olasılık ve İstatistik - erzurum atartürk Üniversitesi olasılık ve raslantı değişkenleri...
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8/9/2019 Olasılık Ve İstatistik - Erzurum Atartürk Üniversitesi Olasılık Ve Raslantı Değişkenleri Ders NotlarI
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CXWSXSI K]K]ÚXDÚ@ĶQCX_Ķ]C_Ķ
EjksĻjĻd vc Xkstjk`tĻ Mcĕnŕdc`jcrnMkbk Akzjk Dky`kd Ķìn`4
www.ccimcrs`etjkrn.oei
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Ñ@_ÑW]úrdìc dky`kdjkrĻ` kzjĻĕĻ hnz ñĕrctni úycjcrn`n dc`mn mcrs `etjkrĻiĻzĻ múzlú` hnr aerikttk
bkzĻrjkykrkd ñĕrc`onyc su`ik yñ`ú`mc ietnvc cticdtcmnr. K`okd mcrs `etjkrĻ`Ļ` hnr dntkpdkmkr mctkyjĻ vc dkpskijĻ bkjc lcjicsn nìn` kyrĻok ñzc`jn hnr ìkjĻŕik lcrcdicdtcmnr. HuìkjĻŕik nsc hnr súrcon lcrcdtnricdtcmnr.Őu k`mkdn bkjnyjc , kikìjkmĻĕĻiĻz ñĕrc`on`n`rkbktjĻdjk tkdnp cmchnjcocĕn hnr ykrmĻioĻ dky`kd ejuŕturikdtĻr. MejkyĻsĻyjk hu mcrs `etjkrĻ`Ļ`hnr mcrs dntkhĻ ycrn`c lcìcicycocĕn vc mcrsn` ñĕrctni úycsn tkrkaĻ`mk` tkvsnyc cmnjc` ycrjn vcykhk`oĻ dky`kdjkrjk hcrkhcr múŕú`újmúĕú`mc dktdĻsĻ`Ļ` kikìjk`mĻĕĻ yñ`mc ejkokĕĻ lñzñ`ú`mc huju`murujkrkd hu `etjkrmk` akymkjk`ĻjikjĻmĻr.
Hu mcrs `etjkrĻ`mk‚Vrehkhnjnty, Xk`mei Qkrnkhjcs k`m _teobkstno Vreocsscs hy Ktbk`ksneus Vkpeujns’, ‚Vrehkhnjnty, Xk`mei Vreocsscs, k`m Cstniktne` ]bcery aer C`ln`ccrs hy Bc`ry _tkrd k`m Geb` \. \eems’, ‚Jcoturc @etcs e` Vrehkhnjnty ]bcery k`m Xk`mei Vreocsscs hy Gck` \kjrk`m Mcpkrtic`t ea Cjcotrnokj C`ln`ccrn`l k`m Oeiputcr
_onc`ocs S`nvcrsnty ea Okjnaer`nk Hcrdcjcy, OK >=
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MCX_ĶÌCX ĶĐĶ
;. EjksĻjĻdk. EjksĻjĻd `cmnr5h. EjksĻjĻĕĻ` djksnd, hkĕĻj arcdk`s vc kdsnyeiktnd tk`ĻijkrĻ
8. Dúicjcr vc Ejkyjkrk. Dúic Ķŕjcijcrnh. Hnjcŕnd ejksĻjĻdo. Deŕujju ejksĻjĻdm. Hkyc„s tcercinc. HkĕĻisĻzjĻd
?. ]cdrkr Cmnjc` Mc`cicjcrk. Hnrjcŕtnrnjinŕ mc`cyjcrh. Ìedju mc`cicjcr
=. Xkstjk`tĻ Mcĕnŕdc`jcrn (]cd Xkstjk`tĻ Mcĕnŕdc`n)k. Hcdjc`mnd mcĕcr vc vkryk`sh. EjksĻjĻd yeĕu`jud vc mkĕĻjĻi ae`dsnye`jkrĻ, Lkussnk` vc mnĕcr yeĕu`jud
ae`dsnye`jkrĻo. Deŕujju MkĕĻjĻijkrm. Hkyc„s tcercin (yeĕu`jud vc mkĕĻjĻi ae`dsnye`jkrĻ`k lc`nŕjctni)
3. ]cd Hnr Xkstjk`tĻ Mcĕnŕdc`n`n` Ae`dsnye`jkrĻk. Xkstjk`tĻ mcĕnŕdc`n l(x)h. l(x)„ n` mkĕĻjĻi ae`dsnye`jkrĻo. Ieic`tjcrm. Dkrkdtcrnstnd ae`dsnye`jkr
1. Ķdn Xkstjk`tĻ Mcĕnŕdc`nk. Hnrjcŕnd mkĕĻjĻi vc yeĕu`jud ae`dsnye`jkrĻh. Ķdn rkstjk`tĻ mcĕnŕdc`n`c knt ae`dsnye`jkr
n. Ķdn rkstjk`tĻ mcĕnŕdc`n`n` tcd hnr ae`dsnye`unn. Ķdn rkstjk`tĻ mcĕnŕdc`n`n` ndn ae`dsnye`u
nnn. Hnrjcŕnd ieic`tjcr, hcdjc`mnd mcĕcrnv. Deŕujju Ķstktnstndjcr
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;. EjksĻjĻĕk Lnrnŕ
EjksĻjĻd tcernsn hcjnrsnzjnĕn` iktciktndscj ejkrkd iemcjjc`icsn njc njlnjc`nr. Cvrc`mclcrìcdjc ŕc` hútú` ejkyjkrĻ` ‚LĶWJĶ’ hnr mcĕnŕdc`n` ae`dsnye`jkrĻ ejmuĕu`u múŕú`úrscd, humcĕnŕdc` vc ae`dsnye`jkrĻ` hnjn`icsn/tk`Ļijk`khnjicsn bkjn`mc ‚hcjnrsnzjndjcr’ ertkmk` dkjdkr.]khnndn hu mcĕnŕdc`n` vc ae`dsnye`jkrĻ` ‚DC_Ķ@’ ejkrkd hnjn`icsn prktndtc iúidú` mcĕnjmnr.Hu`u` nìn` ‚EJK_NJND ]CEXĶ_Ķ’ ‚]CDXKXJK@K@’ ejkyjkrĻ` ertkjkik mkvrk`ĻŕjkrĻ`kemkdjk`Ļr.
‚]CDXKXJK@K@’ ejkyjkr nìn` lñzjcijcr ykpĻjmĻĕĻ`mk vc ñzcjjndjc lñzjcijcrn` skyĻsĻkrtĻrĻjmĻĕĻ`mk hcjjn ertkjkikjkrĻ` skhnt mcĕcrjcrc ykdjkŕtĻĕĻ vc hu ertkjkik mcĕcrn` mkbkse`rkdn ejkyjkrĻ` dktĻjiksĻyjk mkbn mcĕnŕicic c ĕnjnin`mc ejmuĕu lñzjcijc`in ŕtnr.]cdrkrjk`k` ejkyjkrk ñr`cd ejkrkd4 cjcdtre` ykyĻ`ĻiĻ, tcjcae` krkikjkrĻ, rkmkr kjlĻjkiksĻ,dkjntc de`trejú, snstci ìñdicsn, ykzĻ-turk ktĻjiksĻ vh. ejkyjkrĻ vcrchnjnrnz. Hksntìc k`jktikd lcrcdnrsc, hu tcern`n` kikoĻ ejkyjkrĻ` ertkjkik mkvrk`ĻŕjkrĻ`Ļ ejkyjkrĻ` ejksĻjĻdjkrĻ on`sn`mc` tk`Ļijkikd vc dcstnricdtnr.
;.;. EjksĻjĻĕĻ` Iúbc`mnsjndtcdn SylujkikjkrĻ
EjksĻjĻd iúbc`mnsjnd kjk`Ļ`mk hnrìed uylujkik kjk`Ļ`k skbnptnr. Cjcdtrnd-Cjcdtre`ndIúbc`mnsjnĕn kjk`Ļ`k lnrc` uylujkikjkrĻ`mk` hkzĻjkrĻ
Cjcdtrnd mcvrcjcrmc ĻsĻj lúrújtúWkyĻa rkmkr vc rkmye sn`ykjjcrn`n` kjlĻjk`iksĻC`aeriksye` ]cernsnBkhcrjcŕic _nstci ]kskrĻiĻ_nstcijcrn` Lúvc`njnrjnĕn_nstci/Onbkz Bktk/HkŕkrĻsĻzjĻd Erk`jkrĻ vc EjksĻjĻdjkrĻBkhcrjcŕic K ĕjkrĻ, ñr7 Kĕ ]rkanĕn
ŕcdjn`mc sĻrkjk`khnjnr.;.8. Ìc ŕntjn EjksĻjĻd ]k`ĻijkrĻ
EjksĻjĻĕĻ` tk`ĻiĻ lc`cj ejkrkd ? tcicj ykdjkŕĻijk ykpĻjĻr.
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Hu muruimk lñrújmúĕú lnhn tepjki = tk`c ejksĻ se`uì vkr. Cĕcr hnz bcr ndn pkrk`Ļ`mkykzĻ lcjicsn ejksĻjĻĕĻ`Ļ ‚^^’ krĻyerskd, hu muruimk hu ejkyĻ` ejksĻjĻĕĻ ;/= ejur. Cĕcr k` kzhnr pkrk`Ļ` ^ lcjicsn ejksĻjĻĕĻ`Ļ krĻyerskd, hu muruimk ^^,]^ vc ^] se`uìjkrĻ`mk` bcrhnrnhnzni krkmĻĕĻiĻz se`uìjkrĻmĻr. MejkyĻsĻyjk hu ejkyĻ` ejksĻjĻĕĻ hu dcz ?/= ejur.
Djksnd tcern ndn kìĻmk` yctcrsnzmnr7;. Cŕntŕk`sk skbnp ejikyk` ejkyjkr nìn` meĕru se`uì vcricz.Icscjk ktĻjk` ndn zkrĻ` tepjkiĻ`Ļ` ? ejiksĻ ejksĻjĻĕĻ`Ļ krkykjĻi. Hu muruimk iúidú`
tepjkijkr {8,?,=,3,1,,;6,;;,;8} ejkhnjnr. ^k`n ;; akrdjĻ se`uì iúidú`múr. MejkyĻsĻyjktepjkiĻ` ? ejiksĻ ejksĻjĻĕĻ ;/;; mnr mnychnjnrnz. K`okd tkhnndn hu se`uì ^K@JNŐ„tĻr. Ìú`dútepjkiĻ` 8 ejiksĻ vc ? ejiksĻ ky`Ļ ŕk`sk skbnp mcĕnjmnr. ^udkrĻmkdn lnhn hnr tkhje ykpkrkdmeĕru se`uou` ;/;2 ejmu ĕu ìed rkbktjĻdjk lñrújchnjnr.
8. _kyĻjkikykokd dkmkr ìed ejksĻ se`uou ejk` muruijkrmk ykdjkŕĻijkrk hkŕvurikmk`ìñzúi úrctcicz.
;.8.8. HkĕĻj Arcdk`s ]k`ĻiĻ
Hnr K ejkyĻ`Ļ` ejksĻjĻĕĻ`Ļ hkĕĻj arcdk`s yejuyjk hcjnrjcicd hnr mc`cicyn ` dcrctcdrkrjkikdtk` lcìcr. Hu ` mc`cic cs`ksĻ`mk K ejkyĻ`Ļ` lñrújmúĕú muruijkr skyĻjĻr vc hu `K ejkrkd dkymcmnjnr vc K ejkyĻ`Ļ` ejksĻjĻĕĻ
V(K)9 K̀ /` ejkrkd vcrnjnr.
` K:` ejmuĕu`mk` V(K) 6 vc ; krksĻ`mkmĻr (6:V(K) :;). Hu tcernyjc njlnjn prehjcijcrmc` hnrtk`csn bnì hnr zkik` se`suz skyĻmk mc`cic ykpkikykokd ejikiĻzmĻr. Hu muruimk V(K)„ yĻse`ju skyĻmkdn mc`cicjcrmc` dcstnricd muruiu`mk dkjĻrĻz. Mkbk se`rkmk hu erk`Ļ`Ļ` `se`suz ejmuĕu muruimk hnr jnint mcĕcrn`c ykdjkŕtĻĕĻ`Ļ skvu`uruz. ^k`n
V(K) 9 jni (`K /`) ejkrkd vcrnjnr.` ∐
Hu prehjcic rk ĕic` hk ĕĻj arcdk`s ykdjkŕĻiĻ, ejksĻjĻd tcernsn`n lcrìcd mú`ykmkdn ejkyjkrkuylujkikmk ñ`cijn hnr ycrc skbnptnr.
;.8.?. Kdsnyeiktnd ]k`Ļi
Ìe ĕu iemcr` ejksĻjĻd dntkhĻ`mk dujjk`Ļjk` tk`Ļi hu tk`ĻimĻr. K`okd hu tk`ĻiĻykpikmk` ñ`oc tcicj hnrdkì dkvrki úzcrn`mc murikiĻz lcrcdicdtcmnr.
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Ķidk`sĻz Ejky7 EjiksĻ iúidú` ejikyk` ejkymĻr. Dúic tcernsn`mc he ŕ dúicyc tcdkhújcmcr.
Ñr`cd7;. ^kzĻ-]urk ktik mc`cyn
℡9{^,]} Ñr`cd uzkyĻK9{^} ^kzĻ lcjicsn EJK^N
8. 8 dcrc ^kzĻ-]urk ktik mc`cyn℡9{^^,^],]^,]]} Ñr`cd uzkyĻ
K9{^],]^} Hnr ykzĻ vc hnr turk lcjicsn EJK^N?. Hnr mnrc`on` lúoú`ú ñjìic mc`cyn℡9{V7V≣6} ^k`n 6„mk` húyúd hútú` skyĻjkr ñr`cd uzkyĻiĻzĻ ejuŕturuyer.
=. Cjkstnd hnr ìkrpĻŕikmk `údjccr hnr pkrìkoĻĕĻ`, ìkrpĻŕik se`rksĻ kyrĻjĻŕ kìĻsĻ`Ļ lñzjcijcicmc`cyn
℡9{β7 -φ ≭ β ≭ φ} Ñr`cd uzkyĻH9{-φ /=≭ β ≭ φ /8} KyrĻjik kìĻsĻ`Ļ` -φ /= vcφ /8 krksĻ`mk ejiksĻ ejkyĻ.
Dúic Ķŕjcijcrn
p∂ K 7 p, K dúicsn`n` hnr cjcik`ĻmĻr.K⊈H 7 K dúicsn H dúicsn`n` kjt dúicsnmnr (K dúicsn`n` hútú` cjcik`jkrĻ H
dúicsn`n` mc cjcik`ĻmĻr).K9H 7 K„`Ļ` bcr cjcik`Ļ H„`n` vc H„`n` bcr cjcik`Ļ mk K„`Ļ` mk cjcik`Ļ nsc K vc
H dúicjcrn cŕnttnr.
p∂ K, K⊈H, K9H nakmcjcrn`n` dkrŕĻtkrĻ sĻrksĻyjk p∎ K, K⊍ H, K≦H ŕcdjn`mc ykzĻjĻr.∃ 7 Heŕ dúic.℡ 7 Cvrc`scj Dúic (Ñr`cd SzkyĻ).@et7 Bcrbk`ln hnr K dúicsn nìn∃̀ ⊈ K⊈℡ nakmcsn ykzĻjkhnjnr.
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LĶXĶŐ
K vc H bcrbk`ln ndn dúic ejsu` K∧H njc lñstcrnjc`K vc H dúicjcrn`n` hnrjcŕnin K„ykyk mkH„yc hkĕjĻ cjcik`jkr dúicsnmnr.
Hnrjcŕni7 K∧H9{x7x K yk mk xH}
Ñr`cd7 K9{;,8,?} , H9{?,=,2} 90 K∧H9{;,8,?,=,2}K∮H njc lñstcrnjc` K vc H `n` krkdcsntn (dcsnŕnin) K vc H dúicjcrn`n` bcr ndnsn`c
hnrmc` hkĕjĻ cjcik`jkrĻ` dúicsnmnrDcsnŕni7 K∮H9{x7x K vc x H}
Ñr`cd7 K9{;,8,?} , H9{?,3} , K H9{?}Cĕcr K∮H 9∃ nsc yk`n K vc H `n` ertkd cjcik`Ļ yedsk K vc H yc ‚K^XND’ dúicjcr
mc`nr.
K vc H krksĻ`mkdn kyrĻi yk mk KZH hnìnin`mc lñstcrnjc` H dúicsn`n` K yk hkĕjĻ ejupH yc hkĕjĻ ejikyk` cjcik`jkrĻ` dúicsnmnr.
Akrd7 KZH9{x7xK vc x∎H}
Hurkmk KZH vc H `n` kyrĻd ejmuĕu`k yk`n (KZH)∮H96 ejmuĕu`k mnddkt cmn`nz.Ñr`cd7 K9{;,8,?}, H9{?,3} 90 KZH 9{;,8} , HZK9{3}
(KZH)∮H9{;,8}∮{?,3}9∃ (HZK)∮K9{3}∮{;,8,?}9∃ Ko hnìnin`mc lñstcrnjc` K„`Ļ` skjt túijcyc`n K yk hkĕjĻ ejikyk` vc ñr`cd uzkyĻ`k hkĕjĻ
ejk` cjcik`jkrĻ` dúicsnmnr.
]úijcyc`7 Ko 9{x7x ℡ vc x∎K}
Ñr`cd 7 K9{?,3},℡9{;,?,3,
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]krkjĻ Kjk`7 KZH ]krkjĻ Kjk`7 Ko Őcdnj 8. Akrd vc túijcyc` dúicjcrn
DÚIC ĶŐ JCI DSXKJJKXN
;) ]k`Ļijkik DurkjĻ7 K∧K9K , K∮K9K
8) Hnrjndtcjnd 7(K∧H)∧O9K∧(H∧O), (K∮H)∮O 9K∮ (H∮O)
?) Mcĕnŕic 7 K∧H9H∧K, K∮H9H∮K
=)MkĕĻjik 7 K∧
(H∮O)9(K∧
H)∮ (K∧
O), K∮ (H∧
O)9(K∮H)∧
(K∮O)3) Ñzmcŕjnd 7K∧∃∃∃∃9K , K∮∃∃∃∃9∃∃∃∃ , K∧℡9℡ , K∮℡9K
1) ]úijcic 7 K∧Ko9℡ , K∮Ko9∃∃∃∃ , (Ko)o9K , ℡ o9∃∃∃∃, ∃∃∃∃o9℡
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Hu `edtkmk` ntnhkrc` K∧H ycrn`c K+H vc K∮H ycrn`c KH `etksye`u`u mcĕnŕnijn ejkrkddujjk`kokĕĻz.Ñr`cd7
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VTHY 9VT{?}Y + VT{=}Y+………………………….+VT{>}Y
VTKHY9VT{?}Y + VT{=}Y+………………………….+VT{1}Y
Cĕcr VT{;}Y9…………….9VT{;8}Y9;/;8 90 VTKY9;/;8, VTHY9
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HĶJC ŐĶD, DEŐSJJS, ]EVJKI EJK_NJNDJKX QC HK ĐNI_NWJND
Hnjcŕnd ejksĻjĻd7Ķdn K vc H ejkyĻ`Ļ` hnrjndtc lcrìcdjcŕic ejksĻjĻĕĻ`Ļ nakmc cmcr.VTK∮HY vcyk VTKHYŕcdjn`mc lñstcrnjnr. KŕkĕĻmkdn K, H vc O ejkyjkrĻ`Ļ cjc kjkjĻi.K7 {Crzurui mk bcrbk`ln hnr lú`mc sĻokdjĻĕĻ` ;3 eO„`n` úzcrn`mc ejiksĻ}H7 { Crzurui mk bcrbk`ln hnr lú`mc ; oi/i 8 úzcrn ykĕĻŕ ejiksĻ}O7 {K vc H ejkyjkrĻ`Ļ` bcr ndnsn`n` mc ky`Ļ lú`mc lcrìcdjcŕicsn}^k`n O 9 K∮HvcykO 9 KHejsu`.
O„`n` ejksĻjĻĕĻ`k K vc H„`n` ‚Hnjcŕnd ejksĻjĻĕĻ’ vcyk ‚Ertkd ejksĻjĻĕĻ’ mc`nr.
VTOY 9 VTK∮HY 9 VTKHY
Hu dkvrki ndnmc` akzjk ejky nìn` mc lcìcrjnmnr. Ñr`cĕn` bcrbk`ln C,A,L ejkyjkrĻ`Ļ`hnjcŕnd ejksĻjĻĕĻ VTBY 9 VTCALY ejkrkd vcrnjnr.
EjksĻjĻĕĻ` arcdk`s tk`ĻiĻ`mk` bkrcdctjc,
VTKY 9 `K / `, VTHY 9 `H / ` ejikd úzcrc,
VTKHY 9 VTK∮HY 9 K̀H / ` ejur._ñz de`usu K vc H ejkyjkrĻ hnrhnrjcrn`mc` hkĕĻisĻz nscjcr, ertkd ejksĻjĻdjkrĻ hksnt hnr
ìkrpĻi bkjn`n kjĻr7
VTKHY 9 VTKY VTHY
_ñz de`usu K vc H ejkyjkrĻ hnrhnrjcrn`mc` hkĕĻisĻz mcĕnj nscjcr, ertkd ejksĻjĻdjkrĻdeŕujju ejksĻjĻdjkrĻ ykrmĻiĻyjk cjmc cmnjchnjnr7
VTKHY 9 VTK|HY VTHYVTKHY 9 VTH|KY VTKY
Deŕujju ejksĻjĻd7Hnr K ejkyĻ`Ļ`, H ejkyĻ ejmudtk` se`rk ejik ejksĻjĻĕĻ`Ļ nakmc cmcr.^udkrĻmk vcrnjc` K̀H /` K vc `KH /` H erk`jkrĻ`kŕnimn yc`nmc` hkdkjĻi. Hu erk`jkr K vc H
ejkyjkrĻ`Ļ` KH ejkyĻ`Ļ` lcrìcdjcŕicsn krksĻ`mkdn hkĕĻj arcdk`sjkrmĻr. Hu erk`jkr sĻrksĻyjk,sĻokdjĻĕĻ` ;3 eO„`n` úzcrn`mc ejmuĕu lú`jcrmc ykĕĻŕĻ` mk ; oi/i 8„`n` úzcrn`c ìĻdtĻĕĻ lú`jcrn`vc ykĕĻŕĻ` ; oi/i 8„`n` úzcrn`c ìĻdtĻĕĻ lú`jcrmc sĻokdjĻĕĻ` mk ;3eO„`n` úzcrn`mc ejmuĕulú`jcrn` ejksĻjĻĕĻ`Ļ vcricdtcmnr.
Sakd hnr iktciktndscj múzc`jcicyjc,
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Ñr`cd7
^k`mkdn ŕcdnjmc lñstcrnjc` hnr ndnjnbkhcrjcŕic snstcin`mc 6 vc ; scihejjcrn`n`njctnjmnĕn hnr skyĻskj bkhcrjcŕicykpĻjikdtkmĻr. Hu snstcimc ‚^’ kjĻ`k`scihejú, ‚R’ nsc lñ`mcrnjc` scihejúlñstcricdtcmnr.
Őcdnj 1.Ķdnjn bkhcrjcŕic snstcin
Hu snstcin` ñr`cd uzkyĻ ejk` ‚℡’
℡9{(x,y)7 x9 6 vcyk ;, y9 6 vcyk ;}9{(6,6),(6,;),(;,6),(;,;)}
R ejkyĻ R 9 {(x,6),(x,;)} njc vcrnjicdtcmnr.
^ ejkyĻ ^ 9 {(6,y),(;,y)} njc vcrnjicdtcmnr.VT (x,y) 9 (n,g)Y 9 VT x9n Y VT y9g | x9n Y, (n 9 6 , ; vc g 9 6 , ;)
Hu snstcimc bkhcrjcŕic snstcin`mcdn lúrújtúmc` mejkyĻ lñ`mcrnjc` 6 hkzc` ; hkzc` 6ejkrkd kjĻ`khnjicdtcmnr. Ky`Ļ zkik` mk lñ`mcrnjc` ;„mc hkzc` 6 hkzc` ; ejkrkdkjĻ`khnjicdtcmnr.
^kpĻjk` ñjìúijcr `ctnocsn`mc kŕkĕĻmkdn ejksĻjĻdjkr hnjn`icdtcmnr.
VTy9;|x9;Y96.> 4 VTy9;|x96Y96.;VTy96|x9;Y96.; 4 VTy96|x96Y96.>
Ky`Ļ zkik`mk bkhcrjcŕic snstcin`n` tkskrĻiĻ`mk` lcjc` hnr ñzcjjndjc 6 úrctnjicejksĻjĻĕĻ ; úrctnjic ejksĻjĻĕĻ`k cŕnt ejup
VTx96Y9VTx9;Y 9 ;/8„mnr.
HkzĻ hnjcŕnd ejksĻjĻdjkrk hkdkjĻi
VTx96,y96Y 9 VTy96|x96Y VTx96Y96,>* 6,39 6,=3VTx96,y9;Y 9 VTy9;|x96Y VTx96Y96,;* 6,39 6,63
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VTHY9VTH|K; Y.VTK; Y + VTH|K8Y.VTK8Y +………………….+ VTH|K`Y.VTK`Y
Hkzc` VTHY, cŕntjnĕn` skĕ tkrkaĻ`mkdn bcskpjkik`Ļ` meĕksĻ lcrcĕn ertkjkik ejksĻjĻd ejkrkd mkkmjk`mĻrĻjĻr.
Ķspkt7Kn‒jcr kyrĻd ejmuĕu`mk` KnK g9∃ , n≦ g ndc` vc ( ) vc H℡9H9H Kn HKn⊈Kn mnr. MejkyĻsĻyjk (HKn)(HK g)9∃ , bcr n≦ g nìn`.Kdsnyei ?„ú dujjk`Ļrskd (` ejkyk lc`nŕjctnjcrcd)4
VTHY9VTS` n9;HKnY9VTHK; Y+VTHK8Y+……………………+ VTHK`Y9 VTH|K; Y.VTK; Y + VTH|K8Y.VTK8Y +………………….+ VTH|K`Y.VTK`Y
Ñr`cd7Hnr ñ`ocdn ñr`cdjc vcrnjc` ndnjn bkhcrjcŕic snstcin nìn` VTy96Y (6 kjĻ`ik) vc VTy9;Y (;
kjĻ`ik) ejksĻjĻdjkrĻ`Ļ bcskpjkykjĻi.VTy96Y 9 VTy96|x96Y VTx96Y + VTy96|x9;Y VTx9;Y
9 6,> . 6,3 + 6,; . 6,3 96,3VTy9;Y 9 VTy9;|x96Y VTx96Y + VTy9;|x9;Y VTx9;Y
9 6,> . 6,3 + 6,; . 6,3 96,3VTy9;Y n kyrĻok {y96}∧ {y9; }9℡ ejmuĕu`mk` mnddkt cmcrcd,VTy96Y+VTy9;Y9; mc` {Kdsnyei 8} bcskpjkykhnjnrnz.
HkĕĻisĻzjĻd7Hnr H ejkyĻ`Ļ` ertkyk ìĻdiksĻ K ejkyĻ`Ļ` ertkyk ìĻdiksĻ úzcrn`mc bnìhnrctdn ykpiĻyersk, H ejkyĻ njc K ejkyĻndn hkĕĻisĻz ejkymĻr mc`nr. HkĕĻisĻzjĻd, ejksĻjĻĕĻ`ejmudìk ñ`cijn dkvrkijkrĻ`mk` hnrnmnr. KyrĻ ñr`cd uzkyĻ`mk tk`ĻijĻ K vc H ejkyjkrĻ`k ykj`Ļzvc ykj`Ļz
VTKHY9VTKY.VTHY nsc ‚HKĐNI_NW EJK^JKX’ mĻr mc`nr.Lc`cj nakmcsnyjc VTKHY9VTH|KY.VTKY9VTK|HY.VTHY ejmuĕu`mk`, hkĕĻisĻz ejkyjkr nìn`VTH|KY9VTHYVTK|HY9VTKY mnr.MejkyĻsĻyjk hkĕĻisĻzjĻd tk`ĻiĻ K, H ejkyjkrĻ`Ļ` hkĕĻisĻz ejmuĕu murui nìn` K ejkyĻ`Ļ`
H úzcrn`mc vc H ejkyĻ`Ļ` mk K úzcrn`mc bnìhnr ctdnsn ejikmĻĕĻ`Ļ sñyjcicdtcmnr. Hu tk`Ļi mktcicj dkvrkijkrĻiĻzjk ky`Ļ meĕrujtumkmĻr
? akrdjĻ K,H,O ejkyĻ nìn` (V(K)≦6, V(H)≦6, V(O)≦6 ejikd dkymĻyjk) hkĕĻisĻz ejikŕkrtĻŕñyjc vcrnjchnjnr7 VTKHOY9VTKY VTHY VTOYVTKHY9VTKY VTHY , VTHOY9 VTHY VTOY , VTKOY9VTKY VTOY
? ejkyĻ` hkĕĻisĻz ejiksĻ nìn` skmcoc VTKHOY9VTKY VTHY VTOY„`n` yctcrjn ejikmĻĕĻ`k vcbcr ndnjn`n` mc hkĕĻisĻz ejiksĻ lcrcdtnĕn`c mnddkt cmn`nz.
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HK^C_ ]CEXCI Ķ7EjksĻjĻd tcernsn nìn`mc n`ocjc`c` hnr ejky ejkrkd H ejkyĻ`k deŕujju hnr K ejkyĻ (yk`n H
ejkyĻ`Ļ` hnjn`mnĕn bkjmc K ejkyĻ) nìn` ejksĻjĻd mcĕcrn, K ejkyĻ`k deŕujju ejkrkd H ejkyĻ (yk`n KejkyĻ hnjn`mnĕn bkjmcdn H ejkyĻ) nìn` ejksĻjĻd mcĕcrn`mc` akrdjĻmĻr. K`okd hu ndn hnrhnrn`c tcrsdeŕujjujud krksĻ`mk ìed hcjnrjn hnr njnŕdn vkrmĻr vc hu njnŕdnyc (njd kìĻdjkyk` nstktnstndìnĶ̀lnjnz]beiks Hkycs kmĻ`k ktac`) Hkycs ]cercin mc`njicdtcmnr.
Hkycs tcercin`n` nakmc cmnjnŕn7Hkycs tcercin hnr stedkstnd súrcì sĻrksĻ`mk ertkyk ìĻdk`hnr rkstlcjc ejky K njc hnr mnĕcr rkstlcjc ejky H (cĕcr H nìn` dkyhejikiĻŕ ejksĻjĻd vkrsk) nìn`deŕujju ejksĻjĻdjkrĻ vc ikrgn`kj ejksĻjĻdjkrĻ krksĻ`mkdn njnŕdnmnr, yk`n
Hkycs tcercin aeriújú nìn`mc huju`k` bcr hnr tcrnic ñzcj nsnijcr vcrnjicdtcmnr7• V(K) tcrnin`c K nìn` ñ`scj ejksĻjĻd vcyk ikrgn`kj ejksĻjĻd kmĻ vcrnjnr. Hu ñ`scjmnr,
ìú`dú H ejkyĻ bkddĻ`mk ñ`ocmc` bcrbk`ln hnr hnjlnyn nìcricicdtcmnr.• V(K|H) tcrnin vcrnjinŕ H nìn` K„`Ļ` deŕujju ejksĻjĻĕĻ kmĻ`Ļ kjĻr.•
V(H|K) tcrnin vcrnjinŕ K nìn` H„`n` deŕujju ejksĻjĻĕĻ kmĻ`Ļ tkŕĻr.• V(H) tcrnin H ejkyĻ nìn` 'ñ`scj' ejksĻjĻdtĻr vcyk H„`n` ikrgn`kj ejksĻjĻĕĻmĻr vciktciktndscj rejú `erikjnzc cmc` hnr skhnttnr.
Hu ŕcdnjmcdn Hkycs tcercin`n, akzjk iktciktndscj ejikmk`, sczlnyc mkyk`krkdŕñyjckìĻdjkykhnjnrnz7 Hkycs tcercin cĕcr H lñzjcijc`in ŕ nsc, K lñzjcin bkddĻ`mkdn n`k`ìjkrĻ` `cŕcdnjmc lú`ocjjcŕtnrnjchnjcocĕn`n ertkyk ìĻdkrtĻr.
Hkycs tcercin`n` nspkt cmnjnŕn7Hu tcercin nspkt cticd nìn` deŕujju ejksĻjĻd tk`ĻiĻ`mk`ykrkrjk`ĻjĻr. H ejkyĻ hnjn`nrsc K ejkyĻ`Ļ` ejksĻjĻĕĻŕñyjc vcrnjnr7
Ky`Ļŕcdnjmc K ejkyĻ vcrnjinŕ nsc H ejkyĻ`Ļ` ejksĻjĻĕĻŕumur7
Hu ndn mc`djci yc`nmc` múzc`jc`np hnrjcŕtnrnjnrsc,
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VTHY 0 6 vc VTKnY≦ 6, n nìn`
@E]7V TK g|HY„yc K g ‒`n` H de ŕuju`k hk ĕ jĻ ‚se`rkmk`’ ejksĻjĻ ĕ Ļ mc`nr. VTH|K gY„yc nsc H„`n`
K g„yc de ŕuju`k hk ĕ jĻ ‚ñ`ocmc`’ ejksĻjĻ ĕ Ļ mc`nr.
Lc`cjmc prktndtc ‚se`rkmk`’ ejksĻjĻdjkr lñzjcijcr se`uou`mk bcskpjk`Ļr,‚ñ`ocmc`’ ejksĻjĻdjkr nsc lcìin ŕ ñjìúijcrc mkyk`Ļjkrkd dcstnrnjnr (tkbin` cmnjnr.)
Ñr`cd7Ķdn tkhkd mejusu hnsdúvn múŕú`újsú`. ]khkd ; nìn`mc ;6 tk`c ìndejktkjĻ hnsdúvn vc ?6tk`c skmc hnsdúvn huju`muĕu dkhuj cmnjsn`. ]khkd 8 nìn`mc nsc bcr ndn tnp hnsdúvnmc` 86ŕcrtk`c ejmuĕu hnjn`sn`. Cvn` dúìúd ìeouĕu hnr tkhkĕĻ rkstlcjc scìnp hu tkhkdtk` rkstlcjc hnrhnsdúvn scìnp kjsĻ`. Ìeouĕu` hnr tkhkĕĻ mnĕcrn`c vc hnr tnp hnsdúvnyn mnĕcrn`c tcronb cticdtcejmuĕu`k mknr cjninzmc bnìhnr lñstcrlc huju`ikikdtkmĻr. Ìeouĕu` scìtnĕn hnsdúvn`n` skmcejmuĕu lñrújsú`. Ìeou ĕu` hu skmc hnsdúvnyn tkhkd ;„mc` scìinŕ ejiksĻ`Ļ` ejksĻjĻĕĻ`Ļ` `cejkokĕĻ prehjcin hurkmk n`ocjc`icdtcmnr.
_czln njc tkhkd ;„mc skmc hnsdúvn skyĻsĻ`Ļ` ìndejktkjĻ hnsdúvn skyĻsĻ`k lñrc mkbk akzjkejmuĕu lñz ñ`ú`c kjĻ`krkd n`ocjc`c` ejksĻjĻĕĻ` %36„mc` mkbk akzjk ejkokĕĻ bcic` kjlĻjk`Ļr.Hu seruyk ocvkp Hkycs tcercin dujjk`Ļjkrkd dcsn` ejkrkd vcrnjchnjnr.
Ñ`oc seruyu mcĕnŕtnrnp Hkycs tcercin uylujk`khnjcocdŕcdjc sedikd lcrcdicdtcmnr7Ìeou ĕu` hnr skmc hnsdúvn scìinŕ ejmuĕu hnjn`icdtcmnr. E bkjmc hu deŕujjk hnrjndtc tkhkd;„mc` scìni ykpiksĻ ejksĻjĻĕĻ `c ejkokdtĻr5
Hñyjcoc Hkycs tcercin aeriújú`c uyikd nìn`, K ejkyĻ ìeouĕu` tkhkd ;„mc` scìniykpiksĻ4 H ejkyĻ nsc ìeouĕu` hnr skmc hnsdúvn scìicsn ejsu`.Ķstc`njc` ejksĻjĻd hñyjcoc V(K|H)ejkokdtĻr vc hu`u bcskpjkikd nìnŕ̀u ejksĻjĻdjkrĻ` huju`iksĻ lcrcdnr7
• V(K) vcyk bnìhnr mnĕcr hnjln ejikmk` ìeouĕu` tkhkd ;„mc` scìni ykpiksĻ ejksĻjĻĕĻ4Ķdn tkhkd krksĻ`mk tcronb ejikyĻp scìnin` cŕnt ejksĻjĻĕĻ ejmuĕu dkhuj cmnjicdtcmnr.
• V(H) vcyk bnìhnr mnĕcr hnjln ejikmk` ìeouĕu` hnr skmc hnsdúvn scìicsn ejksĻjĻĕĻ4 Mnĕcrhnr nakmc njc hu ìeouĕu` bcr hnr tkhkdtk` hnr skmc hnsdúvn scìic ejksĻjĻĕĻmĻr. HuejksĻjĻd, ñ`oc bcr ndn tkhkdtk` kyrĻ kyrĻ ejkrkd scìnjc` hnr tkhkdtk` hnr skmc hnsdúvnscìic ejksĻjĻĕĻ njc hu tkhkĕĻ scìic ejksĻjĻĕĻ`Ļ` hnrhnrn`c ìkrpĻjiksĻ vc se`rk hu ndnìkrpĻiĻ` tepjk`iksĻ surctnyjc cjmc cmnjnr. ]khkdjkrmk ejk` skmc hnsdúvn`n` skyĻsĻ`Ļ`tepjkik erk`tĻsĻ`mk` hnjn`icdtcmnr dn tkhkd ; mc` hnr skmc hnsdúvn scìic ejksĻjĻĕĻ
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Hñyjcoc ìeouĕu` hnr skmc hnsdúvn scìnin hnjn`mnĕn`c lñrc, tkhkd ;„mc` scìni ykpiksĻ`Ļ`ejksĻjĻĕĻ % 16„mĻr vc sczlninzc lñrc scìtnĕninz %36„mc` mkbk húyúdtúr.
Ñr`cd7Hnr bkhcrjcŕic dk`kjĻ`mk V6 nbtnikjjc 6, V; nbtnikjjc ; lñ`mcrnjicdtcmnr.( V; 9;-V6) Dk`kjmkdn lúrújtú mejkyĻsĻyjkθ ejksĻjĻĕĻ njc 6, ;-θ ejksĻjĻĕĻyjk mk ; kjĻ`Ļyer.KjĻoĻmk ; lñzjcijc`mnĕn`c lñrc ; lñ`mcrnjin ŕ ejik ejksĻjĻĕĻ `cmnr5
Ocvkp7
Dk`kjĻ` ykpĻsĻ yudkrĻmk vcrnjinŕtnr. VTR9;|^9;Y ejksĻjĻĕĻ`Ļ krĻyeruz.
θ
θ θ Ñr`cd7Hnr dk`scr tcstn`c njnŕdn` kŕkĕĻmkdn se`uìjkr tk`Ļijk`Ļyer ejsu`K 7]cstc lñrc tcst cmnjc` dnŕn dk`scr.H 7Dnŕn lcrìcdtc` dk`scr.Ko7]cstc lñrc tcst cmnjc` dnŕn dk`scr mcĕnj.Ho7Dnŕn lcrìcdtc` dk`scr mcĕnj.
kŕkĕĻ kd j ` h j `ĕ ` k k kjĻi
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, ,, , , ,
,
^k`n skmcoc tcstjcrn` ejmuĕu`u sñyjcmnĕn dnŕnjcrn` lcrìcdtc` dk`scr ejik nbtnikjn % 2,<mĻr. Lcrn dkjk` % >;,? júd dĻsĻimk dk`scr tcŕbnsn de`ujk` dnŕn dk`scr ejikykokdtĻr.MejkyĻsĻyjk ‚Ķ̂ Ķ HĶX ]C_] MCĐĶJ’
Hu se`uì ŕkŕĻrtĻoĻ lnhn lñzúdsc mc dk`scr ejk` vc ejikyk`jkrĻ` skyĻsĻ krksĻ`mkdnmc`lcsnzjnd (yk`n dk`scr ejk`jkrĻ` kzjĻĕĻ) tcstn hkŕkrĻsĻz dĻjikdtkmĻr.
_K^IK AEXIÚJJCX Ķ ^crn`c deykrkd ñr`cdjcic7
Hnr terhkmkdn `uikrkjk`mĻrĻjiĻŕ tepjkr ñ`oc ìcdnjnyer, `uikrksĻ dkymcmnjnyer vc se`rktcdrkr terhkyk de`ujuyer. Hu muruimk njd ñr`cdjcicmc ‚`’ tk`c (terhkmk ` tk`c tep ejmuĕuvkrskyĻiĻyjk) akrdjĻ scìc`cd vc ndn`on ñr`cdjcicmc yn`c ‚`’ tk`c akrdjĻ scìc`cd sñz
de`usumur. _e`uì ejkrkd ‚`’ cjcik`Ļ ejk` terhkmk` ‚r’ cjcik`jĻ ‚`r‚ tk`c akrdjĻ sĻrkjk`iĻŕ ñr`cdjcr cjmc cmnjchnjnr.
^crn`c deyikmk` ñr`cdjcic7
^n`c yudkrĻmk hkbscmnjc` muru` nìn` hu scacr ìcdnjc` tepjkrĻ` ycrn`c lcrn de`ujikmĻĕĻ`Ļmúŕú`cjni. MejkyĻsĻyjk njd ìcdni nìn` ‚`’ tep vkrdc` ndn`on ìcdni nìn` ‚`-;’ tep úìú`oúmc ‚`-8’ ŕcdjn`mc kzkjk` skyĻmk tep ejkokdtĻr.
_e`uì ejkrkd7
, ≚ ∘ ; ∘ 8 ∘ ? ∘ + ; 9 ! ∘ ! Hu muruimk ejuŕkhnjcocdŕcdnjjcrn` skyĻsĻ
, 9 9 !! dkmkr ‚akrdjĻ sĻrkjk`iĻŕ’ ñr`cd cjmc cmnjchnjnr.‚`’ cjcik`jĻ hnr pepújèsye`mk` ejuŕ khnjcocd ‚r’ cjcik`jĻ kjt pepújèsye`jkr7
EjksĻjĻdtk ejuŕkhnjcocd tcicj serujkrmk` hnrnmnr. Ñr`cĕn` ;„mc` =„c dkmkr
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;,; 8,; ?,; =,; ^crn`c deykrkd ñr`cdjcicmc nsc;,8 8,8 ?,8 =,8 =x=9;1 lrup cjmc cmnjnr.;,? 8,? ?,? =,? ;,= 8,= ?,= =,=
Hnzc 1 akrdjĻ lrup vcrc` mnznjni nìn` lc`cj hnr aeriúj cjmc cmnjnr. Hu mnznjnic` „n`r„jndeihn`ksye`u mc`nr vc scihejú njc lñstcrnjnr.
‚r’ tk`c cjcik`mk` eju ŕk` pepújksye` nìn` r! tk`c akrdjĻ mnznjni sñz de`usumur.MejkyĻsĻyjk kjt pepújèsye`u nìn` .r! tk`c akrdjĻ sĻrkjkikmk` hkbscmchnjnrnz.Hu muruimk,
.r!9 9 ! 9 !!. ! ≚ ≚ yc ky`Ļ zkik`mk Hn`ei dktskyĻsĻ mc`nr.
Qc 9 !! ! 9 !!. ! ≚ 9 ^k`n
HCX@ESJJN MC@CICJCX Ķ _e`uou ‚V’ ejksĻjĻĕĻ njc hkŕkrĻ{ΰ; 9s} vc q9;-
p ejksĻjĻĕĻ njc hkŕkrĻsĻzjĻd{ΰ; 9a} ejsu` vc tcd hnr mc`cic nìcrc` hnr mc`cicyn cjc kjkjĻi. Humuruimk ñr`cd uzkyĻ℡9{s,a}ejur. Hu mc`cyn tcdrkrjkmĻĕĻiĻzĻ vkrskykjĻi hu muruimk yc`nñr`cd uzkyĻ℡89℡x℡ ejur vc℡8 9{ss, sa, as, aa}. Hu uzky 889= tk`c ejky nìcrnr℡x℡ ìkrpĻiĻ‚DKX]CW^C@ ÌKXVNI’ ejkrkd kmjk`mĻrĻjĻr. Cĕcr ` tk`c ‚hkĕĻisĻz’ mc`cic ykpkrskd
ñr`cd uzkyĻ
℡̀9℡x℡x℡x℡x…………………x℡, ejur vc hu ñr`cd uzky 8` cjcik` nìcrnr.
` ‟ dcrc
9 7 9 8
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ŕ , ∘ ŕ ≚ 4 , ]k`ĻijkmĻĕĻiĻz hu ‚h(d4`,p)’ yc ‚ Hn`ei Dk`u`u’ mc`nr. ‚`’ mc`cic ykpkrkd ‚d’ hkŕkrĻ cjmcctic nbtnikjn krkik ejkyĻ`k mk ‚ Hn`ei Mc`cicjcrn’ mc`nr.
Ñr`cd7 _k`nycmc ;6 hkĕĻisĻz hnt úrctc` kjĻ`ikdtkmĻr. BktkjĻ hnt kjĻ`ik ejksĻjĻĕĻ ( 6 lñ`mcrnjnp
;kjĻ`iksĻ ykmk tki tcrsn ) 6,66; mnr. _k`nycmc ‚C` Kz’ hnr bktk ejik ejksĻjĻĕĻ `cmnr.Ocvkp7
℀ ; ∘ ℀ì ℀
9 ;∘ 6,66;6,>>> 9 ;∘ 6,>>> ≃ 6,6;
Hn`ei dk`u`u njc njlnjn aeriújjcr7
;) ‚d’ ykmk mkbk kz hkŕkrĻ ejik nbtnikjn
9 4 ,9 8) C` kz ‚d’ hkŕkrĻ ejik nbtnikjn
9 4 ,9 9 ; ∘ ?) C` kz ‚g’ c` akzjk ‚d’ hkŕkrĻ ejik nbtnikjn
9 4 ,9 Ñr`cd7
℡9{;,8,?,=} , K9{;,8} , H9{8,?} , O9{?,=}
9 | | Cŕntjnĕn kokhk skĕjk`Ļr iĻ5 E`k hkdkjĻi
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XK_]JK@]N MCĐĶŐ DC@JCX Ķ Xkstlcjc mcĕnŕdc`7 Mcĕcrn mc`cymc` mc`cyc mcĕnŕc` se`uìjkr njc hcjnrtnjc` mcĕnŕdc`c
mc`nr. Ñr`cĕn`, Hnr etepkrdtk skkt 6>.66„mk pkrd bkjn`mcdn krkìjkrĻ` skyĻsĻ, bnjcsnz hnr zkrĻ`ktĻjĻŕĻ`mk úst yúzc lcjcocd skyĻ hnrcr rkstlcjc mcĕnŕdc` mcĕcrnmnr.Xkstjk`tĻ mcĕnŕdc`n dkvrkiĻ erngn`kj ñr`cd (ejksĻjĻd) uzkyĻ`Ļ` ycrn`c ejkyjkrĻ` rkdki
dúicjcrn`mc` ejuŕturmuĕu hnr uzkyĻ dujjk`ikiĻzĻ yk`n rccj(X) cdsc`mc ìkjĻŕkhnjicinzniúidú` dĻjkokdtĻr.
Ñr`cd uzkyĻ ‚℡’ ejk` hnr B mc`cyn`n cjc kjkjĻi.℡ ñr`cd uzkyĻ`Ļ` cjcik`jkrĻ (ΰ) Bmc`cyn`n` rkstlcjc se`uìjkrĻmĻr. Cĕcr bcr hnr cjcik` (ΰ) nìn` rccj hnr skyĻ dujjk`Ļrskdΰ„jcr vcrccj(X) cdsc` krksĻ`mk hnr dkrŕĻjĻd hujik durkjĻ hujuruz. Hu durkjĻ R(ΰ) njc lñstcrcjni. Hñyjchcjnrjn dĻsĻtjkikjkrk skbnp durkjk ‚XK_JK@]N MCĐĶŐDC@Ķ’ mnyeruz. MejkyĻsĻyjk ‚R(.)’ ykmk dĻskok ‚R’ rkstlcjc mcĕnŕdc`n℡ meic`n`mc ìkjĻŕk` vc rccj cdsc`mc mcĕcrjcr kjkhnjc` hnrae`dsnye`mur.
Őcdnjmcdn kìĻdjk`mĻĕĻ lnhn hnr cŕjcŕtnric se`uou`mk R rccj cdsc`mc mcĕcrjcr kjikdtkmĻr.Ñzcjjndjc4
R 7 rksjk`tĻ mcĕnŕdc`n x7 rccj skyĻjkrmkdn dkrŕĻjĻĕĻñzcj hnr ejkyk dkrŕĻjĻd lcjicdtcmnr vc hnz hu ejkyk hnr ejksĻjĻd mcĕcrn ktkikd nstcrnz. HuejksĻjĻd, mnr vc hu ejksĻjĻĕk ‚EJK_NJND MKĐNJNI
AE@D Ķ̂E@S (VMA)’ mcrnz
R(.)
R(ΰ)X
K
℡
K
.
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Hu muruimk ktĻjk` ndn pkrk`Ļ` mk ykzĻ ejik ejksĻjĻĕĻ V(x96), hnrn`n` turk hnrn`n` ykzĻejik ejksĻjĻĕĻ V(x9;), vc bcr ndnsn`n` mc ykzĻ ejik ejksĻjĻĕĻ V(x98), njc nakmc cmcrnz.
Cĕcr ‚ejksĻjĻd mkĕĻjĻi ae`dsnye`u`u (VMA)’ ìnzicyc ìkjĻŕĻrskdŕñyjc hnr lrkand cjmccmcrnz.
Xkstjk`tĻ mcĕnŕdc`n` tk`ĻiĻ`mk csks ejk` ñr`cd uzkyĻ`Ļ` bcr hnr ñĕcsn dkrŕĻjĻĕĻ`mk XccjskyĻ cdsc`n`mc (X) hnr dkrŕĻjĻd hujkhnjicsnmnr.
Ñr`cd7 _Ļ`Ļatkdn ñĕrc`onjcr nìn`E ; 7 Heyu ;36oi„mc` dúìúd ejk`jkr
E87 Heyu ;1>oi„mc` dúìúd ejk`jkr ŕcdjn`mc tk`Ļijk`iĻŕ ejkyjkrĻ cjc kjkjĻi.E?7 Heyu ;16oi„mc` húyúd ejk`jkr
R7 Hnr ñĕrc`on`n` heyu`u` uzu`juĕu nsc
R(E ; )9K; 9{x | x : ;36} V(R : ;36) Hu ejkyjkr mcĕnŕdc`R(E8)9K89{x | x : ;1>} V(R : ;1>) ejksĻjĻdjkrmĻr.R(E?)9K?9{x | x 0 ;16} V(R 0 ;16)
Xkstjk`tĻ mcĕnŕdc`n rccj cdsc`mc kjmĻĕĻ mcĕcrjcrc lñrc ‚súrcdsnz’ vc ‚súrcdjn’ rkstjk`tĻmcĕnŕdc`n ejikd úzcrc ndnyc kyrĻjĻrjkr.
Cĕcr R rkstjk`tĻ mcĕnŕdc`n`n` X mcdn mcĕcr dúicsn skyĻjkhnjnr ejkrkd se`suz hnr dúic nsc
A(x)9V(R≭x)
;
6,
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Hnr ìed muruimk bcr hnr ejksĻjĻd nìn` tcd tcd ejksĻjĻd ae`dsnye`u VT.Y ‒yn ykzikd lkrnp ejkhnjnr.Hu`u` nìn` VMA (ejksĻjĻd mkĕĻjĻi ae`dsnye`u`u) dujjk`ikyk nbtnykì muykrĻz.
VMA 7 Ax(x)9VT R≭ x Yŕcdjn`mc tk`Ļijk`Ļr. @E]7 Xkstjk`tĻ mc ĕ n ŕdc`jcrn`n` húyúd ‚R’ ( ^,W vs. ejkhnjnr ), hu rkstjk`tĻ mc ĕ n ŕdc`n`n`kjmĻ ĕ Ļ mc ĕ crjcrn` nsc dúìúd ‚x’ njc lñstcrnjmn ĕ n`c mnddkt cmn` hu muruimk A x(y) ykzikiĻz `etksye`mk bcrbk l̀n hnr prehjci eju ŕturikz. Ax(y)9VTR ≭ yY ejur.
EjksĻjĻd MkĕĻjĻi Ae`dsnye`u`u` Ax(x)„n` ñzcjjndjcrn7
;) Ax(∐)9; Ax(-∐)96
8) 6≭ Ax(x)≭;
?) x; ≭ x8 ↘ Ax(x; )≭ Ax(x8)=) V(x; : R ≭ x8) 9 Ax(x8) - Ax(x; )≣ 6
Ñr`cd7Hnr hnjlnskykrĻ` bkaĻzksĻ`mkdn hnt mnznsn`n n`ocjcycjni. Cĕcr hnt ‚;’ nsc R9; vc ‚6’ nsc
R96 mĻr. Hntn` 6 ejik ejksĻjĻĕĻ`Ļ` ‚q’, ; ejik ejksĻjĻĕĻ`Ļ` mk ‚;-q’ ejmuĕu`u vkrskykjĻi.Hu muruimk℡9{6,;} ejur. Hu ejksĻjĻĕĻ` mkĕĻjĻi ae`dsnye`u Ax(x)„n bcskpjkykjĻin) x : 6 7 Hu muruimk {R≭ x}9ǰ mnr vc Ax(x)96nn) 6≭ x : ;7 Hu muruimk {R≭ x}9q mur.nnn) x≣ ; 7 Hu muruimk {R≭ x}9℡ ejur yk`n V(R≭;)9;
Ax(x)
;
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Ñr`cd7
Hnr rkstjk`tĻ mcĕnŕdc`n kŕkĕĻmkdn ejksĻjĻd mkĕĻjĻi ae`dsnye`u`k skbnptnr.
6 , ∘∐ ∘; Ax(x) 9 + , ∘ ; : : ; ; , ; ≭ : ∐
k) R 9;/=
h) R 0 ?/=
Ñ`oc hu ejksĻjĻd mkĕĻjĻi ae`dsnye`u`u ìnzcjni
Ax(x)
;
6,3
x-; 6 ; 8
‚k’ ŕĻddĻ`Ļ ìñzicd nìn` 9 jni↘ ∘ ,6 : ^k`n ejksĻjĻd mkĕĻjĻi ae`dsnye`u`u` x `edtksĻ`mkdn sejmk` jnintn`n tk`ĻijkykjĻi.Hurkmkŕu`u nmmnk cmnyeruz V{R9x}9 A(x)-A(x-)
Ķ spkt7
∘ cjc kjkjĻi
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EJK_NJND ^EĐS@JSD AE@D_Ķ̂E@S (pma)
EjksĻjĻd yeĕu`jud ae`dsnye`u.Ñzcjjndjcrn7
∐
∘ ∐ ∘ ∐ a(x)„n` yeruiu7
∘ ^ctcrn`oc dúìúd hnr x mcĕcrn nìn`
∘ mejkyĻsĻyjk dúìúdx nìn ̀VTx:R ≭ x+ xY9A(x) x cĕcr x↘6„k lnmcrsc
VTR9xY9a(x)x↘6 ejur.
HCDJC@MĶD MCĐCX
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R ∐
∐
Dkrcscj hcdjc`mnd mcĕcr cĕcr rkstlcjc mcĕnŕdc`ninz hnr mnrc`ì úzcrn`mcdn lcrnjnin nakmccmnyersk mnrc`on` úzcrn`mc bkrok`k` lúì hu dkrcscj hcdjc`mnd mcĕcrjc erk`tĻjĻmĻr.
Icrdczn ieic`t mnyc tk`Ļijkykok ĕĻiĻz ieic`tjcrmc rkstlcjc mcĕnŕdc`mc` hcdjc`mndmcĕcrn` ìĻdkrtĻjmĻdtk` se`rk cjmc cmnjc` ieic`tjcrmnr.
R R x x
`9; nìn` icrdczn ieic`tn` ‚6’ ejkok ĕĻ kìĻdtĻr.`98 nìn` nsc ejmudìk ñ`cijnmnr. Hu`k ‚vkryk`s’ mc`njnr.
x x x∐
∐
∘ ∘ 8 ∘ 8 ∘ 8
∘
mcĕcrn mc stk`mkrt skpik ejkrkd kmjk`mĻrĻjĻr.Ñr`cd7
k) Őcdnjmc vcrnjc` ejksĻjĻd yeĕu`judae`dsnye`u`u` lcìcrjn hnr ‚pma’ ejiksĻ nìn` ‚k’
`c ejikjĻ5h) CTxYo) 8 m)
Ìñzúi7 k’ ŕĻddĻ`Ļ` ìñzúiú nìn` ejksĻjĻd
a(x)
k
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9 ∐
∐
9 ;= 9
| ;
3 ; 9
∘
∐
∐ ;= 9;| 9 ;; ; 3 ∘ ; 9 ; =;
⇘ 9 ; =; ∘ 9 ; =; ∘ ;; ⇘ 9 = 9 ⇘ 9 . Ñr`cd7
KŕkĕĻmk ejksĻjĻd mkĕĻjĻi ae`dsnye`u vcrnjc` R rkstjk`tĻ mcĕnŕdc`n`n` ejksĻjĻd yeĕu`judae`dsnye`u`u ìnzn`nz.
ĕ MejkyĻsĻyjk ejksĻjĻd mkĕĻjĻiae`dsnye`u`u` túrcvn hnzc ejksĻjĻdyeĕu`jud ae`dsnye`u`u vcrnr.
Ocvkp7
A(x)
;
x8
a(x)
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Mnĕcr hnr tk`Ļimk rcot (mndmñrtlc`) ae`dsnye`u dujjk`Ļjkrkd vcrnjnr.
j i↘∐
Őcdnjmc k↘∐ lnttnĕn`mc mndmñrtlc` lc`nŕjnĕnkzkjkrkd 6„k lnmcr vc yúdscdjnĕn krtkrkd∐„klnmcr.
R9R n `edtksĻ`mkdn súrcdsnzjn ĕ n` túrcvn7
Cjninzmc hnrni hkskikd ae`dsnye`u ejsu`.
h r i hkskikd ae`ds ye`u ; ≣ 66 6
_ejmk lñstcrmnĕninz ae`dsnye` S(x-xn) yk`n hnrnihkskikd ae`dsnye`u`u` x n dkmkr dkyiĻŕ ŕcdjnmnr.(Őcdnj h)
_kĕmkdnŕcdnjmc (ŕcdnj o) nscŕcdnj h `n` x↘6 lnmcrdc` hnr
krcot(kx)
k
x
-;/8k ;/8k
S(x)
;
x6
Őcdnj kS(x-xn)
;
x6 xn
Őcdnj h
A(x)
;
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| ≚ 9 j i∁↘ ;∁ ∘ ∁
| 9 ∘ . ^k`n x 9 xn `edtksĻ`mkdn hnr súrcdsnzjnĕn` túrcvn x 9 xn `edtksĻ`mk tk`ĻijĻ vc yúdscdjnĕn súrcdsnzjnĕn njc meĕruerk`tĻjĻ ejk` hnrν ae`dsnye`u njc vcrnjnr.
9 ≭ 9 jmuĕu`u b tĻrj y r d y`Ļ ŕcd jmc ŕcdj `mc y z h j r z.
M j yĻsĻyj . Ñr`cd7
↘VMA nsc
_Ļd Dujjk`Ļjk` EjksĻjĻd ^e ĕ u`jud Ae`dsnye`jkrĻ7
;) S`naeri pma (múzlú` mkĕĻjĻi)
a(x)
;/ x
xxn- x/8 xn+ x/8
a(x)
;/8
x6
A(x)
;
;/8
x6
;a(x)
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Hu ejksĻjĻd yeĕu`jud ae`dsnye`u`u` VMA„sn`n ìnzcocd ejurskd
9 6 ∘ ∘ ; ≣
ĕ κ + 8 . 9 ∘ ;8 8) Cxpe`k`snycj pma (Ústscj mk ĕ ĻjĻi)(³06)
; Ertkjkik mcĕcr 9 ³Qkryk`sϏ89³ 8
Ústscj mkĕĻjĻiĻ` ejksĻjĻd yeĕu`judae`dsnye`u`u` VMA„sn`n ìnzcrscd
A(x)
;
x6 k h
a(x)
;/³ -
x6
∘
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?) @erikj pma (Lkussnk` mk ĕ ĻjĻi)
; 8
@(³,Ϗ; 8)
@(³,Ϗ88)
³
Ertkjkik mcĕcr CTxY9³Qkryk`s 9Ϗ8 Hu ejksĻjĻd yeĕu`jud ae`dsnye`u`u` VMA„n nsc
;8;+ cr 8 ; 8 Hu n`tclrkjn kjikd zer ejmuĕu nìn` ‚cra’ ae`dsnye`u`mk` ykdjkŕĻi ykpkokĕĻz.
cr ≚ ;8
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3) ≭ cr 1)
0 ∘ cr KŕkĕĻmkdnŕcdnjmc, deyu ikvn ertkjkikmk` hnr stk`mkrt skpik mkbk dúìúdtúr. Hnr `erikjmkĕĻjĻi nìn` hu (deyu ikvn) cĕrn`n` kjtĻ`mk dkjk` kjk`, tepjki kjk`Ļ` %12'n`n dkpskr.Ertkjkikmk` ndn stk`mkrt skpik krkjĻĕĻ`mk `edtkjkr nìn` cĕrn`n` kjtĻ`mk dkjk` kjk` (kìĻd, ertkvc deyu ikvn kjk`) tepjki kjk`Ļ` %>>.
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98(6,;>;)
96,?2811x6,?28 9 83,62≂ 83 dnŕn 86 njc =6 krksĻ`mk `et kjĻr.
=) Xkyjcnlb mk ĕ ĻjĻi (k 0 6) vc (≣̀ 6 )
Ertkjkik mcĕcr ³9 Qkryk`s
Hu pma bkhcrjcŕicmclcondicjn (akmn`l) dk`kjĻ` iemcjjc`icsn`mc dujjk`ĻjĻr.
a(x)
,
xx
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a(x)
V(R96)
x
A(x)
;
x6 ; 8 ?
‚`’ c ĕcr ìed húyúdsc
ĕ .
1) Vensse` mk ĕ ĻjĻiĻ (k06 ) 7
Vensse` mkĕĻjĻiĻ, ejksĻjĻd durkiĻ vc nstktnstnd hnjni dejjkrĻ`mk hnr kyrĻd ejksĻjĻd mkĕĻjĻiĻ ejup,hcjjn hnr skhnt hnrni zkik` krkjĻĕĻ`mk icymk`k lcjic skyĻsĻ`Ļ` ejksĻjĻĕĻ`Ļ nakmc cmcr. Huzkik` krkjĻĕĻ`mk ertkjkik ejky icymk`k lcjic skyĻsĻ`Ļ` hnjn`mnĕn vc bcrbk`ln hnr ejkyjk e`ubcic` tkdnp cmc` ejky krksĻ`mkdn zkik` akrdĻ`Ļ`, ñ`ocdn zkik` akrdjkrĻ`mk` hkĕĻisĻzejuŕtuĕu dkhuj cmnjnr.
Vensse` mkĕĻjĻiĻ ìed dcrc hcjnrjn skhnt zkik` krkjĻĕĻ hnrnijcrn huju`k` prehjcijcrcuylujk`ikdjk hcrkhcr, mnĕcr hnrniscj krkjĻdjĻ prehjcijcrc mc (yk`n hnrni uzkdjĻd, kjk` vcykbkoni nìcrc` prehjcijcrc mc) hkŕkrĻ njc uylujk`khnjnr
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! ;
Őnimn hu`jkrĻ` `ksĻj cjmc cmnjmnĕn`c hkdkjĻi. Ķ spkt7
⇘ ;
⇘ ≤ ; ≧ ; . 9
⇘ 9 ! ∘ ! ! ; ∘ 9 ;! ∘ ; ∘ 8………… ∘ ;. ;∘
9 ;! ; ∘ 9 ! ; ∘ ]kyjer scrnsn`mc`
⇘ ;! ⇘ ! .
Ñr`cd7Hnr hnjlnskykrmk ;6 hn` tk`c cjcdtre`nd cjcik` vkrmĻr. Bcr hnr cjcik` mnĕcrn`mc`hkĕĻisĻz hnr yĻjmk hezujik ejksĻjĻĕĻ (p9;6-=) ejkrkd vcrnjnyer. HnjlnskykrĻ` yĻjse`u`mkìkjĻŕĻyer ejik ejksĻjĻĕĻ `cmnr5 Hnr cjcik` hezujmuĕu`mk hnjlnskykrĻ`mk hezujmuĕu dkhujcmnjnyer.Ocvkp7
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9 7 hnrni zkik`mkdn ejkyjkrĻ` skyĻsĻύ7krkjĻĕĻ` uzu`juĕu (t , t +ύ )
4 , + 9 ! ]rkand iemcjjcicjcrn`mc ìedìk dujjk`Ļjk` ye ĕ u`jud nakmcsn
Ñr`cd7 Őcdjn`mc vcrnjc` hnr VMA nìn` pma yn huju`uz.
Ocvkp7
a(x)
A(x) 9V(R≭x)
;
1/2
3 / 2
= / 2
8/2
-; 6 ; 8
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;
= + = +
∘ ; + 8= +
; tepjkiĻ` n`tclrkjn n`tclrkjjcrn` tepjkiĻ`k cŕnttnr. MejkyĻsĻyjk
+ ;= 9 ;= + ; 9 ;=.; 9 ;=
= 9;= 9 ;=.; 9 ;= ∘ ; ; ; 9 ;.; 9 ;
8= 9 ;= ∘ 8 ;=.; 9 ;= ; ;
_e`uìjkrĻ` bcpsn`n tepjkrskd se`uou` ; ejiksĻ lcrcdnr.
+ + + ; Cĕcr A(x)9V(-;≭ x≭ ; ) serujiu ŕ ejskymĻ ocvkp ;/= + ;/= + ;/2 9 3/2 ejkokdtĻ.
Ñr`cd7k) pma lrkanĕn vcrnjc` ae`dsnye`u` VMA ‒sn`n ìnzn`nzh) V(=:x≭2,3) mcĕcrn`n huju`uz.
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Ocvkp7
k) Bcr hnr krkjĻdnìn` kyrĻ kyrĻ bcskpjkik ykpkrskd
; ≭ x :? KrkjĻĕĻ nìn` A(x)„n ykzkokd ejurskd.
6,8 ∘ ;
6 ì ∘∐ ; 9 6 . MejkyĻsĻyjk skmcoc ndn`on n`tclrkj dkjĻr. 9 6,8 ∘ ; 9 6,88 ∘
6,88 ∘ ∘ ;8 ∘ ; 96 88∘ ;8 ŕ 8ŕ
a(x)
6,=
6,8
x6 ; 8 ? = 3 1 < 2 >
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3≭x:< KrkjĻĕĻ nìnÀ(x)
9 9 6 +6,8 ∘ ;+ 6+ 6,8 ∘ 3
9 6,=+ 6,8 9 6,1 . KrkjĻĕĻ nìn` A(x)
9 9 6,2 + 6,89 6,2 +6,8 . >≭ x:∐ KrkjĻĕĻ nìn` A(x)
9 9 6,2 + 6,89 6,2 +6,8 9 ; .
Bcr krkjĻd nìn` bcskpjkik ykptĻd hu`jkrĻ ìnzcjni ìnznjc` lrkand hnzmc` nstc`c` VMA ejkokdtĻr.
A(x)
;
6,2
6,1
6 =
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9 6,=+ 6,8,96,3^k mk
V( = : x≭ 2,3) 9 A(2,3) - A(=) 9 6,> - 6,= 9 6,3 ejur.
DEŐSJJS ^E ĐS@JSDJKX vc MKĐNJNIJKX
‚℡’ Cvrc`scj dúicsn`c knt hútú` ΰ cjcik`jkrĻ`Ļ nìcrc` hnr O ejkyĻ`Ļ vc hu℡ dúicsn`n`cjcik`jkrĻ`Ļ rccj cdsc`c ktkyk` hnr R rkstjk`tĻ mcĕnŕdc`n cjc kjkjĻi. Hu muruimk yn`c hucvrc`scj dúic`n` kjt dúicsn ejk` hnr H dúicsn tk`ĻijkykjĻi{ ΰ 7 R (ΰ )≭ x } , {ΰ 7ΰ H }
H ejkyĻ`k hkĕjĻŕkrtjĻ mkĕĻjĻi
≚ , ŕ . HurkmkVTR ≭ x,HY ………………….{ R≭ x } vc H ejkyĻ`Ļ` hnjcŕnd ejksĻjĻĕĻmĻr. Cĕcr x↘∐ {R ≭∐ } dcsn` ejkyk mc`d ejur vcA(∐ ,H)9; ejur.
A(x|H) @erikj hnr mkĕĻjĻi ae`dsnye`u`u` hútú` ñzcjjndjcrn`c skbnptnr.Deŕujju yeĕu`jud ae`dsnye`u mk ky`Ļŕcdnjmc
≚ . Ñr`cd7
H ejkyĻ`ĻH ≚ R ≭ ;6} ŕcdjn`mc tk`ĻijkykjĻi vc A(x|H)„yn bcskpjkykjĻi.n) x≣;6 nìn`{R≭;6} ejkyĻ{R≭x} ejkyĻ`Ļ` hnr kjt dúicsnmnr. MejkyĻsĻyjkVR ≭ ;6,R ≭ x 9 VR ≭ ;6 9 A ;6 ejur. AxH V R ≭ x R ≭
;6V R ≭ ;6 9 ;
nn) x≭ ;6 nìn`{R≭ x} ejkyĻ{R≭ ;6} ejkyĻ`Ļ` kjt dúicsnmnr mejkyĻsĻyjkVTR≭ ;6, R ≭ xY9VTR≭ xY9ATxY
AH V R ≭ xA
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A(x) A(x|H)
; ;
6,3
x x6 ;6 86 6 ;6
HĶX XK_JK@]N MCĐĶŐDC@Ķ@Ķ@ AE@D_Ķ̂E@JKXNIúbc`mnsjnd uylujkikjkrĻ`mk djksnd hnr prehjci4 hnr snstcin` lnrnŕn`c uylujk`k` lnrnŕ nìn`ìĻdĻŕĻ` bcskpjk`iksĻmĻr. Hñyjcsn hnr snstcin` lnrnŕn hnr rkstjk`tĻ mcĕnŕdc`n nsc ìĻdĻŕĻ mklc`cjjndjc hnr rkstjk`tĻ mcĕnŕdc`n ejkokdtĻr. Hu muruimk cĕcr lnrnŕ rkstjk`tĻ mcĕnŕdc`jcrn nìn`VMA vcyk pma nakmcjcrn mc hnjn`nyersk snstcin` ìĻdĻŕĻ`mkdn rkstjk`tĻ mcĕnŕdc`n nìn` VMA vcykpma bcskpjk`khnjnr.Hu muruiu hnr ñr`cdjc kìĻdjkykjĻi.Hnr mnrc`ì (X) cjcik`Ļ úzcrn`mc` kdk` kdĻi (N) húyúdjúĕú V njc nakmc cmnjc` indtkrmk hnr lúì
ejuŕturikdtkmĻr.
≚ Hurkmk V vcyk V(.) hkzc` mc V(N) nakmcsn njctk`Ļijk`k` vc ae`dsnye` ejkrkd kmjk`mĻrĻjk` húyúdjúdbcr N mcĕcrn nìn` ejuŕk` lúoú hcjnrjcr.Hu muruimk, cĕcr N hnr rkstjk`tĻ mcĕnŕdc`n ejkrkdtk`Ļijk`iĻ ŕ nsc, V9N8X ŕcdjn`mc tk`ĻijĻ húyúdjúdyc`n hnr rkstjk`tĻ mcĕnŕdc`n ejuŕturikdtkmĻr. E zkik`
ìñzújicsn lcrcdc` seru7Qcrnjc` hnr durkjĻ (ae`dsnye`) vc pma„na x(x) tk`Ļijk`k` R rkstjk`tĻ mcĕnŕdc`n nìn`, yc`n^9l(x) rkstjk`tĻ mcĕnŕdc`n` pma n `c ejkokdtĻr5
X
N
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]cd hnr rkstjk`tĻ mcĕnŕdc`n`n` (ae`dsnye`u)7
R(6,;) krkjĻĕĻ`mk tk`ĻijĻ u`naeri hnr rkstjk`tĻ mcĕnŕdc`n nakmc cmnjsn`. Qc ^98R+? yc`nrkstjk`tĻ mcĕnŕdc`n nìn` Ay(y)95
≭ y 8R+ ? ≭ y} 9 R ≭ ;/8y∘ ?} ej o dtĻr. MejkyĻsĻyjkAy 9 Ay ∘?8 na mcsnyjc dej yjĻdj cjmc cmnjchnjnr. Hurkmk` bkrcdctjc,^„`n` pma„n nìn`7a y 9 mmyAy 9 mmyAy ∘?8 ejur.
9 ∘ ?8 ⇘ 8 9 9 ∘ ?
8 9 .
9 .;8 ⇘ 9 ∘?8 .
;8 .
Lc`cjjcic^9kR+h ŕcdjn`mc pma„n a x(x)ejk` súrcdjn hnr R rkstjk`tĻ mcĕnŕdc`nk 0 6 nìn`
^ ≭ y} 9 R+ h ≭ y} 9 R ≭ y ∘ h ⇘ h h
a x(x) a y(y)
;6,3
6 ; ? = 3
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V R : y ∘h 9 V R ≭ y ∘ h Hnzni nìn` lcrcdjn mcĕnj kik meĕru
≣ ∘ 9 0 ∘ .Nvc NNcŕntjndjcrn`mc`
≭ ∘ + ≣ ∘ 9 ; ⇘
≣ ∘ 9 ; ∘ ≭ ∘ ⇘ 9 ;∘ 9 ≦ 6 ejur.Hnjcŕdc ae`dsnye`7 a(x) vc l(x) vcrnjsn` a(l(x)) nakmcsn ael(x) ejkrkd lñstcrnjnr. Qc hu`k hnjcŕdc ae`dsnye` kmĻvcrnjnr.
. Icscjk7 9 + ; 9 ? + ; 9 ? +; + ; 9 1? + ; 9 8? + ;.? .
Jn`ccr Ejikyk` Hnr ae`dsnye` ñr`cĕn7VMA„n Ax(x) ejkrkd tk`ĻijĻ súrcdjn hnr rkstjk`tĻ mcĕnŕdc`n nìn`̂ 9R 8 ejsu`.
≭ } 9 ≭ } 9 ∘ ≭ ≭ } Hu muruimk ≭ } 9 9 ∘ ∘ . . ĕ. y 0 6 nìn4̀
9 9 ;8 ∘ ∘;8 ∘
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y l(x)9^
y+my
my
a x(x)
mx; mx8 mx? xmx; +x; x; x8- mx8 x8 x? mx?+x?
| | 9 : ≭ + + ∘ : ≭ + : ≭ +
⇘ + + 9
9 + + 9 9 9 ; 9 9 9 ; . ⇘ 9 + +
↘9 ≦6
y
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Őcdjn`mc tk`Ļijk`k` dúicjcrn lñz ñ`ú`c kjkjĻi Bcr ndn muruimk mk pma tk`ĻiĻ`mk` V T CnY 9a x (xn).|mxn |ejkrkd ykzĻjĻr._e`uì ejkrkd
.| | 9 .| x| . Qc hu cŕntjnĕn` bcr ndn tkrkaĻ`Ļ |my| njc hñjcrscd,
9 . 9 . .
Hu muruimk y9l(x) nakmcsn`n` dñdjcrn nìn` my/mxn9l„(x) kjkrkd
9 | | ŕcdj `mc lc`cj h r `ds y ` cjmc c xn9 xn(y) , l„(xn)≦ 6Hu cjmc cmnjc` nakmc l(x) nakmcsn`n` hnrìed dñdú ejmuĕu`mk` hu tkrz prehjcijcrn` ìñzúiú`údejkyjkŕtĻrk` hnr yñ`tcimnr. Cĕcr vcrnjc` hnr y nìn` ^9l(x) nakmcsn`n` dñdú yedsk a y(l)96ejkokdtĻr.Ñr`cd7
R7T6,;Y u`naeri mkĕĻjĻik skbnp ejsu`. ^98R+?ŕcdjn`mc tk`Ļijk`k` ae`dsnye` nìn`a y(y)95l(x) 9 8x+? 8x + ? 9 y↘ x 9 (y-?)/8
l„(x)98
∘ ?8 8 9 ∘ ?8 8 ejur. ^9l(x) n` hcdjc`mnd mcĕcrn vc stk`mkrt skpiksĻ
9
9 9 ∘ ejmuĕu`u hnjnyeru
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8
8 st `m rt s i yĻ y z rs d + 8 ∘ ∘ ∘ 8
∘ jmuĕu`u ĕrc`i ŕt d.M j yĻsĻyj jur. Ñr`cd7 pma„n bcskpjk`kokd ae`dsnye` uylujkikjkrĻ`mk s ` túrú hnr ae`dsnye` djksnd hnruylujkikmĻr. Hurkmk R rkstjk`tĻ mcĕnŕdc`n`n`
;8 , ∘ 6 ĕ . ŕcdjn`mc u`naeri mkĕĻjmĻĕĻ dkhuj cmnjcocdtnr.Ìñzúi N7 (Djksnd ykdjkŕĻijk ìñzcrscd) ñ`oc hu ae`dsnye`u` ŕcdjn`n ìnzcjni.
y9sn`x
y
sn`-; (y) φ-sn`-; (y) x
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9 ;8 ;;∘ + ;8 ;
; ∘ 9 ; ;
; ∘ 6 ; r jĻĕĻ `.
h) -; ≭ y≭ 6 nìn ̀Hc`zcr nŕjcijcr ykpĻjĻrsk se`uìŕu bkjn kjĻr.
; ;; ∘ ;6 ĕ Ìñzúi NN7 (lc`cj ìñzúi) ^9sn`(x) nìn` pma 9 5
;8 , ∘ 6 ĕ Hurkmk l(x) ae`dsnye`u l(x) 9 sn`(x) ejkrkd tk`Ļijk`kokdtĻr. Hu muruimk^-sn`(x)96 mc`djcin`n` dñdjcrn nìn`.
k)
6 ∘
KyrĻok
oes . Lc`cj aeriúj uykrĻ`ok l„(x) nakmcsn`n` dñdjcrmcdn mcĕcrn bcskpjk`ikjĻmĻr. Hu`u` nìn`4
9 oes; 9 oes
oes; 9 oes ∘ oes.oes s `.s ` ∘oes . ‚6’ ejmuĕu`u hnjnyeruz tkhn. @et7
sn`(Ѹ )9 y ↘ Ѹ 9sn`-; (y)oes(sn`-; (y))9oes(Ѹ )9 ;∘
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9 9 ; ∘ _e` ejkrkd
; ; ∘ + ∘ ; ∘ 9 ; ;;∘ 9 ; ;;∘ ≭ ≭ ; h) y : 6 ndc` mc hc`zcrŕcdnjmc nŕjcijcr tcdrkrjk`kokdtĻr.Ñr`cd7
y98x+; ejkrkd vcrnjinŕ ejsu`. a x(x) a y(y)
y98x+;
8/?;/?
x y
-8 6 ; -? 6 ?
+
y 9 - ? nìn` a x(-8) 9 6y 9 ? nìn` a x(;)98/?a y(y) 9 (;/8) (8/?) 9 ;/? a y(?) 9 ;/?
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Ñr`cd7a x(x)↘@(6,;) yk`n ³96 vcϏ89; ejkrkd vcrnjnyer.
≭ m yc t `ĻijĻ jsu` jmuĕu`u b tĻrj y r d ]k`ĻijĻ ejmuĕu krkjĻĕk lñrc ñ`oc lrkanĕn`nìnzcrscd yk`mkdnŕcdnj ejuŕur.Ìñzúic lcìcjni7V{ ^≭ y } 9 V{ R≭ y }↘ x 0 6 9 Ax(y)V{ ^≭ y } 9 V{ R≭ 6 } 9 ;/8 ejur. Lkusnk`ejmuĕu`mk` mejkyĻ.Őnimn Ax(y) vc a(y) yn ìnzjni.
HĶJC ŐĶD MKĐNJNI QC ^E ĐS@JSD AE@D_Ķ̂E@JKXN
Hnr ejksĻjĻd uzkyĻ`mk hnrmc` ìed rkstjk`tĻ mcĕnŕdc`n tk`Ļijkikd iúidú`múr. Ñr`c ĕn`7{ R≭ x ,^ ≭ y } 9 { R≭ x } ∮ { ^ ≭ y } dúicsn ΰ ℡ ejkyjkrĻ`mk` ejuŕsu` vc R(ΰ)≭x,
^( ΰ)≭y ejsu`. Hu muruimk { R≭ x , ^ ≭ y } ejkyĻ njc tk`ĻijĻ `edtkjkr x„ vc y„ múzjcijcrn`mckjttk vcrnjc ŕ̀cdnjmcdn tkrkjĻ hñjlcyc mc`d lcjnr.H lñ i ĕ j l jŕ lú j jk kd ì ji ŕ K` kd ` l ` j i k h `jk
a x(y)
yAx(y) a(y)
;6,3
6,3
y y
h ` k` jk ( ∐ ∐) 9 6 jk kd
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vc hu`mk` mejkyĻ Axy(-∐,-∐) 9 6 ejkokdtĻr.nn) Ñtc yk`mk` { R≭ ∐ } vc { ^ ≭ ∐ } dúicjcrn mc dcsn` ejkyjkrĻ nakmc cttndjcrn nìn`
{ R≭ x , ^ ≭ ∐ } 9 { R≭ x } vc{ R≭ ∐, ^ ≭ y } 9 { ^ ≭ y } ejkokĕĻ`mk`Axy(x,∐) 9 Ax(x)Axy(∐,y) 9 Ay(y) ejur.
nnn) Cĕcr Axy(x,y) súrcdjn vc túrcvn kjĻ`khnjc` hnr ae`dsnye` ejkrkd tk`Ļijk`iĻŕsk hnjc ŕndejksĻjĻd ye ĕ u`jud ae`dsnye`u
, , ℀.
MejkyĻsĻyjk a xy(x,y) mx my 9VT x : R≭ x + mx , y : ^≭ y + myY ejkokĕĻ nìn` túi x vc ymcĕcrjcrn nìn` a xy(x,y)≣ 6 ejur.
nnnn) ^udkrĻmkdn cŕntjnĕn n`tclrkj kjkrkd nakmc cmcrscd
, , cŕntjnĕn cjmc cmnjnr. Hu muruimk Axy(x,y) ae`dsnye`u `clktna ejikyk` hnr ae`dsnye`u` n`tclrkjnejkrkd tk`Ļijk`mĻĕĻ`k lñrc kzkjk` dkrkdtcrmc ejkikz. ^k`n, ( x; , y; ) vc ( x 8 , y 8 ) ndn tk`cskyĻ ìnatn ejikd úzcrc x; ≭ x 8 , y; ≭ y 8 ŕkrtĻ mk skĕjk`ĻrskAxy(x; , y; )≭ Axy( x8 , y8 ) ejikjĻmĻr.KyrĻok hnr súrcdsnzjnd `edtksĻ`mk Axy(x,y) skĕmkdn vc ústtcdn mcĕcrjcrn kjĻr. ^k`n
, j i↘
↘ , .
]úi hu ñzcjjndjcrn ñzctjcrscd N) Axy(∐,∐)9; , Axy(x, -∐) 9 Axy(-∐,y) 9 Axy(-∐,-∐) 96
Axy(∐,y) 9 Ay(y) 4 Axy(x,∐) 9 Ax(x) NN) x; ≭ x8 , y; ≭ y8 nsc Axy(x; , y; )≭ Axy( x8 , y8 ) 9 j i↘↘ , . { x; : R ≭ x8 , y; : ^ ≭ y8} dúicsn`n` cjcik`jkrĻ`Ļ lñz ñ`ú`c kjkjĻi.
Axy(x8 ,y8) 9 {;}+{8}+{?}+{=}ejur. Hnz hu tepjkiĻ` bcr hnr
y„(x,y)
;y8
VT R ≭ ^ ≭ Y 9 A ( ) A ( ) A ( ) + A ( ) j
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VTx; :R ≭ x8 , y; :^ ≭ y8 Y 9 A xy(x8 ,y8) - A xy(x8 ,y; ) - A xy(x; ,y8) + A xy(x; ,y; ) ejur. Ax(x) vcAy(y) ae`dsnye`jkrĻ hnr hnjcŕnd mkĕĻjĻi ae`dsnye`u`mk` cjmc cmnjnrsc hu`jkr
ikrgn`kj mk ĕ ĻjĻi ae`dsnye`jkrĻ ejkrkd kmjk`mĻrĻjĻrjkr. ^k`n7
,∐ , ∐, ,
Ikrgn`kj yeĕu`jud ae`dsnye`jkrĻ nsc
ejkokdjkrĻ ì `yud rĻm dcŕntjndjcrn` túrcvn kjĻ`mĻĕĻ`mk,
,∐ , ∐, , Hurkmk` bkrcdctjc hkzĻ se`uìjkr7 N) a (x,y)≣ 6 , túi x vc y mcĕcrjcrn nìn`
, ; NNN) a xy(x , y) hnr ejksĻjĻĕĻ nakmc cticzdc`7a xy(x , y)mymx 9 VT x : R≭ x+mx , y : ^ ≭ y+my Y hnr ejksĻjĻd nakmc cmcr.HKĐNI_NW XK_JK@]N MCĐĶŐDC@JCXĶ
R vc ^ lnhn ndn rkstjk`tĻ mcĕnŕdc`n, bcr x,y mcĕcr ndnjnsn nìn` { R≭x } vc{ ^≭y } ejkyjkrĻhnrhnrn`mc` hkĕĻisĻz nscjcr dc`mnjcrn mc hkĕĻisĻz ejkrkd kmjk`mĻrĻjĻrjkr. Mkbk ñ`ocdntk`Ļijkrmk` K vc H lnhn ndn ejky VTKHY 9 VTKYVTHYŕkrtĻ skĕjk`iksĻ bkjn`mc hnrhnrjcrn`mc`hkĕĻisĻz ejkrkd tk`Ļijk`iĻŕjkrmĻ. Hurkmk
KH 9 { R≭ }∮{ ^≭ } k`ĻiĻ
jk kd Ļ
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, ejkokdtĻr.Hu se`uìjkr ĻŕĻĕĻ`mk túrcv kjĻ`krkd
cjmc cm j rd hur m ` HkĕĻisĻz ejkyjkr nìn`ŕkrtjĻ ejksĻjĻd yeĕu`jud ae`dsnye`jkrĻ`Ļ` ikrgn`kj ejksĻjĻdyeĕu`jud ae`dsnye`jkrĻ`Ļ vcrcocĕn se`uou`u cjmc cmcrnz. KyrĻok mkbk ñ`oc cjmc cttnĕninz hnrse`uìtk`…7VTx; :R ≭ x8 , y; :^ ≭ y8 Y 9 A xy(x8 ,y8) - A xy(x8 ,y; ) - A xy(x; ,y8) + A xy(x; ,y; )VTx; :R ≭ x8 , y; :^ ≭ y8 Y 9 A x(x8).A y(y8) - A x(x8).A y(y; ) - A x(x; ).A y(y8) + A x(x; ).A y(y; )
9 A x(x8) T A y(y8) - A y(y; ) Y - A x(x; ) T A y(y8) - A y(y; ) Y9 T A x(x8)- A x(x; ) Y T A y(y8) - A y(y; ) Y
VTx; :R ≭ x8 , y; :^ ≭ y8 Y9 VTx; :R ≭ x8Y.VT y; :^ ≭ y8 Y se`uou cjmc cmnjnr.Ñr`cd7
KŕkĕĻmk vcrnjc` a xy ae`dsnye`u nìn` R vc ^ rkstjk`tĻ mcĕnŕdc`jcrn`n` hkĕĻisĻz ejmuĕu`ulñstcrn`nz.
, ; 8 ; 8 ; 8 ejkokĕĻ nìn` R vc ^ hkĕĻisĻz ejkokdjkrmĻr.Ñr`cd7
, , 6 ; 6 ;6 , m ĕcr muruij rŕcdjn`mc tk`Ļijk`iĻŕ ejsu`.Ocvkp7
N) K95
, ; 90
E kik` ; 6
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E kik` ;8 , 6 ;6 ,m ĕcr muruij r
Hc`zcr ŕcd jmc + ;8 , 6 ;6 ,m ĕcr muruij r NNN)Axy(x,y) 9 5Axy(x,y) 9 VT R≭x,^≭y Y tk`ĻiĻ ykpĻjĻrsk sñz de`usu ejksĻjĻĕĻ bcskpjkikd nìn` akrdjĻ muruijkrĻlñz ú`ú`c kjikd lcrcdnr. Hu`k lñrc ,
k) x≣ ; vc y≣ ; ejiksĻ muruiu`mk…4
, ;
h) 6 : x ≭ ; , y ≣ ; ejiksĻ muruiu`mk…4
9 + 9 ‸8 ‸ ‸ ‸
;8 ‸ ;8 , ;8 . o) 6 : y≭ ; , x ≣ ; ejiksĻ muruiu`mk…4
y„
(x,y)
;
x„;
y„
(x,y)
;
x„;
y
y
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9 +
9 + 8 ejkrkd huju`ur.
:6 ,y ` `bcr mcĕcr vcy 6 , x `bcr mcĕcr ejiksĻ muruiu`mk…4Axy(x,y) 9 6 ejur.
a) VT R+ ≭̂ ; Y 95 Ķakmcsn`n` `c k`jkik lcjmnĕn`c hkdkjĻi.
R+^9; me ĕrusu`u` ústú R+^0; hñjlcsn`n hcjnrtnyer
R+^9; me ĕrusu
R+^9; me ĕrusu`u` kjtĻ R+^:; hñjlcsn`nhcjnrtnyer
; 9 ‸. 9 + ‸ ; ∘ ;
y„
;
x„;
y„
;(x,y)
; x„
y„
;
x„;
( x,y)
ĶDĶXK JK@]N MCĐĶŐDC@Ķ@Ķ@ ]CD AE@DĶ̂E@S T W9l(R,^)Y
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ĶDĶ XK_JK@]N MCĐĶŐDC@Ķ@Ķ@ ]CD AE@D_ĶE@S T W9l(R, )Y
Ìe ĕu iúbc`mnsjnd uylujkikjkrĻ`mk hnr W rkstjk`tĻ mcĕnŕdc`n hcjjn hnr krkjĻdtk R vc ^lnhn ndn (bkttk mkbk ìed) rkstjk`tĻ mcĕnŕdc`n`c hkĕjĻ ejkrkd nakmc cmnjchnjnr. Icscjk7
- Hnr duvvctjc`mnrnon lnrnŕn`mcdn W sn`ykjn, R lnhn hnr nŕkrct vc hu`mk` hkĕĻisĻz ejkrkdtk`Ļijk`k` ^ lúrújtú on`sn`mc` W9R+^ ŕcdjn`mc tk`Ļijk`khnjnr.
- Ķdn nŕkrctn ìkrpk` hnr snstcimc, R hu ìkrpĻoĻ`Ļ` hnr lnrnŕn`mcdn, ^ mc hu ìkrpĻoĻ`Ļ` mnĕcrlnrnŕn`mcdn sn`ykj ejikd úzcrc W9R ŕ̂cdjn`mc tk`ĻijĻ ìĻdĻŕ sn`ykjn W„`n` pman `ksĻjtk`Ļijk`khnjnr5
- R vc ^ meĕrujtujkrĻ`mk hkĕĻisĻz bkrcdct cmc` hnr pkrìkoĻd nìn` R vc ^ njc hucdsc`jcrmcdn bkrcdctn nakmc cmc` rkstjk`tĻ mcĕnŕdc`jcrn tk`Ļijk`Ļrsk W9(R8+^ 8);/8 ŕcdjn`mctk`ĻijĻ vc tepjki ycr mcĕnŕtnricyn nakmc cmc` W rkstjk`tĻ mcĕnŕdc`n`n` pman `ksĻjbcskpjk`khnjnr5
Lc`cj ejkrkd W9l(R,^) ŕcdjn`mc tk`Ļijk`kokĕĻiĻz hu túr prehjcijcrn` ìñzúiú mkbkñ`oc ìñzmúĕúiúz ^9l(R) ŕcdjn`mc tk`ĻijĻ prehjcijcrn ìñzúiú`mc` akrdjĻ mcĕnjmnr. Hurkmkñyjc hnr Oz ìñzúi dúicsn huju`ikjĻmĻr dn hu`u` nìn`
{ W≭ z },{ (R,^) ∂ Oz } ejkyjkrĻ ky`Ļ muruiu nakmc cticjnmnr. Hu muruimk
,. ∂ ejkokdtĻr. Hu túr prehjcijcrn ìñzúiú ^ 9 l(R) ŕcdjn`mc tk`Ļijk`k` prehjcijcrn` ìñzúiú`mc`mkbk dkrikŕĻdtĻr. K`okdn) ykrmĻioĻ cdsc` , nn) dkrkdtcrnstnd ae`dsnye`jkrĻ` dujjk`ĻiĻ njc huprehjcijcr ìñzújchnjnrjcr. Ñ`oc hkzĻ ñr`cdjcrn n`ocjcycjni.Ñr`cd7
W9R^ ŕcdjn`mc R vc ^ rkstjk`tĻ mcĕnŕdc`jcrn on`sn`mc` tk`ĻijĻ W rkstjk`tĻ mcĕnŕdc`nnìn` ìñzúi dúicsn O z yn cjmc cmn`.A z(z) vc a z(z) 95
y9z/x vcyk x9z/y
y9z/x vcyk x9z/y
{ W≭ z } njc nakmc cmnjc` ìñzúi dúicsn
y
x
∐ ∐ j
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, ∘ ∘∐, ∐, ∘ , ejur. Hurkmk` bkrcdctjc,
9 95 serusu`u `ksĻjocv j rĻz5 ^udkrĻmk cjmc cmnjc` Az(z) cŕntjnĕn nìn`,, ≚ ,hcj rs z `tclr j ` t `Ļij yĻ y
, ∘ ∘∐, ∐, ∘ , . 9 ∘ ∘∐ + ∐ ∘ 9 ∘ 90 9 ; ∘ ; 9 ; , ; ,
; , ejkrkd huju`ur.
Ñzcj hnr murui ejkrkd R vc ^ hkĕĻisĻz vc hc`zcr ejikd úzcrc
9 ≚ + ejkrkd tk`Ļijk`iĻŕ ejsu`.Humuruim, ; 0 . ; + ; + 90 9 ;∘j i .
y
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Ìñzúi dúicsn y9z-x c ĕrnsn`n` seju`mk dkjk`hñjlc ejkokĕĻ`k lñrc
9 90 9 ∘ ∘ ∘∐ ejur.
Hur m , ≚ ejkrkdt `Ļij `mĻĕĻ`Ļ d huj cm y Hurkmk Az(z) ae`dsnye`u`u cjmc cticd nìn` , mcĕcr ` bcs j i d lcrcd r. z i `
, 0
∘ , se`uoucjmc cm j r. Ìe ĕu muruimk R vc ^ hnrhnrn`mc` hkĕĻisĻz ndn rkstjk`tĻ mcĕnŕdc`n ejkrkd tk`ĻijĻ
ejkhnjnrjcr. E zkik`
, ŕcdj `mc t `ĻijĻ hu m y`Ļ z jc ae`dsnye`jkrĻ`Ļ` de`vejúsye`u ejkrkd mk hnjn`nr.Lñrújmúĕú lnhn hu tnp prehjcijcrn tk`Ļijkrmk` ykrkrjk`krkd ìñzicd bcr zkik` nìn`
yy9z-x vcykx9z-y
x
a x(x) a y(y)
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;
;/8x y
; 8
h) a x(x) ae`dsnye`u`u` y cdsc`n`c lñrc snictrnsn`n kjĻp lrkanĕn` pkrkictrcsn`n
mcĕnŕcjni vc lrkanĕn z dkmkr ñtcjcycjni. Hnz hu`u`jk a x(z-y) yn cjmc cmcrnz.a x(z-y)
;
y
z-; z
o) z : 6 nìn` a x(z-y) njc a y(y) ìkdĻŕk` dĻsiĻ`k hkdkjĻi
a x(z-y) ;
;/8 a y(y)
z-; z 8
m) 6 : z : ; nìn`
a x(z-y) ;
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9 ;.;8 ;8 ;8
a) 8 : z : ? nìn`
;.;8 ;8 ? ∘ 8 .
l) z 0 ? a z(z) ìkdĻŕk` kjk` ejikmĻĕĻ nìn` 6 ejur.6 4…………………z:6
6,3z 4………………....6:z:;
a z(z) 9 6,3 4…………………;:z:8 cjmc cmnjnr.
(?-z)/8 4……………8:z:?
6 4 ………………....z0?
a z(z) `n` ejksĻjĻd yeĕu`jud ae`dsnye`u`u ìnzcrscd
VT z:8 Y 90 ?/=VT z:? Y 90;
cjmc cmcrnz.
; a x(z-y)
;/8 a y(y)
z-; z 8
; a x(z-y)a y(y)
;/8
z-; 8 z
a z(z)
¹
Ķ D Ķ XK_JK@]N MCĐĶŐ DC@ Ķ @ Ķ @ Ķ D Ķ AE@D_ Ķ ^E@S7
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Q9l(x,y) vc \9b(x,y) tnpn`mcdn prehjcijcrmnr
R + Q
Hurkmk { kn } mcĕcrjcrndkzk`oĻ nakmc ctsn`jcr.
^ + \+
^udkrĻmkdnŕcdnj njc hcjnrjn hnr bkhcrjcŕic snstcin ( stcrye tcicj-hk`tjĻ snstci eju ŕuiulnhn ) nakmc cmnjicdtcmnr. Qcyk { kn } mcĕcrjcrn ; ejkrkd scìnjmndjcrn`mc, hcjnrjn hnr bkhcrjcŕicsnstcin`mc Q vc \ sn`ykjjcrn ( sĻrksĻyjk R vc ^ sn`ykjjcrn`n` akrdĻ vc tepjkijkrĻ ) lñ`mcrnjc`sn`ykjn tcspnt cticdtc dujjk`ĻjĻr.
DĻskok Q9l(x,y) vc \9b(x,y) ae`dsnye`jkrĻ tk`ĻijĻ ndc`, a xy(x,y) hnrjcŕnd ejksĻjĻdyeĕu`jud ae`dsnye`jkrĻ njc túrcvn kjĻ`khnjc` l(x,y) vc b(x,y) ae`dsnye`jkrĻ nìn` Avw(v,w) vca vw(v,w)95
Hu túr prehjcijcrn` ìñzúijcrn`mc
, , ∃,
,
. 9 ∃,, t `Ļij i sĻ`m ` ym j ` v9l(x,y) vc w9b(x,y) ejikd úzcrc tk`Ļijk`k`x9ϓ(v,w) vc y9ς (v,w) ae`dsnye`jkrĻ on`sn`mc`
≚ ,℀ , ⋥ ,.
⋻ ⋠℀ ⋥ ℀ i tr s `c
l(x,y)9vb(x,y)9w mñ`úŕúi iktrnsn`n` gkoehyc` iktrnsn,
⋥
+
+
Ìñzúi7 R9ϓ( )98 3
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R9ϓ(v,w)98v-3w vc^9 ς (v,w)9-v+?w ejkokdtĻr. 8, ∘3, ∘; 9 ? ej r d bcs j `Ļr. Hu muruimk
9 8 ∘3∘; ? 90 9 8∜? ∘ ∘3∜∘; 9 ; , 9 ;
8∘;
8 +
, 9 ; ;8 ∘;8 8 ∘ 3 ∘ ? 90 , 9 ;8 ∘;83 ∘ 81 + ?= ejur. ^k`n hu mñ`úŕúi dercjksye`u ejikyk` ndn rkstjk`tĻ mcĕnŕdc`n`n, dercjksye`u ejk` yc`n
ndn rkstjk`tĻ mcĕnŕdc`n`c mñ`úŕtúriú ŕtúr.
Őu k`k dkmkr lñrmúĕúiúz aeriújjcr Q9l(x,y) vc \9b(x,y) ae`dsnye`jkrĻ hnrchnrae`dsnye` ejmudjkrĻ zkik` ykzĻjkhnjnrjcr. K`okd (x; ,y; ), (x8,y8), (x?,y?)…………… (x̀,y`)lnhn `-tk`c Q9l(xn,yn) , \9b(x n ,yn) dñdú`ú` ejiksĻ muruiu`mk ìñzúi4
, 9 , , Őcdjn`mc cjmc cmnjnr. Hurkmkdn ‚n’ n`mnsn gkoehyc` iktrnsn`n`n. dñdtcmcĕcrjc`mnrnjcocĕn`n nakmc cmcr.Ñr`cd7
, ℀, tk` , 0 6
tk` , 6 ejkrkdvcr js `.
Cĕcr R vc ^ hnrhnrn`mc` hkĕĻisĻz vc hc`zcrŕcdnjmc mcĕnŕnyersk vc
, ∘ ejkrkdvcr ji ŕsc a vw(v,w)95Ìñzúi7
Ñ
^9Qsn`(w)9ς (v,w) mĻr.
∘
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,
, ∘
a vw(v,w)9 v.a xyTv.oes(w) ,v.sn`(w)Y
, ;8 ∘ ;8 , 0 6,∘8 86, m ĕcr muruij r a vw(v,w)9a v(v).a w(w) ejkokdtĻr.
^k`n Q vc \ hnrhnrn`mc` hkĕĻisĻz rkstjk`tĻ mcĕnŕdc`jcrnmnr. Hurkmk Q Xkyjcnlb, \ mk 8φ krkjĻĕĻ`mk u`naeri mkĕĻjiĻŕ ndn rkstjk`tĻ mcĕnŕdc`nmnr.
DKXKD]CXĶ_] ĶD AE@D_Ķ̂E@7
Hnr R rkstjk`tĻ mcĕnŕdc`n`c knt dkrkdtcrnstnd ae`dsnye`
Őcdjn`mc tk`ĻijĻmĻr. Hu tk`Ļi c gwx tcrnin`mcdn dujjk`Ļjikyk` (-) nŕkrctn mĻŕĻ`mk
a x(x)ae`dsnye`u Aeurncr trk`saeriu ejkrkd mk kmjk`mĻrĻjkhnjnr.Dkrkdtcrnstnd ae`dsnye` hkĕĻisĻz rkstjk`tĻ mcĕnŕdc`jcrn`n` tepjkiĻ`Ļ` bcskpjk`mĻĕĻ
prehjcijcrmc sĻdìk dujjk`ĻjĻr. Mkbk ñ`oc lñriúŕ ejmuĕuiuz ndn rkstjk`tĻ mcĕnŕdc`n`n`ae`dsnye`u ŕcdjn`mcdn uylujkikjkrĻ` hnr ñr`cĕn W9R; +R8 ŕcdjn`mc tk`ĻijĻ hnr tepjkiae`dsnye`u nìn` lcrcdjn ejk` nŕjcijcr ykpĻjkokd ejursk,
∜ ∘ Őcdjn`mc tepjki rkstjk`tĻ mcĕnŕdc`n ejksĻjĻd yeĕu`jud ae`dsnye`u vc on`sn`mc` nakmc cmnjnrjcr. ^udkrĻmkdn lnhn tk`Ļijk`iĻŕ n`tclrkj nŕjcin`c de`vejúsye` kmĻ
vcrnjnr vc k`kjntnd ejkrkd bcskpjkikd bcr zkik` prktnd ejikykhnjnr.K`okd yudkrĻmkdn lnhn tk`ĻijĻ dkrkdtcrnstnd ae`dsnye`jkr on`sn`mc` hu de`vejúsye`
nŕjcin hksnt hnr ìkrpik nŕjcin`c mñ`úŕtúrújcocdtnr. ^k`n7
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; hu
` .
mcrcocmc` i ic`t vcr r.
`9; nìn` n`tclrkjn` se`uou4
; 9 . ejur. HĶJC ŐĶD DKXKD]CXĶ_] ĶD AE@D_Ķ̂E@
, ,…………. . .. 9 . ... 9
9 ;8 .. . .. ∜
Cŕntjndjcrn yudkrĻmk ykpĻjk` tk`Ļijkik`Ļ` hnjcŕnd ejksĻjĻd mkĕĻjĻi ae`dsnye`u nìn` ìedheyutju lc`nŕjctnjinŕ bkjnmnr.
HĶXJCŐĶD IEIC@]JCX W9l(x,y) vc
, . 9 Cĕcr R vc ^ hnrhnrn`mc` hkĕĻisĻzsk z9l(x) vc w9b(^) mc hkĕĻisĻzmĻr.CT l(x) b(y) Y 9 CT l(x) YCT b(^) Y ejur.
DEQKX^K@_
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R vc ^ ndn rkstjk`tĻ mcĕnŕdc`n ejikd úzcrc
, ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ejkrkd tk`Ļijk`Ļr.
, ji jĻmĻr. , hur m d j ŕd d rcj sy ` d ts yĻs -;≭ ≭; krksĻ`mkmĻr.Cĕcr R vc ^ hkĕĻisĻz nscOev(x,y) 9CTxyY-CTxY.CTyY
9 CTxY.CTyY- CTxY.CTyY9 6 vc 9 6 ejur.
R vc ^ hkĕĻisĻz nsc Oev(x,y)96 , 9 6 ejur. ^k`n dercjksye` yedtur.↘ Cĕcr Oev(x,y)96 nsc bcr zkik` R vc ^ hkĕĻisĻzmĻr. Kik↘Cĕcr 9 6 nsc R vc ̂ bcr zkik` hkĕĻisĻz ejikz.
Ñr`cd7 W9kR+h^ nsc (Ϗz)8 ‒yn,Ϗx , Ϗy vc on`sn`mc` nakmc cmn`nz.Ocvkp7 CTzY9CTkx+hyY9kCTxY+hCTyY
9k³ x + h³ y Qc Ϗ8 9CT( z-³ z )8Y 9 CT{(kx+hy)-(k³ x+h³ y)}8Y
9CT{ k(x-³ x)+h(y-³ y)}8Y9k8CT( x-³ x)8Y + h8CT(y-³ y)8Y + 8khOev(xy)9 k8(Ϗx)8 + h8(Ϗy)8 + 8kh (Ϗx Ϗy)
Ñzcj ejkrkd R vc ^ hkĕĻisĻz nsck 9 h 9 ; nsc W9R+^ 90 (Ϗ )8 9 (Ϗ )8 + (Ϗ )8 ejur
]cerci7 R; , R8, R?………………R@ ejksĻjĻd mkĕĻjĻi ae`dsnye`jkrĻ a ; (x; ), a 8(x8),a?(x?) à(x`) ejkrkd tk`Ļijk`iĻŕ `’ tk`c hnrhnrn`mc` hkĕĻisĻz rkstjk`tĻ mcĕnŕdc`n
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a ?(x?),…………… a (x ) ejkrkd tk Ļijk iĻŕ ‚ tk c hnrhnrn mc hkĕĻisĻz rkstjk tĻ mcĕnŕdc ntk`Ļijk`iĻ ŕ ejsu`jkr. Hu`jkr nìn`4
6 ejikdzcrc ≚ + + + t `Ļij `sĻ` Qcrnjc` hnrγ 06 mcĕcrn nìn` yctcrjn se`suz ‚`’ skyĻ nìn` (Ϗd ) mcĕcrn (Ϗd ) 0 γ _ ̀ d9;,8,……….` ŕkrtĻ`Ļ skĕjkr vc hu muruimk ≚ ……………..+ / @erikjnzc tepjkiĻ `erikj mkĕĻjĻi VMA„sn`c ykdĻ`skrj i↘ ………………..̀erikjm ĕĻjĻi ji dt tmĻr. ]cerci7 6 ; 9 6,;8……….. ejikd úzcrc tk`ĻijĻhkĕĻisĻz vc hc`zcr mkĕĻjĻiĻ ejk` R; , R8, R?………………R@ rkstjk`tĻ mcĕnŕdc`jcrn vcrnjsn` ≚ ; tk`Ļijk`Ļrsk
j i↘ 6,; ejkokdtĻr.
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;) Hnr zkr ktĻjiksĻ mc`cyn`mc ejuŕkhnjcocd se`uìjkr n9;,8,?,=,3,1 ejsu`. hu mc`cyjc njnŕdnjnejkrkd R9n8 rkstjk`tĻ mcĕnŕdc`n tk`Ļijk`Ļyer.
k) Hu rkstjk`tĻ mcĕnŕdc`n`c njnŕdn` ejksĻjĻd mkĕĻjĻi ae`dsnye`u`u (VMA) ìnzn`nz.h) A(;8) mcĕcrn`n huju`uz.o) V(3:R≭
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m)
c)
a)
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l)
b)
?) Úrctnjc` hnr mnrc`on` úzcrn`mc lñstcrnjc` bktk mcĕcrn`n` úzcrn`mc ejiksĻ nbtnikjn p96.68 mnr.Hu muruimk úrctni hk`mĻ`mk` rkslcjc ;6 tk`c mnrc`ì scìtnĕninzn múŕú`cjni.
R(d)7 d tk`c mnrc`on` hezud ejiksĻ rkstjk`tĻ mcĕnŕdc`n ejsu` vc hu rkstjk`tĻ mcĕnŕdc`n4
≭ ≭ ≭
mcĕcrjcrn`n kjsĻ`.k) VTx9gY yn g9;,8,?,= nìn` bcskpjkyĻ`.h) EjksĻjĻd mkĕĻjĻi ae`dsnye`u (VMA) A(x)„n ìnzn`.
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=) EjksĻjĻd yeĕu`jud ae`dsnye`u (pma) kŕkĕĻmk vcrnjc` ae`dsnye` nìn`4
k) d ‒ yĻ huju`uz.h) CTxY ‒ n huju`uz.o) Qkryk`sĻ huju`uz. ( 8)
m) EjksĻjĻd mkĕejĻi ae`dsnye`u`u (VMA) huju`uz.c) V(6:R≭;) ejksĻjĻĕĻ`Ļ huju`uz.
3) KŕkĕĻmk VMA n vcrnjc` R rkstjk`tĻ mcĕnŕdc`n nìn`,
k) EjksĻjĻd yeĕu`jud ae`dsnye`u`u huju`uz.
h)
CTxY n huju`uz.o) Qkryk`sĻ ( 8) huju`uz.m) V(6:R≭;) ejksĻjĻĕĻ`Ļ huju`uz.
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c) V(6:x≭;)9A(;)-A(6)9
3)k)
h) o) m) V(6:x≭;)9A(;)-A(6)9 ;-6.396.3Ìñzúijú _ct-8
;)
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R(d)7 d tk`c mnrc`on` hezud ejiksĻ
9
:≭
:≭
:≭
9
???888;;;66
)(
d d d d
d R
k) VTx9gY yn bcskpjkyĻ`, g96,;,8,?h) A(x) ‒n ìnzn`.o) a(x)„ n ìnzn`.
?)
^udkrĻmk R rkstjk`tĻ mcĕnŕdc`n`n` pma n vcrnjinŕtnr. Hu`k lñrc4
k) d95h) VTR98Y95
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h)
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h)
o)
?)k)
?=8=6
81.6
3.66?
868
88;.6
)8(3.6)8(8
)?(;.6;)(
8
?
?
6
8
8
8
?
6
+
∘∘+9
++∘
++9
∘+++++⇘9∤ ∤ ∤ ∤∤∐
∐∘
∘
∘ ∘
d d
dx x xd
xd xd x x a ν ν
l)
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l)
?83.?
?8.6;3.;=3.6
?8.6(63.6=?2663.6?.6
83.6
6?
8;.6
86
?63.6
?;.6
)8(3.6;.6)8(63.6)?(;.6)(YT
88
?
?
?
6
8
?
6
8
8
9∘9+∘9
+∘+
+∘
∘+∘9
9++
∘
++
∘99
∘+++++99 ∤ ∤ ∤ ∤ ∤∐
∐∘
∘
∘ ∘
x x x x x
x x
mx x x xmxmx x xmx x x x xa R C ν ν
b)
?8.;;
?8.62.?
8>.6)Y?
;1=(6T63.6>.6
83.6
6?
?;.6
86
?8
=63.6
?;.6
)8(3.6;.6)8(63.6)?(;.6)(YT
8??=
8
6
8
8
8
8?
6
88?
?
888
9+9
++∘∘+9
9++
∘
++
∘99
∘+++++99 ∤ ∤∤∤∤∘∘
∐
∐∘
x x x x x
x x
xm x x x x x xmx x a x R C ν
>6?3183.;6
?8.;;
?83.?
?8.;; 88 9∘9
∘9Ϗ
n)
g) A(;)96 ?
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g) A(;)96.?
=)
k) T Y8;;8;8;
68;;)( 8 / 68 / 8 /
6
8 / 9⇘9⇘9∘∘⇘9∐
∘⇘9⇘9 ∘∐∘∘∐
∐∘
∐
∘
∤ ∤ d d ccd dcmxdcmx x a x x
h) 8 / 8 / 8 / 8 / 8 / )(8;)()( t t x
t
x
t
ct R Vcct
ccmx x a t R V ∘∘∐∘∘∐
∘
∐
9≣⇘+9∐
∘999≣ ∤∤
o)
V(R≣t+s | R≣t)9t)V(R
t)Rs+tV(R≣
≣≣
V(R≣t+s , R≣t)9 V(R≣t+s)⇘ V(R≣t+s | R≣t)9 8 / 8 / 8 / )(
t)V(R s)tV(R s
t
st
cc
c ∘∘
+∘
99≣
+≣
m) Bcpsn Cŕnt (c-3)
Ìñzúijú _ct-?
c) V(^9?)95
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c) V( 9?)95a) ax(R)95, ay(y)95l) CTRY95, CT^Y95b) 55, 88 99 y x Ϗ Ϗ
Ocvkp-;71 tk`c 6„mk` akrdjĻ ejksĻjĻĕĻ ejk` murui vkr. Hu muruijkr4 (;,;), (;,?), (8,;), (8,?), (=,;) vc (=,?).
;9∤∤ mxmy axy ejikjĻ. Hurkmkdn ejkyjkr nìn` n`tclrkj tepjkik mñ`úŕcocĕn`mc`4
∛∛ 9 ;)( xy axy ejikjĻ.
k) oT;.;+;.?+8.;+8.?+=.;+=.?Y9; 90 o9;/82h) ^:R ejk` ? ñr`cd `edtksĻ vkr.
V(^:R)9 { }( ) { }( ) { }( ) ( )82;2?.=;.=;.8
82;?,=;,=;,8 9++9++ VVV
o) ^0R ejk` 8 ñr`cd `edtksĻ vkr.
V(^0R)9 { }( ) { }( ) ( ) 82>
?.8?.;82;
?,8?,; 9+9+ VV m) ^9R ejk` tcd ñr`cd `edtksĻ vkr.
V(^9R)9 { }( ) ( )82;;.;
82;;,; 99V
c) y9? ejk` ? ñr`cd `edtksĻ vkr.
V(^9?)9 { }( ) { }( ) { }( )828;
82;8
821
82??,=?,8?,; 9++9++ VVV
a)
9
9
9
9
mnlcr x x x
x ax=8;
,,,,
6< / =< / 8< / ;
)( 99
9
mnlcr y y
y ay ?;
,,,
6= / ?= / ;
)(
Ñr`cĕn`4
R98 nsc { }( ) { }( ) 822?,8;,8)8( 9+9 VV ax l) CTRY9? CT^Y93/8
b)23,;6 88 99 yx Ϗ Ϗ
Ocvkp 87
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Ocv p 8
∛∛∘9 9
99 3
3
;6
;6
?? 6),(YT x y
y x axy xy R^ C
?) R vc ^ { }?,8,; mcĕcrjcrn`n kjk` múzlú` yeĕu`jud ae`dsnye`u`k skbnp rkstjk`tĻ mcĕnŕdc`jcrnejsu`. Hu`k lñrc4
k) Q98R+8^ 90 av(v)95h) \9R-^ 90 aw(w)95
Ocvkp ?7Hu prehjcic ndn ìñzúi ŕcdjnyjc ykdjkŕĻjkhnjnr.
k-;)R vc ^ múzlú` mkĕĻjĻik skbnp ejmudjkrĻ`mk(yk`n bcr ? mcĕcrn ejik ejksĻjĻdjkrĻ cŕntejmuĕu`mk v rkstjk`tĻ mcĕnŕdc`n`mc hu >muruiu kjik ejksĻjĻĕĻ cŕnt yk`n ;/> ejkokdtĻr.)
Hu`k lñrc 4
Ñr`cĕn`4= mcĕcrn ; dcrc kjĻ`khnjmnĕn`mc` ;/>2 mcĕcrn ? dcrc kjĻ`khnjmnĕn`mc` ?/>
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Ñ`oc W9R+^ yn hujkjĻi
N) W:8
az(z)96, z:8
NN) z98 hu muruimk skmcoc z-; vc ; úst ústc lcjcocd az(z)9>;
?;*
?; 9 , z98
z9? z-; vc 8, z-8 vc ; úst ústc lcjcocd az(z)9>8
?;*
?;
?;*
?; 9+ , z9?
z9= z-; vc ?, z-8 vc 8, z-? vc ; úst ústc lcjcocd az(z)9>?
>;
>;
>; 9++ , z9=
hc`zcr ŕcdnjmc, az(z)9>8 ,z93 vc az(z)9
>; ,z91
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h ŕĻddĻ hc`zcrŕcdnjmc ìñzújúr.
=) R vc ^ rkslcjc mcĕnŕdc`jcrn nìn`4
T Ymnlcr x x
x ax 8,6,,68;)( ∂∘9 vc T Ymnlcr y y y ay ;,6,,688)(
∂ ∘9 nsc,
\9R+^ njc vcrnjc` \ rkstjk`tĻ mc ĕnŕdc`n nìn` aw(w) yn huju`uz.
Ocvkp =7De`vejúsye` ejkrkd ocvkp kŕkĕĻmkdn lnhn huju`ur.
≭≭∘+∘
≭≭∘
≭≭+∘
9
mnlcr
wwww
ww
wwww
w aw
,6
?8,1;
8?
8>
8>
8;,8;
1<
;6,1;
8?8
)(?8
?8
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3) R T6,8Y krkjĻĕĻ`mk ^ nsc T6,8Y krkjĻĕĻ`mk S`naeri mkĕĻjĻik skbnp rkstjk`tĻ mcĕnŕdc`jcrnejsu`. Hu`k lñrc4
yudkrĻmkdn snstcimc d mcĕcrn`c skbnp mnrc`ìjcr dkrkrjĻjĻd tcstn`mc` lcìnrnjnyerjkr vc snstciìĻdĻŕĻ`mk W03 ejursk hezud, W≭3 ejursk skĕjki ejmudjkrĻ`k dkrkr vcrnjnyer. d lnrnŕn sĻokdjĻdtejcrk`s mcĕcrn ejup d98 ejkrkd hnjn`nyer, snstcimc hu`u 366O mc tcst cmcrcd ìĻdĻŕk 8d ejkokdŕcdnjmc vcrnjicsn tkskrjk`iĻŕ. K`okd R vc ^ lúrújtújcrn yúzú`mc` snstci ìĻdĻŕĻ W98(d+R+^)ejuyer. Hu`k lñrc snstcic lnrc` mnrc`ìjcrn` tejcrk`s mcĕcrjcrn`n` snstci ìĻdĻŕĻ`mkdn ertkjkikmcĕcrn vc lnrc` hnr mnrc`on` hezud ejik nbtnikjn `cmnr5 (V(W03)95)@E]7 R vc ^ hkĕĻisĻz.
Ocvkp 37Ñ`oc R vc ^ `n` de`vejúsye`u`u kjkrkd Q9R+^ yn hujikd nìn` skmcoc Q rkstjk`tĻmcĕnŕdc`n`n` hkŕjk`lĻì hntnr mcĕcrn`n 8 njc ñtcjcyn` se`rk W98\ rkstjk`tĻ mcĕnŕdc`n`n huju`.
_e`uì4
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2?
1=6?
21=?
?8?
;82;1;
?2=?3;821=
?;
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)8 / )(cxp(8
;)8 / Yoessn`8oessn`
oessn`8sn`oesTcxp(8
;),(
8888
88888
88888
Ϗ φϏ
Ϗ β β β β
β β β β φϏ
wv
R^ ^ R
R^ ^ R wv avw
+∘9
∘+
+++∘9
8)
+∘∘
∘
∘
9 )8(;8;cxp
;8;),( 88
8888 y pxy x y x axy
ϋ Ϗ ϋ φϏ
z9kx+hyw9ox+my
k)
azw(z,w)95h) k,h,o,m `n` bk`ln deihn`ksye`u nìn` z vc w hkĕĻisĻz ejur.
oW9koR+ho^ mW9kmR+hm^
k\9koR+km^ h\9hoR+hm^
oz-kw9^(ho-km) mW-h\9R(km-ho)
km hok\ oW ^ ∘
∘9
km homW h\
hokm h\ mW R
∘
∘9
∘
∘9
),( w z^ ϓ 9 ),( w z R ρ 9
;//~ ∘∘∘
∈∈∈∈ khh
khm
wz ρρ
∘++∘+ 88;;; 88888888 kowzwk zohmwzwh zm
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T Y
T Y
+∘+∘
∘+∘+
∘∘
∘
∘
9
+∘+∘
+∘++
∘∘
∘
∘
9
+∘∘∘∘∘∘∘9
))((8)8()8(;*
);(8;cxp
;8;
)(8)(8)()(;*
);(8;cxp
;8;
8*);(8cxp;8),(
888888
8888
88
888888
8888
888888
km hohm kowzkhkhwom m o z
km ho
km howzkhwomzhm kowzkhwm o z
km ho
khwkmwzhowzomzkm howv avw
ϋ ϋ ϋ
ϋ Ϗ ϋ φϏ
ϋ ϋ Ϗ ϋ φϏ
ϋ ϋ Ϗ ϋ φϏ
h) ko+hm9 ϋ (ho+km) 90 hkĕĻisĻz ̀ erikj ejur.
?) R vc ^ hkĕĻisĻz mcĕnŕdc`jcrn R~T6 ?Y vc ^~T6 =Y S`naeri mkĕĻjĻik skbnpjcr. W9R+ŕ̂cdjn`mctk`Ļijk`mĻĕĻ`k lñrc
k) 5)( 9w xρ
h) 5)( 9w yρ o) 5)( 9w zρ m) az(z)95c) CTzY95a) 58 9 zϏ
k)
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9
9
∘
9∘999 ∤
∘
wocw
w
wc
gcc
gw g
cc gw
c gw
mxcw
gw
x
gw gw gw gw
gw gwx gwx x
8
?sn`*)(
8?
)8?sn`(
88?);(?
;6?
?;
?;)(
8?
8??
6
8?
8?
8?
?
ρ
ρ
h) Hc`zcr ŕcdnjmc )8(sn`)( 8 wocw gw y 9ρ
o)
T Y;;8
;
8818
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h) ∤
∐
∘9 gwx
)()( khj k`t ↝
∘
λλ ;
)(
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h) ∤∐∘ mwcw x ax x )()( ρ tkhjemk` gwct u +λ )(
ww ∘↘ hnzni nìn` gw
ct u t ∘
↝∘
λ λ ;)(
≣+9
∘∘
mnlcr
xcc x ax x x
,6
6,8)(8=
)(8;
3.68886.3)V(x ;8
8=
3.6
8= ∘∘∘∘∐
∘∘ +9∐
∘∘
9+9≣ ∤ ccccmxcc
x x x x
o)
=
8)8(YT
=
=6 6
8=
6
8=
∘
99
⇘99
9+9+9
∘
∘
∐ ∐∘∘
∐
∘∘
∤ ∤∤
x
x
x x x x
cvmxmu
cmv xu
mx xcmx xcmxcc x xC
2;
6;1=8
==88
====
6
= 9∐
∘
∘
9∘
∘
∘
9∘∘∘∘∐
∘
∤∤ x x x x
x c xcmxc xc xc
Ky`Ļŕcdnjmc mnĕcr tkrkatk` mk28 lcjcocd vc CTzY9
2?
m)
T Y
?1;86)8()=(
8;6
)(; 88
99+9
9∘+
∘
99
9w gw
g gw g
gww
mwm
g xC xρ
a)888
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1=;;
;13
1=86
28
1==
6)8(8
)=(=;
6)(;YT
8
?
8
?
8
88
8
88
9
99+9
9
∘
+∘
99
9
Ϗ
ρ w gw
g gw g
gww
mwm
g xC x
3)R~S(6,;) , ^~S(6,;) vc R vc ^ hk ĕĻisĻz ejsu`.
W9R+^\9R-^ ejsu`. z vc w `u` hk ĕĻijĻ akdkt njnŕdnsnz ejmuĕu`u lñstcr.
8,8w z
yw z
x ∘
9
+
9 ejur. 6:z:8 , -;:w:;z+w≭8, z-w≭8 z0|w|
8;
8;
8;
8;
8;
8;
8;
8;
8;
),(~ 9∘
∘9
∘
99
z x
z y
w x
z x
w z g
ν ν
ν ν
ν ν
ν ν
),(),(~),( nn y x axyw z gw z azw 9
:≭∘≭+::∘::9
mnlcr
ww zw zw zw z azw,6
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Ļ
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Ļ
;) Hnr zkr ndn dcrc ktĻjĻyer.Ķjd ktĻŕtk ? lcjnrsc ndn ktĻŕ se`uou`mk tepjkiĻ` < ejik nbtnikjn`cmnr5
8)
^udkrĻmkdnŕcdnjmc lñstcrnjc` ? scihejú` (;,8 vc ?) njctnjmnĕn hnr bkhcrjcŕic dk`kjĻ`mklc`cjmc (ertkjkik mkvrk`Ļŕjkr lñz ñ`ú`c kjĻ`mĻĕĻ`mk) ?, 8 ‒ mc`, 8„ mc ;„ mc` 8 dkt mkbkakzjk sĻdjĻdtk lñ`mcrnjnyer.
VT^9;|R9;Y9;-λ, VT^98|R9;Y9λ /8, VT^9?|R9;Y9λ /8 ejkrkd vcrnjinŕ vc mnĕcr ejkhnjcocd
nbtnikjjcrjc hcrkhcrŕcdnj úzcrn`mc lñstcrnjinŕtnr. Hurmk VT^9;|R9;Y ; lñ`mcrnjmnĕn`mc ;kjĻ`iksĻ deŕujju nbtnikjnmnr. Hu`k lñrc, kjĻoĻ tkrkatk ^9; kjĻ`mĻĕĻ`k lñrc ; lñmcrnjinŕ ejik nbtnikjn ̀ cmnr5
?) Hnr terhkmk 8 dĻriĻzĻ vc ? hcykz tep huju`ikdtkmĻr. Hu terhkmk` hnr tep ìcdnjnp rc`lndkymcmnjnyer vc ycrn`c lcrn deyujuyer. Hu`k lñrc ykpĻjk` = ìcdni se`uou`mk c` kz 8dĻriĻzĻ tep ìwdnjinŕ ejik ejksĻjĻĕĻ `cmnr5
=) Hnr ìnat zkr ;6 dcrc ktĻjĻyer. Hu zkrjkrĻ` tepjkijkrĻ`Ļ` c` kz hnr dcrc < ejik nbtnikjn `cmnr5
3) Hnr sĻ`Ļatk= crdcd vc ? dĻz ñĕrc`on vkrmĻr. Hu ñĕrc`onjcr tcdcr tcdcr sĻ`kvk kjĻ`Ļyerjkr.Ñĕ ` j ` h d d h dĻŕ dj ` l ` h Ļ kjkik jk Ļ`k k kjĻ`ik jk ĻjĻdjk Ļ
ykmk kjtĻsĻ turk skmcoc hnrn ykzĻ) mcvki cmnyer. AkrdjĻ ejk`Ļ ktk` bkdci ejkrkd
hcjnrjc`nyer Hu`k lñrc njd turmk (yk`n njd mc`cicmc) bkdcin` hcjnrjc`icsn ejksĻjĻĕĻ
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hcjnrjc nyer. Hu k lñrc njd turmk (yk n njd mc cicmc) bkdcin hcjnrjc icsn ejksĻjĻĕĻ`cmnr5 (eyu`oujkrĻ` skyĻsĻ`Ļ < kjkrkd prehjcin ìñzú`.)2) K`cstczn mc dujjk`Ļjk` eteiktnd hnr `cacs múzc`jcic kylĻtĻ (H), VH ejksĻjĻĕjk hezujuyer.
Hu hezujik, nzjcic snstcin (I) hu hezudju ĕu kjlĻjkikzsk, bkstk`Ļ` ñjúiú`c schcpvcrnyer.Ķzjcic snstcin (I) mc V I ejksĻjĻĕĻyjk hezujuyer.Ķdn snstcin` hezujikjkrĻtkikic` hk ĕĻisĻz ejkyjkrmĻr. Bkstk`c`n` preacsñrú cĕcr VI 0VH nsc nzjcic onbkzĻdujjk`ik`Ļ` tkikic` lcrcdsnz ejmu ĕu`u skvu`uyer. Hu prcañsñrc dktĻjĻriĻsĻ`Ļz. ̂ edske`k hnr ejksĻjĻd dursu kjiksĻ`Ļ iĻ tkvsnyc cmcrsn`nz5 VI 96.; vc VH96.63 ejkrkd ocvkhĻ`ĻzĻmcstcdjcynon nŕjcin lñstcrn`.
>) KŕkĕĻmkdnŕcdnjmcdn snstcimc ;8 uìjkrĻ krksĻ`k hkĕjk`k` 3Q ‒ jud hnr pnjjc ‚K’ kipújúykdĻjikd nstc`icdtcmnr. K`kbtkrjkrĻ` bcrhnrn nìn` dkpkjĻ de`uimk ejik nbtnikjn ‚p’ mnr vcbcpsn hnrhnrn`mc` hkĕĻisĻz ìkjĻŕikdtkmĻr. p96.= ejkrkd jkihk`Ļ` yk`ik nbtnikjn`nbcskpjkyĻ`Ļz.
Ļ
;) ^98
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k) CT Y95 (Ocvkp7 ;?)h) 59 yϏ (Ocvkp7 =18< )o) V(^0-;?) (Ocvkp7 6.3)
8) ^9R 8+? vc R≂T-8,8Y krkjĻĕĻ`mk múzlú` mkĕĻjĻik skbnpsc4k) CT^Y95 (Ocvkp7 ;?/?)
h) 59
yϏ (Ocvkp7
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96/96