X+ ′F ¨ Fw ) !!7+ 9K" "'4 ￠ 1 M6/ˇ J,F,X1 M6 ¨ +
0,X s,X? a 3? E? s,X? a + z,F 1 M6K¨C–/,X,/ ·!8 ¨ 1 M6K¨C–C^ a ¨ + C^
1 M6K¨C–,X? a!7? Z1 M6,X+c / z ¨ +,X? a?* 1 M6,X+c / z 9G￡ z?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
10
| | | | " " " " E U h h h h
+ G￡ ¨ ￠+ 9K Au1k+ !currency1Au1k? 8?6 ￠+ "? ¨ E? ’ M2 a,X?
1,+ z+ z+ z+ z
M2 M4E￥,X1 M6 A ¨ B ‘ C ¨ W,X+
UU,U,UU DD +-
) !& + 9K ￠ B /ˇ7˙ C ¨ + o,,X s1 b+ 6,X ￡ ¨ G
UlE DD -=
lEU
l
DD -=
z G
l
U
E
l
D
D
-=
1-63 +,X — D
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
11
¨?,X DU ¨ "?,X ¨ + b Dl,XKS z? ¨ E
l
,X? J?,?
D,¨? b ˇ ) G￡ ¨? n J z ¨ W -?G￡ ¨? a1 bA? G￡ D" J1 M6,X"'4,X
— D ¨ " 1 M6,X"'4 ¨ J* grad ></ z ¨ b,
n
n
U
UgradU
-= =
2,+ z + z + z + z U,X G2ˇ,X G2ˇ,X G2ˇ,X G2ˇ
’ l,"'4? ¨ + A?,X)[? ¨ z G+,X z? ¨ B-?G￡,X n ¨ -?G￡4–,X G￡
-?G￡ D ¨? k + z,X? a
n
U
n
U
EE
n
D
D
D
D
=-==
’ Dn C_ bL ¨
n
U
n
U
limE
0n
==
fi D
D
+ +,X z ‘ a ¨
gradUE -=
GM-+ ˇ ) &,X+ z,X? a D 1 bA?& + z,X? a ¨ + z,X,? ¨ + L!:,X
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
12
Oxy,¨ $ 2ˇ ¨ E?* A? $ 2ˇ,X G￡ 9></ ¨ G
U
e
z
U
e
y
U
e
x
U
eEeEeEE
zyx
zzyyxx
- =
-
-
-=
++=
E?1k0? ¨,¨ $ 2ˇ
zyx
e
z
e
y
e
x?
+
+
=
) $ 2ˇ
jq
jqq
e
sinr
1
e
r
1
e
r
r
+
+
=
-?+,X ¨?" k+ z,X?
3 A|A A|A A|A A|A
1 ˉ + U? x ¨ y ¨ z ˉ G￡ ¨ G￡ ¨ -?G￡ ¨ G￡ ¨ )+ E=- U 6 4 D E
x
E
y
‘
E
z
6 J r ¨ M-+ JM2 ‘ < ˇ a,X-?G￡ ¨ W NO$ C ) C? n) ¨ ·5 E,X G￡ J? ( 0,X? 6 *
G￡ D U 9 ￡ mM-+ ¨ J+ k -?G￡? ˉ ¨ + M-+ – \,X?B n,X?
2 ˉ M-+,X) C? n) ￠ g ￥ n _ —?,X ¨ · g ￥ n _? V ZM-+ o E? (M? ¨ o ¨ J) #
F & L 6 * G￡ D ￡ mM-+,X !? M-+ o ¨ 5U" M-+ o C Z?
7˙ b D,X K ’ 6 a ¨ E?,b o,X K ’ 6 a ¨ GL } bP? n)? + + 9K,B n,X+ D @ a ¨?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
13
Z+ 9KK¨,f 0*,X G?!currency1 _E? Y ¨ G? Z g ￥ n _,X <F… C? ￡ mM-+,X? /
0E,E =·=
e
r
+ ) C? n)? k E ‘ U,X G2ˇ ¨ G E=- U ¨? 9 P? n)
0
2
U
e
r
-=
/?" R / ¨,¨ $ 2ˇ ¨? m
0
2
2
2
2
2
2
z
U
y
U
x
U
e
r
=
+
+
-?+ 9K ¨ E E "A? / ¨ z? k +?
3 ˉ + D,X +,XL? 5 &,XE ‰ G ¨ 5 + L? 5 &,XE ‰ \?,X ˇ a? ¨ + ˇ ) & F 0 +
,XL?&? ^+,XL?&,·0KE · L$?,X+ 9K {*,X+ E / + 9K,X 1? r
2
￡ ¨
·L$E ˇ a?&,X+ ′ L? ¨ ·L$E +,X1? ¨ ·5 ^+,XL?&,E
¨ ’+ 9K ·L$ ¨ ·L$E J? 1 E?/? ¨ <Q’,E?,B n&
0 +,XL?& ¨ H?6 ^ ·L$E 0 +,XL? 5 &? ·L$E? ˉ?
h h h h h* _ h* _ h* _ h* _
[ _ _ _ _ ] " + J U $,X+ +,X?
[ ],+ J U $,X & $ s& O ¨ r? l ¨
)
rr
1
rr
1
(
4
q
)r(U
0 -+
-
-
-
=
pe
+ D,,X4{ D ) ¨ E￥ k
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
14
q
q
cos
r
l
1rrr
cos
r
l
1rrr
+?-
-?-
-
+
1-64 + J U $,X+ +,X
^? a? 9 ¨ J +9 2 P?L NM ¨
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
15
2
0
r4
cosql
)r(U
pe
q
=
5 <% + J U $,X ¨ P &,X+
3
0
r4
rp
)r(U
pe
=
+ A? a ‘+ z +,X G2ˇ a?" k+ z
q
jq
q
pe
q
pe
jqq
eEeE
e
r
sinp
4
1
e
r
cosp2
4
1
e
U
sinr
1
e
U
r
1
e
r
U
UE
rr
3
0
r
3
0
r
+=
+=
-
-
-=- =
=
=
=
0E
r
sinp
4
1
E
r
cosp2
4
1
E
3
0
3
0
r
j
q
q
pe
q
pe
+ J U $,X?KS4 ¨ q=0 ¨
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
16
3
0
r
r
p2
4
1
EE
pe
==
+ J U $,X VM6 ¨ q=p/2 ¨
3
0
r
p
4
1
E
pe
-=
+ J U $,X+ z 3? m
5
0
3
0
r4
r)rp(3
r4
p
E
pepe
+-=
[ _ _ _ _ ] " M6+ 9K z s,X?+ ;X,?EH4,X + + ¨,? X R?
[ ] 0 ¨ EH4 ˇ & Z,X+
-
=
S0
'rr
dS
4
1
)z(U
s
pe
E?G ¨ dS=rdfdr ¨ ‰r-r’‰=(z
2
+r
2
)
1/2
,
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
17
[ ]
[ ]
[ ]
++
-+
=
-+=
+
=
+
=
zzR
2
,zzR
2
zzR
2
zr
rdr
2
zr
rd
dr
4
1
)z(U
22
0
22
0
22
0
R
0
22
0
R
0
2
0
22
0
e
s
e
s
e
s
e
s
js
pe
p
Z>0
Z<0
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
18
1-65?+,?EH4,X+ ‘+ z
¨ + z E
z
e
z
U
UE
-=- =
,
e
zR
z
1
2
e
zR
z
1
2
E
z
22
0
z
22
0
+
+
+
-
=
e
s
e
s
Z>0
Z<0
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
19
’ Rfi? ¨ G k ·L$+ GM6,X+ z
-
=
z
0
z
0
e
2
e
2
E
e
s
e
s
1-66 `,?EH,X+ Z,X G2ˇ
[ _ _ _ _ ]?+ ¨ z r ¨ Y? X R
1
‘ R
2
,X) ˙ ¨ " J+ ‘+?
Z>0
Z<0
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
20
[ ] V / ¨ * P? n) "?+ ¨ a" +?
0NK¨ K¨ ¨ 0 X r,XP? M6 ¨
)R(r,
r
e
)RR(
3
e
r4
Q
E
2
2
r3
1
3
2
0
r
2
0
1
-==
e
r
pe
)Rr(R,e)
r
R
r(
3
E
21r
2
3
1
0
2
￡￡-=
e
r
)R(r,0E
13
￡=
B U? r ˉ ‘ E,X G2ˇ ¨+ /? k U? r ˉ
)R(r,
r
1
)RR(
3r
dlr
)RR(
3
)r(U
2
3
1
3
2
0
3
2
1
2
2
r
0
1
-=
--=
￥
e
r
e
r
)Rr(R ),
2
r
r
R
R
2
3
(
3
dlr)
r
R
1(
3
)R(UdlEdlE)r(U
21
23
12
2
0
r
R
3
3
1
0
2
R r
R
12
2
2
2
￡￡--=
--= - -=
￥
e
r
e
r
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
21
)R(r ),RR(
2
)R(U
dlEdlEdlE)r(U
1
2
1
2
2
0
12
R R
R
r
R
3
2 1
2 1
￡-==
- - -=
￥
e
r
A|A U
3
r,X G2ˇ?-? ¨ ) ˙ Y0N7( 1 ’ ¨ J+ ˙ Y><M6,1?
/￥,T?,+,,? 1 0·,0N,XM-+ 05?Fz
22
1-67?+ ) ˙,X+ ‘+
/ u / u / u / u
1? + 9K \ &+ 9K \ &+ 9K \ &+ 9K \ &
b?+ (,X(= ’?+ ’ 7 ’ +?/?+ 9K !7+ 9K ‘ B + 9K? /?+ 9K,y /?+ 9K,? 7 ’ + ¨ + 9K
+ $+ 9K,X H D) ¨ + 9K,XE? (M,+ 9K,XG￡ $ + 9K6 * ¨ 3?6 > #\&A ¨ 6 ￠ (= ’E@/ˇ?,
(= ’ ¨ 2ˇ4? Y ¨ ·A ￥*,X(=) E / ¨ A?2ˇ4? Y+ 9K,X+ G￡,X? D ‘ – !8? ¨ + 9K,?A
G￡?
2? g ￥ n _ g ￥ n _ g ￥ n _ g ￥ n _
g ￥ rP‘ ￥) ¨? M-!6& + 9KK¨,f 0* o? a!7!currency1 b!￡ & + 9K+ G￡ ¨?!currency1 b?& + 9K K¨C–/,X G ¨ J"?&
+ 9K,XE?4?
r
2
12
21
12
e
r
4
1
F
pe
=
g ￥ n _>< + 0* o o? g ￥ n _,X S* 8 ¨ M-+ 9K ‘C–/ 10
7
m- 10
17
m 8 ¨?
3 · t s) · t s) · t s) · t s)
1u ’2ˇ,& + 9K ¨ !￡?& + 9K K¨,X 0* oF? ￠ g ￥ n _ 5 ˇ & + 9K?,X o1 b J,
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23
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26
gradUE -=
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