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P(r,è,0)

z

r

z

Ñ(r,è,z)

è eè

er

ez

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er, eθ, ezV$^$\4nB_'] \3Y4U�hCU&V

U&hC\X&_C¸aW j3Z<nB`&\3ie\"nB`aW Z7\3`qte\&b7vQUaW \Z7Y4Z<nqp g V,w t WeZ7k<hCZ7_*W m�^$\3f"Z7f4U&X&hC\3f4Un&W mu[>fa]$W U&X&`aW [>hCm�XaW V3U&f4Z7g V$n&W [>hCm�g \ U&c3X&_Cmog _on&W mu[>V3`BnB_CZ,W V3U&hCmo_*S U&V>W

er = ex cos θ + ey sin θ

eθ = −ex sin θ + ey cos θ

ez = ez.¾¿k<`Bs Z<p}nClQU|[>fa]$W U&X&`aW [3vQU~Z7f4UBnB_'n*V b,g hCUBlQU|V$^ ] \$^$\>W _*S�n*\3f4mfa^$\4] \&beW zZ7g \3Y4ma�7j4n*V3U�nC\ief4Z,W [>j"^$`&j4r$]4p g V-^$V3`&\3f4Z,W c3À&_*W>[>fa]$W U&X&`aW [3s:Z7f4g g _'nB`aS V,�^7w k1w>n*\"g V b,UBp4n&W [>j:^$_CXaS \�b,Y4`BlÑV$^$j"hCU&V3Uu_Cf&teY&b,`&V3g g \"V b7l¤b,j:p4] _'[3nB`aW z[>\3Y®`&_CY4g V$n*\3m�hCk<_*W<á��=FÜ ã>×>Û7Ü áeæ¶Ý��eÞ�Þ�Öaè>Û7Ø�â��µ�U:g$S V|Z7f4U&c3`Bnqp Z<pZ<n*\|[>V3`BnB_CZ,W V3U&jZ7Y4Z<nqp g V{Z7f4UBnB_'nCV b,g hCUBlQU:hCk<_*Wenqpg \3`&i,s

f(x, y, z) = f(x2 + y2, z)_*S U&V>W�ieV3U&_C`Bj�j4n&W¼�yV3U´V$]4] c3¸&\3f4g _

Z7f4UBnB_'n*V b,g hCUB_CmQ\oV3`aW teg j3m¤nClQU�V3U&_C¸&c$`Bnqp4nClQU·g _'nCV4r$]4p4nqvQU·teV�_'] V$nqnCl¤te_*S[>V$nBc-g$S V

f(r2, z)w

�>w 7 GQâNÜ�Û7Ü á�D�� Ý��eã3è>Öaè â1ç¤Þ�D4ã>Ö��@1·¢�_|[>c te_~Z<p g _*S \P (x, y, z)

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(ρ, θ, ϕ)¹ºr$] h'^$_u¢�k>s g V{»4w¼»4»B½'w

t W�[>V3`BnB_CZ,W V3U&hCm|[>V>W�\>W�Z7ieV>W `aW [>hCm}Z7f4UBnB_'nCV b,g hCU&_Cm{Z7f4U&X&hC\ UBn*V>WTg _�n&W mZ7k<hCZ7_*W m

x = ρ cos θ sinφ, y = ρ sin θ sinφ, z = ρ cosφ.¹�»4w � ½

t W�Z7ieV>W `aW [>hCm{Z7f4UBnB_'n*V b,g hCU&_Cm{Z7f4U&X&hC\ UBn*V>WTg _�n&W m}[>V3`BnB_CZ,W V3U&hCm�g _{n&W mZ7k<hCZ7_*W m 1

ρ2 = x2 + y2 + z2,

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y

z

x

P(ñ,ö,è)eñ

eè

eö

ñö

è

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φ = cos−1

(

z

(x2 + y2 + z2)1/2

)

θ = tan−1 y

x u`Bs Z,W g \�_*S U&V>W�U&V}fa^$\a] \abeS Z7\3f4g _®n&W m-Z7k<hCZ7_*W m"^$\3f�Z7f4U&X&hC\3f4U-nCV�g \ U&V4zXaW V>S V®XaW V3U&Y4Z7g V$nCV

eρ, eθ, eφg _�n*V:g \ U&V3XaW V>S V�XaW V3U&Y4Z7g V$nCV

ex, ey, ez.[>c4z

U&\ UBnCV3m�k<`Bs Z<p:nC\3f"¢�k1w�»4w¼»4»

eρ = sinφ cos θex + sin θ sinφey + cosφez

eθ = −ex sin θ + ey cos θ

eφ = cosφ cos θex + cosφ sin θey − sinφez.

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b,Y4`BlÚV$^$j�nqpg c3À&V

mhCk<_*W<nqp-g \3`Bies

U(x, y, z) = − Gm√

x2 + y2 + z2,

j4n*V3U�p´g c3À&Vm

r3`aS Z<[>_'n*V>W�Z<nqp U�V3`&k>sÌnC\3f£Z7f4Z<nqs g V$nC\3m{Z7f4UBnB_'nCV b,g hCUBlQU$�(x, y, z)

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[>V>WG

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U(ρ) = −Gmρ

,

j$^$\3fρ_*S U&V>W>p"V$^$j3Z<nCV3Z<p:n*\3f:Z<p g _*S \3f

PV$^ j-nqp U�V3`&k>s®nClQU�V3¸Bj UBlQU$w

ô ¥ ¦�§<¨F© ª7«q¬®­ ¯�°4°4±7© ²q¬

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ag4^$\3`&_*S1U&V|V3U&V4z

] f&te_*S7lQmu_C¸Bs m 1a0 = cosαex + cosβey + cos γez

j$^$\3fα, β, γ

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Ox,Oy[>V>W

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cosα�cosβ

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] h�b,\ UBn*V>W$Ý��eã��FÞ�Ø è Ùeã>âÌá�â1è>Ö �@Y�Qßeã>ݤÖ�ei�"nC\3f

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e

0

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a

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;¾ [<S UBp Z<p�fa]$W [>\3YÌZ<p g _*S \3f�Z<n*\³_'^3S ^$_CX&\£g4^$\3`&_*SQU&V£^ _C`aW b,`&V3ie_*S�Z<nC\³V3X&`&V4zU&_*W V$[>j-Z7Y4Z<nqp g V~Z7f4UBnB_'nCV b,g hCUBlQU

OxyV$^$j"n*\XaW c3U&f4Z7g V

r(t) = x(t)ex + y(t)eyp:n*V3k<Yanqp4n*V-[>V>W>p"_'^3W nBc3k<f4U&Z<p:^$_C`aW b,`&c3ie\ Uqn*V>W>V$^$j-n&W m�Z7k<hCZ7_*W m

ô$õhA�öo÷3¨eù&ú<þ,§>ù&§�¨1÷3°aù4²*ùB§��1þ,«q°�]�° ô��

v =dr

dt= xex + yey

γ =dv

dt= xex + yey.

µ�U�g _'] _'nqs Z7\3f4g _�nC\~S XaW \"^$`&j4r$]4p g V-Z7_o^$\4]$W [>hCm�Z7f4UBnB_'n*V b,g hCU&_Cm�teVhCk<\3f4g _

r = r(t)er.

�T�$�3�3�|�a�@���$�>¤{!'!*�*�&���a�C�&��� ó �*��ëq���a�C� � êG���*� ó$# � ð,� ¡t g _'nCV3Z7k>p g V$n&W Z7g j3mTn'lQUyg \ U&V3XaW V>S lQU�XaW V3U&f4Z7g c$n'lQU�V$^$jun&W mT[>V3`BnB_CZ,W V3U&hCm

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er(t) = ex cos θ(t) + ey sin θ(t)

eθ(t) = −ex sin θ(t) + ey cos θ(t).µ�Ufa^$\ tehCZ7\3f4g _:j4n&WFn*\}Z7Y4Z<nqp g V�nClQU-^$\a]$W [3vQU~Z7f4UBnB_'nCV b,g hCUBlQU-^$_C`aW Z<nB`&h zie_'n*V>W<g _yb7lQUaW V$[3s�n*V3k<Yanqp4n*V

ω = dθ/dt,p®n*V3k<Yanqp4n*V-Z<nC\U&hC\"Z7Y4Z<nqp g V"teV

_*S U&V>W

v =dr

dt=dr

dter + r

derdt

=dr

dter + rωeθ.

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γ =dv

dt= (r − rω2)er + (2rω + r

d2θ

dt2)eθ.

0ÌW V{_CUBXaW V3iehC`&\3f4Z7V~_CieV3`&g \ab7s"nClQU:^$V3`&V$^$c3UBl Z7k<hCZ7_'lQU-_*S U&V>W,p|\3g4V$]4s~[>fBz[3]$W [3s:[<S UBp Z<p|¹

r = θ =Z<nCV te_C`&c~½

ô A ¦�§<¨F© ª7«q¬®­ ¯�°4°4±7© ²q¬

v = rωeθ

γ =dv

dt= −rω2er.

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Oxyz[>V>W

Ox′y′z′V3`&k7W [>c

(t = 0)nCV3fan&S À&\ UBn*V>W¼wF¢Tnqp|Z7f4U&hCk<_*W V�n*\{X&_CYanB_C`&\�Z7Y4Z<nqp g V

[<W U&_*S nCV>W&[>V$nBcug4s4[>\3m·n*\3foc3¸&\ U&VOx

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P (x, y, z, t)V$^$j�n*\�hCU&V�Z7Y4Z<nqp g V:Z<n*\�c$]4] \,w

t W,g _'n*V3Z7k>p g V$n&W Z7g \>S,nC\3f®�NV$]$W ] V>S \3f®_*S U&V>W

x′ = x− vt

y′ = y

z′ = z

t′ = t

ô$õhA�öo÷3¨eù&ú<þ,§>ù&§�¨1÷3°aù4²*ùB§��1þ,«q°�]�° ô�a

RTS U&V>WyieV3U&_C`&jIj4n&WypIX&_CYanB_C`BpÑfa^$j te_CZ<p�n*\3fEinstein

X&_CU³Z7f4g&r3V3XaS À&_*Wyg _n&W m~Z7k<hCZ7_*W m~V3fanBhCmaw t W�g _'nCV3Z7k>p g V$n&W Z7g \>S�nC\3f{�NV$]$W ] V>S \3f{fa^$\ teh'nC\3f4U�c$^$_*W z`Bp®n*V3k<Yanqp4nCV-XaW c3X&\3Z<p monC\3f®i,l�nBj3maw t

Einsteinb,_CUaS [>_Cf4Z7_�nC\3f4m�^$V3`BV$^$c3UBl

g _'n*V3Z7k>p g V$n&W Z7g \3Y4m[>c3U&\ UBnCV3m"k<`Bs Z<p}n'lQU�g _'n*V3Z7k>p g V$n&W Z7g4vQU�n*\3fLorentz^$\3f�^$`&\R©a^ s `Bk7V3UynClQU�fa^$\ tehCZ7_'vQUon*\3f>w µ�UohCU&V�b,_�b,\ U&j3m�Z7f4U&h r$p�Z<nqp UuV3`&k>s

n*\3f�Z7f4Z<nqs g V$nC\3moZ7f4UBnB_'n*V b,g hCUBlQUOxyz

[>V>W$g _'n*V3Z7k>p g V$n&S À&_'nCV>W3Z<n*\:Z7Y4Z<nqp&zg V

Ox′y′z′g _on&W mob,`&V3g g$W [>hCmuZ7k<hCZ7_*W m

x′ = α(x− vt)

y′ = y

z′ = z

t′ = βx+ γt

t fa^$\4] \&beW Z7g j3mTnClQU�V3f&teV>S `&_'nClQU�Z<n*V te_C`BvQU�teVubeS U&_*W4h'nBZ,W4vQZ<nB_�U&V�_C¸&V3Z7ieV4z]$W Z>te_*Sep-X&_CYanB_C`Bp"fa^$j te_CZ<p-n*\3f

Einstein.µ�U�[>V>W,Z<nCV|X&Y4\~Z7f4Z<nqs g V$nCV|hCU&V

i,l�nB_*W U&juZ<s g Vu¸&_'[<S UBp Z7_¤j4n*V3UTnCVuX&Y4\uZ7f4Z<nqs g V$nCVun*V3fan&S À&\ Uqn*V3U·[>V>WBXaW V3XaS X&_'n*V>Wg _�nqp U-S XaW V~n*V3k<Yanqp4n*V|^$`&\3m�ja] _Cm�n&W m®XaW _Cf&teY4U&Z7_*W m�teV{W Z7k<Y4\3f4U:n*V3fanBj4k<`&\ U&V\>W7Z7k<hCZ7_*W m

x′2 + y′2 + z′2 − c2t′2 = 0

x2 + y2 + z2 − c2t2 = 0c3`&V

x′2 + y′2 + z′2 − c2t′2 = x2 + y2 + z2 − c2t2

x′2 − c2t′2 = x2 − c2t2.

x�c3U&\ UBn*V3mok<`Bs Z<p:nClQU�g _'n*V3Z7k>p g V$n&W Z7g4vQULorentz

teVhCk<\3f4g _

(α2 − c2β2)x2 − 2(vα2 + c2βγ)xt− (c2γ2 − v2a2)t2 = x2 − c2t2

s

α2 − c2β2 = 1

vα2 + c2βγ = 0

c2γ2 − v2α2 = c2.ª Y4U&\ UBn*V3m�nC\Z7Y4Z<nqp g V"r3`aS Z<[>\3f4g _

γ =1

√

1− v2

c2

, β = − v/c2√

1− v2

c2

, α2 = γ2

ô�s ¦�§<¨F© ª7«q¬®­ ¯�°4°4±7© ²q¬

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^$V>S `&U&\3f4U�nqvQ`&V-nqp"g \3`&i,s

x′ =x− vt√

1− v2

c2

y′ = y

z′ = z

t′ =t− vx/c2√

1− v2

c2t W7g _'n*V3Z7k>p g V$n&W Z7g \>S,V3fan*\>S7c$]4] V3¸&V3U�nqp U�V3UBn&S ]4p d,p:g V3mobeW V-nC\ U�[>j3Z7g \,w��V3`&V$nqp `&\3Y4Uqn*V>WNU&hCV�ieV>W U&j3g _CU&V}j$^ lQm-XaW V3Z<n*\a]4s{n*\3f{k<`&j U&\3f|[>V>WNZ7f4Z<n*\4]4sn*\3f}g4s4[>\3f4ma��beW VÌ^$V3`&c3X&_*W b,g V�X&Y4\}b,_�b,\ U&j4n*V}^$\3f}Z7f4g&r3V>S U&\3f4Unqp�k<`&\ UaW [3sZ<n&W b,g4s

t1 A(x0, y0, z0, t1)[>V>W

t2V3UBn&S Z<nC\>W k<V�Z<n*\"S XaW \®Z<p g _*S \®Z<nC\:Z7Y4Z<nqp g V

Oxyz,Z<n*\�Z7Y4Z<nqp g V

O′x′y′z′teV�V$^$hCk<\3f4Uyk<`&\ UaW [>c:¹�X&W V3Z<n*\4]4son*\3f�k<`&j U&\3f ½

∆t =∆t′

√

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xy\�^ ] _C\ UBh'[3nBp g4Vunqp m·b,`&V3i1W [3s m·V3U&V$^$V3`&c$Z<n*V3Z<p m·g _·nqpuk<`Bs Z<ponClQU�W Z7\3Z<nCV t<zg$W [3vQU®[>V3g4^$fa]4vQU®^$\3f"^$_C`aW b,`&c3deV3g _os XBp_*S U&V>W,j4n&Weg4^$\3`&_*SeU&V~_'^$_'[3n*V te_*S,[>V>WZ7_�^$_C`aW Z7Z7j4nB_C`&_Cm�XaW V3Z<nBc3Z7_*W maw1¾ÔZ7f4U&c3`Bnqp Z<p

w = f(x, y, z)g4^$\3`&_*S1U&V~^$V4z

`&V3Z<n*V te_*Seg _�n&W m:W Z7\3Z<n*V teg$W [>hCm®_'^$W iec3U&_*W _Cmf(x, y, z) = k.

��W V|^$V3`&c3X&_*W b,g Vp{Z7f4U&c3`Bnqp Z<p

f(x, y, z) = x2 + y2 + z2 g4^$\3`&_*SNU&V�^$V3`&V3Z<n*V te_*SFg _®n&W m-\4zg j$[>_CUBnB`&_Cm�Z7ieV>S `&_Cm

x2 + y2 + z2 = kZ<n*\�Z7Y4Z<nqp g V�n'lQUÌ[>V3`BnB_CZ,W V3UBvQU

Z7f4UBnB_'n*V b,g hCUBlQU$w�{âFÛ��F×>Ö$Ü ç¤Þ�â���W��� 7 H�Ö4×7Üh�FÝ�è>Ö�è7Ü �Ü Ý¤ÙFÝ�è â�·Þ�Ü á�D��:á�âFÞFÕeß=eÖ��:è����"Ý��@Yã��FÛ3è��FÝ����

f(x, y) =√

9− x2 − y2çQÜ�â

k = 0, 1, 2, 3�

� Õ��Fã3è��FÝ���� t W�W Z7\3Z<n*V teg$W [>hCm´[>V3g4^$Ya] _Cm´fa^$\4] \&beS À&\ UBn*V3W�V$^$j nqpÚZ7k<hCZ<pf(x, y) = k

sx2 + y2 = 9 − k2 w·RTS U&V>W·ieV3U&_C`&j´j4n&WQbeW V¶n&W m�n&W g hCm{nC\3f

k = 0, 1, 2\>WeW Z7\3Z<nCV teg$W [>hCm�[>V3g4^$Ya] _Cm�_*S U&V>W7\3g j$[>_CUBnB`&\3W<[>Ya[3] \>W7g _u[>hCUBnB`&\

n*\Z<p g _*S \(0, 0)

[>V>W<V$[3n&S U&V √9− k2

¹ºr$] h'^$_u¢�k1wF»4w å4å ½'w

-2

0

2

-2

0

2

10

15

20

25

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�T�$�3�3���a� é4é$��� êG�B� �4�q��ê�� �f = x2 + y2 �B�B�eëB�1� �aëq�&���Cñ ó � � # �o�B� ó �%�"!*ê������*�u�&���*��qê��'� ëCîB�

(0, 0) ¡

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i) f1(x, y) = ln(1− x2 − y2), ii) f2(x, y) =1

1− x2 − y2,

iii) f3(x, y) = ln(y − x).

È ßeÝ����

ô$õW�°Éo÷3þ,«q°4²q¬®§<¨eª>ú<¨1²B© ¬ ¥��

• (i) u`Bp Z,W g \$^$\>W vQUBn*V3mo^$\a]$W [>hCm�Z7f4UBnB_'n*V b,g hCU&_Cm&�

x = r cosφ, y = r sinφ�

poZ7f4U&c3`Bnqp Z<pf1(x, y, z)

^$V>S `&U&_*W*nqp�g \3`&i,sf1(r) = ln(1−r2)

[>V>Wq\3`aS À&_'n*V>WbeW V

0 ≤ r < 1w�RQ^$\3g hCUBlQm®n*\{^$_CXaS \{\3`aW Z7g \3Y|nqp m"V$^$\4nB_'] _*S n*V>WFV$^$j}j4] V

n*V"Z<p g _*S V:^$\3fur3`aS Z<[>\ UBnCV>W3Z<n*\"_CZ<l�nB_C`aW [>j:nC\3f�[>fa[3]$W [>\3Y�XaS Z<[>\3f�V$[3n&S U&V3mr = 1.

• (ii)0�_:nC\ U�S XaW \}nB`&j$^$\{r3`aS Z<[>\3f4g _:j4n&WN[>V>WNp}Z7f4U&c3`Bnqp Z<p

f2hCk<_*WN^$_CXaS \

\3`aW Z7g \3Y-\4] j$[3]4p `&\-n*\|_'^3S ^$_CX&\(xy)

_'[3nBj3munqp m�^$_C`aW iehC`&_*W V3mun*\3f"[>Ya[3] \3fr = 1.

• (iii)��W V�U&V�\3`aS À&_�n*V>WFp{Z7f4U&c3`Bnqp Z<p

f3teV}^$`&h'^$_*WNU&VÌW Z7k<Y4_*W

y − x > 0XBp4] V3XBsy > x

w�RQ^$\3g hCUBlQm�nC\Ä^$_CXaS \�\3`aW Z7g \3Y nqp mIZ7f4U&c3`Bnqp Z<p mf3V$^$\4nB_'] _*S nCV>WQV$^$j¶ja] V³nCV¶Z<p g _*S V³nC\3fÌp g$W _'^3W ^$hCX&\3f£^$\3f}r3`aS Z<[>_'n*V>W�^$c3UBl

V$^$j-nqp U�_Cf&te_*S Vy = x

w

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f(x, y) =1√x− y

+1√x+ y

+ 2.

y

x0

y=x

y=-x

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f(x, y)^$`BV beg4V$naW [3s�teV ^$`&h'^$_*W®nC\

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x > y[>V>W

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x = y[>V>W

x = −y ½'w��WVÂ�{â��1Û3Ö��TÖ$ØNè Ù³Õ1Ö4×7Ø ÙÑÙFÛ7Ü Ý¤Þ�Ù1ß´è����|Ý��eã��FÛ3è��FÝ����

f(x, y, z) =

√1− z2

3−√

4− x2 − y2.

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[>V>W4− x2 − y2 ≥ 0

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z2 ≤ 1s −1 ≤ z ≤ 1

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RQ^$\3g hCUBlQma��nC\}^$_CXaS \

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x2 + y2 = 4[>V>W7Y4de\3m�å>w

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1 ≤ z ≤ 1.U RQnBZ,W,n*VZ<p g _*S V(x, y, z)

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√

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(x, y)_CUBnBj3m

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z =√

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cos(x2 + y2) ≥ 0,[>V>W

2kπ − π/2 ≤ x2 + y2 ≤2kπ + π/2 k = 1, 2,

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k = 0, 1, 2,w¼w¼w¼w@¹ºr$] h'^$_u¢�k1wF»4w å�/3½

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U = Ax2+By2

2− x2y2 à�ä1Õ1Ù��

A,B�·Öaè7Ü á�D��~Ý�è â�·Ö4Û�D������{â

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cos(x2 + y2) ¡

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x2

a2+y2

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x = 0[>V>W

y = 0_*S U&V>W1^$V3`&V4r3\4] hCma�F_CUBv \>W�nC\3g hCm:nqp m"g _�n*V�_'^$S ^ _CX&V

z =

¥�s ¦�§<¨F© ª7«q¬®­ ¯�°4°4±7© ²q¬

p_*S U&V>W�\>W�_']4] _*S de_*W m

(x2/(2a2cp)) + (y2/(2b2cp)) = 1��nClQU´\$^$\>S lQU£\>W

p g$W c3¸&\ U&_CmyV3f4¸&c3U&\3f�[>V t,vQm�V3f4¸&c3U&_*W nC\pw$¾ g \3`&i,s�nqp mo_'^3W iec3U&_*W V3m�[>V>W$n'lQU

W Z7\3Z<nCV teg$W [3vQU�[>V3g4^$fa]4vQU�ieV>S U&_'n*V>W>Z<n*\~¢�k>s g V{»4w å�¦>w

DFEJÒ¡Ó£K�L�P¤K�É>J�ÊÌM�JºLÚÎ�NNKQÓ�����£�V®r3`&_�te_*S<n*\"^$_CXaS \-\3`aW Z7g \3Y®nClQU�^$V3`&V$[>c$nCl¶Z7f4U&V3`Bnqs Z7_'lQU�1

i) f(x, y) = 3y2 − 9x+ 5 lnx2,

ii) f(x, y, z) = z(x2 + y2 − 1)1/2 + ln[z2(4− x2 + y2)],

iii) f(x, y) = x(1− y)/(y2 − 2y + 1),

iv) f(x, y, z) = (x2 + y2 − z)1/2 + ln(z2 + x2 + y2).

��-,���£�V®r3`&_�te_*S<[>V>W7U&V-Z7k<_CXaW V3Z<nB_*S<n*\-^$_CXaS \-\3`aW Z7g \3Y®nClQU�Z7f4U&V3`Bnqs Z7_'lQU�1

i) f(x, y) = x/(y2 − 4x),

ii) f(x, y) = [6− (2x+ 3y)]1/2,

iii) f(x, y) = [x2 + y2 − 4] ln (16− x2 − y2),

iv) f(x, y) = (x2 − y2)1/2 + (x2 + y2 − 1)1/2.

��WV��Ô£�V¶r3`&_�te_*S�[>V>W�U&VIZ7k<_CXaW V3Z<nB_*SynC\I^$_CXaS \I\3`aW Z7g \3Y¶nqp mÌZ7f4U&c3`Bnqp Z<p m 1f(x, y) = x ln(y − 1) + (1− x2)1/2

w��Z���.£�VÌr3`&_�te_*S�n*\³^ _CXaS \³\3`aW Z7g \3YÌ[>V>WQU&V³Z7k<_CXaW V3Z<nC\3Y4U}\>WTW Z7\3Z<nCV teg$W [>hCm[>V3g4^$Ya] _Cm�nqp muZ7f4U&c3`Bnqp Z<p m

f(x, y) = y/(x2 + y2)w

��-m���£�V®r3`&_�te_*S<[>V>W7U&V-Z7k<_CXaW V3Z<nB_*S<n*\-^$_CXaS \-\3`aW Z7g \3Y®nClQU�Z7f4U&V3`Bnqs Z7_'lQU�1

i) f(x, y) = ln(x+ y),

ii) f(x, y) = arccos(xy),

iii) f(x, y) = ln(x2 + y),

iv) f(x, y) = arcsin(x+ (xy)1/2.

��W����£�V®r3`&_�te\3Y4U�nCV-^$_CXaS V-\3`aW Z7g \3Y®nClQU�Z7f4U&V3`Bnqs Z7_'lQU�1

i) f(x, y) = ln(a− x2 + y2) + (x2 + y2 − b),

ii) f(x, y) = (x2 − y2)1/2 + (x2 + y2)1/2.

ô$õZ�;Õ�¨eª>ú<¨1²B© ¬��F© §�ü�\3¨� ¥R�

��J}<��£�Vμ�`&_�te\3Y4U�nCV-^$_CXaS V-\3`aW Z7g \3Y®nClQU�Z7f4U&V3`Bnqs Z7_'lQU�1i) f(x, y, z) = ln(xyz),

ii) f(x, y, z) = (1− x2 − y2 − z2)1/2.��WÑ���£�V{r3`&_�te\3Y4U~\>W¤W Z7\3Z<n*V teg$W [>hCm-[>V3g4^$Ya] _Cm"n'lQU�Z7f4U&V3`Bnqs Z7_'lQUu = xy[>V>W

v = x2y2 w��WÖ���£�V®r3`&_�te\3Y4U�\>W,W Z7\3Z<n*V teg$W [>hCm�_'^3W iec3U&_*W _CmonClQU�Z7f4U&V3`Bnqs Z7_'lQU�1i) V = x+ y + z,

ii) U = ln(1− x2 − y2 − z2).