aQBi': R!R/û`Bp#H2X PM TQb2f:R⇤⇥R!R/û}MB2 T`f(x, y) ='(y/x)X JQMi`2` [m2 f pû`B}2 H `2HiBQM ,
x@f
@x(x, y) +y@f
@y(x, y) = 0X
1t2`+B+2 N ,
@f
@x(x, y) = @
@x'(y/x) = y
x2 ·'0(y/x)T` /û`BpiBQM /ǶmM2 +QKTQbû2X
@f
@y(x, y) = @
@y'(y/x) = 1
x·'0(y/x) .QM+ x@f
@x(x, y) +y@f
@y(x, y) = y
x·'0(y/x) + y
x·'0(y/x) = 0X _2K`[m2 ,
_û+BT`Q[m2K2Mi- QM T2mi KQMi`2` [m2 bB f pû`B}2 H `2HiBQM x@f
@x(x, y) +y@f
@y(x, y) = 0- HQ`b BH 2tBbi2 ' i2HH2 [m2 f(x, y) ='(y/x)
1M 2z2i- 2M TQbMi g(r,✓) =f(rcos✓, rsin✓)- QM Q#iB2Mi ,r·@g
@r =r Å
cos✓@f
@x+ sin✓@f
@y ã T` H `ĕ;H2 /2 H +?ŗM2X
.QM+ f 2bi bQHmiBQM bbBg 2bi bQHmiBQM /2 r· @g
@r = 0- /QM+ @g
@r = 0X .QM+ g(r,✓) = (✓)- +Ƕ2bi ¨ /B`2 f(x, y) = (arctan(y/x)) ='(y/x)X .2 THmb- bBf 2biC1- BH 7mi '/2 +Hbb2C1X
*H+mH2` H2b /û`Bpû2b T`iB2HH2b /ǶQ`/`2 k /2b 7QM+iBQMb bmBpMi2b ,
URV , f(x, y) =x2(x+y) UkV , f(x, y) = cos(xy)
1t2`+B+2 Ry ,
PM i`Qmp2 @f
@x(x, y) = 3x2+ 2xy 2i @f
@y(x, y) =x2X SmBb @2f
@y@x(x, y) = 2x= @2f
@y@x(x, y)U+H+mH Qm #B2M h?X /2 a+?r`xV 1M}M- @2f
@x@x(x, y) = 6x+ 2y 2i @2f
@y@y(x, y) = 0
_2K`[m2 U?Q`b T`Q;`KK2- TQm` H2b THmb KQiBpûbV , QM TT2HH2 >2bbB2MM2 /2fH Ki`B+2 /2 iBHH2 k bmBpMi2 ,
Hf(x, y) =
Ü @2f
@x@x(x, y) @2f
@y@x(x, y)
@2f
@y@x(x, y) @2f
@y@y(x, y) ê
GQ`b[m2f 2bi /2 +Hbb2C2- +2ii2 Ki`B+2 2bi bvKûi`B[m2 Ui?X /2 a+?r`xV `û2HH2- /QM+ XXX /B;QMHBb#H2 T` i?X bT2+i`H 5 a2b pH2m`b T`QT`2b /ûT2M/2Mi /2 (x, y)X 1M (x0, y0)mM TQBMi +`BiB[m2 Urf(x0, y0) = 0V ,
ě bB H2b /2mt pH2m`b T`QT`2b bQMi bi`B+i2K2Mi TQbBiBp2b- H2 TQBMi +`BiB[m2 2bi mM KBMBKmK- ě bB H2b H2b /2mt pH2m`b T`QT`2b bQMi bi`B+i2K2Mi Mû;iBp2b- H2 TQBMi +`BiB[m2 2bi mM KtBKmK-
ě HǶmM2 2bi bi`B+i2K2Mi TQbBiBp2b- HǶmi`2 bi`B+i2K2Mi Mû;iBp2- H2 TQBMi +`BiB[m2 MǶ2bi MB mM KtBKmK- MB mM KBKBMmK UQM T`H2 /2 TQBMi +QH Qm TQBMi b2HH2VX
ě aB HǶmM2 /2b pH2m`b T`QT`2b bǶMMmH2- QM M2 T2mi `B2M /B`2 bMb ûim/2 THmb TT`Q7QM/B2X 1t2KTH2 ,
A+B-r(f)(x, y) = (3x2+ 2xy, x2) = (0,0),x= 0, y2RX hQmb H2b TQBMib /2 HǶt2 /2b Q`/QMMû2b bQMi /2b TQBMib +`BiB[m2bX 1M mM TQBMi (0, y)- H >2bbB2MM2 2bi û;H2 ¨
Hf(0, y) =
Å2y 0 0 0 ã
.Mb +2 +b- QM M2 T2mi Tb +QM+Hm`2 /2 H Mim`2 /2b TQBMib +`BiB[m2bX RR
SQm` H /2mtBĕK2 7QM+iBQM- QM i`Qmp2 H >2bbB2MM2 ,
Hf(x, y) =
Ü @2f
@x@x(x, y) @2f
@y@x(x, y)
@2f
@y@x(x, y) @2f
@y@y(x, y) ê
=
Å y2cos(xy) xycos(xy) xycos(xy) x2cos(xy) ã
aQB2Mif: (x, y)7!f(x, y)/2 +Hbb2C2 2ig: (r,✓)7!f(rcos✓, rsin✓)X CmbiB}2` [m2 g 2bi /2 +Hbb2C2 2i 2tT`BK2`
@2f
@x2 +@2f
@y2 2M 7QM+iBQM /2b /û`Bpû2b T`iB2HH2b /2gX
1t2`+B+2 RR ,
PM TQb2g(r,✓) =f(rcos✓, rsin✓)X HQ`b ,
@g
@r(r,✓) = cos✓@f
@x(rcos✓, rsin✓) + sin✓@f
@y(rcos✓, rsin✓)
@g
@✓(r,✓) = rsin✓@f
@x(rcos✓, rsin✓) +rcos✓@f
@y(rcos✓, rsin✓) m Tbb;2- 2M +QK#BMMi H2b HB;M2b /m bvbiĕK2- QM T2mi +H+mH2` ,
@f
@x(rcos✓, rsin✓) = cos✓@g
@r(r,✓) 1
rsin✓@g
@✓(r,✓)
@f
@y(rcos✓, rsin✓) = sin✓@g
@r(r,✓) +1
rcos✓@g
@✓(r,✓) 1i i`Qmp2` HǶ2tT`2bbBQM /m ;`/B2Mi 2M TQHB`2 ,
rf(rcos✓, rsin✓) = (cos✓@g
@r(r,✓) 1 rsin✓@g
@✓(r,✓),sin✓@g
@r(r,✓) +1
rcos✓@g
@✓(r,✓)) SQm` H2 GTH+B2M ,
@2g
@r2(r,✓) = cos✓ Å
cos✓@2f
@x2(rcos✓, rsin✓) + sin✓ @2f
@y@x(rcos✓, rsin✓) ã
X + sin✓
Å
cos✓ @2f
@x@y(rcos✓, rsin✓) + sin✓@2f
@2y(rcos✓, rsin✓) ã
@2g
@r2(r,✓) = cos2✓@2f
@x2(rcos✓, rsin✓) + 2 sin✓cos✓ @2f
@y@x(rcos✓, rsin✓) + sin2✓@2f
@2y(rcos✓, rsin✓) 1i @2g
@✓2(r,✓) = rcos✓@f
@x(rcos✓, rsin✓) rsin✓@f
@y(rcos✓, rsin✓) X +r2sin2✓@2f
@x2(rcos✓, rsin✓) 2r2sin✓cos✓ @2f
@x@y(rcos✓, rsin✓) X +r2cos2✓@2f
@y2(rcos✓, rsin✓) HQ`b ,
@2g
@r2(r,✓) + 1 r2
@2g
@✓2(r,✓) = f(rcos✓, rsin✓) 1 r
Å@f
@x(rcos✓, rsin✓) +@f
@y(rcos✓, rsin✓) ã
@2g
@r2(r,✓) + 1 r2
@2g
@✓2(r,✓) = f(rcos✓, rsin✓) 1 r
@g
@r(r,✓)X 6BMH2K2Mi ,
f(x, y) =@2g
@r2(r,✓) + 1 r2
@2g
@✓2(r,✓) +1 r
@g
@r(r,✓).
Rk
aQBif:R2!R mM2 7QM+iBQM /Bzû`2MiB#H2 ?QKQ;ĕM2 /2 /2;`ûn2N+Ƕ2bi@¨@/B`2 pû`B}Mi 8t2R,8(x, y)2R2, f(tx, ty) =tnf(x, y)X
RX JQMi`2` [m2
x@f
@x+y@f
@y =nfX
kX PM bmTTQb2n>1X JQMi`2` [m2 H2b /û`Bpû2b T`iB2HH2b /2f bQMi 2HH2b mbbB ?QKQ;ĕM2b- T`û+Bb2` H2m` /2;`ûX 1t2`+B+2 Rk ,
PM +QMbB/ĕ`2 H 7QM+iBQM f:R2!R/û}MB2 T`
f(x, y) =
®sin(xy)
|x|+|y| bB(x, y)6= (0,0)
0 bBMQMX
RX f 2bi@2HH2 +QMiBMm2 \ kX f 2bi@2HH2 /2 +Hbb2 C1\
1t2`+B+2 Rj ,
RX |sin(xy)|⇠(0,0)|xy| /QM+|f(x, y)|⇠ |xy|
|x|+|y| 6|y| |x|
|x|+|y| 6|y| i2M/ p2`b y HQ`b[m2(x, y)i2M/ p2`b (0,0)X
S` +QKT`BbQM-f i2M/ p2`b y û;H2K2Mi 2if 2bi +QMiBMm2 2M(0; 0)X S`iQmi BHH2m`b-f 2bi +QMiBMm2 T` i?ûQ`ĕK2b QTû`iQB`2b bm` H2b 7QM+iBQMb +QMiBMm2bX
6BMH2K2Mif 2bi +QMiBMm2 bm`R2X kX .2 KāK2-f 2bi C1 bm`R2\ {(0,0)}X
P` TQm`x >0 2iy >0- @f
@x(x, y) =y(x+y) cos(xy) sin(xy)
(x+y)2 X
HQ`b @f
@x(1/n,1/n) =2/n2·cos(1/n2) sin(1/n2)
4/n2 !`6= 0 = @f
@x(0,0)X .QM+ @f
@x MǶ2bi Tb +QMiBMm2 2M y 2if MǶ2bi Tb C15
aQBif:R2!R H 7QM+iBQM /û}MB2 T`
f(x, y) =
® xy3
x2+y2 bB(x, y)6= (0,0)
0 bBMQMX
RX JQMi`2` [m2f 2bi /2 +Hbb2C1 bm` R2X
kX JQMi`2` [m2 @x@y@2f (0,0) 2i @y@x@2f (0,0)2tBbi2Mi 2i /Bzĕ`2MiX ZmǶ2M /û/mB`2 \
1t2`+B+2 R9 ,
RX S` +QKTQbBiBQM-f 2biC1 bm` R2\ {(0,0)}2i @f
@x(x, y) = y3 x2+y2
2x2y3
(x2+y2)2 2i @f
@y(x, y) = 3xy2 x2+y2
2xy4 (x2+y2)2X .2 THmb- @f
@x(0,0) = 0 = @f
@y(0,0)T` /û`BpiBQM /2b 7QM+iBQMb T`iB2HH2b Qm T` imt /Ƕ++`QBbb2K2MiX 1M}M- @f
@x(x, y) 6|y| y2
x2+y2 + 2|y| Å xy
x2+y2 ã2
!0 HQ`b[m2(x, y)i2M/ p2`b (0,0)X .2 KāK2- QM KQMi`2 [m2 @f
@x(x, y) !0X
.ǶT`ĕb H2 +Qm`b-f 2bi /QM+ /2 +Hbb2 C1 bm`R2X kX S` +QMi`2- 1
y Å@f
@x(0, y) ã
= 12i 1 x
Å@f
@y(x,0) ã
= 0 /QM+ @2f
@x@y(0,0)6= @2f
@y@x(0,0) 2i T` i?X /2 a+?r`x-f MǶ2bi Tb /2 +Hbb2C2 bm` R2X
Rj
