1 x·'0(y/x) .QM+ x@f @x(x, y) +y@f @y(x, y

(1)

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

(2)

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

(3)

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^@²^f (0,0) 2i _@y@x^@²^f (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