Relación de Problemas L1

 

Jntkj³ntlhns ntlhns pnrn bn Khdgdj³ Khdgdj³ľn y bn Kjprksn. Cdmbk orncd ck Khdgdj Khdg dj³ ³ľn y Ncjlglstr Ncjlg lstrnhl³ nhl³dg dg ck kjprksns. Bkhhl³ d dg g >.

>.  Enbbn

kb ornclkgtk ck bns `ughldgks6

x,y,z)) :  x = yz n.   ` > (x,y,z x,y,z)) :  Bg  Bg((xyz xyz)) m.   ` = (x,y,z x,y,z)) :  k xyz h.   ` 8 (x,y,z u,v,w)) :   uv c.   ` 3 (u,v,w w =.   Hnbhubk C ` ((x1 ) Hnbhubk  C v ` 

 x 1  : (>, pnrn x pnrn (>, >) y  v  : (8, (8, =), pnrn bn sloulkgtk `ughl³dg6 dg6 ` ( ` (x> , x= ) :

  Bg x> x=

 y,, rkspkhkjprksn prdcuhk cds mlkgks utlblzngcd jngd ck dmrn y jntkrln prljn kg hngtlcncks x hngtlcncks  x  k  y tlvnjkgtk. Bn `ughl³ddg g ck prdcuhhl³dg dg vlkgk cncn pdr bn sloulkgtk kxprksl³ddg6 g6 8.  [gn

q  : Bg((x= + y  y)), xy >/= )  : (q > , q = ) : (Bg Xnmlkgcd quk bn kjprksn kst³n utlblzngcd >1 uglcncks ck jngd ck dmrn y >11 uglcncks ck jntkrlns prljns, q  pnrn kstds glvkbks. ´^u³k lg`drjnhl³ hnbhubk bn jntrlz fnhdmlngn ck bn `ughl³dg dg ck prdcuhhl³dg  dg   q  pnrn dg dg khdg³djlhn djlhn rkhdok kstn jntrlz4 kjprksn prdcuhk ug mlkg   N, n pnrtlr ck cds `nhtdrks   @ >   y   @ = , quk utlblzn kg hngtlcncks   x   k   y , rkspkhtlvnjkgtk, sko³ ug ug bn sloulkgtk `ughl³ddg g ck prdcuhhl³ddg6 g6 3.   [gn

  x= y   + xy 8 Kg kstk lgstngtk, bn kjprksn utlblzn > uglcnc ck   @ >   y = ck   @ = , y sk kst³n pbngtkngcd bn pdslmlblcnc ck lghrkjkgtnr ug pdhd bn hngtlcnc utlblzncn ck  @ > , y clsjlgulr kb trlpbk bn hngtlcnc utlblzncn ck  @ = . ´Ks kstd nhdgskfnmbk4 q (x, y ) :

;.  Hnbhubk

bn jntrlz eksslngn ck bn `ughl³dg dg ` ( ` (x,y,z x,y,z)) :

0.  Xkn

  8yz x=

bn `ughl³ddg `  g  `    ck﬎glcn6

⊍ ]8 ↝ ]=  x∝y  ` ( ` (x,y,z x,y,z)) : k z ∝ x , Bg Bg(> (> + x + x= z y ) `   6  C

. Hnbhubk bn ckrlvncn clrkhhldgnb ck `  ck  `  kg  kg kb pugtd (>, (>, >, =) sko³ ug ug bn clrkhhl³dg dg ckb vkhtdr v vkhtdr  v :  : (> (>,, >, >) k lgtkrprktk kb rksubtncd. >

 

?.  [gn

kjprksn kst³n hdjkrhlnbl hdjk rhlnblzngcd zngcd ug gukvd nrt³ nrt³ľhubd huyn `ugh `ughl³ l³dg dg ck ckjngcn kstljncn ks6 c( p, r) :  k

r p

slkgcd r bn rkgtn rkgtn jkcln ck bds hdgsu hdgsujlcdr jlcdrks ks pdtkghlnbk pdtkghlnbkss y p kb prkhld prkhld ugltnrld ck clhed mlkg. Xl hdgslckr hdgslckrnjds njds quk, kg kb jdjkgtd nhtunb,  r  : >1 y p y  p :  : =, jkclcds kg jlbbdgks ck kurds, cktkrjlgnr6 n) Bn kbnstlhlcnc kbnstlhlcnc ckjngcn-p ckjngcn-prkhld rkhld y ckjngcn-r ckjngcn-rkgt kgtnn ck clhed mlkg. m) Xl supdgkjds quk kb prkhld ckb mlkg nujkgtn hdg bn rkgtn kg ugn prdpdrhl³dg   ξ, ´kg qu³k rkbnhl³ddg g nujkgtn d clsjlguyk bn ckjngcn4 5.  Cncn

bn `ughl³dg dg

= 8

` ( ` (x,y,z x,y,z)) :  Bg

x y  z

cdgck

 x : t   :  t  :  k  yz : : k  =∝t = t

∍` (t)  t : pnrn t  : > utlblzngcd bn ]kobn ck bn Hnckgn hnbhubk   ∍` ( ∍t   pnrn

/= , xhds

∝ >))

o (u, v ) :  u >/= v =

ngnblhk sl sk pukckg hdjpdgkr, y kg hnsd n﬎rjntlvd, hnbhubk bn ckrlvncn pnrhlnb ck bn `ughl³dg hdjpukstn hdg rkspkhtd n su skougcn vnrlnmbk, kg kb pugtd (>, >), npblhngcd bn rkobn ck bn hnckgn. >1.  Kg

ug prdhksd prdcuhtlvd, n pnrtlr ck trks jntkrlns prljns, J>, J= y J8, sk dmtlkgkg cds mlkgks skjltkrjlg skjlt krjlgncds ncds M> y M= jkcln jkclngtk gtk bn sloulk sloulkgtk gtk `ughl³ ddg g ck prdcuhhl³dg6 dg6 x





k + Bg  Bg((z = )

 ` (x,y,z x,y,z)) : xz y (u, v ) :  ` (

y

cdgck x, y, z sdg bns hngtlcncks ck jntkrln prljn utlblzncns ck J>, J= y J8 rkspkhtlvnjkgtk, jlkgtrns quk quk u y v sdg sdg bn bnss hn hngt gtlc lcnc ncks ks ck M> y M=. M=. N pn pnrt rtlr lr ck M> y M= sk pr prdcu dcuhk hk ug mlkg mlkg ﬎g ﬎gnb nb,, M jk jkcln clngt gtkk bn `ughl³ dg6 dg6   (u + v  + v))= o (u, v ) : 8 Hnbhubk bn prdcuhtlvlcnc jnrolgnb ck bns trks jntkrlns prljns, J>, J= y J8 sl kg kb jdjkgtd nhtunb utlblznjds ugn uglcnc ck hncn ugn ck kbbns. >>.  Kg

ug prdhksd prdcuhtlvd, n pnrtlr ck trks jntkrlns prljns, N, M y H, sk dmtlkgkg cds mlkgks skjltkrjlgncds M> y M= jkclngtk bn sloulkgtk `ughl³ddg g ck prdcuhhl³ddg6 g6 =

=

x y  + y   x + y   , Bg Bg(  ` (x,y,z x,y,z)) : ( (u, v ) :  ` ( z z   + >) 8

=

 

cdgck x, y, z sdg bns hngtlcncks ck jntkrln prljn utlblzncns ck N, M y H rkspkhtlvnjkgtk, jlkgtrns quk u y v sdg bns hngtlcncks ck M> y M=. N pnrtlr ck M> y M= sk prdcuhk ug mlkg ﬎gnb, M jkclngtk bn `ughl³dg6   (u + v  + v))= o (u, v ) : v u Hnbhubk bn prdcuhtlvlcnc jnrolgnb ck bns trks jntkrlns prljns, N, M y H sl kg kb jdjkgtd nhtunb utlblznjds ugn uglcnc ck hncn ugn ck kbbns.

∝

>=.  Cncn

bn `ughl³dg6 dg6

¿  b bggxy ↝v   : (∝8, =), kg kb pugtd (>,>) n) Hnbhubk Hnbhubk bn ckrlvncn ckrlvncn clrkhhldgnb clrkhhldgnb sko³ u ug g bn clrkhhl³dg dg ∝ ` (x, y) : (x + >) y = +

m) Hnbhubk Hnbhubk kb cksnrrdbbd cksnrrdbbd ck Snybdr Snybdr ck orncd = nbrkckcdr ckb pugtd (>,>). >8.  Cncn

bn `ughl³ddg `  g  ` ((x, y) :  k ∝xy hnbhubk kb cksnrrdbbd ck Snybdr ck drckg cds kg kb pugtd (>, 1).

>3.  Cncn

bn `ughl³ddg g

 x + xy =

` ( ` (x, y ) :

x

∝y

n) Kst³ uclksk uclksk su edjdokgklcnc, cktkrjlgngcd, kg su hnsd, kb orncd. m) Cktkrj Cktk rj³³ľgksk ľgk sk hu³ hu ngtd ³ngtd vnbk6 x >;.   [gn

∍`  ∍`   + y  +  y ∍x ∍x

kjprksn prdcuhk ug mlkg utlblzngcd cds `nhtdrks prdcuhtlvds kg hngtlcncks x k y, ck nhukrcd Hdgtkstk rnzdgncnj rnzdgncnjkgt kgtkk n bn sloulkgtk sloulkgtk prkougtn6 prkougtn6 Xl hdg bn sloulkgtk kxprksl³dg6 dg6   q (x, y ) : (x + y  +  y))k x/y . Hdgtkstk nujkgtnjds kb cdmbk bds `nhtdrks prdcuhtlvds, ´hu³ngtd ngtd nujkgtnr³n bn prdcuhhl³dg4, dg4, ´pdr ´pd r qu³k4 k4

∖ 

>0.  [gn

kjprksn prdcuhk ug mlkg, q, utlblzngcd cds `nhtdrks prdcuhtlvds kg hngtlcncks x k y, ck nhukrcd hdg bn sloulkgtk kxprksl³ddg6q  g6q (x, y ) : y :  ykkx/y . Cktkrjlgk6 n) Xl bn `ughl³dg dg q ks edjdo³ kkgkn gkn y, kg hnsd n﬎rjntlvd ck qu³ k orncd. Lgtkrprktk kb rksubtncd. ´Qdcr³ ´Qdcr³ľn hdghbulr kb tlpd ck rkgcljlkgtd n kshnbn quk prkskgtn4 m) Hnbhubk Hnbhubk bn jntrlz eksslngn eksslngn ck bn `ughl³ `ughl³ddg g q kg kb pugtd (8, 8). >?.  [gn kjprksn prdcuhk cds mlkgks utlblzngcd trks `nhtdrks prdcuhtlvds kg hngtlcncks x, y, z, rkspkhtlvnjkgtk. Bn `ughl³dg dg ck prdcuhhl³dg dg vlkgk cncn pdr bn sloulkgtk kxprksl³ddg6 g6 x,y,z)) x,y,z)), q = (x,y,z q (x,y,z x,y,z)) : (q > (x,y,z )) : (Bg (Bg((x= z + y  +  y)), y

 x + x ) =

=

q = (x,y,z x,y,z)) Ngnblhk sl bn `ughl³dg dg ck prdcuhhl³ddg g ckb skougcd mlkg ((q  )) ks ugn `ughl³ddg g edjdo³kgkn kgkn y, kg hnsd n﬎rjntlvd n﬎rjn tlvd,, ck qu³k orncd. orncd. Lgtkrprk Lgtkrprktk tk bds rksub rksubtncds tncds.. Qdr dtrn pnrtk, bns hngtlcncks hngtlcncks kjpbkncns kjpbkncns ck bds cds `nhtdrks kvdbuhldgng kg kb tlkjpd ck nhukrcd hdg bns sloulkgtks kxprksldgks6 x(t) : > + t +  t= y (t) : 8k=t  = z (t) : t cktkrjlgk bn vnrlnhl³ddg g ck bn prdcuhhl³dg dg ck njmds mlkgks hdg rkspkhtd nb tlkjpd, utlblzngcd bn rkobn ck bn hnckgn, sl t : >. 8

 

>5.  [gn

kjprksn ckclhncn n bn prdcuhhl³dg dg ck ug mlkg utlblzn kg su prdhksd prdcuhtlvd cds prdcuhtds skjltkrjlgncds,   X >   y   X = , kg hngtlcncks   u   y   v  rkspkhtlvnjkgtk, y dmtlkgk ugn hngtlcnc ckb mlkg ﬎gnb,   w, skjltkrjlgncds, sko³ u ug g bn sloulkgtk sloulkgtk kxprksl³ kxprksldg6 ³dg6 w  : `   :  ` ((u, v ) :

  u= v = >0

n) Cktkrj Cktkrjlgk lgk kb tlpd ck rkg rkgclj cljlkg lkgtds tds ck ksh kshnbn nbn quk juks jukstrn trn clh clhen en `ughl³ `ughl³ dg dg k lg lgtkr tkrprk prktk tk kb rks rksubt ubtncd ncd dmtkglcd. m) Xupdgon Xupdgon quk bn kjprksn ncqulkrk ncqulkrk ugn j³ nqulgn nqulgn jkcln jkclngtk gtk bn hunb dmtlkgk bds cds prdcuhtds skjltkrjlskjltkrjl X >   y  X = , n pnrtlr ck trks jntkrlns prljns6   J > ,  J =   y  J 8 , quk bn kjprksn utlblzn kg hngtlcncks gncds, X  gncds, x, y y z rkspkhtlvnjkgtk. Clhen trngs`drjnhl³dg dg vlkgk cncn pdr bn sloulkgtk kxprksl³dg6 dg6  o((x,y,z x,y,z)) : (x= + (z (u, v ) :  o (z

∝ >)8, xy>/=)

Cktkrjlgk bn prdcuhtlvlcnc jnrolgnb ck hncn ugn ck bns trks jntkrlns prljns kg kb mlkg ﬎gnb, sl kg kstk jdjkgtd bn kjprksn kst³n utlblzngcd bns sloulkgtks hngtlcncks ck bns jlsjns6 x : >, y : 3, z : =.  X >   y  X = , kg hngtlcncks u y kjprksn prdcuhk ug mlkg N, utlblzn utlblzngcd gcd cds prdcuh prdcuhtds tds skjltkr skjltkrjlgnc jlgncds ds X  v, rkspkhtlvnjkgtk, slkgcd su `ughl³dg dg ck prdcuhhl³ddg6 g6 > ,  J =   y  J 8 , quk utlblzn kg hngtlcncks x, y, z, rkspkhtlvnjkgtk, sko³ u ug g bn sloulkgtk `ughl³ddg g ck prdcuhhl³dg dg hdgfugtn6  ` (x,y,z x,y,z)) : ( (u, v ) :  ` (

xz =   + bg z,xy= z ) y

n) Xl kg kstk jdjkgtd jdjkgtd sk kst³ kst³n utlblzngcd > uglcnc ck hncn jntkrln prljn, cktkrjlgk qu³k hngtlcnc sk kst³ n prdcuhlkgcd ckb mlkg N.  ` ,,  ` = 4 m) ´^u³ k rkgcljlkgtds ck kshnbn prkskgtn bn skougcn `ughl³ddg g hdjpdgkgtk ck ck `  h) Xl bds prkhlds ugltnrlds ugltnrlds ck bns trks jntkrlns jntkrlns prljns ks kb jlsjd, ´ck hu³nnbb hdgvkg hdgvkgcr³ cr³ľľnn j³ns ns nujkgtnr bn hngtlcnc utlblzncn4 =1.  Cncn

bn rkbnhl³dg6 dg6 8

y  : = x + 8 n) Cktkrjlgk Cktkrjlgk sl ks pdslmbk n﬎rjnr n﬎rjnr bn kxlstkghln ck y hdjd `ughl³ dg dg ljpb³ ljpb³ľhltn ck x kg ug kgtdrgd ckb pugtd (>, =). m) Cktkrjlgk Cktkrjlgk sl bn `ughl³ `ughl³dg dg ckb npnrtncd npnrtncd ngtkrldr ngtkrldr ks hrkh hrkhlkgt lkgtkk d ckhrkhlkg ckhrkhlkgtk tk y h³ dghnvn dghnvn d hdgvkxn kg x : >. ` (x, y ) :   xy = +  yx = , ngnblhk bn kxlstkghln ck bn `ughl³dg dg ljpb³ ljp b³ľľhlt h ltnn   y   :   π(x) n pnrtlr bn `ughl³dg  dg   ` ( ck bn hurvn hurvn ck glvkb glvkb quk pnsn pdr kb pugt pugtdd (>, >). Nsljls Nsljlsjd, jd, sl kxlstk clhen `ughl³ `ughl³ddg g ljpb³ľhltn ľhltn cktkrjlgk su prljkrn y skougcn ckrlvncn. =>.   Cncn

==.  Xkn

bn `ughl³dg dg ck prdcuhhl³dg dg ck ugn kjprksn6 q (x,y,z x,y,z)) :  xy + y = z

cdgck nhtunb x, y y sk z kst³ rkprkskg rkprk skgtng tng bns ugn hngtlcn hngtlcncks cks ck utlblznc utlblzncns ckcktrks `nhtd `nhtdrks rks prdcuhtlvds. prdcuhtlvds. Xnmlkgcd Xnmlkgcd quk kg kb jdjkgtd n utlblzngcd uglcnc hncnns ugd kbbd, hnbhubk6

3

 

n) Bns prdcuhtlvlcnc prdcuhtlvlcncks ks jnrolgnb jnrolgnbks ks ck clhe clheds ds `nhtdr `nhtdrks. ks. ´Xl tuvlkrng tuvlkrng kb jlsjd prkhld, prkhld, hu³nnbb bk nhdgskf nhd gskfnr³ nr³ľľnn nujkgtnr nb kjprksnrld4 m) Bn rkbnhl³ rkbnhl³ dg dg jnrolgnb j nrolgnb t³kkhglhn hglhn ck ssustltuhl ustltuhl³³ddg g ckb prljkr `nhtdr pdr kb skougcd kxpblhngcd su slogl﬎hncd. =8.  Xkn

bn `ughl³dg dg ck utlblcnc ck ug hdgsujlcdr6 [ ( [ (x, y ) :

∝ x  =>y8 ,

x, y 7  1

cdgck x k y rkprkskgtng bns hngtlcncks hdgsujlcns ck cds mlkgks   N   y   M . Kstuclk Kstuclk kb hrkhljlkgt hrkhljlkgtdd ck bns hurvns ck lgcl`krkghln hdjd `ughldgks ck x ck  x,, ns³ ns³ľ hdjd bn hdghnvlcnc d hdgvkxlcnc hdgvkxlcnc ck bns jlsjns. ∖ 

 q (x, y ) :  k x y , slkgcd x slkgcd  x  k  y  bns hngtlcncks quk utlblzn ck `ughl³dg dg ck prdcuhhl³ddg g ck ugn kjprksn ks ks q  7  1. Qnrn ug glvkb ck prdcuhhl³ddg i cds `nhtdrks prdcuhtlvds, x,y prdcuhtlvds,  x,y 7 1. g  i  ﬎fd, rkspdgcn rkspdgcn n bns sloulkgtk sloulkgtkss hukstldgks hukstldgks66 =3.   Bn

∍y n) Cktkrjlgk, Cktkrjlgk, sl ks pdslmbk, bn rkbnhl³ rkbnhl³ddg g jn jnrolgnb rolgnb t³kkhglhn hglhn ck su sustltuhl stltuhl³dg ³d g ck x pdr y ( ∍x (x))

m) ´Xdg bns hurvns hurvns ck glvkb ckb npnrt npnrtncd ncd ngtkrldr ngtkrldr hrkhlkg hrkhlkgtks tks d ckhrkhlkg ckhrkhlkgtks4 tks4 ´sdg h³dghnvns dghnvns d hdgvkxns4 =;.  Bn

kjprksn @BNSDXN prdcuhk bntns ck `nmncn kg hngtlcnc q (jlbks ck bntns), utlblzngcd hdjd `nhtdrks prdcuhtlvds prdcuht lvds bkoujmrks bkoujmrks y henhlgns henhlgns kg hngt hngtlcnck lcnckss x k y (jlbk (jlbkss ck ilbds), rkspkhtlv rkspkhtlvnjkg njkgtk. tk. [gn lgvkstlonhl lgvkstlonhl³³ddg g lgtkrgn en pkrjltlcd dmtkgkr bn sloulkgtk rkbnhl³ddg g kgtrk bns vnrlnmbks hltncns6 +> xyq+> k∝xyq

bg(xy xy)) ∝ (=/q  ∝ bg( (=/q ) + > : 1

n) ´Ck﬎gk ´Ck﬎gk kstn rkbnhl³ rkbnhldg ³dg bn prdcuhhl³ddg g ck bntns hdjd `ughl³dg dg ck bns hngtlcncks ck bds `nhtdrks sl q : >, x : y : >4 Kg hnsd n﬎rjntlvd, hnbhubnr bns prdcuhtlvlcncks jnrolgnbks kg clhed pugtd. m) Xl sk tuvlksk bn pdslmlblcnc ck nujkgtnr, nb jlsjd prkhld, bn hngtlcnc utlblzncn ck nbougd ck bds `nhtdrks ´hu³ nnbb kb kbko kolr³ lr³ľľn4 n4 =0.   Bn

kjprksn GN]NHEL, GN]NHEL, X.N. prdcuhk ug rk`rkshd rk`rkshd ck gnrng gnrngfn fn kg hngtlcnc hngtlcnc q (jlbks ck bltrds), bltrds), utlblzngcd hdjd `nhtdrks prdcuhtlvds noun hnrmdgntncn y fnrnmk ck gnrngfn kg hngtlcncks x k y (jlbks ck bltrds), rkspkhtlvnjkgtk, slkgcd bn `ughl³dg dg ck prdcuhhl³ddg6 g6 q (x, y ) : ;x + 0y 0y Hnbhubk bn rkbnhl³dg dg jnrolgnb t³kkhglhn hglhn ck sustltuhl³ ddg g ckb fnrnmk ck gnrng gnrngfn fn pdr kb noun hnrmdgnt hnrmdgntncn ncn.. Xl clsjlguyk bn hngtlcnc ck fnrnmk ck gnrngfn utlblzncn, ´tkgcr³n quk nujkgtnr kg jnydr d jkgdr jkclcn bn ck noun hnrmdgntncn hnrmdgntncn pnrn jngtkgkr jngtkgkr kb glvk glvkbb ck prdcuhhl³ ddg4 g4 =?.   Bds

onstds kg pumblhlcnc pumblhlcnc (o) ck ugn kjprks kjprksnn ckpkgckg ckb glvk glvkbb ck vkgtns vkgtns (v), ckb prkhld ckb prdcuhtd ( p, p : 1) y ck bds rkstngtks hdstks (h), sko³ ug ug bn kxprksl³ddg6 g6



o(v,p,h v,p,h)) :  bg(>1  bg (>1/h /h)) + (vp ( vp>/= )/>1 ´H³ddjd jd ckmk vnrlnr kb glvkb ck vkgtns kg `ughl³ddg g ckb prkhld pnrn quk bds onstds kg pumblhlcnc pkrjngkzhng hdgstngtks4 =5.  Xk

snmk quk bn rkbnhl³ddg g kgtrk bn hngtlcnc ck prdcuhhl³dg dg ck ug mlkg, q, y bns hngtlcncks x k y ck bds `nhtdrks prdcuhtlvds utlblzncds ks 6 qk xy

∝ xbgy ∝ kq = : 1

Xnmlkgcd quk, pnrn x : y : >, bn prdcuhhl³dg dg ks q : >, hnbhubk, pnrn kstds glvkbks6 n) Bns prdcuhtlvlcncks prdcuhtlvlcncks jnrolgnbks jnrolgnbks ck bds cds `nhtdrks `nhtdrks prdcuhtlv prdcuhtlvds. ds. ;

 

m) Bn rkbnhl³ rkbnhl³ dg dg jnrolgnb t³ kkhglhn hglhn ck sustltuhl³ dg dg ck y pdr x, ck jngkrn quk sk jngtkgon kb glvkb ck prdcuhhl³ dg. dg. =) +  xy ∝ > : 1

bg( bg(x y z

cdgck z ks kb g³ ujkrd ujkrd ck pdrt³ntlbks, ntlbks, x hng hngtlcnc tlcnc ck helps k y bn hngtlcnc hngtlcnc ck pbnhns mnsk. Kg kstk jdjkgtd, jdjkgtd, x  : y  z  : =.  :  y :  : >, >, z n) Hnbhubk Hnbhubk bn rkbnhl³ dg dg jnrolgnb t³kkhglhn hglhn ck ssustltuh ustltuhl³ l³dg dg ck bn hngtlcnc ck helps pdr bn hngtlcnc ck pbnhns mnsk. m) Hnbhubk Hnbhubk bn rkbnhl³ dg dg jnrolgnb t³ kkhglhn hglhn ck sustltuhl³ dg dg ck bn hngtlcnc ck pbnhns mnsk pdr bn hngtlcnc ck helps. h) ´Kxlstk ´Kxlstk nbougn rkbnhl³ rkbnhl³ ddg g kgt kgtrk rk bns cds rkbnhldgks rkbnhldgks jnrolgnbks jnrolgnbks ck sustltuhl³ sustltuhl³ dg4 dg4 81.  Bn

hngtlcnc ckjngcncn ck ug mlkg (c), su prkhld (p), y bn rkgtn ck bds lgclvlcuds quk bd hdgsujkg (r) vkrl﬎hng bn sloulkgtk khunhl³dg6 dg6

∖ 

rp  + pskg  pskg((rc =Bg rp +

  c

 : ; r = p Kg kstk lgstngtk, bds vnbdrks ck bns vnrlnmbks sdg6 c : ;9 p : >/=9 r : =. Cktkrjlgk sl bn khunhl³dg ngtkrldr ck﬎gk n bn ckjngcn hdjd `ughl³dg dg ljpb³ ljpb³ľhltn ckb prkhld prkhld y bn rkgtn, kg ug kgtdrgd ckb pugtd (;, >/=, =). Kg hnsd n﬎rjntlvd, hnbhubk bn kbnstlhlcnc ck bn ckjngcn-prkhld y bn kbnstlhlcnc ck bn ckjngcn-rkgtn, njmns kg (>/=, =). >1) +

∝

∝

K>.  Cncn

bn `ughl³dg6 dg6

o (u, v ) :  u = + v = bg( u) bg(u

¿

n) Kspkhl﬎quk Kspkhl﬎quk su cdjlgld y hnbhubk hnbhubk su jntrlz jntrlz eksslngn. m) Ngnblhk Ngnblhk sl ks pdslmbk hdjpdgkr bn `ughl³ `ughl³ ddg g o hdg bn sloulkgtk `ughl³ddg6 g6 ` ( ` (x, y ) :

=

x= + ;y ;y = ,

 x  x +  + >

y Kg hnsd n﬎rjntlvd, hnbhubk kb ornclkgtk ck bn `ughl³ddg g hdjpukstn kg kb pugtd (= (=,, >).



 

K=.  Cncn

bn `ughl³ddg g

=

` ( ` (x, y ) :

y + > x

  ,k

∖ 

y¿ x



n) Kspkhl﬎q Kspkhl﬎quk uk su cdjlgld. cdjlgld.  `  kg m) Hnbhubk Hnbhubk bn jntrlz jntrlz fnhdmlngn fnhdmlngn ck ck `   kg kb pugtd (>, (>, 1).

∝↝

` (> h) Xl n pnrtlr pnrtlr ckb ckb pug pugtd td (>, (>, 1) gds jdvkjds kg bn clrkhhl³ddg g ckb vkhtdr v   : (=, (= , >), hnbhubk C hnbhubk Cv `  (>,, 1) k lgtkrprktk kb rksubtncd. snmk quk bn rkbnhl³dg dg kgtrk bn hngtlcnc ck prdcuhhl³dg dg ck ug mlkg,   q , y bns hngtlcncks hngtlcncks   x   k   y  ck bds rkhursds prdcuhtlvds utlblzncds ks6 K8.  Xk

 Bg((x y q  8 +  x   + Bg y

¿ ¿ ∝ >) ∝ q = : 1 0

 

Xnmlkgcd quk x : y : >, q : =, ngnblhk sl bn khunhl³ddg g ngtkrldr ck﬎gk q hdjd `ughl³dg dg ljpb ljpb³³ľhltn ck x k y. Kg hnsd n﬎rjntlvd, dmtkgon bn prdcuhtlvlcnc jnrolgnb rkspkhtd n hncn ugd ck bds rkhursds kg kb pugtd (>, (>, >). Xl sk tuvlksk bn pdslmlblcnc ck nujkgtnr nb jlsjd prkhld bn hngtlcnc utlblzncn ck nbougd ck bds rkhursds ´hu³nnbb kbkolr³ľn4 ľn4 Fustl﬎quk Fustl﬎q uk su rkspukstn. rkspuks tn. K3.  Cncn

bn `ughl³ddg g ` ( ` (x, y ) :

kx¿

,

∝> =

y



n) Kspkhl﬎q Kspkhl﬎quk uk su cdjlgld. cdjlgld.

∖ y  x



 `  kg m) Hnbhubk Hnbhubk bn jntrlz jntrlz fnhdmlngn fnhdmlngn ck ck `   kg kb pugtd (1, (1, >).

 ∝↝

` (1 h) Xl n pnr pnrtl tlrr ckb pugt pugtdd (1, (1, >) gds jdvkjds kg bn clrkhhl³ddg g ckb vkhtdr v   : (> (>,, =), hnbhubk hnbhubk   Cv `  (1,, >) k lgtkrprktk kb rksubtncd.

∝

snmk quk bn rkbnhl³dg dg kgtrk bn hngtlcnc ck prdcuhhl³dg dg ck ug mlkg,   q , y bns hngtlcncks hngtlcncks   x   k   y  ck bds rkhursds prdcuhtlvds utlblzncds ks6 q = Bg Bg((x y q  8) 3 : 1 x K;.  Xk

 ∝

¿ ¿ ∝ ∝

Xnmlkgcd quk x : >, y : q : =, ngnblhk sl bn khunhl³ddg g ngtkrldr ck﬎gk q hdjd `ughl³dg dg ljpb ljpb³³ľhltn ck x k y. Kg hnsd n﬎rjntlvd, dmtkgon bn prdcuhtlvlcnc jnrolgnb rkspkhtd n hncn ugd ck bds rkhursds kg kb pugtd (>, (>, =). Xl sk tuvlksk bn pdslmlblcnc ck nujkgtnr nb jlsjd prkhld bn hngtlcnc utlblzncn ck nbougd ck bds rkhursds ´hu³nnbb kbkolr³ľn4 ľn4 Fustl﬎quk Fustl﬎q uk su rkspukstn. rkspuks tn.

?