Linear Algebra | jiblm.org

8/9/2019 Linear Algebra | jiblm.org

1/31

LHYSDKG HF ND[YNS !CKRIM GIKSDNDJ ND AK]EIAK]NBR

Dh$ ;0' %Rip}$ 3005.

Gndiks Kgjicsk

Mkvnm A$ Bgkso

RYD^ Diz Pkg}|


2/31

Bhd}id}r

Kbodhzgimjiaid} nnn

]h }ei Ndr}syb}hs nv

; Vib}hsr nd 2!Rpkbi ;

3 Gndiks Rpkbir :

2 Nddis Pshmyb} Rpkbir ;9

= Gndiks ]skdrfhsak}nhdr 3;

nn


3/31

Kbodhzgimjiaid}

]enr jynmi zkr zsn}}id ydmis }ei kyrpnbir hf RYD^ Diz Pkg}|$ ]ei ky}ehs jsk}ifygg~

kbodhzgimjir As$ Ekss~ Gybkr' Ls$ kdm }ei Imybk}nhdkg Kmvkdbiaid} Fhydmk}nhd fhs }eins

rypphs} nd }ei psipksk}nhd hf }ei ndbgymim jskpenbr$ Ei kgrh znreir }h kbodhzgimji }ei

eksm zhso hf }ei akd~ RYD^ Diz Pkg}| r}ymid}r zehri iffhs}r kdm fiimckbo ekvi gim }h

}ei sidim jynmi }ek} nr dhz cifhsi ~hy$

nnn


4/31

]h }ei Ndr}syb}hs

Gndiks kgjicsk nr k }hpnb }ek} bkd ci }kyje} k} akd~ mnffisid} givigr' mipidmndj yphd }ei

rhpenr}nbk}nhd hf }ei kymnidbi$ ]eiri dh}ir zisi ndn}nkgg~ mivighpim fhs k hdi riair}is

rhpehahsi!lydnhs givig gndiks kgjicsk bhysri fhs k jshyp hf r}ymid}r zeh ekm k fkangnks!

n}~ zn}e vib}hsr fsha ayg}n!vksnkcgi bkgbygyr kdm pe~rnbr$ Cy} }eiri r}ymid}r ekm ekm dh

psnhs bhysri nd vib}hs kgjicsk pis ri' kdm ekm andnakg iupisnidbi pshvndj }eihsiar$ ]ei~bhapgi}im B0$>6=9=;%j.!9$5>0;:%i.!;$66==3%c$06.!9$5>0;:%6==3%s.!=$36;0=%n.0$2$99>6>%}.0$>6=9=;%n.0$>6=9=;%b.!p:%i.!;$66==;:%i.!;$66==3%r.2$96;;%


5/31

]h }ei Ndr}syb}hs v

~hy }h }endo bksifygg~ kchy} }ei midn}nhd hf rpkd$ Nd }eiri dh}ir N ekvi sr} midim

}ei rpkd hf k dn}i ri}' gi} }ei r}ymid}r zhso zn}e }enr dh}nhd k cn}' kdm }eid iu}idmim }ei

midn}nhd }h kgg ri}r$ Cy} N ekvi fhydm' zn}e rhai bgkrrir zeh ksi r}syjjgndj zn}e }eibhdbip} hf kd kcr}skb} gndiks rpkbi' }ei iu}idmim midn}nhd cibhair }hh kcr}skb}$ Fhs }ehri

bgkrrir N sibhaaidm han}}ndj }ei iu}idmim midn}nhd kdm ahmnf~ndj hs han}}ndj }ei fiz

rycri{yid} pshcgiar }ek} mipidm yphd n}$ Kgahr} kgg hf zek} nr mhdi eisi bhdbisdr hdg~

dn}i ckrir kdm dn}i!mnaidrnhdkg gndiks rpkbir' kdm }eisifhsi mhir dh} mipidm yphd }ei

iu}idmim midn}nhd hf rpkd$

Pshcgia 20 bkd gikm }h nd}isir}ndj mnrbyrrnhdr rndbi }ei~ mhd} ~i} ekvi }ei }hhgr }h

rhgvi n}' cy} }ei~ ksi bis}kndg~ nd k phrn}nhd }h }endo kchy} n} kdm akoi k sikrhdkcgi bhdlib!

}ysi' pks}nbygksg~ nf }ei~ ekvi mhdi Pshcgia 3>$ Pshcgia 26 nr k r}ip nd }ei snje} mnsib}nhd'

kdm n} nr yrykgg~ psh}kcgi }h mnrbyrr zei}eis hs dh} Pshcgia 20 nr rhgvim c~ rhgvndj Pshc!

gia 26$ Pshcgia 20 ksnrir kjknd kr Pshcgia =5' zeisi dhz }ei~ kppsibnk}i }ei rhgy}nhd kr

kd ikr~ kppgnbk}nhd hf }ei Ckrnr ]eihsia$ Fhs iupisnidbir gnoi }enr n} nr naphs}kd} }h pkrr

hy} }ei }iu} kr n} nr diimim' sk}eis }ekd nd k rndjgi pkboi}$ Kf}is }ei~ pshvi Giaak =5 kdm]eihsia =0' N gnoi }h sknri }ei nrryi kr }h zei}eis }enr zhygm gikm }h k pshhf }ek} ivis~ gnd!

iks rpkbi %kr' fhs iukapgi'B_0';Q. ekr k ckrnr$ A~ nd}id}nhd nr }h sknri }enr {yir}nhdcy} dh}psirr n} ydgirr rhai r}ymid}%r. }koi k sikg nd}isir} nd n}$ ]ei bekp}is idmr zn}e kppgnbk}nhdr

hf Jkyrrnkd ignandk}nhd' zenbe zkr nd}shmybim zn}ehy} k dkai nd Bekp}is ;$

K bid}skg }eiai hf }ei }ensm bekp}is nr }ek} r}k}iaid}r zenbe ksi }syi nd S3 kdm S2 kdm

bkd ci iupsirrim nd }ei gkdjykji hf nddis pshmyb}r ksi jidiskgg~ }syi nd kgg nddis pshmyb}

rpkbir$ ]enr }eiai nr k dnbi nggyr}sk}nhd hf }ei pshbirr hf ak}eiak}nbkg kcr}skb}nhd$ ]ei

aknd iffhs} hf }ei bekp}is nr }h pshvi }ei Jska!Rbeanm} ]eihsia' zenbe nr cyng} yp nd akd~

r}ipr$ Iksg~ nd }ei bekp}is zi pshvi }ei Bkybe~!Rbezks}| Ndi{ykgn}~ yrndj k rnapgi }snbo

}ek} kppiksr kr k enjeg~yd!nd}yn}nvi skccn}!hy}!hf!k!ek}'kdm }eid yri n} }h pshvi }ei ]snkdjgi

Ndi{ykgn}~$ Kr kd kppgnbk}nhd hf }ei Jska!Rbeanm} ]eihsia zi ji} vis~ r}sknje}fhszksm

pshhfr hf }eiri }zh siryg}r fsha hys miipis ydmisr}kdmndj hf }ei r}syb}ysi hf }eiri rpkbir$

]ei fhys}e bekp}is nd}shmybir }ei dh}nhd hf nrhahspenb rpkbir }eshyje }ei hcrisvk}nhd

}ek} kgg }zh!mnaidrnhdkg nddis pshmyb} rpkbir ghho lyr} gnoi S3$ ]enr gikmr }h }ei dh}nhd

hf k gndiks }skdrfhsak}nhd' zenbe N cignivi rehygm ci midim c~ napgnbk}nhdr kdm }eid

pshvid %Giaak :=. }h ci i{ynvkgid} }h i{yk}nhdr$ Zi bhdbgymi c~ riindj ehz gndiks

}skdrfhsak}nhdr hf dn}i!mnaidrnhdkg rpkbir ksi sipsirid}im c~ ak}snbir$

N ekvi }kyje} }ei igiaid}ks~ visrnhd hf }enr bhysri %Be ;'3'2. }h bghri }h 20 r}ymid}r

kdm }ei kmvkdbim visrnhd %Be 3'2'=. }h kr fiz kr ;0' ch}e k} RYD^ Diz Pkg}|$ N jnvi }zh

iukar' k anm}isa kdm k dkg' ikbe bhyd}ndj 39) hf }ei jskmi$ Bgkrr psirid}k}nhdr kdm

r}ymid} phs}fhgnhr ikbe bhyd} kdh}eis 39)$

Bgkrr }nai nr psnaksng~ hsjkdn|im kshydm bgkrr psirid}k}nhdr$ N psirid} midn}nhdr kdm

r}k}iaid}r hf pshcgiar kdm }eihsiar nd kd nd}iskb}nvi mnrbyrrnhd zn}e }ei bgkrr$ N }s~ }h

ah}nvk}i }ei bhd}id}' sknri {yir}nhdr' kdm bhddib} nmikr$ ]ei r}ymid}r ksi }eid gif} }h rhgvi

}ei pshcgiar kdm pshvi }ei }eihsiar hd }eins hzd' hy}rnmi hf bgkrr' zn}ehy} bhdryg}ndjh}eis rhysbir$ K} }ei r}ks} hf bgkrr r}ymid}r akso hd k reii} zenbe n}iar }ei~ ksi sikm~

}h psirid}$ N behhri r}ymid}r }h psirid} zeh ekvi psirid}im }ei gikr}$ Fhs bgkrrir zn}e

hvis 30 idshggim' N dhsakgg~ ekvi riviskg r}ymid}r py} yp zhso rnayg}kdihyrg~$ ]eid N jh

}eshyje ikbe pshhf(rhgy}nhd ndmnvnmykgg~ zn}e }ei bgkrr$ R}ymid}r akoi jhhm anr}koir }ek}

ksi jidiskgg~ ndr}syb}nvi }h }eins bgkrrak}ir$ K r}ymid} psirid}ndj kd ndbhssib} pshhf ekr

}ei hpphs}ydn}~ }h bhssib} n} fhs }ei diu} bgkrr aii}ndj$

Mkvnm A$ Bgkso Lhysdkg hf Nd{yns~!Ckrim Giksdndj nd Ak}eiak}nbr


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]h }ei Ndr}syb}hs vn

Kgg r}ymid}r ksi si{ynsim }h oiip k phs}fhgnhbhdrnr}ndj hf k dkg bhssib} rhgy}nhd }h ikbe

pshcgia kdm pshhf hf ikbe }eihsia$ Fhs }ei phs}fhgnh zhso' }ei~ bkd bhdryg} ikbe h}eis hs

ai' kdm }ei~ bkd akoi ayg}npgi k}}iap}r }h ji} n} snje}$ Fhs zikois r}ymid}r }ei pshbirrhf zsn}ndj yp k pshhf(rhgy}nhd fsha }eins hzd bgkrr dh}ir bkd ci kr aybe hf k mnrbhvis~

iupisnidbi kr kd~$ Cy} }ei~ ksi knmim c~ }ei fkb} }ek} }ei~ ekvi kgsikm~ riid }ei zhso mhdi

nd bgkrr$

Phs}fhgnhr ksi si{ynsim }h ci sikmkcgi' bhapgi}i kdm bhssib}$ Kbbhsmndjg~' }ei dkg

phs}fhgnh jskmi nr in}eis 39) hs 0)$ N ekvi }eia }ysdim nd pisnhmnbkgg~ rh }ek} N bkd sibhsm

}eins pshjsirr$ Nd mhndj rh N behhri rkapgi n}iar }h sikm bksifygg~? }ei dyacis hf }eia

cindj mi}isandim c~ }ei }nai N ekvi }h mh rh$ Hbbkrnhdkgg~ N han} k rigib}im n}ia fsha

bgkrr psirid}k}nhd kdm ekvi kgg r}ymid}r mh n} }eiarigvir fhs }ei phs}fhgnh$

N hffis }eiri dh}ir kr k r}ks}ndj phnd} fhs }ei ndr}syb}hs zeh zhygm gnoi }h }ikbe kd

nd{yns~!ckrim bhysri nd gndiks kgjicsk$ ]eiri dh}ir ekvi ivhgvim hvis akd~ n}isk}nhdr kdm

dhz zhso zigg fhs ahr} hf }ei ak}eiak}nbr r}ymid}r k} RYD^ Diz Pkg}|' k ahmisk}ig~

bhapi}n}nvi = ~iks r}k}i bhggiji$ Mipidmndj hd ~hys kymnidbi' ~hy ak~ zigg diim }h ahm!nf~ }eia' in}eis kmmndj ahsi ak}isnkg kdm ahsi mnfbyg} pshcgiar hs ronppndj rhai hf

}ei ak}isnkg kdm ggndj nd rhai ikrnis pshcgiar$ Sijksmgirr hf ~hys kymnidbi' ~hy ak~

behhri }h sipgkbi rhai pks}r hf }enr jynmi zn}e }hpnbr' iukapgir hs pshcgiar }ek} ak}be

~hys pisrhdkg nd}isir}r$

Mkvnm Bgkso

bgksom@dizpkg}|$imy

Rip}iacis 3005



7/31

Bekp}is ;

Vib}hsr nd 2!Rpkbi

K vib}hs nr kd hsmisim }snpgi P 1 %k;' k3' k2. zeisi k;' k3 kdm k2 ksi nd }ei ri} S hf sikgdyacisr$ ]ei rpibnkg vib}hs 0


8/31

Vib}hsr nd 2!Rpkbi 3

k mnsib}nhd kdm k gidj}e$ NfP sipsirid}r hdi hf }eiri pe~rnbkg {ykd}n}nir' n}r mnsib}nhd }iggr

yr ehz n} nr hsnid}im kdm n}r gidj}e jnvir yr n}r akjdn}ymi$ ]ei gidj}e hs akjdn}ymi hfP nr

bkggim }ei dhsa hfP kdm nr midh}im c~ P$

Nd }enr bhysri ~hy zngg giksd kchy} gndiks kgjicsk c~ rhgvndj k bksifygg~ mirnjdim ri!

{yidbi hf pshcgiar$ Ydgnoi h}eis ak}eiak}nbr bhysrir ~hy ekvi ekm nd }ei pkr}' rhgvndj k

pshcgia nd }enr bhysri zngg kgzk~r ekvi }zh bhaphdid}r$

;$ Fndm k rhgy}nhd$

3$ Iupgknd ehz ~hy odhz }ek} ~hys rhgy}nhd nr bhssib}$

]enr zngg eigp psipksi ~hy }h yri ak}eiak}nbr nd }ei fy}ysi' zeid }eisi zngg dh} ci rhaihdi

k} ekdm }h }igg ~hy nf ~hys rhgy}nhd nr bhssib}$ Nd }ehri rn}yk}nhdr' n} zngg ci yp }h ~hy }h

mi}isandi zei}eis hs dh} n} nr bhssib}$

hys sr} }kro zngg ci }h dm k zk~ }h bhapy}i }ei dhsa hf k vib}hs yrndj k zigg odhzd

}eihsia fsha jihai}s~$

s

s

sk c

b

P~}ekjhsikd ]eihsia

k3 / c3 1 b3

Nd }ei fhgghzndj }zh pshcgiar' bksifygg~ mskz kdm gkcig }ei sigivkd} snje} }snkdjgir }h rehz

ehz ~hy ksi yrndj }enr }eihsia$

Pshcgia ;$ Yri Fnjysi ;$; }h dmP kdm }eid yri P }h dm[$Pshcgia 3$ Dhz gi}P


9/31


s0

N[

NV

NY

Fnjysi ;$3< Vib}hsr nd }ei mnsib}nhd hf[$

Pshcgia =$ Ikbe hf }eiri kgjicsknb bhdbip}r ekr k dnbi jihai}snb nd}ispsi}k}nhd$ ]h rii

}enr' gi}P 1 %k;' k3'0. kdm[ 1 %c;' c3'0.$ Mskz kd u~!pgkdi %}ei phnd}r zn}e |!bhhsmndk}i0.' kdm ndmnbk}i nd n} }ei ghbk}nhdr hf phnd}rP' [' %;.P' 2[ kdmP / [$

Fhs vib}hsr P kdm [' }ei mnffisidbi [ P nr }ei ydn{yi vib}hs }ek}' zeid kmmim }h P'jnvir yr [$ Zi bkd mirbsnci }ei mnffisidbi ch}e jihai}snbkgg~ kdm kgjicsknbkgg~' ikbe nd

}zh mnffisid} zk~r$

Pshcgia 9$ Mskz kd hsnjnd 0 kdm }eid mskz }zh vib}hsr %kr ksshzr. P kdm[ iakdk}ndjfsha n}$

%n. Mskz kd ksshz }ek} sipsirid}r }ei mnffisidbi [ P$%nn. Mskz %;.P kdm }eid mskz [ / %;.P$

Pshcgia 6$ Gi}P 1 %k;' k3' k2. kdm[ 1 %c;' c3' c2.$

%n. Fndm }ei bhhsmndk}ir [ P 1 % ' ' . hf }ei vib}hs ~hy zhygm diim }hkmm }h P }h ji}[$

%nv. Rehz ehz }h bhapy}i }ei bhhsmndk}ir [ / %;.P 1 % ' ' .$Pshcgia :$ Gi}P kdm[ ci kd~ }zh vib}hsr$ Fndm k mirbsnp}nhd fhs }ei gndi P[ }eshyje P

kdm[ c~ nmid}nf~ndj }ei vib}hs phnd}ndj fsha P }h [ kdm kmmndj rbkgks ayg}npgir hf n} }h k

rndjgi phnd} hd n}$

Hdi naaimnk}i kppgnbk}nhd hf }eiri nmikr ksnrir zeid zi rhgvi k r~r}ia hf gndiks i{yk!

}nhdr$ Fhs iukapgi' rypphri zi zkd} k jhhm mirbsnp}nhd hf }ei ri} hf kgg rhgy}nhdr u' ~ kdm

| }h }ei r~r}ia hf rnayg}kdihyr gndiks i{yk}nhdr

2u 3~ | 1 0 _;Q3u / ~ / | 1 ;0 _3Q

u / =~ / 2| 1 30 _2Q

Nf zi }endo hf k rhgy}nhd }h }enr r~r}ia kr k phnd} %u'~'|.' }eid zi ksi krondj fhs k jihai}snbmirbsnp}nhd hf }ei ri} R hf kgg hf n}r rhgy}nhdr$

]ei naphs}kd}hcrisvk}nhd }h akoi nr }ek} zi bkd }skdrfhsa k r~r}ia hf gndiks i{yk}nhdr

nd}h k diz r~r}ia }ek} ekr iukb}g~ }ei rkai rhgy}nhdr c~ bekdjndj lyr} hdi hf }ei i{yk}nhdr

nd hdi hf }zh zk~rQ nr hc}kndim c~ kmmndj = }nair _5Q }h _2Q$

Zi bkd ignandk}i _:Q rndbi n} nr rk}nrim c~ ivis~ phnd} %u'~'|. kdm }eisifhsi bhd}sncy}irdh}endj }h }ei rhgy}nhd$ Zi hc}knd _;0Q kdm _;;Q c~ rhgvndj _5Q kdm _>Q fhs ~ kdm u$

0u ;=~ ;0| 1 60 _=Q0u :~ 9| 1 20 _9Qu / =~ / 2| 1 30 _2Q

0u :~ / 9| 1 20 _6Q0u 0~ 0| 1 0 _:Qu / =~ / 2| 1 30 _2Q

0u ~ / 9:| 1 20

:_5Q

u / =~ / 2| 1 30 _2Q

0u / ~ / 9:| 1 20

:_5Q

u / 0~ / ;:| 1 30

:_>Q

~ 1 209|:

_;0Q

u 1 30|:

_;;Q

]ei rhgy}nhdr }h }ei r~r}ia {_;0Q' _;;Qq ksi hc}kndim c~ behhrndj lyr} kd~ vkgyi fhs | kdm}kondj u 1 30|

:kdm ~ 1 209|

:$ Nf zi gi} P


11/31


Fsha Pshcgia : zi rii }ek} R nr }ei gndi }ehyje P 1 % 30:

' 20:

' 0. nd }ei mnsib}nhd hf[ 1

% ;:

' 9:

' ;.$

Rhgvi ikbe hf }ei fhgghzndj r~r}iar nd }ei rkai zk~' kdm jnvi k jihai}snb mirbsnp}nhd

hf }ei rhgy}nhd ri}$ Ci rysi }h rycr}n}y}i ~hys rhgy}nhdr nd}h }ei hsnjndkg i{yk}nhdr }h rii nf

}ei~ ksi bhssib}$

Pshcgia 5$ =u ~ / 2| 1 9 kdm 2u ~ / 3| 1 :$Pshcgia >$ :u ;;~ 3| 1 2 kdm 5u 3~ / 2| 1 ;$Pshcgia ;0$ 2u 9~ /| 1 3 kdm 6u / ;0~ 3| 1 2$

Zi rk~ }ek} k vib}hs P nr k gndiks bhacndk}nhd hf vib}hsr P;' P3' $ $ $ ' Pd nf }eisi ksi

dyacisr };'}3' $ $ $ '}d rybe }ek}

P 1 };P; / }3P3 / / }dPd$]ei rpkd hf }ei ri} C .' %='3'6.q6$ C1 {%='2':.' %;' 9'2.q:$ C1 {%;'0' 0.' %0';' 0.' %0'0' ;.q5$ C1

{%;'0' 0.' %2'0' 2.' %0'0' ;.

q>$ C1 {%;'3'2.' %0' 0'0.' %3'='6.q

K ri} C hf vib}hsr nr rknm }h rpkd S2 nf RpkdC1S2? }ek} nr' nf ivis~ vib}hs nr k gndiksbhacndk}nhd hf }ei vib}hsr nd C$ K bid}skg pshcgia fhs yr zngg ci }ek} hf dmndj hy} zeid

k ri} C rpkdr S2$

Fhs ikbe hf }ei fhgghzndj fhys pshcgiar' mibnmi zei}eis hs dh} {P' [' Sq rpkdr S2$]h rehz }ek} n} mhir' ~hy zngg diim }h rehz ehz kd kscn}sks~ phnd} K


12/31


iupsirrim kr k gndiks bhacndk}nhd hf{P' [' Sq' }ek} nr' fhs ivis~ behnbi hfK }eisi iunr} u'~ kdm | rybe }ek}

K 1 uP /~[ /|S$

]h rehz }ek} n} mhir dh}' ~hy diim }h iuencn} k pks}nbygks vib}hs K }ek} nr dh} nd n}r rpkd'

}ek} nr' fhs rhai behnbi hfK }eisi mh dh} iunr} u' ~ kdm | rybe }ek}

K 1 uP /~[ /|S$

Pshcgia ;3$ Gi}P 1 %;' 0'0.' [ 1 %0' 9'0.' S 1 %0'0'3.$Pshcgia ;2$ Gi}P 1 %;' 0';.' [ 1 %0' ;'0.' S 1 %;';';.$

Pshcgia ;=$ Gi}P 1 %3' 0';.' [ 1 %;';' ;.' S 1 %0'2'3.$Pshcgia ;9$ Gi}P 1 %;';' 3.' [ 1 %2';' 9.' S 1 %2'9'=.$Pshcgia ;6$ Hf }ei psivnhyr fhys pshcgiar' ghho k} }ei hdir zenbe mh rpkd S2$ Fhs ikbehf }eiri' nr }eisi ahsi }ekd hdi behnbi hf u' ~ kdm | hs nr }ei behnbi ydn{yi8

K ri} C hf vib}hsr nr bkggim k ckrnr fhsS2 nf ivis~ vib}hs bkd ci zsn}}id nd hdi kdm hdg~

hdi zk~ kr k gndiks bhacndk}nhd hf vib}hsr nd C$

Pshcgia ;:$ Gi}N


13/31

Bekp}is 3

Gndiks Rpkbir

K gndiks rpkbi nr k ri} G hf hclib}r bkggim phnd}r rybe }ek}' fhs kgg P kdm [ nd G kdm ivis~sikg dyacis k' }eisi ksi

k ydn{yi phnd} P / [ nd G bkggim }ei rya hfP kdm [ kdm k ydn{yi phnd} kP nd G bkggim }ei rbkgks pshmyb} hfk kdm P

rybe }ek} }ei fhgghzndj kunhar ehgm$

_Kmmn}nhdQ

;$. P / [ 1 [ / P fhs kgg phnd}r P kdm ['

3$. P / %[ / S. 1 %P / [. / S fhs kgg phnd}r P' [ kdm S' kdm

2$. }eisi nr k phnd} 0 rybe }ek} P / 0 1 P fhs ivis~ phnd} P$

_Rbkgks Pshmyb}Q Fhs kgg phnd}r P kdm [ kdm kgg sikg dyacisr k kdm c'

=$. k%P / [. 1 kP / k['

9$. %k / c.P 1 kP / cP'

6$. k%cP. 1 %kc.P kdm

:$. kP 1 0 nf kdm hdg~ nfk 1 0 hs P 1 0$

]eihsia ;>$ Fhs ivis~ phnd}[ zi ekvi }ek}[ / %;.[ 1 0$

_End}< ]h pshvi }enr }eihsia' kppg~ Kunha : }kondj k


14/31

Gndiks Rpkbir 5

]eihsia 3;$ Fhs ivis~ pkns hf phnd}r P kdm[ kdm kgg sikg dyacisr k' zi ekvi k%P[. 1kP

k[$

]eihsia 33$ Fhs ivis~ phnd}P zi ekvi ;P 1 P$

]h midi k pks}nbygks gndiks rpkbi' zi ayr} rpibnf~ zek} n}r phnd}r ksi' zek} }ei rya

hf }zh phnd}r nr kdm zek} }ei rbkgks pshmyb} hf k dyacis kdm k phnd} nr$ Beibo }ek} ikbe hf

}ei fhgghzndj iukapgir nr k gndiks rpkbi c~ visnf~ndj Kunhar ; }h :$

;$ ]ei rpkbi S3 bhdrnr}r hf kgg hsmisim pknsr hf sikg dyacisr$ NfP 1 %k' c. kdm [ 1%b' m.' }eid P / [ kdm sP ksi midim kr

%k' c. / %b' m.


15/31

Gndiks Rpkbir >

=$ Fhs ikbe phrn}nvi nd}ijis d' gi} Pd_0';Q midh}i }ei ri} hf kgg fB_0';Q rybe }ek} fnr kphg~dhankg fydb}nhd hf mijsii girr }ekd d$ I{ynvkgid}g~' f

Pd_0';Q nf }eisi ksi sikg

dyacisr k0' k;'$$ $ ' kd; rybe }ek}' fhs kgg u _0' ;Q'f%u. bhdmn}nhdr }h visnf~ }ek} A nr k

gndiks rpkbi$ Yrndj }ei fkb} }ek} A nr bhd}kndim nd k ri} G }ek} zi kgsikm~ odhz }h ci k

gndiks rpkbi' zi dm }ek} aybe girr nr dibirrks~$

]eihsia 32$ K rycri}A hf k gndiks rpkbi G nr k rycrpkbi hfG nf kdm hdg~ nf 0 A kdmA nr bghrim ydmis ch}e kmmn}nhd kdm rbkgks ayg}npgnbk}nhd$

]eihsia 3=$ Gi}A ci k rycri} hf k gndiks rpkbi G$ ]eid A nr k rycrpkbi hf G nf kdm

hdg~ nf 0 A kdm' fhs ivis~ P kdm[ nd A kdm ivis~ pkns k kdm c hf dyacisr' kP / c[ nrkgrh ndA$

K zikg}e hf diz kdm nd}isir}ndj gndiks rpkbir ksnri kr rycrpkbir hf fkangnks gndiks

rpkbir$

Pshcgia 39$ Fhs ikbe hf }ei fhgghzndj behnbir hfA' in}eis visnf~ }ek}A nr k rycrpkbi hf

}ei jnvid rpkbi hs rehz }ek} n} fkngr }h rk}nrf~ hdi hf }ei midndj pshpis}nir hf k rycrpkbi$

%n. A bhdrnr}r hf kgg %u'~. S3 rk}nrf~ndj 9u 2~ 1 0$%nn. A bhdrnr}r hf kgg %u'~. S3 rk}nrf~ndj 9u 2~ 1 =$

%nnn. A bhdrnr}r hf kgg %u'~'|. S2 fhs zenbe | 1 0$%nv. A bhdrnr}r hf kgg %u'~'|. S2 fhs zenbe | 0$%v. A bhdrnr}r hf kgg mnffisid}nkcgi fydb}nhdr ndB_0' ;Q$

%vn. A bhdrnr}r hf kgg fydb}nhdr fnd B_0' ;Q rybe }ek}2f / 9f 3f1 0$%vnn. A bhdrnr}r hf kgg fydb}nhdr fnd B_0';Q rybe }ek}2f / 9f 3f1 3$%vnnn. A bhdrnr}r hf kgg fydb}nhdr fnd B_0' ;Q rybe }ek}f% ;

=. 1 0$

%nu. A bhdrnr}r hf kgg fydb}nhdr fnd B_0' ;Q rybe }ek}f%u. nr sk}nhdkg fhs kgg u _0';Q$



16/31

Gndiks Rpkbir ;0

Vnizndj k gndiks rpkbi kgjicsknbkgg~ ekr hdi naaimnk}i kmvkd}kji$ Rhai phnd}r bkd ci

jidisk}im kr ryar hf pshmyb}r hf h}eis phnd}r' }eisic~ akondj }eia nd k ridri simydmkd}

nd }ei psiridbi hf }ehri h}eis phnd}r$ ]h akoi }enr nmik psibnri' zi rk~ }ek} k phnd} P nrk gndiks bhacndk}nhd hf k dn}i ri} C


17/31

Gndiks Rpkbir ;;

Pshcgia 29$ Ehz akd~ phnd}r bkd ci nd k gndiksg~ ndmipidmid} rycri} hfS38 Ehz bkd

zi nmid}nf~ gndiksg~ ndmipidmid} rycri}r hfS3 jihai}snbkgg~8

Pshcgia 26$ Mhir B_0';Q ekvi kd nddn}i gndiksg~ ndmipidmid} rycri}8 _ Dh}i< ^hy mhd}diim fhsaygkr fhs fydb}nhdr? lyr} jskper zngg mh$Q

]ei nmikr hf gndiks ndmipidmidbi kdm rpkddndj ri}r bhacndi }h jnvi yr hdi hf }ei

bid}skg bhdbip}r hf gndiks kgjicsk$ K rycri} C hf k gndiks rpkbi G nr k ckrnr fhs G nf ikbe

phnd} hfG bkd ci iupsirrim nd hdi kdm hdg~ hdi zk~ kr k gndiks bhacndk}nhd hf phnd}r

hfC$ Nd hys }isandhghj~' }enr nr }ei rkai kr rk~ndj }ek} C nr k ckrnr fhs G nf n} nr gndiksg~

ndmipidmid} kdm n} rpkdr G$

Pshcgia 2:$ Jnvi k jihai}snb mirbsnp}nhd hf }ei rycri}rC hf S3 kdmS2 }ek} fhsa ckrir$

Mh kgg ckrir fhs hdi hf }eiri rpkbir ekvi }ei rkai dyacis hf phnd}r8

Pshcgia 25$ Fndm k ckrnr fhs ikbe hf }ei fhgghzndj rpkbir$

%n.S: %nn.P>_0' ;Q %nnn.P_0';Q$

Giaak 2>$ Rypphri C nr k gndiksg~ ndmipidmid} rycri} hfG kdm P nr k phnd} hfG dh} nd

Rpkd%C.$ ]eid C{Pq nr kgrh gndiksg~ ndmipidmid}$]eihsia =0$ C nr k ckrnr fhsG nf kdm hdg~ nf n} nr k akunakg gndiksg~ ndmipidmid} rycri}

hf G' }ek} nr' n} nr gndiksg~ ndmipidmid} cy} nr dh} k pshpis rycri} hf kd~ h}eis gndiksg~

ndmipidmid} ri}$

Giaak =;$ Rypphri C rpkdr G kdm P nr k phnd} hfC rybe }ek} P Rpkd%C{Pq.$ ]eidC{Pq kgrh rpkdr G$]eihsia =3$ C nr k ckrnr fhsG nf kdm hdg~ nf n} nr k andnakg rpkddndj ri} fhsG' }ek} nr' n}

rpkdr G cy} dh pshpis rycri} hfC rpkdr G$

Zi odhz }ek}S;= ekr k ckrnr hf ;= phnd}r$ Mh ~hy }endo }ek} }eisi anje} kgrh ci k ckrnr

fhs S;= zn}e ;2 phnd}r' hs ak~ci ;: phnd}r8 Zi zngg dhz rii }ek} }eiri ondm hf r}skdji

}endjr bkd divis ekppid$

Sipgkbiaid} Giaak =2$ Rypphri }ek}C rpkdr G' }ek} [ G kdm }ek} P nr k phnd} hfCrybe }ek}' zeid [ nr zsn}}id kr k gndiks bhacndk}nhd hf phnd}r hfC' }ei bhifbnid} hf P nr

dh} |ish$ Gi}C ci }ei ri} hc}kndim fshaC c~ sipgkbndj P zn}e [$ ]eidC kgrh rpkdr G$

Fydmkaid}kg Giaak ==$ Bhdrnmisndj hdg~ dn}i rycri}r hfG' dh gndiksg~ ndmipidmid}

ri} ekr ahsi phnd}r }ekd k rpkddndj ri}$

Gndiks Ndmipidmidbi ]eihsia =9$ Rypphri }ek}G ekr k ckrnr zn}e k dn}i dyacis d hf

phnd}r$ ]eid }ei fhgghzndj ksi kgg }syi$

%n. Dh gndiksg~ ndmipidmid} ri} bhd}kndr ahsi }ekd d phnd}r$

%nn. Ivis~ gndiksg~ ndmipidmid} ri} zn}e d phnd}r nr k ckrnr$

%nnn. Ivis~ gndiksg~ ndmipidmid} ri} nr bhd}kndim nd k ckrnr$



18/31

Gndiks Rpkbir ;3

Rpkddndj ]eihsia =6$ Rypphri }ek}G ekr k ckrnr zn}e k dn}i dyacis d hf phnd}r$ ]eid

}ei fhgghzndj ksi kgg }syi$

%n. Dh rpkddndj ri} bhd}kndr fizis }ekd d phnd}r$

%nn. Ivis~ rpkddndj ri} zn}e d phnd}r nr k ckrnr$

%nnn. Ivis~ rpkddndj ri} bhd}kndr k ckrnr$

Ckrnr ]eihsia =:$ Nf G ekr k ckrnr zn}e k dn}i dyacis d hf phnd}r' }eid ivis~ ckrnr fhs

G ekr d phnd}r$

Pshcgia =5$ Rehz }ek} dh dn}i ri} rpkdrB_0' ;Q$

Nf k gndiks rpkbi G ekr k dn}i ckrnr' }eid }ei mnaidrnhd hfG nr }ei dyacis d hf phnd}r

nd ivis~ ckrnr fhs G$ ]ei fhgghzndj }eihsia bkd ci yrim }h dm k ckrnr fhs k rpkbi G nf zi

ksi jnvid k dn}i rpkddndj ri} fhs G$ Yri }ei Sipgkbiaid} Giaak }h pshvi n}$

]eihsia =>$ Gi}C ci k rpkddndj ri} fhsG kdm gi}P ci nd C$ Krryai }ek}[ nr hc}kndim

fsha P in}eis

%n. c~ kmmndj }h P k gndiks bhacndk}nhd hf }ei siakndndj igiaid}r hfC hs

%nn. c~ ayg}npg~ndj P c~ k dhd|ish dyacis$

Gi}C ci hc}kndim fsha C c~ sipgkbndj P zn}e [$ ]eid C kgrh rpkdr G$

]enr }eihsia zngg jnvi yr kd kgjhsn}ea }h bkd dm k ckrnr fhs }ei rycrpkbi hf Sd

rpkddim c~ kd~ jnvid dn}i ri} hf phnd}r$ Kd a!c~!d ak}snu A ekr a shzr' ikbe k phnd}

hfSd$ ]ei ak}snu A nr rknm }h ci shz simybim nf }ei sr} dhd|ish id}s~ nd ikbe shz nr k ;'

kdm }ei id}snir kchvi kdm cighz }ek} ; ksi kgg 0$

Giaak 90$ ]ei dhd|ish shzr hf k simybim ak}snu ksi gndiksg~ ndmipidmid}$

]eihsia 9;$ Gi} R ci k ri} hf a phnd}r hfSd kdm gi}G 1 Rpkd%R.$ Gi} A ci }ei a!c~!dak}snu hc}kndim c~ r}kbondj }ei phnd}r hf R nd k bhgyad kdm gi} A ci }ei simybim ak}snuhc}kndim fsha A c~ sipik}im kppgnbk}nhdr hf ]eihsia => %bkggimJkyrrnkd ignandk}nhd.$

]eid }ei dhd|ish shzr hf A fhsa k ckrnr fhsG$

Pshcgia 93$ Fndm k ckrnr fhs }ei rycrpkbi hf S9 rpkddim c~ }ei phnd}r %0'9' ;'3'2.'%;'0' ;'6' =.' %0'0' ;'3' 2. kdm%;' 9'3'5':.$

Pshcgia 92$ Fndm k ckrnr fhs }ei rycrpkbi hf S2 rpkddim c~ }ei phnd}r %5'2' 9.' %9';'>.'%;'2'

;. kdm%3'

;'9.$

Pshcgia 9=$ Fndm k ckrnr fhs }ei rycrpkbi hf S= rpkddim c~ }ei phnd}r %6'3';'2.'%9'3'0' ;.' %;'0' ;'3. kdm%2'3'3' 2.$

Ehz bkd zi mirbsnci }ei rhgy}nhd ri} R hf }ei rndjgi gndiks i{yk}nhd

=u ~ / 2| 1 98



19/31

Gndiks Rpkbir ;2

Zi bkd kdrzis }enr iukb}g~ kr zi mnm }ei r~r}iar hf gndiks i{yk}nhdr nd Bekp}is ; c~

dh}nbndj }ek} kd~ vkgyir hf~ kdm | zngg mi}isandi k ydn{yi vkgyi hfu rybe }ek} %u'~'|. nr k

rhgy}nhd$

R1{%u'~'|. wu 1 9/~2|=

q1{% 9/~2|

='~'|. w~'| Sq

1{% 9=

'0'0. / %~=

'~'0. / % 2|=

' 0'|. w~'| Sq1{% 9= '0'0. /~% ;= ';' 0. /|% 2= '0';. w~'| Sq$

Nf zi }koi [


20/31

Gndiks Rpkbir ;=

Pshcgia 9:$

u / ~ | 6y 1 =3u / 9~ ;=y 1 362u / :~ | 3=y 1 26



21/31

Bekp}is 2

Nddis Pshmyb} Rpkbir

]ei nddis pshmyb} %hs mh} pshmyb}. hf }zh phnd}rP 1 %k' c. kdm [ 1 %b' m. ndS3 nr midimc~

P [ 1 %k' c. %b' m.


22/31

Nddis Pshmyb} Rpkbir ;6

]enr hcrisvk}nhd ryjjir}r }ek} zi ghho k} h}eis gndiks rpkbir zenbe ekvi k rnangks nddis

pshmyb}$ ]h mh rh' zi r}ks} c~ r}k}ndj }ei irrid}nkg pshpis}nir hf }ei nddis pshmyb}$ C~ kd

nddis pshmyb} rpkbi zi aikd k gndiks rpkbi G nd zenbe }eisi nr k zk~ }h ayg}npg~ kd~phnd} P kdm kd~ phnd} [ }h hc}knd k dyacis P [ %bkggim }ei nddis pshmyb} hfP kdm [. rh}ek}' fhs kgg P' [' S G kdm kgg k S' }ei fhgghzndj kmmn}nhdkg kunhar ehgm$

5$. P P 0' kdm P P 1 0 nf kdm hdg~ nfP 1 0$>$. P [ 1 [ P

;0$. k%P [. 1 %kP. [;;$. P %[ / S. 1 P [ / P S

]ei dhsa hf k phnd} P nr midim }h ci

P


23/31

Nddis Pshmyb} Rpkbir ;:

]eihsia 6;$ %]snkdjgi Ndi{ykgn}~.

P / [ P / [

P

[P / [

0

s

s

sZi ksi kbbyr}haim }h yrndj }ei r}kdmksm ckrir

{n'lq 1 {%;'0.' %0';.q kdm {n'l' oq 1 {%;' 0'0.' %0';'0.' %0'0';.q

fhs C3 kdm C2$ Zi zngg dhz rii ehz rnangks r}kdmksm ckrir bkd ci bhdr}syb}im fhs kd

kscn}sks~ nddis pshmyb} rpkbi G' kdm nd }ei pshbirr hf mhndj rh rii zek} n} nr }ek} nr rpibnkg

kchy} }eiri ckrir$

Nd C3 }ei i{yk}nhd P [ 1 P[bhr%. }iggr yr }ek} P kdm [ ksi hs}ehjhdkg %pis!pidmnbygks. nf kdm hdg~ nfP [ 1 0$ Kbbhsmndjg~' fhs P' [ G' zi midi P kdm [ }h cihs}ehjhdkg nfP [ 1 0$ K rycri} C hfG nr hs}ehjhdkg nf ivis~ pkns hf mnr}ndb} phnd}r nd Cksi hs}ehjhdkg$

Giaak 63$ Ivis~ hs}ehjhdkg ri}C dh} bhd}kndndj 0 nr gndiksg~ ndmipidmid}$

P~}ekjhsikd Giaak 62$ Nf P nr hs}ehjhdkg }h [' }eid

P3 / [3 1 P / [3$0

[P

P / [ss s

K ydn} vib}hs nr k phnd} Y zenbe ekr dhsa ;$ Phnd}r P kdm [ ksi rknm }h ci nd }ei rkai

mnsib}nhd nf' fhs rhai k 4 0' zi ekvi P 1 k[$

Giaak 6=$ Fhs ivis~ phnd} P h}eis }ekd 0 ' }ei phnd}Y


24/31

Nddis Pshmyb} Rpkbir ;5

Dh}nbi }ek} }eiri ksi iukb}g~ }ei ydn{yi bhifbnid}r diimim }h iupsirr K kdm C kr gndiks

bhacndk}nhdr hf }enr ckrnr.' P3 1 %3'2.q %kd kscn}sks~ ckrnr. kdm

%nn. C3 1

{P; 1 %

;3

'

23

.' P3 1 %

23

'

;3

.

q%kd hs}ehdhsakg ckrnr.$

Fhs ikbe hf }eiri }zh ckrir'

%k. bhapy}i }ei nddis pshmyb}r k;K' k3K' k;C' k3C hfK kdmC zn}e sirpib} }h }enr ckrnr$

%c. Nr K 1 k;KP; / k3KP38 Nr C 1 k;CP; / k3CP38

%b. Nr K C 1 k;Kk;C / k3Kk3C8

%m. Nr wwKww 1

k3;K / k33K8

]eiri iupisnaid}r gikm yr }h }ei fhgghzndj }eihsiar$

]eihsia 66$ Gi}C1 {P;' P3'$$ $ ' Pdq ci kd hs}ehdhsakg ckrnr fhsG kdm gi} P G ci kphnd} hf

G$ Fhs n 1 ;'3' $ $ $ ' d' gi} kn 1 P Pn$ ]eid k;' k3' $ $ $ ksi }ei bhifbnid}r fhsP zn}esirpib} }h C' }ek} nr'

P 1 k;P; / k3P3 / / kdPd$]eihsia 6:$ Gi}C 1 {P;' P3'$$ $ ' Pdq ci kd hs}ehdhsakg ckrnr fhsG ' gi}P 1 k;P; /k3P3 / / kdPd kdm[ 1 c;P; / c3P3 / / cdPd$ ]eid

%n. P [ 1 k;c; / k3c3 / / kdcd'

%nn. P 1

k3; / k33 / / k3d$

]eiri dnbi pshpis}nir hf hs}ehdhsakg ckrir gikm yr }h kro }zh {yir}nhdr kchy} nddis

pshmyb} rpkbir$

Mhir ivis~ dn}i!mnaidrnhdkg nddis pshmyb} rpkbi ekvi kd hs}ehdhsakg ckrnr8 Nf rh' nr }eisi k pskb}nbkg ai}ehm }h dm hdi8

]h kdrzis }eiri {yir}nhdr' gi} A ci k rycrpkbi hf kd nddis pshmyb} rpkbi G kdm gi}

C1 {P;' P3'$$ $ ' Poq ci kd hs}ehdhsakg ckrnr fhs A$ Fhs ikbe phnd} P G' zi midi }eihs}ehjhdkg pshlib}nhd hf P nd A }h ci }ei phnd}

PA


25/31

Nddis Pshmyb} Rpkbir ;>

NfP nr k phnd} nd A' }eid ]eihsia 66 }iggr yr }ek} PA nr P n}rigf$ Kr kdh}eis iukapgi'

bhdrnmis }ei hs}ehdhsakg ckrnr C 1

{n 1 %;'0'0.'l 1 %0';' 0.

qfhs k rycrpkbi A hfS2$ Nf

P 1 %k' c' b. S2' }eidPA 1 %P n' P l' P o. 1 %k' c' 0.

nr }ei bghrir} phnd} hfA }h P$

Giaak 65$ Krryai }ek}C 1 {P;' P3'$$ $ ' Poq nr kd hs}ehdhsakg ckrnr fhs k rycrpkbi AhfG$ Gi}P ci kd~ phnd} hfG$ ]eid

%n. [A fhs hs}ehdhsakg ckrir$

Giaak 6>$ Krryai }ek}C1 {P;' P3'$$ $ ' Poq nr kd hs}ehdhsakg ri} }ek} mhir dh} rpkdG$Gi}

A


26/31

Nddis Pshmyb} Rpkbir 30

_End}< Fndm kd hs}ehdhsakg ckrnr fhs }ei rycrpkbi A


27/31

Bekp}is =

Gndiks ]skdrfhsak}nhdr

Rypphri }ek} ~hy jh }h cy~ k diz bks' kdm }ek} ~hy dm }zh bksr fhs rkgi hf }ei rkai akoi'ahmig' bhghs kdm psnbi zn}e }ei rkai hp}nhdr$ ]ei ksi dh} gn}iskgg~ }ei rkai bks' cy} kr fks

kr ~hy }ei bhdryais ksi bhdbisdim' }ei~ ksi ndmnr}ndjynrekcgi$ ]ei bhapkd~ }ek} ~hy

ksi cy~ndj fsha ak~ pshmybi }ehyrkdmr hf ndmnvnmykg bksr ivis~ ~iks' cy} hdg~ k ekdmfyg

hf irrid}nkgg~ mnffisid} ondmr$ hys }kro nr }h behhri kahdj }ei ekdmfyg hf mnffisid} ondmr'

dh} kahdj }ei }ehyrkdmr hf mnffisid} ndmnvnmykg bksr$

]ei rkai ekppidr zn}e gndiks rpkbir kdm nddis pshmyb} rpkbir$ Nd hsmis }h ydmisr}kdm

}eiri r}syb}ysir' hys }kro nr }h sibhjdn|i }ei irrid}nkgg~ mnffisid} ondmr$

Fhs iukapgi' }ei nddis pshmyb} rpkbi S3 ekr }ei r}kdmksm hs}ehdhsakg ckrnr {n 1%;'0.'l 1 %0';.q$ Fhs kscn}sks~ phnd}r

P 1 kn / cl kdm [ 1 bn / ml

nd S3' kmmn}nhd' rbkgks ayg}npgnbk}nhd kdm nddis pshmyb} ksi jnvid c~

P / [ 1 %k / b.n / %c / m.l' sP 1 skn / scl' P [ 1 kb / cm$

Dhz bhdrnmis kd~ h}eis }zh mnaidrnhdkg nddis pshmyb} rpkbi G$ C~ }ei Jska!Rbeanm}]eihsia'G ekr kd hs}ehdhsakg ckrnr {n'lq$ Fhs kscn}sks~ phnd}r

P 1 kn / cl kdm [ 1 bn / ml

nd G' kmmn}nhd' rbkgks ayg}npgnbk}nhd kdm nddis pshmyb} ksi jnvid c~

P / [ 1 %k / b.n / %c / m.l' sP 1 skn / scl' P [ 1 kb / cm$

Zek} zi rii nr }ek}' kg}ehyje }ei phnd}r hfS3

kdm G ak~ ci vis~ mnffisid} ri}r hf hclib}r'kr nddis pshmyb} rpkbir S3 kdm G ksi kcrhgy}ig~ ndmnr}ndjynrekcgi$ ]ei~ ksi gnoi }zh dizsim ]h~h}k Bhshggkr zn}e r}kdmksm }skdranrrnhd' knsckjr' dh KB' kdm k skmnh kdm }kpi mibo

}ek} ksi cindj rhgm k} }ei rkai psnbi$ Nd ak}eiak}nbkg }isandhghj~ zenbe zi zngg midi

cighz' zi rk~ }ek} S3 kdm G ksi nrhahspenb kr nddis pshmyb} rpkbir$Nd }enr ridri zi }endo hfS3 kr cindj }ei hdg~ }zh mnaidrnhdkg nddis pshmyb} rpkbi$ K

}~pnbkg kppgnbk}nhd hf }enr bhdbip} zkr jnvid k} }ei idm hf }ei gkr} bekp}is$ Krryai zi zkd}

}h pshvi rhai }eihsiar kchy} }zh %gndiksg~ ndmipidmid}. phnd}r P kdm [ hf kd kscn}sks~

3;


28/31

Gndiks ]skdrfhsak}nhdr 33

nddis pshmyb} rpkbi$ C~ yrndj }ei fkb} }ek} }ei rycrpkbi }ei~ rpkd nr nrhahspenb }h S3' zi

bhygm rnapg~ pkskpeskri pshhfr }ek} }ei }eihsiar zisi }syi nd S3$

]ei naphs}kd} }hhg fhs mirbsncndj }ei ndmnr}ndjynrekcngn}~ hfS3 kdm G kchvi nr }eibhssirphdmidbi

kn / cl kn / cl

ci}ziid }eins phnd}r$ K }skdrfhsak}nhd fsha k gndiks rpkbi G }h k gndiks rpkbi G nr kfydb}nhd ] }ek} pknsr ikbe phnd} P hfG zn}e k phnd} ]%P. hfG$ K }skdrfhsak}nhd ] nr kgndiks }skdrfhsak}nhd nf

%n. P / [ 1 S napgnir }ek} ]%P. / ]%[. 1 ]%S. kdm

%nn. kP 1 R napgnir }ek} k]%P. 1 ]%R.$

ss

ss

S

[

P

Rk

/

G

ss

ss

]%S.

]%[.

]%P.

]%R.k

/

G

Giaak :=$ Fhs k }skdrfhsak}nhd ] < G G fsha G }h G ' }ei fhgghzndj ksi i{ynvkgid}$

%n. ] nr k gndiks }skdrfhsak}nhd$

%nn. Fhs kgg P' [' k zi ekvi ]%P / [. 1 ]%P. / ]%[. kdm ]%kP. 1 k]%P.$

%nnn. Fhs kgg P' [' k' c zi ekvi ]%kP / c[. 1 k]%P. / c]%[.$

_Pshvi %n. nf kdm hdg~ nf %nn. nf kdm hdg~ nf %nnn.$Q

Gi} ] < G G ci k gndiks }skdrfhsak}nhd$ ]ei dygg rpkbi hf] nr }ei rycri} D%].


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Giaak :6$ Nf ] < G G nr k gndiks }skdrfhsak}nhd' }eidD%]. nr k rycrpkbi hf G kdmS%]. nr k rycrpkbi hfG$

Sibkgg }ek} ] < G G nr hdi!}h!hdi nf fhs kgg P' [ G' zi ekvi P 1 [ napgnir ]%P. 1]%[.' kdm ] nr hd}h G nf fhs kgg P G }eisi nr k P G rybe }ek} ]%P. 1 P$Giaak ::$ K gndiks }skdrfhsak}nhd ] < G G nr hdi!}h!hdi nf kdm hdg~ nfD%]. 1 {0q'kdm ] nr hd}h nf kdm hdg~ nfS%]. 1 G$

K gndiks }skdrfhsak}nhd ] < G G nr bkggim k gndiks rpkbi nrhahspenra nf n} nr hdi!}h!hdi kdm hd}h$ K gndiks rpkbi nrhahspenra ] < G G nr kd nddis pshmyb} rpkbi nrh!ahspenra nf n} ekr }ei kmmn}nhdkg pshpis}~ }ek}

%nnn. P [ 1 ]%P. ]%[. fhs kgg P' [ G$Zi rk~ }ek} G kdm G ksi nrhahspenb nf }eisi nr kd nrhahspenra ]


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Gndiks ]skdrfhsak}nhdr 3=

N} }ysdr hy} }h ci bhdvidnid} }h yri }ei a ; bhgyad ak}snu

b;b3$$$

ba

}h midh}i }ei phnd} P 1 b;P; / b3P3 / / baPa hfG kdm' rnangksg~' }h yri k d; bhgyadak}snu }h midh}i k phnd} hfG$ Nd }enr dh}k}nhd }ei gndiks }skdrfhsak}nhd ] nr jnvid rnapg~c~ ak}snu ayg}npgnbk}nhd$

Ak}snu ]eihsia 50$ Gi} ] < G G ci k gndiks }skdrfhsak}nhd fsha kd amnaidrnhdkggndiks rpkbi G }h kd dmnaidrnhdkg gndiks rpkbiG zn}e ckrir C kdmC kr kchvi$ ]eid ]nr jnvid c~ ak}snu ayg}npgnbk}nhd kr

]

b;b3$$$

ba

1

b;k;; / b3k;3 / / bak;ab;k3; / b3k33 /

/ bak3a

$$$

b;kd; / b3kd3 / / bakda

1

k;; k;3 $ $ $ k;ak3; k33 $ $ $ k3a

$$$

kd; kd3 $ $ $ kda

b;b3$$$

ba

hs' r}k}im ahsi bhdbnrig~' ]%P. 1 A]P$ Bhdvisrig~' nf A nr kd~ d a ak}snu' }eid }eiak}snu i{yk}nhd ]%P.


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Zek} nr }ei ak}snu fhs }ei bhaphrn}nhd Y ] < G G8

Bhaphrn}nhd ]eihsia 53$ AY] 1 AYA] ' }ek} nr' ayg}npgnbk}nhd hf ak}snbir bhssi!rphdmr iukb}g~ }h bhaphrn}nhd hf }ei bhssirphdmndj gndiks }skdrfhsak}nhdr$

]enr }eihsia kdrzisr }zh {yir}nhdr$ N} iupgkndr ze~ zi behhri }enr bhapgnbk}im zk~

hf ayg}npg~ndj ak}snbir$ N} kgrh iupgkndr ze~ }enr bhapgnbk}im ayg}npgnbk}nhd rehygm }ysd

hy} }h ci krrhbnk}nvi$ N} nr cibkyri bhaphrn}nhd hf fydb}nhdr nr kgzk~r krrhbnk}nvi

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