~ ?yvD ?Dy
? ??
?±s# ?¨
~ ?yvD ?Dy
),(),( yxfyxxf+
xyxf
x
≈ ),(
),(),( yxfyyxf+
yyxf
y
≈ ),(
=íf
x y￥
ê±s
=íf
x y￥
ê9
?Bíf
±sD?9
D±s￥1"¤
B a ?±s￥?l
~ ?yvD ?Dy
?Tf
),( yxfz = ? ),( yx ￥
#×
=
μ?li
! ),( yyxxP?+?+
′
1?
#×
=￥
?iB?5??
?￥f
′-μ
),(),( yxfyyxxf+?+
1f
?
P
??1M
9
yx,￥ ?9
:1 z?
'
z? = ),(),( yxfyyxxf+?+
?9
￥à
Q
~ ?yvD ?Dy
?Tf
),( yxfz = ? ),( yx ￥ ?9
),(),( yxfyyxxfz+?+=? V[V
U1
)(ρoyBxAz +?+?=? ? BA,?G ??
yx,7?D yx,μ1
22
)()( yx?+?=ρ
5?f
),( yxfz = ? ),( yx V±s
yBxA?+? ?1f
),( yxfz = ? ),( yx ￥
?±s :1 dz' dz yBxA?+?
?±s￥?l
~ ?yvD ?Dy
f
?
u× D
=ò?)) V±s 5
??f
D
= V±s,
?Tf
),( yxfz = ? ),( yx V±s,5
f
?? ??,
Y
L
),(ρoyBxAz +?+?=?
,0lim
0
=?
→
z
ρ
),(lim
0
0
yyxxf
y
x
+?+
→?
→?
]),([lim
0
zyxf?+=
→ρ
),( yxf=
#f
),( yxfz = ? ),( yx ) ??,
~ ?yvD ?Dy
= a V±￥Hq
? ?
A1Hq ?Tf
),( yxfz = ?
),( yx V±s5?f
? ),( yx ￥
ê?
x
z
a
y
z
Ai Of
),( yxfz = ? ),( yx ￥ ?±s
1
y
y
z
x
x
z
dz?
+?
=
~ ?yvD ?Dy
￡
?Tf
),( yxfz = ? ),( yxP V±s,
∈?+?+
′
),( yyxxP P￥
?
#×
)(ρoyBxAz +?+?=? 9? ?,
? 0=?y
H
T ˉ? ? N
H || x?=ρ,
),(),( yxfyxxf+ |),(| xoxA?+=
A
x
yxfyxxf
x
=
+
→?
),(),(
lim
0
,
x
z
=
] ? V¤
.
y
z
B
=
~ ?yvD ?Dy
Bíf
?￥?
i ±si
íf
￥ò
ê?
i ?±si
è ?
.
00
0
),(
22
22
22
=+
≠+
+
=
yx
yx
yx
xy
yxf
? )0,0( )μ
0)0,0()0,0( ==
yx
ff
~ ?yvD ?Dy
])0,0()0,0([ yfxfz
yx
+,
)()(
22
yx
yx
+?
=
?T I
n? ),( yxP
′
"°L xy = tí? )0,0(
5
ρ
22
)()( yx
yx
+?
22
)()( xx
xx
+?
=,
2
1
=
a
ü
?
" 0→ρ 7 t? 0,0→ρ
?
H
),(])0,0()0,0([ ρoyfxfz
yx
≠+
f
? )0,0( )? V±,
~ ?yvD ?Dy
a
ü íf
￥ò
ê?
i i?
￡ ?
±si
? ?
sHq ?Tf
),( yxfz = ￥
ê
?
x
z
a
y
z
? ),( yx ??5?f
?
),( yx V±s
￡
),(),( yxfyyxxfz+?+=?
)],(),([ yyxfyyxxf?++?+=
)],,(),([ yxfyyxf++
~ ?yvD ?Dy
),(),( yyxfyyxxf?++?+
xyyxxf
x
+?+= ),(
1
θ )10(
1
<<θ
?B?Z ?|
=?¨ ?ì μ °?′? ?
xxyxf
x
+?=
1
),( ε
G
ê?
￥ ???
O? 0,0 →?→? yx
H 0
1
→ε,
?
1
ε 1 yx,￥f
,
~ ?yvD ?Dy
xxyxf
x
+?=
1
),( ε yyyxf
y
+?+
2
),( εz?
21
21
εε
ρ
εε
+≤
+? yx
Q
,0
0
→?
→ρ
#f
),( yxfz = ? ),( yx ) V±,
] ?
),(),( yxfyyxf+
,),(
2
yyyxf
y
+?= ε ? 0→?y
H 0
2
→ε,
~ ?yvD ?Dy
8
: ?±s1
.dy
y
z
dx
x
z
dz
+
=
?±s￥?l Vw<? ?í# ?í[
f
.dz
z
u
dy
y
u
dx
x
u
du
+
+
=
Yèá
ìü=íf
￥ ?±s??
￥
?
ê±s-?q
Y?1=íf
￥±sˉ
?F
?F
e ?
?Fe ?9
a¨?=í[
f
￥ f ?
~ ?yvD ?Dy
è 9
f
xy
ez = ? )1,2( )￥ ?±s,
3
,
xy
ye
x
z
=
,
xy
xe
y
z
=
,
2
)1,2(
e
x
z
=
,2
2
)1,2(
e
y
z
=
.2
22
dyedxedz +=
p ?±s
~ ?yvD ?Dy
è pf
)2cos( yxyz?= ?
4
π
=x π=y
4
π
=dx π=dy
H￥ ?±s,
3
),2sin( yxy
x
z
=
),2sin(2)2cos( yxyyx
y
z
+?=
dy
y
z
dx
x
z
dz
),
4
(
),
4
(
),
4
(
π
π
π
π
π
π
+
= ).74(
8
2
ππ +=
~ ?yvD ?Dy
è 9
f
yz
e
y
xu ++=
2
sin ￥ ?±s,
3,1=
x
u
,
2
cos
2
1
yz
ze
y
y
u
+=
,
yz
ye
z
u
=
p ?±s
.)
2
cos
2
1
( dzyedyze
y
dxdu
yzyz
+++=
~ ?yvD ?Dy
è
k￡f
=
≠
+=
)0,0(),(,0
)0,0(),(,
1
sin
),(
22
yx
yx
yx
xy
yxf
? )0,0( ?? O
ê?
i ?
ê?
? )0,0(
? ??7 f ? )0,0( V±,
±
^ ?μ1?l)
?
ê?
3s
)0,0(),( ≠yx )0,0(),( =yx )
,
~ ?yvD ?Dy
￡
7
,cosθρ=x
,sinθρ=y
5
22
)0,0(),(
1
sinlim
yx
xy
yx
+
→
ρ
θθρ
ρ
1
sincossinlim
2
0
=
→
0= ),0,0(f=
#f
? )0,0( ??,
=)0,0(
x
f
x
fxf
x
→?
)0,0()0,(
lim
0
,0
00
lim
0
=
=
→?
x
x
] ?
.0)0,0( =
y
f
~ ?yvD ?Dy
? )0,0(),( ≠yx
H
=),( yxf
x
,
1
cos
)(
1
sin
22322
2
22
yxyx
yx
yx
y
++
+
?? ),( yxP °L xy = t? )0,0(
H,
),(lim
)0,0(),(
yxf
x
xx →
,
||2
1
cos
||22||2
1
sinlim
3
3
0
=
→
xx
x
x
x
x
?i ,
~ ?yvD ?Dy
[ ),( yxf
x
)0,0( ? ??,
] ? V￡ ),( yxf
y
)0,0( ? ??,
)0,0(),( fyxff=?
22
)()(
1
sin
yx
yx
+?
=
))()((
22
yxo?+?=
# ),( yxf ? )0,0( V±,0
)0,0(
=df
~ ?yvD ?Dy
íf
?? a V? a V±￥1"
f
V±
f
??
ê?
??
f
V?
~ ?yvD ?Dy
?±s í
9
?￥?¨
??l
Hμí
?
T
?? O?
ê?
￥
??=íf
yxyxfyxf
yxPyxfz
yx
=
,),(),,(
),(),(
.),(),( yyxfxyxfdzz
yx
+?=≈?
9 V?
.),(),(),(
),(
yyxfxyxfyxf
yyxxf
yx
+?+≈
+?+
~ ?yvD ?Dy
è 9
02.2
)04.1( ￥í
′,
3,),(
y
xyxf =
!f
.02.0,04.0,2,1 =?=?== yxyx |
,1)2,1( =fQ
,),(
1?
=
y
x
yxyxf,ln),( xxyxf
y
y
=
,2)2,1( =
x
f
,0)2,1( =
y
f
?
T¤
02.0004.021)04.1(
02.2
×+×+≈,08.1=
~ ?yvD ?Dy
íf
?±s￥à
Q
íf
?±s￥ pE
íf
?? a V? a V±￥1"
?i DBíf
μv uY
? al2
~ ?yvD ?Dy
f
),( yxfz = ? ),(
00
yx ) V±￥ sHq
^,
1 ),( yxf ? ),(
00
yx ) ??
2 ),( yxf
x
′
a ),( yxf
y
′
? ),(
00
yx ￥
#×i
3 yyxfxyxfz
yx
′
′
),(),(
? 0)()(
22
→?+? yx
H
^í kl
4
22
)()(
),(),(
yx
yyxfxyxfz
yx
+?
′
′
,
? 0)()(
22
→?+? yx
H
^í kl
,
± I5
~ ?yvD ?Dy
B aA b5
a
!
x
y
ez =
5 =
x
z
@ @ @ @ @ @ @ @ @ @ @ @ @
=
y
z
@ @ @ @ @ @ @ @ @ @ @ @ =dz @ @ @ @ @ @ @ @ @ @ @ @
a ? )ln(
222
zyxu ++=
5
=du @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a ?f
x
y
z =
? 1,2 == yx
2.0,1.0?=?=? yx
H
f
￥ ?9
=?z @ @ @ @ @ @ @ ?±s =dz @ @ @ @ @ @ @ @
a ?f
y
x
xyz +=
5 xz ￥
ê9
=? z
x
@ @ @ @ @ @ @ @ @ @ @ =
→?
x
z
x
x 0
lim @ @ @ @ @ @ @ @ @ @ @ @ @ @
5
~ ?yvD ?Dy
= a pf
)1ln(
22
yxz ++= ?,1=x
2=y
H￥ ?±s
? a9
33
)97.1()02.1( + ￥í
′
1 a
!μBí?ó?? ?
? ￥CD?￥¨ (1
cm1.0
=ú1 cm20
=??1 cm4
p ? ? T8￥
í
′
? a?¤B v ???r1￥
HHésY1 m1.063±
m1.078±
?
H￥C?1
0
160±
k p ???
￥í
′
i p 'μMμ
B a ?¨ ?±s￡
ü e￥Mμ??òy0￥M
μ-
￥Mμ??$"
#"
￥M
μ-
~ ?yvD ?Dy
t a pf
),( yxf
=+
≠+
+
+
=
0,0
0,
1
sin)(
22
22
22
22
yx
yx
yx
yx
￥
ê?
iù? ? )0,0( )
ê?
￥ ???#
f
),( yxf ￥ V±?
~ ?yvD ?Dy
B a a )(
1
,
1
,
2
dydx
x
y
e
x
e
x
e
x
y
x
y
x
y
x
y
a
222
)(2
zyx
zdzydyxdx
++
++
a
a
y
yx
y
y
1
,)
1
( +?+
= a dydx
3
2
3
1
+ ? a
1 a
3
cm3.55
? a %.30.1,m6.27,m2128
22
t a ),(),,( yxfyxf
yx
′′
)0,0( ) (? ??
),( yxf ?
) V±
5s?
? ??
?±s# ?¨
~ ?yvD ?Dy
),(),( yxfyxxf+
xyxf
x
≈ ),(
),(),( yxfyyxf+
yyxf
y
≈ ),(
=íf
x y￥
ê±s
=íf
x y￥
ê9
?Bíf
±sD?9
D±s￥1"¤
B a ?±s￥?l
~ ?yvD ?Dy
?Tf
),( yxfz = ? ),( yx ￥
#×
=
μ?li
! ),( yyxxP?+?+
′
1?
#×
=￥
?iB?5??
?￥f
′-μ
),(),( yxfyyxxf+?+
1f
?
P
??1M
9
yx,￥ ?9
:1 z?
'
z? = ),(),( yxfyyxxf+?+
?9
￥à
Q
~ ?yvD ?Dy
?Tf
),( yxfz = ? ),( yx ￥ ?9
),(),( yxfyyxxfz+?+=? V[V
U1
)(ρoyBxAz +?+?=? ? BA,?G ??
yx,7?D yx,μ1
22
)()( yx?+?=ρ
5?f
),( yxfz = ? ),( yx V±s
yBxA?+? ?1f
),( yxfz = ? ),( yx ￥
?±s :1 dz' dz yBxA?+?
?±s￥?l
~ ?yvD ?Dy
f
?
u× D
=ò?)) V±s 5
??f
D
= V±s,
?Tf
),( yxfz = ? ),( yx V±s,5
f
?? ??,
Y
L
),(ρoyBxAz +?+?=?
,0lim
0
=?
→
z
ρ
),(lim
0
0
yyxxf
y
x
+?+
→?
→?
]),([lim
0
zyxf?+=
→ρ
),( yxf=
#f
),( yxfz = ? ),( yx ) ??,
~ ?yvD ?Dy
= a V±￥Hq
? ?
A1Hq ?Tf
),( yxfz = ?
),( yx V±s5?f
? ),( yx ￥
ê?
x
z
a
y
z
Ai Of
),( yxfz = ? ),( yx ￥ ?±s
1
y
y
z
x
x
z
dz?
+?
=
~ ?yvD ?Dy
￡
?Tf
),( yxfz = ? ),( yxP V±s,
∈?+?+
′
),( yyxxP P￥
?
#×
)(ρoyBxAz +?+?=? 9? ?,
? 0=?y
H
T ˉ? ? N
H || x?=ρ,
),(),( yxfyxxf+ |),(| xoxA?+=
A
x
yxfyxxf
x
=
+
→?
),(),(
lim
0
,
x
z
=
] ? V¤
.
y
z
B
=
~ ?yvD ?Dy
Bíf
?￥?
i ±si
íf
￥ò
ê?
i ?±si
è ?
.
00
0
),(
22
22
22
=+
≠+
+
=
yx
yx
yx
xy
yxf
? )0,0( )μ
0)0,0()0,0( ==
yx
ff
~ ?yvD ?Dy
])0,0()0,0([ yfxfz
yx
+,
)()(
22
yx
yx
+?
=
?T I
n? ),( yxP
′
"°L xy = tí? )0,0(
5
ρ
22
)()( yx
yx
+?
22
)()( xx
xx
+?
=,
2
1
=
a
ü
?
" 0→ρ 7 t? 0,0→ρ
?
H
),(])0,0()0,0([ ρoyfxfz
yx
≠+
f
? )0,0( )? V±,
~ ?yvD ?Dy
a
ü íf
￥ò
ê?
i i?
￡ ?
±si
? ?
sHq ?Tf
),( yxfz = ￥
ê
?
x
z
a
y
z
? ),( yx ??5?f
?
),( yx V±s
￡
),(),( yxfyyxxfz+?+=?
)],(),([ yyxfyyxxf?++?+=
)],,(),([ yxfyyxf++
~ ?yvD ?Dy
),(),( yyxfyyxxf?++?+
xyyxxf
x
+?+= ),(
1
θ )10(
1
<<θ
?B?Z ?|
=?¨ ?ì μ °?′? ?
xxyxf
x
+?=
1
),( ε
G
ê?
￥ ???
O? 0,0 →?→? yx
H 0
1
→ε,
?
1
ε 1 yx,￥f
,
~ ?yvD ?Dy
xxyxf
x
+?=
1
),( ε yyyxf
y
+?+
2
),( εz?
21
21
εε
ρ
εε
+≤
+? yx
Q
,0
0
→?
→ρ
#f
),( yxfz = ? ),( yx ) V±,
] ?
),(),( yxfyyxf+
,),(
2
yyyxf
y
+?= ε ? 0→?y
H 0
2
→ε,
~ ?yvD ?Dy
8
: ?±s1
.dy
y
z
dx
x
z
dz
+
=
?±s￥?l Vw<? ?í# ?í[
f
.dz
z
u
dy
y
u
dx
x
u
du
+
+
=
Yèá
ìü=íf
￥ ?±s??
￥
?
ê±s-?q
Y?1=íf
￥±sˉ
?F
?F
e ?
?Fe ?9
a¨?=í[
f
￥ f ?
~ ?yvD ?Dy
è 9
f
xy
ez = ? )1,2( )￥ ?±s,
3
,
xy
ye
x
z
=
,
xy
xe
y
z
=
,
2
)1,2(
e
x
z
=
,2
2
)1,2(
e
y
z
=
.2
22
dyedxedz +=
p ?±s
~ ?yvD ?Dy
è pf
)2cos( yxyz?= ?
4
π
=x π=y
4
π
=dx π=dy
H￥ ?±s,
3
),2sin( yxy
x
z
=
),2sin(2)2cos( yxyyx
y
z
+?=
dy
y
z
dx
x
z
dz
),
4
(
),
4
(
),
4
(
π
π
π
π
π
π
+
= ).74(
8
2
ππ +=
~ ?yvD ?Dy
è 9
f
yz
e
y
xu ++=
2
sin ￥ ?±s,
3,1=
x
u
,
2
cos
2
1
yz
ze
y
y
u
+=
,
yz
ye
z
u
=
p ?±s
.)
2
cos
2
1
( dzyedyze
y
dxdu
yzyz
+++=
~ ?yvD ?Dy
è
k￡f
=
≠
+=
)0,0(),(,0
)0,0(),(,
1
sin
),(
22
yx
yx
yx
xy
yxf
? )0,0( ?? O
ê?
i ?
ê?
? )0,0(
? ??7 f ? )0,0( V±,
±
^ ?μ1?l)
?
ê?
3s
)0,0(),( ≠yx )0,0(),( =yx )
,
~ ?yvD ?Dy
￡
7
,cosθρ=x
,sinθρ=y
5
22
)0,0(),(
1
sinlim
yx
xy
yx
+
→
ρ
θθρ
ρ
1
sincossinlim
2
0
=
→
0= ),0,0(f=
#f
? )0,0( ??,
=)0,0(
x
f
x
fxf
x
→?
)0,0()0,(
lim
0
,0
00
lim
0
=
=
→?
x
x
] ?
.0)0,0( =
y
f
~ ?yvD ?Dy
? )0,0(),( ≠yx
H
=),( yxf
x
,
1
cos
)(
1
sin
22322
2
22
yxyx
yx
yx
y
++
+
?? ),( yxP °L xy = t? )0,0(
H,
),(lim
)0,0(),(
yxf
x
xx →
,
||2
1
cos
||22||2
1
sinlim
3
3
0
=
→
xx
x
x
x
x
?i ,
~ ?yvD ?Dy
[ ),( yxf
x
)0,0( ? ??,
] ? V￡ ),( yxf
y
)0,0( ? ??,
)0,0(),( fyxff=?
22
)()(
1
sin
yx
yx
+?
=
))()((
22
yxo?+?=
# ),( yxf ? )0,0( V±,0
)0,0(
=df
~ ?yvD ?Dy
íf
?? a V? a V±￥1"
f
V±
f
??
ê?
??
f
V?
~ ?yvD ?Dy
?±s í
9
?￥?¨
??l
Hμí
?
T
?? O?
ê?
￥
??=íf
yxyxfyxf
yxPyxfz
yx
=
,),(),,(
),(),(
.),(),( yyxfxyxfdzz
yx
+?=≈?
9 V?
.),(),(),(
),(
yyxfxyxfyxf
yyxxf
yx
+?+≈
+?+
~ ?yvD ?Dy
è 9
02.2
)04.1( ￥í
′,
3,),(
y
xyxf =
!f
.02.0,04.0,2,1 =?=?== yxyx |
,1)2,1( =fQ
,),(
1?
=
y
x
yxyxf,ln),( xxyxf
y
y
=
,2)2,1( =
x
f
,0)2,1( =
y
f
?
T¤
02.0004.021)04.1(
02.2
×+×+≈,08.1=
~ ?yvD ?Dy
íf
?±s￥à
Q
íf
?±s￥ pE
íf
?? a V? a V±￥1"
?i DBíf
μv uY
? al2
~ ?yvD ?Dy
f
),( yxfz = ? ),(
00
yx ) V±￥ sHq
^,
1 ),( yxf ? ),(
00
yx ) ??
2 ),( yxf
x
′
a ),( yxf
y
′
? ),(
00
yx ￥
#×i
3 yyxfxyxfz
yx
′
′
),(),(
? 0)()(
22
→?+? yx
H
^í kl
4
22
)()(
),(),(
yx
yyxfxyxfz
yx
+?
′
′
,
? 0)()(
22
→?+? yx
H
^í kl
,
± I5
~ ?yvD ?Dy
B aA b5
a
!
x
y
ez =
5 =
x
z
@ @ @ @ @ @ @ @ @ @ @ @ @
=
y
z
@ @ @ @ @ @ @ @ @ @ @ @ =dz @ @ @ @ @ @ @ @ @ @ @ @
a ? )ln(
222
zyxu ++=
5
=du @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a ?f
x
y
z =
? 1,2 == yx
2.0,1.0?=?=? yx
H
f
￥ ?9
=?z @ @ @ @ @ @ @ ?±s =dz @ @ @ @ @ @ @ @
a ?f
y
x
xyz +=
5 xz ￥
ê9
=? z
x
@ @ @ @ @ @ @ @ @ @ @ =
→?
x
z
x
x 0
lim @ @ @ @ @ @ @ @ @ @ @ @ @ @
5
~ ?yvD ?Dy
= a pf
)1ln(
22
yxz ++= ?,1=x
2=y
H￥ ?±s
? a9
33
)97.1()02.1( + ￥í
′
1 a
!μBí?ó?? ?
? ￥CD?￥¨ (1
cm1.0
=ú1 cm20
=??1 cm4
p ? ? T8￥
í
′
? a?¤B v ???r1￥
HHésY1 m1.063±
m1.078±
?
H￥C?1
0
160±
k p ???
￥í
′
i p 'μMμ
B a ?¨ ?±s￡
ü e￥Mμ??òy0￥M
μ-
￥Mμ??$"
#"
￥M
μ-
~ ?yvD ?Dy
t a pf
),( yxf
=+
≠+
+
+
=
0,0
0,
1
sin)(
22
22
22
22
yx
yx
yx
yx
￥
ê?
iù? ? )0,0( )
ê?
￥ ???#
f
),( yxf ￥ V±?
~ ?yvD ?Dy
B a a )(
1
,
1
,
2
dydx
x
y
e
x
e
x
e
x
y
x
y
x
y
x
y
a
222
)(2
zyx
zdzydyxdx
++
++
a
a
y
yx
y
y
1
,)
1
( +?+
= a dydx
3
2
3
1
+ ? a
1 a
3
cm3.55
? a %.30.1,m6.27,m2128
22
t a ),(),,( yxfyxf
yx
′′
)0,0( ) (? ??
),( yxf ?
) V±
5s?
