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yx
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xy
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y
x
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x
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z
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2
2
2
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xy
xy
y
x
42
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x
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0
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0
0
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x
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0
0
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y
x
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y
x
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+
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x
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x
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yx
z
+
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x
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z
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xyxyxyPQ?++?+?=L
pn? bW?
#× a u×?à
Q
{ }
n
RPPPPPU ∈<=,||),(
00
δδ
+
y1?
HL1
à a
ü
a
bW
?W￥ ?
3,2,1=n
=? aH?? a u× a
??à
Q9 V?l
#×
!
?1
~ ?yvD ?Dy
= a íf
￥à
Q
! D
^
ü
￥B??" ?T?
??
DyxP ∈),( M
z?vB?￥E59μ ??￥′
?5? z
^M
yx,￥=íf
:1
),( yxfz =
:1 )(Pfz =
? 2≥n
H níf
d?1 íf
,
íf
?]"μ?l× a′× a1M
a
yM
?à
Q,
?
1 V?l ?í# ?í[
f
~ ?yvD ?Dy
è p ￥?l×
2
22
)3arcsin(
),(
yx
yx
yxf
=
3
>?
≤
0
13
2
22
yx
yx
>
≤+≤
2
22
42
yx
yx
p?l×1 }.,42|),{(
222
yxyxyxD >≤+≤=
~ ?yvD ?Dy
=íf
￥m?
),( yxfz =
!f
),( yxfz = ￥?l×1 D? ?i
|?￥ DyxP ∈),( ?￥f
′1
),( yxfz = ?"[ x1US a y1:U
S a z1
US bWü ??B? ),,( zyxM
? x |R D
B M?
H¤B? bW?"
}),(),,(|),,{( Dyxyxfzzyx ∈= ???"?
1=íf
￥m?,
?/:m
~ ?yvD ?Dy
=íf
￥m?Yè
^Bf w
,
~ ?yvD ?Dy
x
y
z
o
xyz sin=
è ?,
m? ?·m,
2222
azyx =++
è ?,
Pm o
,
}.),{(
222
ayxyxD ≤+=
222
yxaz=
.
222
yxaz=
?′s|,
~ ?yvD ?Dy
?l
!f
),( yxfz = ￥?l×1
),(,
000
yxPD
^
? ?T? ?ió?￥?
ε 9i ?
δ
P¤?
a??
T
δ<?+?=<
2
0
2
00
)()(||0 yyxxPP ￥B M
??μ ε<? |),(| Ayxf ? ?5? A 1f
),( yxfz = ?
0
xx →
0
yy →
H￥ K
:1 Ayxf
yy
xx
=
→
→
),(lim
0
0
? a íf
￥ K
)(),(
0
PPAyxf →→
,
~ ?yvD ?Dy
a
ü
1?l? ￥Z
T
^ ?i￥
0
PP →
2=íf
￥ K9?=× K );,(lim
0
0
yxf
yy
xx
→
→
3=íf
￥ K
E5DBíf
?
~ ?yvD ?Dy
è p￡
￡
0
1
sin)(lim
22
22
0
0
=
+
+
→
→
yx
yx
y
x
0
1
sin)(
22
22
+
+
yx
yx
22
22
1
sin
yx
yx
+
+=
22
yx +≤
,0>?ε
,εδ =?
?
Hδ<?+?<
22
)0()0(0 yx
ε<?
+
+ 0
1
sin)(
22
22
yx
yx
e2
? ?
~ ?yvD ?Dy
è p K,
)sin(
lim
22
2
0
0
yx
yx
y
x
+
→
→
3
22
2
0
0
)sin(
lim
yx
yx
y
x
+
→
→
,
)sin(
lim
22
2
2
2
0
0
yx
yx
yx
yx
y
x
+
=
→
→
?
yx
yx
y
x
2
2
0
0
)sin(
lim
→
→
u
u
u
sin
lim
0→
,1=
22
2
yx
yx
+
x
2
1
≤,0
0
→?
→x
.0
)sin(
lim
22
2
0
0
=
+
∴
→
→
yx
yx
y
x
yxu
2
=
~ ?yvD ?Dy
è ￡
ü ?i
￡
26
3
0
0
lim
yx
yx
y
x
+
→
→
|
,
3
kxy =
26
3
0
0
lim
yx
yx
y
x
+
→
→
626
33
0
3
lim
xkx
kxx
kxy
x
+
=
=
→
,
1
2
k
k
+
=
′
k￥?]7M
# K?i
~ ?yvD ?Dy
?i ,
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
lb
lb
~ ?yvD ?Dy
1
7 ),( yxP kxy = t_? ),(
000
yxP ?
K′D kμ15 Vy K?i
2 s
??] tíZ
T
P ),(lim
0
0
yxf
yy
xx
→
→
i
?
??M?N
H9 Vy ),( yxf ?
),(
000
yxP ) K?i
?? K ?i ￥ZE
~ ?yvD ?Dy
?l
! níf
)(Pf ￥?l×1?"
0
,PD
^
? ?T? ?ió?￥?
ε
9i ?
δ
P¤?
a??
T
δ<< ||0
0
PP ￥B M? DP∈ ?μ
ε<? |)(| APf ? ?5? A1 níf
)(Pf
?
0
PP →
H￥ K:1
APf
PP
=
→
)(lim
0
níf
￥ K ?¨?f
￥?
Tμ
~ ?yvD ?Dy
! níf
)(Pf ￥?l×1?"
0
,PD
^
? O DP ∈
0
?T )()(lim
0
0
PfPf
PP
=
→
5? níf
)(Pf ?
0
P) ??
!
0
P
^f
)(Pf ￥?l×￥
? ?T
)(Pf ?
0
P)? ??5?
0
P
^f
)(Pf ￥
W?,
1 a íf
￥ ???
?l
~ ?yvD ?Dy
è )
f
=
≠
+
+
=
)0,0(),(,0
)0,0(),(,
),(
22
33
yx
yx
yx
yx
yxf
(0,0))￥ ???
3
|
,cosθρ=x
θρsin=y
)0,0(),( fyxf?
)cos(sin
33
θθρ +=
ρ2<
~ ?yvD ?Dy
ερ <<? 2)0,0(),( fyxf
#f
(0,0)) ??,
),0,0(),(lim
)0,0(),(
fyxf
yx
=
→
,0>?ε,
2
ε
δ =?
?
H
δ<+<
22
0 yx
~ ?yvD ?Dy
è )
f
=+
≠+
+
=
0,0
0,
),(
22
22
22
yx
yx
yx
xy
yxf
(0,0)￥ ???
3 | kxy =
22
0
0
lim
yx
xy
y
x
+
→
→
222
2
0
lim
xkx
kx
kxy
x
+
=
=
→
2
1 k
k
+
=
′
k￥?]7M K?i
#f
(0,0))? ??
~ ?yvD ?Dy
> u×
??f
￥?é
μ?> u× %
￥ í ??f
%
à
|¤
￥Kv′Kl′òBQ
μ?> u× %
￥ í ??f
?
T %
|¤
??]￥f
′5
%
|¤o??
′-W￥ ?′à
BQ
1Kv′Kl′? ?
2o′? ?
~ ?yvD ?Dy
í?f
? í [
T#'?f
üVμKQ￥
15
ˉ??
?￥ V
¨B?
T0
V
U￥ íf
? í?f
B M í?f
?l u×
=
^ ??￥
?l u×
^·c ?l×
=￥ u×> u×
~ ?yvD ?Dy
è ,
11
lim
0
0
xy
xy
y
x
+
→
→
p
3
)11(
11
lim
0
0
++
+
=
→
→
xyxy
xy
y
x
e
T
11
1
lim
0
0
++
=
→
→
xy
y
x
.
2
1
=
).()(lim
)()(
)()(lim
00
0
0
0
PfPfP
PfPfP
PfPf
PP
PP
=
→
→
) ???
^?
￥?l×￥
=?5
^
O
^?f
H ?TB?1 p
~ ?yvD ?Dy
íf
K￥à
Q
íf
??￥à
Q
> u×
??f
￥?é
?i tíZ
T￥ ?i?
1 al2
íf
￥?l
~ ?yvD ?Dy
?? ),( yx "í
H
ü
wL t_?
? ),(
00
yx
Hf
),( yxf ? t_? A
?
? Ayxf
yxyx
=
→
),(lim
),(),(
00
$
± I5
~ ?yvD ?Dy
± I53s
?
,
è
,
)(
),(
242
23
yx
yx
yxf
+
=
)0,0(),( →yx
|
,kxy =
2442
223
)(
),(
xkx
xkx
kxxf
+
=
0
0
→?
→x
?
^ ?i ,),(lim
)0,0(),(
yxf
yx →
ey1 ? |
,
2
yx =
244
26
2
)(
),(
yy
yy
yyf
+
=
.
4
1
→
~ ?yvD ?Dy
B a A b5
a ?
y
x
xyyxyxf tan),(
22
+=
5 ),( tytxf @ @ @ @
a ?
xy
yx
yxf
2
),(
22
+
=
5 =? )3,2(f @ @ @ @ @ @ @ @ @ @
=),1(
x
y
f @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a ? )0()(
22
>
+
= y
y
yx
x
y
f
5 =)(xf @ @ @ @ @ @ @ @
a ?
22
),( yx
x
y
yxf?=+
5 =),( yxf @ @ @ @ @ @ @ @ @
f
)1ln(
4
22
2
yx
yx
z
= ￥?l×
^ @ @ @ @ @ @ @ @ @ @
5
~ ?yvD ?Dy
af
yxz?= ￥?l×
^ @ @ @ @ @ @ @ @ @ @ @ @ @ @
af
x
y
z arcsin= ￥?l×
^ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
af
xy
xy
z
2
2
2
2
+
= ￥W?
^ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
= a p/
ò K
a
xy
xy
y
x
42
lim
0
0
+?
→
→
a
x
xy
y
x
sin
lim
0
0
→
→
a
2222
22
0
0
)(
)cos(1
lim
yxyx
yx
y
x
+
+?
→
→
~ ?yvD ?Dy
? a ￡
ü 0lim
22
0
0
=
+
→
→
yx
xy
y
x
1 a ￡
ü K
yx
xy
y
x
+
+
→
→
11
lim
0
0
?i
~ ?yvD ?Dy
B a a ),(
2
yxft a
12
13
),( yxf
a
x
x
2
1+
a
y
y
x
+
1
1
2
a{ }xyyxyx 4,10),(
222
≤<+<
a{ }yxyxyx ≥≥≥
2
,0,0),(
a{ }xyxxyx ≤≤?>,0),(
{ }xyxxyx?≤≤<∪,0),(
a{ }02),(
2
=? xyyx
= a a
4
1
a a ∞+
5s?
~ ?yvD ?Dy
?i ,
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
~ ?yvD ?Dy
43 26
3
0
0
lim
yx
yx
y
x
+
→
→
,
26
3
m?
yx
yx
z
+
=
?i ,
