diferansiyel denklemler - arzu erdem ders notları
TRANSCRIPT
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Amj mjhiradsjyi` midb`if`ir det`arj
Arzu Irmif
l >33:/>313 Kuz medifj fugidmjs`jb det`ar1
1
Baydab`ar?
1. Bay`iy ^ilterys- \urviy eh App`jlan`i Ada`ysjs>. [j``jaf Neyli adm ^jlgarm MjSrjfa - I`ifidtary mjffiiridtja` iquatjeds adm neudmary va`ui pren`ifs9 Akustes >33:
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Ledtidts
@jst eh Hjkuris v
@jst eh ]an`is vjj
Lgaptir 3. Kjrjs 11. Fatifatjbsi` femi` ir 1>. Ikrj aj`isjdjd mjhiradsjyi` midb`if`irj 9
Lgaptir 1. Mjhiradsjyi` midb`if ir vi ed`ard lezuf`irj ;. Njrjdlj firtinimid AMM 1y+l>
herfudmabj AMM 1:
13. @jdiir AMM >111. Nirdeu``j midb`ifj >=1>. ^jllatj Midb`ifj >8
171=. Jdtikrasyed L arpad >. @jdiir gefeoid AMM jljd tifi` tierif`ir 9;>=. \anjt batsay` gefeoid `jdiir AMM 83>9. Gefeoid e`fayad AMM 8>>8. Laulgy-Iu`ir midb ifj ;3
Lgaptir 8. \anjt batsay`Jbjdlj Firtinimid Mjhiradsjyi` Midb`if`irjd Yyku`afa`ar ;, , 6 sanjt 6 3
mjhiradsjyi` midbi`ifj j`i jhami imj`fjstjr.
1.>. Ibe`eoj? ^amyeabtjh atb urud`ir. ^amyeabtjvjtj, yubsib atef akr`b` (uradyuf fjdira`j kjnj)i`ifidt`irjd br`fas sedulu i`mi imj`jr. Zapay ramyeabtjvjtj bjfya, tp vi dub`iir idiroj kjnj a`ad`arma lebbu``ads`mr. Habat dub`iir idirojdjd idmustrjsi` bu``adf sed mirili mjbbat kiribtjrjyer. L udbu ramyeabtjh atbfammi`ir pepu`asyed alsdmad tig`jbi e`usturfabtamr. ^amyeabtjh nezu`fadd fatifatjbsi` jhamisj, nezu`faj`i eradt` e`arab jhami imj`jr vi mjhiradsjyi` midb ifj
mY
mt 6 bY, b6 sanjt 43
nuramaYfammidjd biyfit addmabj rayeabtjjb eradd kestirfibtimjr. Nu midb`ifjd lezufu
Y(t) 6 Y3ixp (b (t t3))sib`jdmi jhami imj`jr.
1.. bura`-`ar e`arab ma am`admr`fabtamr. Bip`ir kizikid`irjd das` garibit ittjkj serusudud livand virfjstjr adlabdimid serusudud livan maga sedra Ka`j`ie Ka`j`ij vi Diwted tarahdmad virj`fjstjr. Diwtedud yirlibjfjbura`da keri, kudis vi kizikid`ir arasdma libjf buvvitj
H 6
r 3. KJ^J\
e`arab jhami imj`fjstjr. Nurama K kidi` yirlibjfj sanjtj f vi F srasy`a kizikid vi kudisjd but`i`irjmjr.Ney`ili kizikid`irjd libjf`irj a`tdma kudisjd garibitjdj jgfa` imirsib
fm>x
mt> 6
r. bura` e`arab jhami imj`fjstjr. Nu pren`ifjd jdtikra` a`dfas j`i i`mi imj`id pren`if mi Bip`irpren`ifj e`arab am`admr`fstr.
1.=. Mudya yabddma sirnist musfi garibitj. Mudya uzirjdmi, yirlibjfjdjd sanjt e`mukudu varsa-yarab sirnist musfi garibitjdj i`i a`a`f. f 6 sanjt but`idjd akr`k g yubsib`jb t zafad k :71 lf/sd>surtudfi jvfisjdj kestirfib uziri, surtudfi buvvitj
H 6 fke`arab kestirj`jr vi Diwted midb`ifj
m>g
mt> 6 k
e`arab yaz`r. Nudud lezufu
g6 k
> t> + l1t + l>
e`arab nu`udur. Nurama l1,, l> biyfi sanjt`irmjr.
1.9. \ekutfa (sdfa) jljd Diwted femi`j. \ekutfa (sdfa) e`ay, gayatfzd gir a`addma bu`-`ad`ad njr js`ifmjr. Gavay sekutfa js`if`irj, hrd stfa vn. Diwtedud sekutfa buraf e`arab am`admr`adfemi`
m
mt 6b (] t) , b6 sanjt 43
j`i jhami imj`jr. Nurama ](t) sekutu`ad disdidjd gjlnjr itbj kestirfimjkj slab`btr(meyuf debtas). (t) biyfit addmabj slab`b vit zafad kestirfibtimjr.Njr njdadd b`jfa tarahdmad st`mkd varsaya`f. G(t) j`i slab`kd artf eraddA (t) j`i slab`kd mikjsjferadd kestirjrsib, yubarmabj midb`ifj
mmt
6b (] t) + G(t) + A (t)e`arab muzid`iyinj`jrjz.
1.8. Fibadjb tjtrisjf vi \arbal`ar. Zaprab`ard ruzkarma gsrts, suyud ma`ka`adfas nu femi``irjljd naz fizjbsi` e`ay`armr. Id tifi` sa`df garibitj, njr yiri sanjt as` e`ad njr nenjdmid akr njr ljsjfjd j`irjkirj garibitjmjr. Geebi yasasda keri nas`adkl addmad bars taraha e`ad garibitj
H1 6 by, b6 sanjte`arab yaz`r. Nuramay yirmikjstjrfiyj kestirjr. \urtudfi buvvitjdj mi
H> 6 ` mymt
e`arab yazanj`jrjz. h(t) j`i tep`af ms buvvitj (ruzkar) kestirjrsib, Diwtedud >. bura`da keriH6H1+ H>+ h(t)
vi femi`
fm>y
mt> + `
my
mt + by 6 h(t)
1.;. Asha`t`ard lebfisj. Asha`t`ard pirjyemjb njljfmi lebfisj asakmabj mjhiradsjyi` midb`if j`i jhamiimj`jr
m=u
mx= 6h
nurama u ye`ud muz pezjsyedudmad lebfisj sedulu i`mi imj`id yir mikjstjrfisj vi h firbizbal buvvitjdjdyekud`ukumur.
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1. FA]IFA]JB\I@ FEMI@@I^ vxx+ (a+ n) vt+ anv 6 3,l> 6
1
L@, a6
K
L, n6
^
@
L6 bapasjti, @6 bidmjdj jdmub`ifi, ^6 risjs tad s, K6 balab.
1.13. Faxwi`` midb`ifj. I`ibtrefakditjb a`ad > nj`isidi sagjptjr. Ii`ibtrjb a`ad kestirid vibter viGfakditjb a`ad kestirid vibter. Nuda keri femi` njr ljht midb`ifmid e`usur?
I
t 6 l ( G) =o, I6 =
G
t 6 l ( I) , G6 3
o vi i`ibtrjb abf vi yub`idfi yekud`ub`ar, l < 1313 sb gzd kestirfibtimjr. Nu midb`if sjstifjdjfizjbti kidi`mi
1
l
I
t 6 lur`G =
l o, mjvI6 =
1
l
G
t 6 lur`I, mjvG6 3
sib`jdmi kestirj`jr.
1.11. Davjir-\tebis Midb`ifj. Zapsbad fammi`irjd abf nu tur midb`if`ir j`i jhami imj`jr
vt+ (v.) v+1p6 vv
p nasdl, yekud`ub, v absbadd gzd kestirfibtimjr.
1.1>. \u`afa(jrrjkatjed) sjstif`irjdjd femi``ifisj. \u`afa sjstif`irjdjd fatifatjbsi` femi`j
L() t6 (B() x)x+ (B() (z 1))z \()sib`jdmimjr. Nurama teprabtabj dif nasdl, L() teprakd su bapasjtisj, B() gjmre jb j`itbid`jkjd meyuferad, \() baydab hedbsjyedu, t zafad, x yatay ibsid, z mjbiy ibsidj kestirfibtimjr.
1.1
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= 3. KJ^J\
1.19. Hjdadsta fatifatjbsi` femi`. Fatifatjbsi` fidadstabj tifi` la`sfa`ar ma`ka`adad steb fiyat-`armr vi asakmabj midb`if j`i jhami imj`jr
ut+1
>A>x>uxx+ Nxux Lu 6 3, A , N , L 6 sanjt.
1.18. Nuyuyid tufer femi`j. Nu femi` `jdiir e`fayad mjhiradsjyi` midb`if j`i jhami imj`jr
ut 6 h(u) (ulx)xlt 6 k (l, p)
pt 6 g (u, l) Bpu, l,psrasy`a gasta`b` gulridjd bedsadtrasyedu, gulri ms svs, pretiaz (idzjf`irjd parla`adfasd sak`ayadidzjf krunu).
1.1;. Ma`ka midb`ifj. ]i` vn. kjnj fammi`ir uzirjdmibj ma`ka`adfa garibitj
utt b> (x) u6 H(x, t)j`i jhami imj`jr.
1.17. Mjhiradsjyi` Midb`if`irjd ]arjgj. Mjhiradsjyi` midb`if`irj tadfamad vi ed`ard lezuf`irj gabbdnj`kj sagjnj e`famad edli njraz tarjgjdmid nagsimi`jf. Mjhiradsjyi` midb`if`ir bedusu j`b e`arab Jsaal Diwted(18=
1;>;) vi Ketthrjim [j`gi`f @ijndjz(18=81;18) tarahdma 1; yuzy`ma la`s`faya nas`adfstr. Diwted, Jdkj`tirimi nuyuyup, ]rjdjty be`ioj- Lafnrjmki mi ikjtjf a`fstr vi 188: ma @ulasjad (tud prehisset`ub gjzfit`irjdjdnak` e`muku ) prehiseru e`fustur. Mivrjf yaratad la`sfa`ar gisap`afama vi fibadjb prn`if`irmi, 1889y`dma kirlib`isfjstjr. Fatifatjb lafjsa tarahdmad banu` kerfisjdi rakfid, Diwted i`istjrj`ir gabbdma lebgassas e`mukudmad la`sfa`ard 187; yi bamar nasfafstr. 187; y`dma leb ud`u bjtan Sgj`esepgjai Datura`jsSrjdljpja Fatgifatjla na`fstr. Diwted mjhiradsjyi` midb`if`ir j`i la`sfa`ard yuruturbid,gisap`afamavi fibadjbtibj tifi` ki`jsfi`irj Iu`ir tarahdmad sak`adm. Diwted, 1. firtinimid mjhiradsjyi` midb`if`irjmy/mx6 h(x), my/mx6 h(y), vimy/mx6 h(x, y) herfudma sdadmrm. \edrasdma jsi h(x, y)x vi y djd pe`j-defu e`mukudma, sirj`irj bu``adarab lezuf yedtifj ki`jstjrmj. Diwtedud abtjh la`sfa`ar 18:3 `ard nas`ardmased nu`mu vi maga edli i`mi itfjs e`muku sedul`ard yayd`adfas vi muzid`idfisj la`sfa`ard kirlib`istjrmj.
18:8 ma Nrjtjsg Fjdt ti tibrar prehiser e`mu. 1;39 mi seva`yi e`arab j`ad imj`mj vi [istfjdstir Anniy mikefu`mu.@ijndjz, >3 yasdma @ijpzjk mi A`tmerh udjvirsjtisjdmi, fi`ezefi a`addma mebtera la`sfasd tafaf`am Gayatneyudla, njrbal a`admabj la`sfa`ar j`i fisku` e`mu. Edli`jb`j a`ad`ar arasdma fatifatjb varmr ludbu >3`j yas`ardma nu a`adma la`sfa`ar kirlib`istjrfjstjr. Diwted mad njraz sedra e`fasda rakfid mjhiradsjyi`midb`if`ir j`i j`kj`j tifi` sedul`ara u`asfstr habat 187=ti Diwted mad edli nas`fstr. Fatifatjbsi` detasy-ed`ar bu``adfa bedusudma leb jyjmjr vi turiv jljd my/mx vi jdtikra` sifne`udud bu``adf`ar eda ajttjr. 18:1y`dma mikjsbid`iri ayrfa yedtifjdj virfjstjr vi gefeoid midb`if`irj, mikjsbid`irjdi ayr`anj`id mjhiradsjyi`midb`if`iri jdmjrkifjstjr. 1. firtinimid mjhiradsjyi` midb`if`irjd lezufudu 18:= y`dma virfjstjr. Gayatdnjr i`lj kjnj yasafstr vi A`fa bra`jyit aj`i`irjdi tavsjyi`irmi nu`udfustur. Nu kerivj sayisjdmi, leb saymakizj muzid`ifjs vi yazsfa`ard mjkir fatifatjblj`iri tasyanj fjstjr. Ezi``jb`i mi Nirdeu``j barmis`iri. Nujsnjr jkj sayisjdmi pib leb pren`if 1; yuzy`ma lezu`inj`fjstjr.Oaben(189=1;39) vi Oegadd(188;1;=7) Nirdeu``j barmis`ir mjhiradsjyi` midb`if`irmi yedtif`ir ki`jstjrjp nud-
`ard uyku`afa a`ad`ard kidjs`itfjs`irmjr. Oaben, 187; mi Nasi` mi prehiser e`fustur. Oegadd, barmisjdjd1;39 y`dma e`fisjdmid sedra ayd kerivi kitjrj`fjstjr. Gir jbj amafma bavkal bsbadl vi ezi``jb`i mi bidmjtartsfa`ardma sbla barstr`r`arm. Zjdi mi gir jbjsj mi fatifatjbti leb edif`j ki`jsfi`iri jfza atfs`armr.Gisap`afa`ard ma yarmf j`i fibadjbti mjhiradsjyi` midb if e`arab jhami imj`id pren`if`ir jljd lezuf yedtif`irjki`jstjrfjs`irmjr. 18:3 ma Oalen y 6 Ray a mjhiradsjyi` midb`ifjdj lezfustur vi faba`isjdmi j`bmiha jdtikra` tirjfjdi yir virfjstjr. 18:= ti Oegadd my/mx 6 y/ax. mjhiradsjyi` midb`ifjdjd lezufudu i`miitfjstjr. Ki`jstjrmjb`irj id edif`j pren`if`irmid njrjsj mi nralgjstelgredi pren`ifjmjr.Oegaddd ek`u e`ad Madji` Nirdeu``j(1;331;7>), maga giduz yidj buru`fus e`ad \t. Sitirsnurk abamifjsidkel ittj adlab 1;
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>. IK^J AJ@I\JDJD MJHI^AD\JZI@ MIDB@IF@I^J 9
17 yuzy`d id edif`j fatifatjblj`irjdmid njrjsj mi Oegadd Nirdeu``jdjd ekridljsj e`ad vi Nasi` yabd`ardmayasayad @iedgarm Iu`ir(1;3;1;7; mi arbamas Madji` Nirdeu``j yj tabjp imirib \t. Sitirsnurk akjrfjstjr. 1;>;1;=1 vi 1;881;7< y``ar arasdma \t. Sitirsnurk abamifjsjdmi 1;=11;88 mi Nir`jd abamifjsjdmila`sfstr. Iu`ir nutud zafadd id virf`j fatifatjblj`irjdmi njrjmjr vi tuf la`sf`ar tep`afma ;3 mirkjyj
kilir. J`kj a`ad fatifatjkjd vi uyku`afadd tuf a`ad`ard bapsar. Zasafdd sed 1; y`d ber e`arabkiljrfisjdi rakfid, e`idi bamar la`sf`ard mivaf ittjrfjstjr. Ezi``jb`i mi fibadjkj fatifatjbmi leb jyjbu``adrm. @akradki, Iu`irjd fibadjb uyku`afa`r jljd ada`jzmibj id edif`j la`sfa garibitjd nj`jfjdi uyku-`adm tanjrjdj bu``adm. Mjkir la`sfa`ar j`i njr`jbti 1;. Ikrj aj`isjdjd mjhiradsjyi` midb`if`irj
(x, y) muz`ifjdmi
y6 h(x, l1, l>,...,ld) (>.1)
ikrj aj`isjdj musudi`jf. Bapa` herfama
(x,y,l1, l>,...,ld) 6 3 (>.>)
sib`jdmi yazanj`jrjz. Nurama l1, l>,...,ld uykud parafitri`irmjr. Ikrjdjd bapa` herfudma bapa` hedbsjyed`ardturivj bura`d uyku`ayarab d miha turiv a`anj`jrjz.
x+
yy 6 3, (>.
x> + >
>
xy+
>
y >y> +
yy 6 3,
. . .d
xd + . . . +
yy(d) 6 3.
d nj`jdfiyid`j sjstfimi l1, l>,...,ld parafitri`irjdj yeb imirsib,
H
x,y,y, . . . , y(d)
6 3
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LGAS]I^ 1
Mjhiradsjyi` midb`if`ir vi ed`ard lezuf`irj
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7 1. MJHI^AD\JZI@ MIDB@IF@I^ XI ED@A^JD LE_YF@I^J
ya ma
y(d) 6h
x,y,y, y, . . . , y(d1)
mdy
mxd 6h
x,y,
my
mx,m>y
mx>, . . . ,
md1y
mxd1
(=.=)
]adf =.. firtinimid
y 6h(x,y,y)midb`ifj jljd y yadj lezuf ikrjsjdjd ikrj`jkj nu`udfa`mr. < vi maga yubsib firtinimid AMM jljd nidzirjkieftrjb yeruf`ar yebtur.
]adf =.:.
y1 6 k1(x) , y> 6 k>(x) , . . . , yd6 kd (x) (=.8)
hedbsjyed`ar
y1 6 h1(x, y1, y>, . . . , yd)y> 6 h>(x, y1, y>, . . . , yd). . .yd6 hd(x, y1, y>, . . . , yd)
(=.;)
sjstifjdj ezmis e`arab sak`yersa, (=.8) hedbsjyed`arda sjstifjd lezufu (jdtikra`j) midjr.
Nu murufma ma lezuf`irj bapa` hedbjyed kjnj musudinj`jrjz. (Nbz Yyar=.8).
Yyar =.13. d6 > murufu jljdy1, y> hedbsjyed`ar yirjdi nj`jdfiyidy, z hedbsjyed`ard i`i a`a f. Nuda
keri(=.8) hedbsjyed`ar
y6 k1 (x) , z 6 k>(x)
kiefitrjbsi` e`arab njr ikrj tadf`ar (< neyut`u uzayma). Nu siniptid eturu(=.8) hedbsjyed`ar(=.;)sjstifjdjdjdtikra` ikrjsj mjyi am`admr`r.
Yyar =.11. Kidi` e`arab(=.;) sjstifjdmi nj`jdfiyid hedbsjyed says j`i midb`if says isjt e`fab zerudma
mikj`mjr. Habat lezuf tadf aydmr. Zjdi mi lezufud var`k vi tib`jkj gabbdma njr kidi``ifi yapfa`yz.
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=. ]IFI@ BAX^AF@A^ :
]ierif =.1>. (=.;) sjstifj vi
S(a, n1, n>, . . . , nd) (=.7)
debtasd i`i a`a f. (=.;) sjstifjdmibj h1, h>, . . . , h d d+ 1 mikjsbid j vi mikjsbid`irj x, y1, y>, . . . , yd e`fabuziri, S debtasdd njr E befsu`ukudma, s urib`j vi y1, y>, . . . , yd mikjsbid`irjdi k eri surib`j bsfj turiv`irisagjp hedbsjyed`ar e`sud. E befsu`ukudma(=.;) sjstifjdj vi
k1(a) 6 n1, k> (a) 6 n>, . . . , kd(a) 6 nd (=.:)
besu`udu (nas`adkl besu`u) sak`ayadk1(x) , k> (x) , . . . , kd (x) hedbsjyed`ar varmr vi nu hedbsjyed`ar tibtjr.
]adf =.1 mikirj j`i j`kj`j jsi sdr mikir pren`ifjmidjr.
Erdib =.1=.
y + y 6 3,y (1) 6
6 9
sdr mikir pren ifjmjr (\MS)- (neudmary va`ui pren if NXS).
Yyar =.18. Yyar =.13i keri h1, h>, . . . , h d hedbsjyed`ar E befsu`ukudma yubarmabj besu`ar sak`yersa
(=.;) sjstifjdjdS debtasdmad kilid njr tib jdtikra` ikrjsj varmr.
Ezi` murufma
h(x, y) ,h
y(x, y) (=.13)
hedbsjyed`arS debtasddE befsu`ukudma surib`j jsi
y 6h(x, y)midb`ifjdjd njr tib jdtikra` ikrjsj fivluttur vi nu ikrj S debtasdmad kilir.
Yyar =.1;. Ikir girx Ra, nV , y1, y> Rl, mV jljd|h(x, y1) h(x, y>)| B|y1 y>| (=.11)
isjtsjz`jkjdj sak`ayalab sibj`mi pezjtjh njrBsays nu`udanj`jrsih(x, y).hedbsjyeduy mikjsbidjdi k eri^
(a x n, l y m) ne`kisjdmi @jplgjtz besu`udu sak`ar midjr. Ezi` e`arabh(x, y).hedbsjyeduy mikjsbidjdikeri^ ne`kisjdmi t urivj fivlut vi hy
B (=.1>)besu`u sak`adyer jsi(=.11) besu`u sak`adr. Habat tirsj mekru mikj`mjr. Zadjy mikjsbidjdi k eri turivj fivlute`fayanj`jr habat (=.11) besu`udu sak`ayad hedbsjyed`ar ma varmr. Asakmabj erdib nudud`a j`kj`jmjr.
Erdib =.17. h(x, y) 6 |y| hedbsjyeduy mikjsbidjdi k eri bsfj turivj yebtur adlab @jpslgjtz besu`udu sak`ar.Kirlibtid
||y1| |y>| | |y1 y>| , B6 1sak`adr.
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=. ]IFI@ BAX^AF@A^ 11
midb`ifjdjd kidi` lezufu mikj`mjr. L udbu
l1ixp (x + l>) 6 l1ixp (x)ixp(l>) 6 Bixp x
e`arab yazanj`jrjz.
Yyar =.>:. Kidi` murufma
H
x,y,y, y, . . . , y(d)
6 3 (=.18)
midb`ifj jljd kidi` jdtikra`mid nagsitfib leb fufbud mikj`mjr. L udbu lezufud tib`jkj serusudud edli`jb`ilivap adfas kiribfibtimjr. Erdikjd
y> x=y> 6 3 (=.1;)midb`ifj asakmabj mikir`ir jljd sak`adr?
y 6x>y, y 6 x>yIkir ni``j njr ne`kimid vi ya nas`adkl besu``ardmad nagsimjyersab, (=.18) midb`ifjdjd kidi` lezufudmid nag-sitfib fufbudmur.
]adf =., 3) debtas neyudla
y6 1
>;(x >), 3) , (>, 3) debta`ardma nj`i AMM djd jdtikra` ikrjsjmjr. L udbu nu debta`arma(=.>3) hedbsjyedumx
my 6 y
x
AMM yj sak`ar.
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LGAS]I^ >
Njrjdlj firtinimid AMM
9. y 6h(x) herfudmabj midb`if`irh(x) virj`id J ara`kdma surib`j njr hedbsjyed e`fab uziri
y 6h(x)AMM djd kidi` lezufu
y6 h(x) mxsib`jdmimjr vi ni`jrsjz jdtikra` biyfi sanjtj jlirjr.y (x3) 6 y3
nas`adkl besu`u virj`mjkjdmi lezuf
y6 y3+
xx3
h(t) mt
sib`jdmimjr.
Erdib 9.1.
y6
AMM djd lezufudu nu`uduz.
L ezuf
y 6
mx y 6 x< + l
Erdib 9.>.
y 6
y (;
NMS djd lezufudu nu`uduz.L ezuf
y6 mx y 6 x< + l >; 6 >; + l l6 3 y 6 x. NJ^JDLJ FI^]INIMID AMM
Erdib 9.=.
y 6 8x + 1y (>) 6 3
NMS djd lezufudu nu`uduz.L ezuf
y 6
8x + 1mx y6 x< + x + l 3 6 7 1> +l l6 >3 y6 x< + x + >3
Erdib 9.9.
y 6 ixp (x)y (3) 6 1, y (3) 6 1, y (3) 6 y 6
(ixp (x) + l1x + l>) mx y 6 ixp(x) + l1 x
>
> + l>x + l< 6 1 + l1
y 6 ixp(x) + >x>
8. y 6h(y) herfudmabj midb`if`irh(y) 6 3 surib`j njr hedbsjyed e`fab uziri (Nbz. Yyar=.
AMM djd lezufu
x61
y + l
ya ma
y6 1
l x
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;. MIKJ\BID@I^JDI AZ^J@ANJ@ID AMM 19
mjr. Ikir lezuf(
y 1 6
mx
y 1 6 x + l y6 1 + (x + l)>
;. Mikjsbid`irjdi ayr`anj`id AMM
]ierif ;.1. h(x) , Ra, nV ara`kdma s urib`j, k (y) , Rl, mV ara`kdma s urib`j hedbsjyed`ar vik (y) 6 3 e`sud.
y 6 h(x)k (y)
AMM djd lezufu h(x) mx6
k (y) my
herfudmamr. Ikir lezuf(x3, y3)debtasdmad kiljyersa
xx3
h(x) mx6 yy3
k (y) my
Yyar ;.>.
y 6h(x) k (y)midb`ifj jljdmi nidzir tierif kilir`jmjr. L ezuf
h(x) mx6
1
k (y)my
sib`jdmimjr vi lezuf(x3, y3) debtasdmad kiljyersa xx3
h(x) mx6
yy3
1
k (y)my
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18 >. NJ^JDLJ FI^]INIMID AMM
mjr. Ikirk (y3) 6 jsi lezuf
y6 y3
mr.
Erdib ;.
1
y> 6x les x sjd x + lviy (3) 6 1 besu`udmadl 6 1/> i`mi imj`jr. Ney`ili lezuf
y6 1
>x les x >sjd x + 1 .
y (3) 6 +1 besu`udmad me`ay pezjtjh beb a`dfstr. Ayd midb`ifjd y (3) 6 3 nas`adkl besu` u lezufuy 6 3 mr.
Erdib ;.=.
y 6 yx < , x 6 y
(y> + =)my 6 3
>
x + >+
1
x + >y
y> + =my 6 3
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;. MIKJ\BID@I^JDI AZ^J@ANJ@ID AMM 1;
midb`ifjdjd lezufu
`d (x + >)>
+ `d (x + + => 6 `d l`d(x + >)
>(x + + =)> 6 `d l(x + >)
>(x + + =)> 6 l
(x + >)>
(x + + =>
Erdib ;.8.
x (y+ 1)> mx +
x> + 1
yiymy 6 3
midb`ifjdj lezuduz.
L ezuf Midb`ifj
x
(x> + 1)mx +
yiy
(y+ 1)> my6 3
sib`jdmi yazanj`jrjz. Nuda keri x
(x> + 1)mx +
yiy
(y+ 1)>my 6 3
1
>`d
x> + 1
+
(y+ 1) iy
(y+ 1)> my
iy
(y+ 1)>my 6
iy 6umy
(y+1)> 6mv v6 1(y+1)
l
1
>`d
x> + 1
+
iy
(y+ 1)my
i
y
(y+ 1)+
iy
(y+ 1)my
6 l
midb`ifjdjd lezufu1
>`d
x> + 1
+ iy
(y+ 1)6l
Erdib ;.;.
my
mx6
x
y (1 x>)1/>midb`ifjdj lezuduz.
L ezuf Midb`ifj
ymy x
(1 x>)1/> mx6 3sib`jdmi yazanj`jrjz. Nuda keri
ymy
x
(1 x>)1/>mx 6 3
1
>y> +
1 x>1/> 6 l
midb`ifjdjd lezufu1
>y> +
1 x>1/> 6l
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:. y 6 ha1x+n1y+l1a>x+n>y+l>
HE^FYDMABJ AMM 1:
L ezuf Midb`if gefeoid AMM mjr. u6 y/x y 6 xu y6 u + xu jhamisjdj midb`ifmi yirjdi yazarsab
y 6
y+ x +
y x
y+ x y x
u + xu 6 xu + x + xu xxu + x xu x
u + xu 6
u + 1 +
u 1
u + 1 u 1
u + xu 6
u + 1 +
u 1>
u + 1 (u 1)
6 u + 1 + u 1 + >u> 1
> 6u +
u> 1
xu 6
u> 1
muu> 1 6
mx
x ().jdt
`d
|u +u> 1| 6 `d x + `d l u +u> 1 6lx u6y/x
y
x+
yx
> 1 6 lx
Erdib 7.8.
+ :xy+ 9y>
mx 8x> + =xy my6 3, y (>) 6 8 NMS djd lezufudu nu`uduz.L ezuf Midb`if gefeoid AMM mjr. u6 y/x y 6 xu y6 u + xu jhamisjdj midb`ifmi yirjdi yazarsab
y 6 + :xy+ 9y>
8x> + =xy
u +xu 6 + :ux> + 9u>x>
8x>
+ =ux>
u +xu 6
< + :u + 9u>
8 + =u xu 6 < + :u + 9u
> 8u =u>8 + =u
xu 6 + ) 6 8 (: + : + 6 >l l6 :
>
+
6 :
>x
:. y 6ha1x+n1y+l1a>x+n>y+l>
herfudmabj AMM
a1 n1a> n> 6 3 (:.1)
e`fab uziri
y 6h
a1x + n1y+ l1a>x + n>y+ l>
(:.>)
AMM djd lezufu jljd
x6 W+ g, y6 Z + b (:.
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>3 >. NJ^JDLJ FI^]INIMID AMM
mikjsbid mikjstjrfisj yap`r. Nudu (:.>) midb`ifjdmi yaza`f?
mZ
mW 6h
a1W+ n1Z + a1g + n1b+ l1a>W+ n>Z + a>g + n>b+ l>
(:.=)
mjhiradsjyi` midb`ifjdmia1g + n1b+ l1 6 3 (:.9)
a>g + n>b+ l> 6 3
e`alab sibj`mi silirsibmZ
mW 6h
a1W+ n1Z
a>W+ n>Z
(:.8)
midb`ifjdj i`mi imirjz bj nu midb`if gefeoid midb`ifmjr vi gefeoid midb`if`ir jljd kilir`j e`ad lezufyedtifj bu``ad`r.Ikir (:.1) mitirfjdadt shr jsi yadja>x + n>y6 b (a1x + n1y) jsi (:.>) midb`ifj be`ayla lezu`ur.
z6 a1x + n1y a>x + n>y6 bzyazarab
z 6a1+ n1y y 6z
a1
n1jhami`irjdj (:.>)midb`ifjdmi yazarsab
zn1
6a1n1
+ h
z+ l1bz + l>
mz
a1+ n1h z+l1bz+l>
6mxmikjsbid`irjdi ayr`anj`id AMM i`mi imirjz.
Erdib :.1.
y 6 >x y+ :x
AMM djd lezufudu nu`uduz.L ezuf > 11 < 6 3
e`mukudmad (:.9) midb`ifjdjd lezirsib g 69, b 61 nu`uruz. x 6 W 9, y 6 Z 1 yazmkfzma vimidb`ifmi yirjdi yazmkfzma
mZ
mW 6
>W ZW z1
>z+ 1
z+ mz 6
mW
W
1>
`d
> >z+ 6 `d (W) + `d l `d
> >z+ 6 > ` d (lW) 6 `d 1l>W>
6 `d l1W>
, l1 6 1
l>
> >z+ 6 l1W>
,
z 6 ZW kirj medusufudu viW6x + 9, Z 6y + 1 kirj medusufudu yaparsab
>W> >WZ + 6l1
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13. @JDII^ AMM >1
ya ma> (x + 9)
> > (x + 9) (y+ 1) + < (y+ 1)> 6l1lezufudu i`mi imirjz.
Erdib :.>.(>x + x + x+ z medusufudu bu``adarab z 6 > + ) /< i`mi imirjz.Nud`ar midb`ifmi bu``admkfzma
z >z+ 1
z 6 < (z+ 1)>z+ 1
+ >
z 6 z 1>z+ 1
mikjsbid`irjdi ayr`anj`id AMM i`mi imirjz. >z+ 1
z 1 mz 6
mx> +
z+ < `d(z 1) 6 x + l
lezufudu i`mi imirjz. z 6 >x + (>x + x + )midb`ifjdi mi (13.1) midb`ifjdi bars`b ki`id gefe oid `jdiir midb`if midjr.Gefeoid tirjfj gefe oid AMM j`i barstr``fafa`mr. Nuda rakfid `jtiraturmi nu tirjf bu``ad`fabtamr.
a (x) , n (x) J ara`kdma surib`j hedbsjyed`ar e`sud. Yyar =.>=j bu` adarab (13.1) vi (13.>) AMM djd tiblezufudmid nagsiminj`jrjz. Edli`jb`i (13.>) midb`ifj mikjsbid`irjdi ayr`anj`id AMMmjr vi lezuf
my
y 6
a (x) mx
`d (Ly) 6
a (x) mx, Ly 43
Ly 6 ixp
a (x) mx
,
y 6 l ixp
a (x) mx
, L6 1/l (13.
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>> >. NJ^JDLJ FI^]INIMID AMM
\jfmj (13.1) midb`ifjdjd kidi` lezufudu i`mi imi`jf. L ezufu parafitri`irjd mikjsjfj e`arab am`admr`adyedtifj bu``adarab nu`a`f. (13.1) jd kidi` lezufudu (13.xy6 x< (13.;)
AMM midb`ifjdjd lezufudu nu`uduz.L ezuf (13.1) midb`ifjdi keria (x) 6 >x, n (x) 6x< mjr. Nuda keri (13.8) herfuda keri
y 6 i
>xmx
xxmx)mx + l1
y 6 ix
>
x
mx + l1
y 6 ix
>
x>xix
>
mx+ l1
,
x> 6u, xix>
mx6 mv
>xmx6 mu, v6 1> ix>
y 6 ix> 1>
x>ix> 1> >xix>mx+ l1
y 6 ix>
1
>x>ix
> 1>
ix>
+ l1
y 6
1
>
x> 1+ l1ix> (13.7)
vi (13.;)midb`ifjdjd lezufu(13.7) sib`jdmimjr.
Yyar 13.>. (13.1)midb`ifjy3 6 3nas adkl besu`udu sak`asa nj`i(13.8)lezufu tibtjr. Erdikjd(13.;)midb`if
lezufu(3, 3) debtasdmad kiljyersa(13.7) a keri lezuf
y61
> x> 1
+
1
>ix
>
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13. @JDII^ AMM >x + 1
x
y6 i>x
AMM midb`ifjdjd lezufudu nu`uduz.
L ezuf (13.1) midb`ifjdi keria (x) 6 >x+1x , n (x) 6 ixp(>x) mjr. Nuda keri (13.8) herfuda keri
y 6 i >x+1
x mx
ixp(>x) i(
>x+1
x mx)mx + l1
y 6
i>x
x
ixp(>x)ixp(>x) xmx + l1
y 6
i>x
x
x>
> + l1
,
y 6 xi>x
> + l1
i>x
x (13.1x
> + l1
i>x
x
sib`jdmimjr.
Erdib 13.9. x> + 1
mymx
+ =xy 6 x
y (>) 6 1.
AMM midb`ifjdjd lezufudu nu`uduz.
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>= >. NJ^JDLJ FI^]INIMID AMM
L ezuf (13.1) midb`ifjdi keria (x) 6 =x(x>+1) , n (x) 6 x
(x>+1) mjr. Nuda keri (13.8) herfuda keri
y 6 i
=x
(x>+1)mx
x
(x> + 1)i
=x
(x>+1)mx
mx + l1
y 6
x> + 1> x
(x> + 1)
x> + 1
>mx + l1
y 6
x> + 1
> x==
+x>
> + l1
,
y(>) 6 1 1 6 (9)>
18
= +
=
>+ l1
l1 6 1: (13.1=)
vi midb`ifjdjd lezufu
y6
x> + 1>x=
= +
x>
> + 1:
sib`jdmimjr.
11. Nirdeu``j midb`ifj
y + a(x)y6 n(x)yd (11.1)herfudmabj AMM `iri Nirdeu``j AMM midjr.a (x) , n (x) J ara`kdma surib`j hedbsjyed`ar e`sud. d 6 3 ya ma d 6 1 murufu jljd srasy`a (13.1) vi (13.>)`jdiir AMM yj i`mi imirjz. Nuramad saysdd 3 vi 1 mid harb` muruf`ard i`i a`alakz. (11.1) midb`ifjdj yd
yi ne`mukufuzmiyyd
+ a(x)
yd1 6n(x) (11.>)
midb`ifjdj i`mi imirjz.
z6 1
yd1 6yd+1
medusufudu bu``admkfzma
z 6 (d + 1) ydy yyd
6 z
(d + 1)nu`uruz. Nu jhami`irj (11.>) midb`ifjdmi yazarsab
z(d + 1)+ a(x)z 6 n(x) (11.x
mz
(z 1) 6 >xmx
`d (z 1) 6 x> + l
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11. NI^DEY@@J MIDB@IFJ >9
viz 6 y> medusufudmid
`d
1 y>
y>
6 x> + l
(11.=) midb`ifjdjd lezufumur.
Erdib 11.>.
xmy
mx+ y6 >x8y=
AMM lezufu nu`uduz.L ezuf midb ifjxy = ya ne`i`jf?
y=my
mx+ x1y< 6 >x9
i`mi imirjz.
z 6 y< mzmx
6 l (x) 6 >x< + l1
z6 >x< + l1x< y6 z1/< 6 >x< + l11/x< + l1
1/ mxx
z 6 lx> z 6 l (x) x>
lx> >xx
l (x) x> 6 =x
l 6 =x< l (x) 6 x= + l1 z 6 x> + l1x> y6 z 1/= 6
x> + l1x
>1/= > 6 (1 + l1)1/= l1 6 19
viy= 6x> + 19x>
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>8 >. NJ^JDLJ FI^]INIMID AMM
midb`ifjdjd lezufumur.
1>. ^jllatj Midb`ifj
y 6a (x) y> + n (x) y+ l (x) (1>.1)herudmabj midb if`iri ^jllatj midb if`irj midjr. a (x) 6 3 murufudma (1>.1) midb`ifj `jdiir midb`ifmjr vil (x) 6 3 murufudma (1>.1) midb`ifj Nirdeu``j midb`ifjmjr.Kidi` e`arab ^jllatj midb ifjdj ada`jtjb e`arab lezfib fufbud mikj`mjr. Ikir (1>.1) midb`ifjdjd njr lezufunj`jdjyersa kidi` lezufu i`mi itfib fufbudmur? y1(x) (1>.1) midb`ifjdjd njr lezufu e`sud, yadj
y1 6 a (x) y>1+ n (x) y1+ l (x) (1>.>)
yaz`r.y6 y1+ z (1>...1) midb`ifjdmi bu``adrsab
y1+ z 6 a (x) (y1+ z)> + n (x) (y1+ z) + l (x)y1+ z
6 a (x) y>1+ >a (x) y1z+ a (x) z
> + n (x) y1+ n (x) z+ l (x)
(1>.>) mid hayma`adrsabz 6 (>a (x) y1+ n (x)) z+ a (x) z> (1>.=)
Nirdeuj``j midb`ifjdj i`mi imirjz.
Erdib 1>.1.
xy 6 =x> =x (1>.9)AMM jljd njr ezi` lezufy1 6 Ax + N e`fab uziri kidi` lezufudu nu`uduz.
L ezuf Ezi` lezuf y16 Ax +N (1>.9) ma yirjdi yazarsab
xA + >ANx + N> 6 =x> =xjhamisjdmid
A6 >, N6 3i`mi imirjz. Ney`ili midb`ifjd njr ezi` lezufu
y1 6 >x
mjr. (1>.1) midb`ifjdi keri (1>.9) ma
a (x) 6 1x
, n (x) 6 .x + z
(1>.=) midb`ifj
z 6
= +
Nirdeu``j midb`ifjdi medusur. (11.1) Nirdeu``j midb ifjdjd jhamisjdi keri
a (x) 6 =
mjr. Nuda keri
u61
zmedusufudmid
u +
= ;
vi lezufud (13.8) herfuda keri
u (x) 6 i
( .1. ^jllatj midb`ifjdjd ezi` muruf`ar.1>.1.1. A..
y 6ay> + ny+ lturudmibj midb`if`ir. Nuramaa, n, lsanjt`irmjr. Midb`if Ne`uf8 vi ; tibj yedtif`ir bu``ad`arab lezu`ur.
1>.1.>. N..
y 6ay> + nx>
turudmibj midb`if`ir. Nuramaa, n biyfi sanjt`irmjr.
y61
z
medusufu j`i
y 6 zz>
nu`udur. Nu jhami`irj midb`ifmi yirjdi yazmkfzma
zz>
6 a
z>+
n
x>
vi
z 6 a n z
x
>gefeoid midb`ifj i`mi imj`jr vi Ne`uf7 tibj yedtif j`i lezuf i`mi imj`jr.
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>7 >. NJ^JDLJ FI^]INIMID AMM
1
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1:
e`mukudmad (1
H(x, y) 6 x> y> mx + (y) H(x, y) 6
x + (y) (1xy+ (y) 6 y> >xy
(y) 6 y> (y) 6 y
AMM djd lezufudu nu`uduz.
L ezuf
>x les y+ y
y 6
x< x> sjd y yx
6 >x sjd y
e`mukudmad (1x les y+ y
mx + (y)
H(x, y) 6 x> les y+ x sjd y y
(y) 6 y (y) 6 y>
> + l
sed (y) jhamisjdj midb`ifjdmi yirjdi yazarsab kidi` lezufu
H(x, y) 6 x> les y+ x
>
+ l
sib`jdmi i`mi imirjz. \ed e`arab nas`adkl besu`udmad,
3 6 =>
+ l l6 >
x> les y+ x
> 6 >
lezufmur.
Erdib 1x >ixy sjd>x + >x) mx + (xixy les>x
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. NJ^JDLJ FI^]INIMID AMM
L ezuf
(yixy les>x >ixy sjd>x + >x)y
6(xixy les>x x >x sjd>x + xy les>x)
e`mukudmad (1ixy sjd>x + >x
H
y 6 k (x, y) 6 xixy les>x <
H(x, y) 6
(xixy les>x x) ixy x) ixy x >ixy sjd>x + >x
y (les>x) ixy
>(sjd>x) ixy +
(x) 6yixy les>x
>ixy sjd>x + >x
(x) 6 >x (x) 6 x> + lsed (x) jhamisjdj midb ifjdmi yirjdi yazarsab kidi` lezufu
H(x, y) 6 (les >x) ixy + lsib`jdmi i`mi imirjz.
1=. Jdtikrasyed L arpad
Ikir (1) midb`ifj taf mjhiradsjyi` midb`if mikj` jsi yubarmabj yedtif uyku`adafaz. Ney`i njr murufma(1) midb`ifjdj taf AMM yapalab sibj`mi shrmad harb` njr hedbsjyed j`i larpfa`yz. Zadj,
f (x, y) h(x, y) mx + f (x, y) k (x, y) my6 3 (1=.1)
midb`ifj taf AMM e`alab sibj`mi f (x, y)6 3 hedbsjyedu nu`fa`yz.
]adf 1=.1. Ikir (1) midb`ifj taf mjhiradsjyi` midb`if mikj` habat (1=.1) midb`ifj taf mjhiradsjyi`midb`if jsif (x, y) hedbsjyeduda(1) midb`ifjdjd jdtikrasyed larpad (jdtikratjed halter) midjr.
Erdib 1=.>.
mx+
>x + y
my6 3
AMM taf mikj`mjr. L udbu
h(x, y) 6 , k (x, y) 6 >x + y h
y 6 < + 7xy,
k
x6 > + 8xy
h
y 6
k
xhabatf (x, y) 6 x>y jljd
y> + =xx + 1>x 6
(fk)
x
besu`udu sak`amkdmad tafmr. Nu murufmaf (x, y) 6 x>y jdtikrasyed larpadmr.
Yyar 1=.
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1=. JD]IK^A\ZED L A^SADJ )
i`mi imirjz.
Kidi`mi jdtikrasyed larpadd nu`fab e`mubla zermur. Nu yuzmid (1=.>) jhamisjdj bu``adarab f (x, y) jdti-krasyed larpad samilix vi samili y djd hedbsjyedu e`muku ezi` muruf`arma jdli`iyilikjz.
1=.1. J.Muruf. f (x, y) jdtikrasyed larpad samili x jd njr hedbsjyedu e`sud. Nu murufma (1=.>)jhamisjdmi fy 6 3 mr vi (1=.>)jhamisjdmid
k (x, y)f
x 6
h
y k
x
f
mf
f 6
hy kx
k (x, y)
mx (1=.)jhamisjdmi fx 6 3 mr vi (1=.>)jhamisjdmid
h(x, y)fy
6
h
y k
x
f
mf
f
6
hy kx
h(x, y)
my
(1=.9)
`d f 6
hy kx
h(x, y)
my
f (x) 6 ixp
hy kx
h(x, y)
my
(1=.8)Yyar 1=.9. (1=.9) midb`ifjdmimy jd batsaysy jd njr hedbsjyedu e`fa`mr.
Erdib 1=.8.(x y) mx my6 3
AMM djd lezufudu nu`uduz.
L ezuf
h(x, y) 6 (x y) , k (x, y) 6 1 h
y 6 1, k
x6 3
h
y 6 k
x
e`mukudmad midb`if taf AMM mikj`mjr. Nuda keri jdtikrasyed larpadd nu`a`f. Edli f (x, y) jdtikrasyedlarpad samilix jd njr hedbsjyedu e`fa murufudu jdli`iyi`jf?
hy kx
k (x, y)
6 1
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>. NJ^JDLJ FI^]INIMID AMM
vi samilix jd njr hedbsjyedumur. (1=.=) lezufudmid jdtikrasyed larpad
f (x) 6 ixp
hy kx
k (x, y)
mx
6 ixp
mx
6 ix
e`arab nu`udur. Nuda keri
ix (x y) mx ixmy6 3 (1=.;)midb`ifj taf AMM mjr?
h(x, y) 6 ix (x y) , k (x, y) 6 ix h
y 6 ix, k
x6 ix
h
y 6
k
x
vi lezufu
Hx
6 h(x, y) 6 ix (x y)
H(x, y) 6
(ix (x y)) mx+ (y) (=:) vi (91) de`u herfu``irmid
H(x, y) 6 ix (x 1) yix + (y) (1=.7)H
y 6 k (x, y) ix + (y) 6 ix
(y) 6 3 (y) 6 lsed (y) jhamisjdj (1=.7) midb`ifjdmi yirjdi yazarsab (1=.;) midb`ifjdjd kidi` lezufudu
H(x, y) 6 ix (x 1) yix + l H(x, y) 6 ix (x
y
1) + l
sib`jdmi i`mi imirjz.
Erdib 1=.;.
ymx + (< +
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1=. JD]IK^A\ZED L A^SADJ
ymy
6 i> `d y 6i`dy
>
6y>
e`arab nu`udur. Nuda keriy (< +
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. NJ^JDLJ FI^]INIMID AMM
jdtikra` larpadmr. Nuda keri
+ y+
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1=. JD]IK^A\ZED L A^SADJ ) herfu`udi keri
k (x, y)f
x h(x, y) f
y 6
h
y k
x
f
h
yk
x 6 =xy+ > f
x 6 3,(>xy> + y) f
y 6 (=xy+ >) f
mf
f 6
>
ymy
`d f 6 > `d y f 6 y>
jdtikra` larpadmr. Nuda keri
(>xy> + y)y>mx + (>y< x)y>my6 3midb`ifj taf AMM mjr. h(x, y) 6 (>xy> + y)y>, k (x, y) 6 (>y< x)y>
hy
6 (>xy> + y)y>y
6 1y>
,
k
x 6
(>y< x)y>x
6 1y>
h
y 6
k
x
H
y 6 k (x, y) 6 (>y< x)y>
H(x, y) 6
(>y< x)y> my
6 y> +x
y
+ (x)
H
x 6
y> + xy + (x)
x
6 1
y+ (x)
6 h(x, y) 6 (>xy> + y)y> 1
y+ (x) 6 (>xy> + y)y>
(x) 6 >x (x) 6
(>x) mx
(x) 6 x> + l H(x, y) 6 y> +
x
y + (x) H(x, y) 6 y> +
x
y+ x> + l6 3
lezufmur.
1=.
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. NJ^JDLJ FI^]INIMID AMM
]adf 1=.13. Ikirk (x3, y3) 6 3, h(x3, y3) 6 3
besu`u sak`adyersa(x3, y3) debtasda(1. Ikir AMM jd lezufu tibj` debtamad kilirsi nu tibj` debtaya muk uf debtas midjr. Erdib1=.11mi(3, 3) debtas njr muk uf (demi) debtasmr.
Erdib 1=.1 6l
e`arab i`mi imj`jr. L ezufmi(3, 3) debtasdmad kilfifibtimjr. vi
y6 3, x6 3
lezufud asjfptet`ar mikj`mjr.
]adf 1=.18.Ikir AMM jd tibj` debtasdmabj lezufu asjfptet`ar mikj` jsi vi lezuf tibj` debtamad kilfjyersa
nu tibj` debtaya firbiz (lidtir) debtas midjr. Erdib1=.19mi(3, 3) debtas njr firbiz debtasmr.
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LGAS]I^ )
e`arab i`mi imj`jr. Ezi` e`arab mjb yerudki murufu jljd
`jf
>
y tad 1 +y tad 61y
i`mi imj`jr. Nuda keri(19.>) jhamisjdmid
h
x,y,
1y
6 3
sib`jdi medusur.
Yyar 19.>. Xirj`id ikrjdjd mjhiradsjyi` midb`ifj
S(x, y)mx + Q(x, y)my6 3 (19.
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< 7 6l>, l 4 3 (19.9)
ikj aj`isj, aydE firbiz`j vil yarlap` majri`irmjr. Nu ikrj`iri mjb e`ad ikrj aj`isjdj nu`uduz.
L ezuf (19.9) jhamisjdmi mjhiradsjyi` a`rsab
>xmx + >ymy 6 3 (19.8)
AMM i`mi imirjz. (19.x, Q (x, y) 6 >y
vi (19.=) i keri(19.9) ikrj aj`isjdi mjb ikrj aj`isj
>ymx >xmy6 3 (19.;)e`arab i`mi imirjz. AMM jd lezufu
>ymx >xmy6 3 myy
6 mx
x y6 fx
j`i jhami ij`id mekru aj`i`irjmjr.
18. Fibadjb pren`if`irj
Nj`jdmjkj kjnj Diwtedud jbjdlj garibit badudu,
H 6 fa6 fmv
mt 6f
m>x
mt>
f 6 wk
sib`jdmi jhami imj`jr. Nurama f ljsfjd but`isj, Hljsfi itbj imid sanjt buvvit, a ljsfjd garibitjdjdJvfisj, vgz vi kjmj`id ye` x , w jsi ljsfjd akr`k j`i jhami imj`jr.
Erdib 18.1. But`isjf e`ad njr ljsfjd yirmid e`mubla yubsibti nu`udad njr debtamad J`b gzsz e`arab sirnistmusfiyi nrab`yer. Ljsfi itbj imid yirlibjf buvvitj sanjt vi gava mjridljdjd ljsfjd gz `i eradt` e`mukubanu` imj`mjkjdi keri. girgadkj njr t addma ljsfjd nas`adkl debtasdmad gadkj uzab`b`a e`mukudu vi e admagadkj gz`a garibit itfibti e`mukudu nu`uduz.
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18. FIBADJB S^EN@IF@I^J k
b> + l l6 g f
>k
b>
x 6 fk
b
t +
f
bi
bf t
+ g f>k
b>
Erdib 18.>. Yzirjdmibj medadf`ar j`i njr`jbti njr parasutludud akr`k 183 `n (`jnri) virj`fjstjr. Saras utal`famad edli gava mjridlj gzd yarsda isjttjr. Sarasut 9 sd sedra al`mkdma gava mjridlj gzd barisjdjd9/7j bamar e`uyer. Sarasut al`famad vi al`mbtad sedra parasutludud gzd nu`uduz. (Zirlibjfj jvfisjdjk 6 hjt/sd> e`arab a`dz.)
L ezuf Sarasut al`famad edlibj gzd nu`a`f.
w6 183`n f6 wk
6 183/ 6 9
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= 3 .burafdmad
H 6 fmv
mt 6H1+ H> f mv
mt6 183 v
> 9 mv
mt6 183 v
>
9
mv
mt 6
3
v
> mv
3 v 6 mt
13 `d(3 v) 6 1
13 t `d l 3 v 6 li 113 t v (t) 6 3 li 113t
e`arab nu`uruz. Sarasutludud j`b zafadma gzdd e`fafasdmad
v (3) 6 3
besu`udmad
3 6 3 l l6 3 v (t) 6 3
1 i 113t
parasut al`famad edlibj parasutludud gzd virfibtimjr. 9 sd. sedrabj gz
9>.93->.9-9
133
93
3
-93
-133
-193
->33
x
y
x
y
v (9) 6 3
1 i 1139
e`arab nu`uruz. Nuda keri 9 sd sedrabj mjhiradsjyi` midb`ifj
fmv
mt 6 183 9v
>
7
v (9) 6 3
1 i 1139
6 1>9.:1
? e`arab virinj`jrjz. Midb`ifjd lezufu
mv
18> v> 6 mt
7 1
118 v + 118 + v mv6 mt7 `d
18 + v
18 v 6 =t + `d l 18 + v
18 v 6li=t v 1 + li=t6 18 li=t 1
v 6 18
li=t 1
(1 + li=t) v (9) 6 3
1 i 1139
3
1 i 1139
6 18
li>3 1
(1 + li>3) l6 i>3
v (t) 6 18
1131=>
i>3+=t 11 + 1131=> i
>3+=t , t 9
e`arab nu`udur.
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18. FIBADJB S^EN@IF@I^J =1
) H>, surtudfi buvvitj, akr`kd mjbiy nj`isidj j`i surtudfi buvvitjdjd larpfmr?
H> 6 D6
1
==7les+ H
fmv
mt 6 >= 8
< v
>v (3) 6 3
mjhiradsjyi` midb`ifjdj lezi`jf?
mv=7 1>< v 6
mt
< v
6 t
< v 6 li t< v (3) 6 3
=7 1>
< 6 l v 6
=7 1>
)
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= > = e`fas murufudma sed debtamabj gzdd di e`muku serusuda ki`i`jf?
mx
mt 6 v6
=7 1>
.8h t/ sil
1;. Erad Sren`if`irj
Erad pren`if`irj, njr fizjbsi` nuyub`ubtibj, njrjf zafad jljdmi fiymada ki`id mikjsfi e`arab tarjh imj`jr.
Erdib 1;.1. ^amyeabtjh njr i`ifidtjd nezudfa gz i`ifidtjd fivlut fjbtar j`i mekru eradt`mr. ^ayeabtjhi`ifidtjd erojda` akr`kdd yars 1933 y` jljdmi mak`mkda k eri
(1) =933 y` sedra, ramyeabtjh i`ifidtjd akr`kd nu`uduz.(>) Erojda` akr`kdd %13 uda varfas jljd di bamar zafad kilfisj kiribtjkjdj nu`uduz.
L ezuf x ramyeabtjh i`ifidtjd fjbtard kestirfib uziri mx/mt ramyebatjh nezu`fadd eradd ni`jrtjr. Nu
erad fivlut fjbtar j`i mekru eradt` e`mukudmad
mx
mt 6 bx
fjbtarma a`afa e`mukudmad jsarit dikatjhmjr.
x (3) 6x3
vi 1933 y` sedrabj nezu`fa fjbtar
x (1933) 6x3
>mjr.
mx
mt 6 bx,x (3) 6x3
mjhiradsjyi` midb`ifjdjd lezufu
x6 x3ibt x (1933) 6 x3
>e`mukuda keri
x3>
6 x3i1933b b6 `d >
1933 3.333=8 x6 x3i3.333=8t
ib 6 1>1/1933 x6 x3 ibt 6x3 1
>t/1933
(1) t6 =933 jljd nezudfa
x6 x3
1
>
=933/19336
x37
Nu jsi =933 sidi sedudma nezudfadd 1/7 viya %1>.9 e`mu kudu sey`ir(>)
x6x313
x313
6x3
1
>
t/1933 t6 1933 `d13
`d > =:79
y` sedra kirlib`isjr.
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17. SESY@A\ZED S^EN@IF@I^J =. Diwtedud sekufa burafda keri, sekuyad njr ljsjfjd slab`k, ljsfjd slab`k vi ljsfjbap`ayad ertafd sanjt slab`kdd harb j`i eradt`mr. t6 3mabj93 sanjt slab`b` njr ertafmabj slab`k73
e`mukuda k eri vi9sadjyi sedra ljsfjd slab`k;3i must uk udi keri ljsfjd slab`kdd, t yi nak` hedbsjyedudunu`uduz vi 13 sd sedrabj slab`kd nu`uduz.
L ezuf x,t zafadmabj ljsfjd slab`kd kestirsjd nuda keri mjhiradsjyi` midb`if
mx
mt 6 b (x 93)
x (3) 6 73
vi
x (9) 6 ;3
besu``arfz varmr.
mx
x 93 6 bmt `d (x 93) 6bt + `d l x 93 6 libt x6 93 + libt x (3) 6 73
x 137x> 6mt 133
1
x+
138
1 138x
mx6 mt
`d x `d 1 138x 6 t133
+ `d l x1 138x 6li
t/133
x6 lit/133
1 + l138it/133, x (1:73) 6 139 139 6 li
1:73/133
1 + l138i1:73/133
l 6 138
:i::9
x (>333) 6138
:i::9
i>333/133
1 + 138
:i::9
138i>333/133 11:=:9
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= = . sbbd lezufu jljd?
>.139 6
138
:i::9
it/133
1 + 138
:i::9
138it/133 i1:.7t/133 6 =
: t >381
1:. Barsf Sren`if`irj
Nurama barsf`armabj erad`ar i`i a`dr. \ fammisj ni``j erad`armabj barsf`ar jlirjp, barsf, barstrl j`isanjt tutu`uyer. x j`i \ fammisjdjd fjbtard kestirjrsib mx/mt x mibj mikjsjfjd t yi keri eradd virjr. Nudakeri mjhiradsjyi` femi`j
mx
mt 6kjrjs virj`irj lbs virj`irj
e`arab virj`jr.
Erdib 1:.1. Nas`adklta 93 ka`(ka``ed, `n `jb tuz jlirid lezu`fus tuz`u su ib`idfibtimjr. Barstrl j`i barsf gefeoid tutu`urbid,nasba njr vadamad ayd gz`a barsf nesa`t`fabtamr. Nuda keri, tadbta girgadkj t zafaddmabj tuz eradddimjr5
L ezuf Kjrjs virj`irj? < ka`/sd `jb gz`a tadbd jljdi > `n `jb tuz jlirid lezu`fus tuz`u sukjrjs virj`irj6 >(`n/ka`)
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>3. I@IB]^JB MIX^I S^EN@IF@I^J =9
L ezuf Krjs`ir?
kjrjs virj`irj6 >(`n/ka`)9(ka`/sd) 6 13 `n/sd
kjrjs gz 9 vi lbs gz < e`mukuda keri tep`af gz
9 < 6 >vi ney`ili girgadkj zafadmabj tuz`u su fjbtar, nas`adklta ma 93 e`mukudmad
93+ >t
vi tuz bedsadtrasyedu jsix
93+ >te`arab virj`jr. Ney`ili
ljbjs virj`irj6 x
93 + >t(`n/ka`)t x6 13
`jdiir midb`ifjdj i`mi imirjz.
mx
mt +
tx 6 3 mx
x 6 tmt `d x6 `d(93 + >t) + `d l
x 6 l (93 + >t) x6 l (t) (93 + >t)
gefeoid e`fayad midb`ifjd lezufu e`sud.
l (93 + >t) t)9/> + t
l (t) (93 + >t)
6 13
l 6 13(93 + >t) l6
13 (93 + >t) mt6 7
> (t + >9)
9/>+ l1
x6 7> (t + >9)9/> + l1 (93 + >t) x (3) 6 3besu`udmad
3 6
7
> (>9)9/>
+ l1
(93)
l6 >9333
>
x 6
7
> (t + >9)9/> >9333
>
(93 + >t)
sedul e`arab nu`uruz.
>3. I`ibtrjb Mivri Sren`if`irj
Id nasjt i`ibtrjb mivri`irj, oidirater viya pj` kjnj i`ibtrjb baydak vi idirojyj bu``adad njr rizjster (erdikjdi`ibtrjb afpu`u) (risjstadli) nu`udad njr sirj mivrimjr. Ikir mukfi bapat`rsa njr Jabf rizjsteri mekruabalab vi njr ve`tao musfisjdi sinip e`ulabtr. Zadj rizjsterud jbj uludmabj petadsjyi` harb` e`ulabtr. Nupetadsjyi` harb viya ve`tao mususu jsi Xe`tfitri midj`id njr i`t j`i e`lu`inj jr. Ibitrjb mivri`irjdmibj nasjt njrbura` Bjrlgeffi bura` e`arab ama`admr`r, Nu bura`a keri, i`ibtrjb mivrisjdmibj tuf ve`tao`ard tep`af, tep`afbuvviti isjttjr. ]ep`af buvvitj I(t) j`i kestirjrsib (ifh-i`iltrefetjvi herli)
X@+ X + XL6I(t)
^ rizjster (rijstadli), L bapasjter (lapaljter), J jdmubter (dmulter). J6J(t) i`ibtrjb mivrisjdmibj abf viq6 q(t) bapasjtermibj adj i`ibtrjb yubudu kestirfib uziri
q 6 J
sib`jdmi njr nakdt fivluttur. Egf badududa keri rizjster uzirjdmibj ve`tao musub`uku abf j`i mekrueradt`mr?
X 6 ^J
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= 8 .firtinimid midb`ifj ^L@ midb`ifj e`arab ama`dmr`r.Jdmubterud e`famk murufma mivriyi^Lmivrisj midjr vi midb`if
XL+ X 6 I(t) 1L
q+ ^J6 I(t) J6q
^q+ 1
Lq 6 I(t)
e`ad 1. firtinimid AMM i`mi imirjz.
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>3. I@IB]^JB MIX^I S^EN@IF@I^J =;
Erdib >3.1. \ibj`mi virj`id ^Lmivrisjdmi (a) edli \ adagtardd 1 bedufuda kitjrj`fisj murufudma (n)\ adagtardd > bedufuda kitjrj`fisj murufudma mivrimid kilid abf sjmmitjdj nu`uduz.
I(t)
\
1
>
^
L
L ezuf (a) ikir adagtar njr murufudma jsi ifh-i`iltrefetjvi herli I(t) mjr vi midb`if
^q+ 1
Lq6 I(t) q+ 1
^Lq6
I(t)
^
`jdiir midb`ifmjr vi lezuf
q 6 i
1^L mt
I(t)
^ i
1^L mtmt + l1
q 6 I(t)
^ 1
^Lq6 J
J 6
I(t)
^ 1
^Li t^L I(t)^ i t^L mt + l1(n) sbb jljd jsi I(t) 6 3 murufumur vi midb`ifjd lezufudmi yirjdi yazmkfzma
J6 l1^L
i t^L
e`arab nu`uruz. Nuramabj - jsaritj abfd tirs yedi mekru kjttjkjdj kestirjr.
Erdib >3.>. >3 `ub njr mjridl ti`j, 3.31H(Haram) `b njr bapasjtads e`ad bedmadsater, ifh(i`iltrefetjviherli)sj=3i>t+>3i=t `jb njr uritili sirj e`arab nak`adyer. Nas`adkl zafadma yubu e`famkda k eri girgadkjnjr admabj yubudu nu`uduz5
L ezuf
^6 >3, L6 3.31H, I(t) 6 =3i>t
+ >3i=t
virj`irj vi midb`if?
q+ 1
^Lq 6
I(t)
^q(3) 6 3
`jdiir midb`ifj jljd edli gefeoid midb`ifj i`i a`a`f?
mq
q 6 1
^Lmt6 9mt q6 li9t
vi gefeoid e`fayad midb`ifjd lezufu
q6 l (t) i9t
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= 7 t + >3i=t
l6 =3i3it
l (t) 6
=3
) e`arab nu`uruz. ^s6 ^1+ ^> jsitep`af risjstads virjr.
X 6 J(^1+ ^>) J6 X^1+ ^>
X1 6 J.^1 6 ^1
^1+ ^>X,
X> 6 J.^> 6 ^>
^1+ ^>X
L. besu`udma jsi
X 6 X 6 J1^1 6 J>^> J6 J1+ J>J 6 X
1
^1+
1
^>
X 6 ^1^>
^1+ ^>J
J1 6 X
^16
1
^1
^1^>^1+ ^>
J6 ^>
^1+ ^>J
J> 6 X
^>6
1
^>
^1^>^1+ ^>
J6 ^1
^1+ ^>J
sedul`ard i`mi imirjz.
Erdib >3.
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>3. I@IB]^JB MIX^I S^EN@IF@I^J =:
JL6LmXmt
, Jr 6X
^>, J6
1
^1(X X )
J6 JL+ J 1^1 (X X ) 6 LmXmt + X^>
mXmt
+
1
L^>+
1
L^1
X 6
X
^1
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LGAS]I^ =
1. firtinimid yubsib mirili`j AMM
1.firtinimid yubsib firtini`j AMM kidi` herfu
H(x,y,y) 6 3 (19.1)
sib`jdmimjr. Kidi` e`arab
y 6p
e`arab kestirj`jr. Nuda keri (19.1) midb`ifj
H(x,y,p) 6 3 (19.>)
sib`jdmi jhami imj`jr.
Yyar 19.=. (19.1) midb`ifjdj lezfib kidi`mi leb be`ay mikj`mjr. Xi gatta lezufud e`fas ma kiribfiz.
Nudud jljd id edif`j sedul`ar nurama ni`jrtilikjz.
18. y6 h(x, p) herfudmabj AMM
(19.1) midb`ifjy6 h(x, p) (18.1)
herfudma e`sud.
my
mx6p my6 pmx
vi (18.1) midb`ifjdmid
h
xmx +
h
pmp 6 my 6 pmx mx i ne`irsib
h
x+
h
p
mp
mx 6 p (18.>)
(18.>) midb`ifj p vi x jd midb`ifjmjr vi Ne`uf> mibj lezYf yedtif`irj j`i lezuf i`mi imj`inj`jr.
Erdib 18.1.
y> 1 + y> 6 3 (18.
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9> =. 1. FI^]INIMID ZYB\IB MI^ILI@J AMM
e`arab ma yazanj`jrjz. (18.>) jhamisjdmid
h(x, p) 6
1 p>h
x
+ h
p
mp
mx
6 p
p
1 p>mp
mx 6 p
p
1 +
11 p>
mp
mx
6 3
1 + 1
1 p>mp
mx 6 3, p6 3
11 p>
mp 6 mx, mymx
6 3
arlsjdp 6 x + l, y6 l p 6 sjd (
x + l) , y6 l
y 6 les (x + l) , y6 l(18.=) AMM djd lezuf`irjmjr.
1;. x6 h(y, p) herfudmabj AMM
(19.1) midb`ifjx6 h(y, p) (1;.1)
herfudma e`sud.mx
my 6
1mymx
61
p
vi (1;.1) midb`ifjdmid
h
y+
h
p
mp
my 6
mx
my 6
1
ph
y+
h
p
mp
my 6
1
p (1;.>)
(1;.>) midb`ifj p vi y jd midb`ifjmjr vi Ne`uf> mibj lezuf yedtif`irj j`i lezuf i`mi imj`inj`jr.
Erdib 1;.1.
x 1 + y> 6 3 (1;.) jhamisjdmid
h(x, p) 6 1 p>h
y+
h
p
mp
my 6
1
p
>p mpmy
6 1
p >p>mp6 my
>) lezufudmiyirjdi yazarsab leuzufud parafitrjb kestirjfj e`arab
y6 (p) x + (p) (17.9)
i`mi imirjz.
Erdib 17.1.
y6 >xy + y> ya may 6 >xp+p> (17.8)AMM djd lezufudu nu`uduz.
L ezuf (17.1) jhamisjdi keri
(p) 6 >p, (p) 6 p>
vi (17.
p >p x + >p
p >p 6 >
px >
`jdiir midb`ifjdj i`mi imirjz. (13.1) jhamisjdmid
a (p) 6>
p, n (p) 6 >
vi (13.8) lezufudmid
x 6 i p
pmp
>i
>i`dp>mp+ l16 p>
>p>mp + l
6 p>
>
p1
> (17.7)
i`mi imirjz. (17.;) vi (17.7) , (17.8) d kidi` lezufudud parafitrjb midb`ifjmjr.
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9= =. 1. FI^]INIMID ZYB\IB MI^ILI@J AMM
1:. L`ajraut Midb`ifj
@akradki midb`ifjdmi (p) 6 p murufudma
y6 xy
+ (y) viyay 6 px + (p) (1:.1)
turudmibj midb`if`iri L`ajraut midb`ifj midjr. (1:.1) midb`ifjdj x i keri mjhiradsjyi`jdj a`rsab
p 6 p + xmp
mx+ (p)mp
mx
mp
mx(x + (p)) 6 3 mp
mx6 3
p 6 l (1:.>)
lezufudu i`mi imirjz. Nu (1:.>) yj (1:.1) midb`ifjdmi bu``adrsab (1:.1) midb`ifjdjd kidi` lezufudu
y6 lx + (l) (1:.) jhamisjdi keri
(p) 6 p
p 1 (1:.:)vi (1:.>) medusufudmid
p6 l (1:.13)
i`mi imirjz. Nu (1:.13) yj (1:.7) midb`ifjdmi bu``adrsab (1:.;) midb`ifjdjd kidi` lezufudu
y 6 lx + l
l 1 (1:.11)i`mi imirjz. Ikir (1:.=) sak`adrsa
x + (p) 6 x +p
(p
1)
p
p
(p 1)> 6x p
(p 1)> 6 3 mp
(p 1)> 6 xmx
mp
(p 1)> 6
xmx (8) jhamisjdmid
1p 1 6
x> + l
>
p 6 1 >x> + l
(1:.1>)
i`mi imirjz. Nu (1:.1>) yj (1:.7) midb`ifjdmi bu``adrsab (1:.;) midb`ifjdjd kidi` lezufudu
y 6
1 >
x> + l
x + 1 +
1
1 >x>+l
1
(1:.1
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1:. L@AJ^AY] MIDB@IFJ 99
i`mi imirjz.
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LGAS]I^ 9
Zubsib Firtinimid @jdiir AMM
>1. Kjrjs
]adf >1.1. d. jdlj firtinimid gefeoid e`fayad njr `jdiir AMM kidi` e`arab
a3(x)y(d) + a1(x)y
(d1) + ... + ad 1(x)y+ ad(x)y 6 H(x) (>1.1)
a3(x)mdy
mxd+ a1(x)
md1y
mxd1+ + ad1(x)my
mx+ ad(x) y 6 H(x)
sib`jdmi yaz`r. Nuramaa3(x), a1(x), . . . , ad(x) mikjsbid batsay`armr. Nu batsay`ar viH(x) hedbsjyedu, x jd
njrJ6 Ra, nV ara`kdma tadf`adfs s urib`j hedbsjyed `armr.
]adf >1.>. Ikir, H(x) 6 3 jsi, (>1.1) midb`ifj
a3(x)y(d) + a1(x)y
(d1) + ... + ad 1(x)y+ ad(x)y6 3 (>1.>)
sib`jdj a`r. E zafad nu midb`ifi, d jdlj firtinimid mikjsbid batsay` gefeoid `jdiir AMM midjr.
Erdib >1.. @jdiir gefeoid AMM jljd tifi` tierif`ir
gefeoid e`fayadd.jdlj firtinimid mikjsbid batsay` njr `jdiir mjhiradsjyi` midb`ifjd, yadj(>1.1) midb`ifjdjdkidi` lezufudu nu`fab afalfzmr. Nudud jljd edli (>1.>) gefeoid mjhiradsjyi` midb`ifjd kidi` lezufududnu`udfas jdli`idilibtjr. ((>1.1) mjhiradsjyi` midb`ifjdjd lezufu j`i j`kj`j var`b tierifj ]ierif (=.>>.1. d. jdlj firtinimid njr gefeoid mjhiradsjyi` midb ifjd njrjnjrjdmid harb` f sayma lezufuy1, y>,...,yf e sud. Nurama f dmr. Nu murufma l1, l>,...,lf batsay`ar biyfi sanjt say`ar e`fab uziri,y 6 l1y1+ l>y>+ ... + lfyf hedbsjyedu ma ayd midb`ifjd njr lezufu e`ur.
]adf >>.>. y1, y>,...,yf girgadkj hedbsjyed`ar vi l1, l>,...,lf girgadkj biyfi sanjt say`ar e`sud`ar. Numurufmal1y1+ l>y>+ ... + lfyf jhamisjdiy1, y>,...,yf hedbsjyed`ardd `jdiir befnjdasyedu midjr.
Yyar >>.>.>i keri(>1.>) gefeoid mjhiradsjyi` midb`ifjd lezuf`irjdjd `jdiir befnjdasyedu ma
njr lezufmur.
Erdib >>.=. sjd x viles x hedbsjyed`ary+y 6 3 AMM djd lezufumur. Yyar>>.1=a keriy 6 l1les x +l>sjd x mi midb`ifjd lezufumur.
]adf >>.9. Njra
x
nara`kdma tadf`afs e`ady1, y>,...,yfhedbsjyed bufisj jljd, gipsj shr e`fayadl1, l>,...,lf kjnj sanjt say`ar nu`udanj`jrsi vi xjd nu ara`btabj nutud mikir`irj jljd,
l1y1+ l>y>+ ... + lfyf6 3
jsi, nu hedbsjyed`ara ara`ardma `jdiir nakf` hedbsjyed`ar midjr.
Erdib >>.8. x vi>x hedbsjyed`ar R3, 1V ara kdma jdiir nakf`mr. L udbu
l1x + l>(>x) 6 3 (l1+ >l>) x6 3,x R3, 1V l1+ >l> 6 3 l1 6 >l>Erdikjdl> 6 1, l16 > jljd jhami sak`adfs e`ur.]adf >>.;. Njra x n ara`kdma `jdiir nakf` e`fayad hedbsjyed`ara jsi, `jdiir nakfsz hedbsjyed`ar
midjr. Zadj,l1y1+ l>y>+ ... + lfyf6 3 l16 l> 6 6lf6 3
9;
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97 9. ZYB\IB FI^]INIMID @JDII^ AMM
Erdib >>.7. x vi x> hedbsjyed`ar R3, 1V ara`kdma `jdiir nakfszmr. l1x+ l>x> 6 3 jhamisjdj mjhirad-
sjyi``irsibl1+ >l>x6 3 i`mi imirjz vix j`i nu midb`ifj larparsabl1x + >l>x> 6 3 i`mi imirjz.
l1x + l>x> 6 3 vil1x + >l>x
> 6 3,x R3, 1V
midb`if`irjdj lbartrsabl>x>
6 3,x R3, 1V i`mi imirjz. Nuda keril> 6 3 6 l1 mjr.]ierif >>.:. (>1.>) gefeoid mjhiradsjyi` midb`ifj d `jdiir nakfsz lezufi sagjptjr. y1, y>,...,yd (>1.>)
gefeoid mjhiradsjyi` midb`ifjdjd `jdiir nakfsz lezuf`irj jsi(>1.>) djd kidi` lezufu
l1y1+ l>y>+ ... + ldyd
j`i jhami imj`jr. Nuramal1, l>,...,ld biyfi sanjt say`ar.
Erdib >>.13. sjd x viles x `jdiir nakfsz hedbsjyed`ary+ y6 3 AMM djd lezufumur. Yyar >>.1=a vi]ierif>>.:keriy 6 l1les x + l>sjd x mi midb`ifjd lezufumur.
\jfmj, jbj viya maga haz`a hedbsjyedud gadkj besu``arma `jdiir nakf` viya `jdiir nakfsz e`mukudu arastra`f.
]adf >>.11. Njra x nara`kdma tadf`afs e`ady1, y>,...,yd hedbsjyed`ar(d 1) .firtinmid turivisagjp e`sud`ar.
[(y1, y>,...,yd) 6
y1 y> . . . ydy1 y
> . . . y
d
......
y(d1)1 y
(d1)> . . . y
(d1)d
mitirfjdadtda [redsbjad midjr via x n debtasdmabj mikirj bsala[(y1, y>,...,yd) j`i kestirj`jr.]ierif >>.1>. y1, y>,...,yd (>1.>) gefeoid mjhiradsjyi` midb`ifjdjd lezuf`irj `jdiir nakfsz e`fas jljd
kirib vi yitir besu`[(y1, y>,...,yd) 6 3 e`fa`mr.
Erdib >>.1>.1=. (>1.>) gefeoid mjhiradsjyi` midb`ifjdjd lezufuy (x) e`sud vi
y (x3) 6 3, y (x3) 6 3, . . . , y
(d1) (x3) 6 3
besu`udu sak`yersa nu murufma lezufy (x) 6 3 mr.
]adf >>.19. Yyar>>.1=mibj lezufi asjbar lezuf midjr.
>1.>) gefeoid mjhiradsjyi` midb`ifjdjd asjbar e`fayad h lezufudu nj`jyersab y 6 h vmedus ufu j`i midb`ifj(d 1) . firtiniyi jdmjrkiyinj`jrjz.]ierif >.
a3(x)y+ a1(x)y
+ a>(x)y 6 3 (>)
medus ufudu midb`ifmi yirjdi yaza`f
a3(x) (h v)
+ a1(x) (h v) + a>(x)h v 6 3 a3(x) (h
v+ >hv+ h v) + a1(x) (hv+ h v) + a>(x)h v 6 3
(a3(x)h+ a1(x)h + a>(x)h)
63
v+ a3(x) h v+ (>a3(x) h
+ a1(x) h) v 6 3
j`i(>a3(x) h
(x) + a1(x) h(x)V w6 3 (>
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> mx (>xy + >y6 3 midb`ifjdjd njr lezufu e`fab uziri, firtinidjdjdmjrkidfisj yedtifjdj bu``adarab `jdiir nakfsz lezufu nu`uduz.
L ezuf (> + 1, a1(x) 6 >x, a>(x) 6 > mjr. (> mx
6 ixp >xx>+1 mx
x> mx 6
x>+16u>xmx6mu
ixp muu x>
mx
6
ixp (`du)
x> mx6
u + l
x> mx 6
x>+16u
x> + 1
x> mx6
1 +
1
x>
mx
6 x 1x
+ l6 x 1x
+ l
(>) medusufudmid lezufu
y6 x
x 1
x+ l
6 x> + lx 1
sib`jdmi i`mi imirjz. ]ierif >>.1i keri
y 6 l>x + l< x> + lx
16 A x> 1+ Nx
kidi` lezufmur.
Erdib >y =xy+ =y6 3midb`ifjdjd njr lezufu e`sud. Firtinidjd jdmjrkidfisj yedtifj j`i kidi` lezufu nu`uduz.
L ezufy 6 xv
medusufudu midb`ifmi yirjdi yaza`f.
x> (xv) =x (xv)+ =xv 6 3 x> (>v+ xv) =x (v+ xv) + =xv6 3
x>
(xv >v) 6 3 xv >v6 3 w6v xw 6 >w mw
w 6 >
mx
x `d w 6 > `d x + `d l w6 lx>
w6vv 6 lx> mv6 lx>mx
v 6 lx=.1)
sanjt batsay` gefeoid midb`ifjd lezufuduy6 ifx (>=.>)
e`arab araya`f. Nuda keri
mby
mxb 6fbifx, b6 1, . . ,d
turiv`irjdj (>=.1) midb`ifjdmi yirjdi yazarsab
ifx
a3fd + a1f
d1 + +... + ad 1 f + ad
6 3
midb`ifjdj i`mi imirjz. ifx 6 3 e`mukudmada3f
d + a1fd1 + +... + ad 1f + ad6 3 (>=.=.1. (>=.,...,fd kjnj d tadi bebu e`fa mr. Nu beb`ir njrir njrirmidb`if(>=.1)ma yirjdi bedursa gir mihasdma njr ezi` lezuf i`mi imj`ilibtjr. Nuda keri midb`ifjd beb`irjdikeri lezufu jrmi`iyi`jf.
>=.1. 1. Muruf? Ayrb rii` beb`ir. (>=.=.>. (>=.,...,fdkjnj harb` (ayrb) rii` bebu var jsi(>=.1)midb`ifjdjdkidi` lezufu
y6 l1if1x + l>i
f>x + ... + ldifdx (>=.=)
sib`jdmimjr. l1,l,...,ld biyfi sanjt`irmjr.
Erdib >=. 6 3. Ney`ili midb`ifjd lezufu (f 1) (f >) 6 3 f1 61, f> 6 >. (>=.=)
mad
y6 l1ix + l>i
>x
lezufmur.
Yyar >=.=. Erdib>=.x hedbsjyed`ar `jdiir nakfszmr.
[
ix, i>x
6
ix i>xix >i>x6 ix 6 3
mr.
Erdib >=.9. y
y
1>y6 3, y (3) 6 f 1> 6 3lezuf`irf1,> 6
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>=. \ANJ] BA]\AZJ@J GEFEOID @JDII^ AMM 81
besu` ardmad
l1+ l> 6 6 >
y 6 i=.>. >. Muruf? ]ibrar beb`ir.
]ierif >=.8. (>=. 6 6 fb 6 f) nu murufma(>=.1) midb`ifjdjd tibrar imid beb`iri bars`b ki`id lezufu
l1+ l>x + + lbxb1
ifx (>=.9)
sib`jdmimjr. fb+1, fb+>, . . . , fd njrnjrjdmid harb` rii` beb`ir e`fab uziri(>=.1) midb`ifjdjd kidi` lezufu
y6
l1+ l>x + + lbxb1
ifx + lb+1ifb+1x + lb+>i
fb+>x + ... + ldifdx (>=.8)
herfudmamr.
Erdib >=.;. yjv + 9y+ 8y+ =y 7y6 3 AMM djd lezufudu nu`uduz.L ezuf Midb`ifjd barabtirjstjb midb`ifj
f= + 9f< + 8f> + =f 7 6 3vi midb`ifjd beb`irj >, >, >, 1 mjr. Nuda keri tibrar imid f 6 > bebudi bars`b ki`id lezuf (>=.9) mad
y1 6
l1+ l>x + l
i>x
vif> 6 1 bebudi bars`b ki`id lezufy> 6 l=i
x
mjr. Nuda keri kidi` lezuf(>=.8) mid
y6 y1+ y>6 l1+ l>x + l i>x + l=ixmr.
Erdib >=.7. y(9) >y(=) + y 6 3 midb`ifjdjd lezufudu nu`uduz.L ezuf Mjhiradsjyi` midb`ifjd barabtirjstjb midb`ifj
f9 >f= + f< 6 3vi lezuf`irf1,>,. Muruf? ]ibrar` beb`ir mjr. Nuda keri
1, x, x>, ix, xix
`jdiir nakfsz lezuf`irmjr. ]ierif >x + l + l=ix + l9xi
lezufmur.
>=. 61 e`fab uziri befp`ibs say`ar a+ jn, n6 3 herfudmamr. (a, n rii` say`ar). a jnsaysda jsia + jn djd befp`ibs is`idjkj midjr.
]ierif >=.13. (>=.les nx) (>=.;)
e`arab yaz`r.
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8> 9. ZYB\IB FI^]INIMID @JDII^ AMM
Sreeh. Kidi` lezufub1i
(a+jn)x + b>i(ajn)x
e`arab yazanj jrjz. Nuramaij 6 les + j sjd
Iu`ir herfu`udu bu``adrsab
b1i(a+jn)x + b>i
(ajn)x 6 b1iaxijnx + b>i
axijnx
6 iax
b1ijnx + b>i
jnx6 iax (b1(les nx + j sjd nx) + b>(les nx j sjd nx))6 iax ((b1+ b>)les nx + j (b1 b>)sjd nx)6 iax (l1sjd nx + l>les nx) , l1 6 b1+ b>, l>6 b1 b>
i`mi imirjz.
Erdib >=.11. y 8y+ >9y6 3 AMM djd lezufudu nu`uduz.L ezuf AMM djd barabtirjstjb midb`ifj f> 8f + >9 6 3 mr. Mid`ifjd beb`irj
f 6 8
68 7j
> 6 < =j
a 6 =.;)
y 6 iles =x)
lezufmur.
]ierif >=.1>. (>=.x + + lbxb1
sjd nx + iax
l1+ l>x + + lbxb1
les nx (>=.7)
e`arab yaz`r.
Erdib >=.1y+ y 6 3 AMM djd lezufudu nu`uduz.L ezuf Mjhiradsjyi` midb`ifjd barabtirjstjb midb`ifj
f8 >f< + 1 6 3lezuf`irf1,>,< 6 1, 1>+ j
, 1>+ j
, vi mjkir`irj isjt e`fab uziri 1. Muruf? Ayrb rii` beb`ir & >. Muruf& x les
, i
1>x sjd
, xi
1>x les
, xi
1>x sjd
`jdiir nakfsz lezuf`irmjr. ]ierif >xi
x + lx les
+ l=i
1>x sjd
+ l9xi
1>x les
+ l8xi
1>x sjd
lezufmur.
>9. Gefeoid e`fayad AMM
(>1.1) gefeoid e`fayad midb`ifj i`i a`a`f?
a3(x)y(d) + a1(x)y
(d1) + ... + ad 1(x)y+ ad(x)y 6 H(x)
]ierif >9.1. v hedbsjyedu (>1.1) gefeoid e`fayad midb`ifjd njr lezufu viu hedbsjyedu(>1.>) gefeoidmidb`ifjd lezufu jsiu + v hedbsjyedu ma(>1.1) gefeoid e`fayad midb`ifjd njr lezufumur.
Erdib >9.>. y 6 x hedbsjyedu y+ y 6 x gefeoid e`fayad midb`ifjdjd njr lezufumur. sjd x hedbsjyeduy+y6 3gefeoid AMM djd lezufumur. ]ierif>9.1i keriy 6 x+sjd xhedbsjyedu may +y 6 x midb`ifjdjdlezufumur.
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>9. GEFEOID E@FAZAD AMM 81.>) gefeoid midb`ifjd kidi` lezufu e`ad yl (lefp`ifidtary hudltjed)djd das` nu`udmukudu nj`jyeruz.\jfmj jsi afalfz, gefeoid e`fayad nu midb`ifjd njr ezi` lezufu e`ad vi njr biyfi sanjt say jlirfiyid yp(partjlu`ar jdtikra`) lezufudu vi sedul e`arab (>1.1) gefeoid e`fayad midb`ifjdjd kidi` lezufudu nu`fabtr.yp djd nu`udfas j`i j`kj`j e`arab njrbal fitet ki`jstjrj`fjstjr. Nu fitet`ar, sras j`i,
(j)? Ni`jrsjz Batsay`ar Fitemu(jj)? Sarafitri`irjd Mikjsjfj Fitemu
>9.1. Ni`jrsjz Batsay`ar Fitemu.
]adf >9.) iax, a 6 3(9.=. xx, sjd() , les(>x + /=) YL hedbsjyed`armr.
]adf >9.9. @jdiir nakfsz YL hedbsjyed`ardd bufisjdi bsala YL bufisj midjr.
YL hedbsjyedu YL bufisj1 xd
xd, xd1, . . . , x , 1
> iax {iax}< sjd (nx + l)
viya les (nx + l){sjd(nx + l) , les(nx + l)}
= xdiax
xdiax, xd1iax, . . . , x iax, iax
9 xd sjd(nx + l)
viyax
d
les(nx + l)
{xd sjd(nx + l) , xd les(nx + l) ,xd1 sjd(nx + l) , xd1 les(nx + l) ,
. . . , x sjd(nx + l) , x les(nx + l) ,sjd(nx + l) , les(nx + l)}8 iax sjd(nx + l)
viyaiax les(nx + l){iax sjd(nx + l) , iax les(nx + l)}
; xdiax sjd(nx + l)viyaxdiax les(nx + l)
{xdiax sjd(nx + l) , xdiax les(nx + l) ,xd1iax sjd(nx + l) , xd1iax les(nx + l) ,. . . , x iax sjd(nx + l) , xiax les(nx + l) ,iax sjd(nx + l) , iax les(nx + l)}
Erdib >9.8. h(x) 6x> sjd x hedbsjyedudud YL bufisj
x> sjd x, x> les x, x sjd x, x les x, sjd x, les x
Zedtif(>1.1) gefeoid e`fayad midb`ifjdmi
a3(x)y(d) + a1(x)y(d1) + ... + ad 1(x)y+ ad(x)y 6 H(x)
H(x) hedbsjyedudu YL hedbsjyed`ardd `jdiir befnjdasyedu e`fas murufudma nu yedtifj uku`ayanj`jrjz.
(1) H 6l1u1+ l>u>+ + lfuf e`sud. Nurama u1, u>, . . . , uf YL hedbsjyed`armr.(>) u1, u>, . . . , uf YL hedbsjyed`arda bars`b ki`id YL bufi`irjdj\1, \>, . . . , \ f e`arab ni`jr`iyi`jf.(9.;. y >y ix 13sjd x AMM djd lezufudu nu`uduz.
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8= 9. ZYB\IB FI^]INIMID @JDII^ AMM
L ezuf
y >y
>f
< 6 3
barabtirjstjb midb`ifjdjd lezufudmidf1 6 6 1 yl 6l1iix herfudmamr. \jfmj gefeoide`fayad tirjfi ki`i`jf?
>ix 13 sjd xhedbsjyedu(1)
ix, sjd x, les x
YL hedbsjyed`ardd `jdiir befnjdasyedu e`arab yaz`fstr. Nuda keri YL bufi`irj(>)
\1 6 {ix}\> 6 {sjd x, les x}
(bufi`irjdjd i`ifad`ar j`i gefeoid bsfd hedbsjyed`ar njr njrj j`i ayd mikj`mjr. Nu sbbma kili`jf.(9) < vi =. muruf`ar sak`adfamkdmad njr sedrabj muruf.
(8) Ezi` hedbsjyedu nu bufi`irjd e`usturfus e`muku i`ifa`ard befnjdasyedu sib`jdmi yaz`mkd varsaya`f?
yp 6 Aix + N sjd x + Lles x
yp 6 Aix + N les x Lsjd x
yp 6 Aix N sjd x Lles x
midb`ifmi yirjdi yazarsab
y >y ix 13sjd x Aix N sjd x Lles x > (Aix + N les x< (Aix + N sjd x + Lles x) 6 >ix 13sjd x
=Aix + (=N+ >L)sjd x + (=L >N)les x 6 >ix 13sjd x=A 6 >,
=N+ >L 6 13=L >N 6 3 A6 1
>, N6 >, L6 1
yp 6 1>
ix + > sjd x les xezi` lezuf vi
y 6 yl+ yp 6 l1ii
x 1>
ix + > sjd x les xkidi` lezufmur.
Erdib >9.7. y y6 >x
>
+ i
x
+ >xi
x
+ =i
6 3barabtirjstjb midb`ifjdjd lezufudmid f1 6 1, f> 6 > yl 6 l1ix +l>i>x herfudmamr. \jfmj gefeoide`fayad tirjfi ki`i`jf?
>x> + ix + >xix + =i, ix, xix, i
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>9. GEFEOID E@FAZAD AMM 89
YL hedbsjyed`ardd `jdiir befnjdasyedu e`arab yaz`fstr. Nuda keri YL bufi`irj(>)
\1 6
x>, x, 1
\> 6 {ix}\< 6 {xix, ix}\= 6
iix +Ixix + H iAx + Iix + >xMix + xIix + A + >Mix + >Iix + =xMix + xIix + :H iMix
midb`ifmi yirjdi yazarsab
y y 6 >x> + ix + >xix + =iA + >Mix + >Iix + =xMix + xIix + :H iAx + Iix + >xMix + xIix + Ax> + Nx +
+Ixix + H ix> + ix + >xix + =iA L) + (>N 8A) x + >Ax>
+>MiIxix
+ (>I H) ix
6 >x>
+ ix
+ >xix
+ =iA L 6 3
>N 8A 6 3>A 6 >
>M 6 =
>I 6 >>I H 6 1 A6 1, N 6 , H 6 , I6 1
yp 6 x> + + >x>ix xix x
+ x>
+ + >x>
ix
xix
les x AMM djd lezufudu nu`uduz.L ezuf
yjv + y6 3
gefeoid midb`ifjd lezufuf= + f> 6 3
barabtirjstjb midb`ifjdjd lezufudmidf1,> 6 3, fx+ lles x
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88 9. ZYB\IB FI^]INIMID @JDII^ AMM
hedbsjyedu(1)
x>, sjd x, les x
YL hedbsjyed`ardd `jdiir befnjdasyedu e`arab yaz`fstr. Nuda keri YL bufi`irj(>)
\1 6
x>, x, 1
\> 6 {sjd x, les x}\< 6 {les x, sjd x}
( 6 \< e`mukudmad\1, \> bufi`irj i`i a`dlabtr.(=) \1 6
x>, x, 1
bufisj x, 1 hedbsjyed`ard jlirmjkjdmid vi x, 1 hedbsjyed`ar gefeoid midb`ifjd lezufu
jljdmi e`mukudmad \1 bufisjdjd i`ifad`ard x> j`i larpp yidjmid muzid`irjz? \1 6{x=, x}. Ney`ili nubufidjd i`ifad`ar turmis bsfd lezufudmibj hedbsjyed`armad harb`mr. Zjdi\>bufisjdjd i`ifad`ar, turmisbsfd lezufudmi e`mukudmad, i`ifad`ar x j`i larpa`f? \> 6 {x sjd x, x les x}(9) Zidjmid muzid`idid bufi`ir srasy`a?
\1 6 {x=, x}\> 6 {x sjd x, x les x}
sib`jdmimjr.(8) Ezi` hedbsjyedu nu bufi`irjd e`usturfus e`muku i`ifa`ard befnjdasyedu sib`jdmi yaz`mkd varsaya`f?
yp 6 Ax= + Nx< +Lx> + Mx sjd x + Ix les x
yp 6 >Lx + Iles x + M sjd x + xM les x xIsjd x + =Ax< + yp 6 >L+ 8Nx + >M les x >Isjd x xIles x xM sjd x + 1>Ax>
yp 6 8N+ >=Ax les x >=A =M les x + =Isjd x + xIles x + xM sjd x+>L+ 8Nx + >M les x >Isjd x xIles x xM sjd x + 1>Ax>
6 + = sjd x >les x>=A + >L 6 3
8N 6 3
1>A 6 M 6 >>I 6 = A6 1
=, N6 3, L6
yp 6
1
= x=
+ x sjd x + >x les x
ezi` lezuf vi
y6 yl+ yp6 l1+ l>x + lx les x
kidi` lezufmur.
>9.>. Sarafitri`irjd Mikjsjfj Fitemu.
y + a (x) y6 n (x)herfudmabj midb`if`iri `jdiir midb`ifjdj gatr`aya`f. Midb`ifjd
y + a (x) y6 3
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>9. GEFEOID E@FAZAD AMM 8;
turmis midb`ifjdi bars`b ki`id lezufu
y 6 l ixp
a (x) mx
,
e`arab i`mi itfjstjb.(13.1) jd kidi` lezufudu
y6 l (x)ixp
a (x) mx
.
herfudma arafstb bj nu tierj e`arab parafitri`irjd mikjsjfj fitemu mur. Zedtifj >. firtinmid mikjsbidbatsay` njr AMM jljd ad`ata`f?
a3(x) y+ a1(x) y
+ a>(x) 6H(x) (>9.1)
midb`ifjdj i`i a`a`f. y1 viy>
a3(x) y+ a1(x) y
+ a>(x) 6 3 (>9.>)
gefeoid midb`ifjdjd `jdiir nakfsz lezuf`irj e`sud`ar. Nu murufma
yl 6 l1y1+ l>y>(>9.>) gefeoid midb`ifjdjd njr kidi` lezufumur. Sarafitri irjd mikjsjfj yedtifjdmi l1, l> sanjt`irj yirjdihedbsjyed`ar musudu`ur. Zadj
yp 6 v1(x) y1(x) + v>(x) y>(x) (>9.9.1) midb`ifjdjd njr lezufumur. Nurama v1(x) vi v>(x) hedbsjyed`ardd jhami`irjdj nu`fafzkiribfibtimjr. I`jfjzmi > nj`jdfiyid var vi 1 besu` e`arabma (>9.. njr ib besu`a jgtjyal muyu`fabtamr.
yp 6 v1(x) y1(x) + v>(x) y
>(x) + v
1(x) y1(x) + v
>(x) y>(x) (>9.=)
Nuramay p hedbsjyedudu nu`famad edli maga edli nagsitfjs e`mukufuz >. besu` e`arab
v1(x) y1(x) + v>(x) y>(x) 6 3 (>9.9)
a`a`f. \jfmj hedbsjyedud >. turivjdj a`rsab
yp 6v1(x) y1(x) + v>(x) y
> (x) + v
1(x) y
1(x) + v
>(x) y
>(x) (>9.8)
i`mi imirjz. (>9.9.9.=) , (>9.8) jhami`irjdj (>9.1) midb`ifjdmi yirjdiyazarsab
a3(x) Rv1(x) y1 (x) + v>(x) y
>(x) + v
1(x) y
1(x) + v
>(x) y
> (x)V +
a1(x) Rv1(x) y1(x) + v>(x) y
>(x)V + a>(x) Rv1(x) y1(x) + v>(x) y>(x)V
6 H(x)
v1(x) Ra3(x) y1(x) + a1(x) y1(x) + a>(x) y1 (x)V +
v>(x) Ra3(x) y>(x) + a1(x) y>(x) + a>(x) y> (x)V + a3(x) Rv
1(x) y
1(x) + v
>(x) y
>(x)V
6H(x)(>9.;)
i`mi imirjz. y1 viy> gefeoid midb`ifjdjd `jdiir nakfsz lezuf`irj e`mukudmada3(x) y1 (x) +a1(x) y1(x) +a>(x) y1(x) 6 3 vi a3(x) y> (x) + a1(x) y
>(x) + a>(x) y>(x) besu``ar sak`adr. (>9.;) mi nud`ar mjbbati a`rsab
a3(x) Rv1(x) y
1(x) + v
>(x) y
> (x)V 6H(x) (>9.7)
besu`udu i`mi imirjz.Ney`ili nj`jdfiyid v1(x) viv>(x) hedbsjyed`ar jljd v1(x) y1 (x) + v
>(x) y>(x) 6 3
v1(x) y1 (x) + v
>(x) y
>(x) 6
H(x)a3(x)
(>9.:)
midb`if sjstifjdj i`mi imirjz. Batsay`ar mitirfjdadt
[Ry1(x) , y>(x)V 6
y1(x) y>(x)y1(x) y
>(x)
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87 9. ZYB\IB FI^]INIMID @JDII^ AMM
e`arab i`mi imirjz. y1 viy> `jdiir nakfsz lezuf`ir e`mukudmad [redbjyid[Ry1(x) , y>(x)V 6 3 mr. Ney`ili(>9.:) sjstifjdjd tib lezufu varmr. Xiv 1(x) viv
>(x) hedbsjyed`ard
v1(x) 6 3 y>(x)
H(x)
a3(x) y
>(x)y1 (x) y>(x)y1 (x) y>(x)
6 H(x) y>(x)
a3(x) [Ry1 (x) , y>(x)V (>9.13)
v>(x) 6
y1(x) 3y1(x) H(x)a3(x)y1 (x) y>(x)y1 (x) y>(x) 6
H(x) y1(x)
a3(x) [Ry1(x) , y> (x)V (>9.11)
Ney`ili (>9.13) vi (>9.11) mi jdtikra` a`mkfzmav1(x) viv>(x) hedbsjyed`ard nu`uruz.
Erdib >9.13.y+ y6 tad x
AMM djd lezufudu nu`uduz.L ezuf Gefeoid bsfd lezufu
yl 6 l1sjd x + l>les x
e`mukudmad kidi` lezufuyp 6 v1(x)sjd x + v>(x)les x
e`sud.yp 6 v1(x)les x v>(x)sjd x + v1(x)sjd x + v>(x)les x
Nuramay p hedbsjyedudu nu`famad edli maga edli nagsitfjs e`mukufuz >. besu` e`arab
v1(x)sjd x + v>(x)les x6 3
a`a`f. \jfmj hedbsjyedud >. turivjdj a`rsab
yp
6
v1(x)sjd x
v>(x)les x + v1
(x)les x
v>(x)sjd x
i`mi imirjz. Midb`ifmi yirjdi yazarsab
v1(x)les x v>(x)sjd x6 tad xi`mi imirjz. Ney`ili nj`jdfiyidv1(x) viv>(x) hedbsjyed`ar jljd
v1(x)sjd x + v>(x)les x6 3
v1(x)les x v>(x)sjd x6 tad xmidb`if sjstifjdj i`mi imirjz. Xiv 1(x) viv
>(x) hedbsjyed`ard
v1(x) 6
3 les xtad x sjd x
sjd x les xles x
sjd x
6 sjd x v1(x) 6 les x + l(x) 6
sjd x 3les x tad xsjd x les xles x sjd x 6
sjd> x
les x 6
les> x 1les x
v>(x) 6
(les x sil x) mx 6
(;;).jdtksjd x `d (sil x + tad x) + l=
e`arab i`mi imirjz. Nuda keri kidi` lezuf
yp(x) 6 v1(x)sjd x + v>(x)les x
6 ( les x + l
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>9. GEFEOID E@FAZAD AMM 8:
e`arab i`mi imirjz.
y 6 yl+ yp 6 l1sjd x + l>les x + l + 11f
8 6 3
f
1,>,, i>x + lx + v(x) i
>x + x + v(x) i
>x + :v(x) i>x +
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;3 9. ZYB\IB FI^]INIMID @JDII^ AMM
midb`if sjstifjdj i`mi imirjz. Xiv 1(x) , v>(x) viv
x ii>x x xi x + l=
v>(x) 6
ix 3 ii>x 3
ix
=i>x
ix
ix i>x ii>x x
>ixi>xii>x vx + l8
v>(x) 6
(les x sil x) mx 6
(;;).jdtksjd x `d (sil x + tad x) + l=
? ? ? ? 3 ? e`arab i`mi imirjz. Nuda k eri kidi` lezuf
yp(x) 6 v1(x) ix + v>(x) i
>x + vx +
1
=i>x + l8
i
xix + l=ix + ix + l9i
>x
1
=
ix + l8ii
>x +lx 1
=ix + l8i
x + Lixix
nu`uruz.
>8. Laulgy-Iu`ir midb`ifj
]adf >8.1.
a3xdy(d) + a1x
d1y(d1) +... + ad 1xy+ ady6 3 (>8.1)
midb`ifjdi Laulgy-Iu`ir midb`ifj midjr.
]ierif >8.>. x6 it medus ufu j`i(>8.1) midb`ifj sanjt batsay` `jdiir midb`ifi medus ur.
Erdib >8.y >xy+ >y6 xy
mx> 6
1
x
m>y
mt>mt
mx 1
x>my
mt 6
1
x>
m>y
mt> my
mt
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LGAS]I^ 8
\anjt batsay` Jbjdlj Firtinimid Mjhiradsjyi` Midb`if`irjd
Yyku`afa`ar
>;. \a`df Garibitj
\jstif midki bedufudma njr fjbtar asak mekru libj`jr vi nrab`rsa kjmjp-ki`fi garibitj yapar. Nuda sa`dfgaribitj viy tjtrisjf garibitj midjr. Gava mjridlj jgfa` imj`jrsi, njr nasba miyjs`i surtudfi buvvitj yeb banu`imj`jrsi, ljsjf yubarma vi asakma ayd jfjt ir arasdma njr kjmjp-ki`fi garibitj yaplabtr. Nudanasjt garfedjbgaribit midjr. Ikir surtudfi varsa ljsfjd sa`df kid`jkj kjttjbli aza`lab vi ljsjf njr suri midki bedufuda
ki`jlibtjr. Ljsfjd nu tur garibitjdisirnist seduf`u garfedjb garibitj midjr. Nudud yad sra, ljsfi mivaf`vi sjdusejma` njr hedbsjyed`a jhami imj`inj`id njr garfedjb itbj virj`inj`jr. Nu murufma ljsfjd garibitj zer`adfsseduf`u garfedjb garibittjr.
1? Nasjt Garfedjb Garibit
Geebi yasasda keri libj`id vi nrab`ad njr yayd nuyub`uku, yayd uzafa fjbtar j`i eradt` njr kirj lakrlbuvvittjr. Nuda keri buvvitj,
H6 bxj`i kestirjrjz. Nuda keri mjhiradsjyi` midb`if
fm>x
mt>
6
bx
m>x
mt>
+ 3x6 3, 3 6 b
fe`arab virj`jr.
>? \irnist \eduf`u Garfedjb Garibit
Midki bedufudmadx uzab`kdma nu`uda njr ljsfi itbj imid sedmurulu buvvitjd nuyub`uku
nmx
mt
j`i virj`jr. n 4 3 sanjttjr vi sedmurfi batsays e`arab am`admr`r. \edmurulu buvvitjd yedu, ljsfjd garibityedudi zttr. Nuda keri mjhiradsjyi` midb`ifj
fm>x
mt> 6 bx n mx
mt f m
>x
mt> + n
mx
mt + bx 6 3
e`arab virjrjz.
;
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;= 8. \ANJ] BA]\AZJ@J JBJDLJ FI^]INIMID MJHI^AD\JZI@ MIDB@IF@I^JD YZKY@AFA@A J
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>7. I@IB]^JB MIX^I S^EN@IF@I^J ;9
(>)
> m>x
mt> 6 = mx
mt :7x + les t > m
>x
mt> + =
mx
mt + :7x6 les t
edli gefeoid midb`ifj lezi`jf
> m>x
mt> + =
mx
mt + :7x6 3 >f> + =f + :7 6 3 f> + >f + ; 6 3
f1,>6 j
8 1 xl(t) 6 it
l1les
8t + l>sjd
8t
sjfmj ki`i`jf gefeoid e`fayad midb`ifjd ezi` lezufudi YL yedtifjdj bu``adrsab?
u1 6 les t \1 6 {les t, sjd t} xp(t) 6 A les t + N sjd t xp 6 A sjd t + N les t xp 6 A les t N
> m>x
mt> + =
mx
mt + :7x6 les t > (A les t N sjd t) + = (A sjd t + N les t) + :7 (A les t + N sjd t) 6 les t
(:8A + =N)les t + (=A + :8N)sjd t6 les t :8A + =N6 1=A + :8N6 3 A6 8
9;;, N6
1
>sjd
8t
+ 8
9;;les t +
1
>
8 6 1> 6 11< 7=7
8
x (t) 6 it
8
9;;les
8t 1
1< 7=7
8sjd
8t
+
8
9;;les t +
1
>7. I`ibtrjb Mivri Sren`if`irj
Id nasjt i`ibtrjb mivri`irj, oidirater viya pj` kjnj i`ibtrjb baydak vi idirojyj bu``adad njr rizjster (erdikjdi`ibtrjb afpu`u) (risjstadli) nu`udad njr sirj mivrimjr. Ikir mukfi bapat`rsa njr Jabf rizjsteri mekruabalab vi njr ve`tao musfisjdi sinip e`ulabtr. Zadj rizjsterud jbj uludmabj petadsjyi` harb` e`ulabtr. Nupetadsjyi` harb viya ve`tao mususu jsi Xe`tfitri midj`id njr i`t j`i e`lu`inj jr. Ibitrjb mivri`irjdmibj nasjt njr
^isjstr
Mfi
Baydab
bura` Bjrlgeffi bura` e`arab ama`admr`r, Nu bura`a keri, i`ibtrjb mivrisjdmibj tuf ve`tao`ard tep`af, tep`afbuvviti isjttjr. ]ep`af buvvitj I(t) j`i kestirjrsib (ifh-i`iltrefetjvi herli)
X@+ X + XL6I(t)
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;8 8. \ANJ] BA]\AZJ@J JBJDLJ FI^]INIMID MJHI^AD\JZI@ MIDB@IF@I^JD YZKY@AFA@A J
X@+ X + XL6I(t)
^ rizjster (rijstadli), L bapasjter (lapaljter), J jdmubter (dmulter). J6J(t) i`ibtrjb mivrisjdmibj abf viq6 q(t) bapasjtermibj adj i`ibtrjb yubudu kestirfib uziri
q 6 J
sib`jdmi njr nakdt fivluttur. Egf badududa keri rizjster uzirjdmibj ve`tao musub`uku abf j`i mekrueradt`mr?
X 6 ^J
nurama ^ rizjsterud mjridljmjr vi sanjttjr. Bapasjtermibj ve`tao mususu jsi bapasjtermibj i`ibtrjb yubu j`ieradt`mr vi
XL6 1
Lq
e`arab virj jr. NuramaLbapasjtadstr (lapaljmadli). \ed e`arab jdmubtermibj ve`tao mususu jsi abfd mikjsjfgz j`i eradt`mr?
X@6 @J
@ sanjtjdi jdmjbterud jdmubtads midjr (gidry j`i e`lu`ur) (jdmultadli). Bjrlgeffi bura da keri
@J+ 1
Lq+ ^J6 I(t)
nakdtsd i`mi imirjz. Nurama turiv a`rsab viq 6 J
jhamisjdi keri
@q+ ^q+ 1
Lq 6 I(t)
@J+ 1
Lq+ ^J 6 I(t)
@J+ ^J+ 1L
J 6 I(t) >.firtinimid midb`ifj ^L@ midb`ifj e`arab ama`dmr`r.
Erdib >7.1. Mjridlj 13 e`ad njr mjridl ti`j vi ez jdmubsjyed batsays @ 6 3.> gidry e`ad njr nenjd,i`ibtrefeter buvvitj =3 ve`t vi bapasjtads e`fayad njr mekru abf uritiljdi sirj e`arab nak`adyer. Nas`adkltaabf vi i`ibtrjb yubu e`famkda keri.girgadkj zafadmabj i`ibtrjb yubudu vi abf nu`uduz.
L ezuf Zubarmabj midb`if keri
^ 6 13, @6 3.>, I(t) 6 =3, XL yeb3.>q+ 13q 6 =3 f> + 93f6 3 f1 6 3, f> 6 93
ql(t) 6 l1+ l>i93t
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>7. I@IB]^JB MIX^I S^EN@IF@I^J ;;
ezi` lezuf jljd?
u1 6 1, \1 6 {1} \1 6 {t} qp(t) 6 At 93A6 =3 A6 =
9
qp(t) 6
=
9 t q(t) 6 l1+ l>i93t
+
=
9 t
q(3) 6 3 l1+ l> 6 3, J6 q 6 93l>i93t +=9 J(3) 6 3 93l> 6 =
9 l>6 =
>93 l1 6 =
>93
q(t) 6 =
>93
i93t 1+=
9t, J(t) 6
=
9
1 i93t
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LGAS]I^ ;
@jdiir Mjhiradsjyi` Midb`if \jstif`irj
\u ada bamar njr nj`jdfiyid`j njr mjhiradsjyi` midb`if`irj jdli`imjb. \jfmj jsi jbj nj`jdfiyid hedbsjyedu nu`udadjbj mjhiradsjyi` midb`ifj vi kidi` e`arab d nj`jdfiyid hedbsjyedu nu`udad d mjhiradsjyi` midb`ifj jdli`iyilikjz.Maga edliMjhiradsjyi` midb`if sjstif`irjdj tadtfab j`i nas`aya`f.
>:. @jdiir sjstif tur`irj (Jbj nj`jdfiyid`j jbj midb`if)
x vi y nj`jdfiyid hedbjyed`ar e`fab uziri 1. firtinimid mikjsbid batsay` `jdiir mjhiradsjyi` midb`if sjstifjkidi` e`arab,
a1(t) x+ a>(t) y+ a(t) y+ n< (t) x + n= (t) y6 H>(t)
(>:.1)
sib`jdmi yaz`r. Nurama t nakfsz mikjsbidmjr.
Erdib >:.1.
>x+ x + y 6 t>x (t) y+ a(t) y+ n(t) (>:.>)
sib`jdmi yaz`r.(>:.1) sjstifjdj ezi` e`arab
x 6 a11(t) x + a1>(t) y+ H1(t)y 6 a>1(t) x + a>>(t) y+ H>(t)
(>:.:.>.
x6 t>x + (t + 1) y+ tx t1. firtinimid, sanjt batsay` njr sjstifmjr.
d sayma mjhiradsjyi` midb`ifmid e`usad sjstifj kidi` e`arab,
x1 6 a11(t) x1+ a1>(t) x>+ + a1d(t) xd+ H1(t)x> 6 a>1(t) x1+ a>>(t) x>+ + a>d(t) xd+ H>(t)...xd6 ad1(t) x1+ ad>(t) x>+ + add(t) xd+ Hd(t)
(>:.=)
sib`jdmi yaz`r.
;:
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73 ;. @JDII^ MJHI^AD\JZI@ MIDB@IF \J\]IF@I^J
x6 < m> t + 9
m
tt< 6 17t + 19t> >t@ RkV
Erdib e`sud. h6t>, k6 sjd t jljd
@ RkV 6 @ RkV
e`mukudu k estirjdjz.
L ezuf
@ RkV 6
mt>
+ > sjd t
+ 9m
mt
+ > sjd t
> + > sjd t6 + 17
@ RhV 6
mt>
t>
+ 9m
mt
t> > t>
6 >t> + 13t + 8@ RkV 6
mt>(sjd t) + 9
m
mt(sjd t) > (sjd t)
6 9 les t 9sjd t @ RkV 6 t> + 13t + 8+ >(9 les t 9sjd t)6 + 17
@ RkV 6 @ RkV
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y+
`jdiir sjstifjdj
(>M My 6 t(>M+ M+ 7) y 6 >
j`i kestirinj`jrjz. Nuda keri@1, @>, @M >M@M+ M+ 7
Zedtif(. midb`if jsi @> epirateru j`i larp`p tarah tarahalbart`r?
@=@1x + @=@>y6 @=h1(t)@>@@=y6 @>h>(t)
(@=@1 @>@h> (t) ()sanjt batsay`, tib nj`jdfiyid`j `jdiir AMM i`mi imj`jr. Ne`uf9mabj yedtif`ir bu``ad`arab lezuf nu`udur.Nidzir sibj`mi nj`jdfiyidy hedbsjyedudu nu`fab jljd jsi (. midb`if jsi @1 epirateru j`i larp`p tarah taraha lbart`r?
@M My 6 t(>M+ M+ 7) y 6 >
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7> ;. @JDII^ MJHI^AD\JZI@ MIDB@IF \J\]IF@I^