48 Eng Case Stds

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Cbhtahtsbhtahts Ahj`haar`hj CfsaStueàs

6 Fee`hj sbuhe wfvas 77 Cbkpiax raprasahtft`bh bd sbuhe wfvas fhe sbuhe raact`bh :3 Sahs`t`v`ty bd k`crbplbhas 6?5 Wadrfct`bh 631 Gafk eadbrkft`bh 614 Eaact`bh bd f gafk 7?: Gucni`hj bd cbiukhs 74< Kfx`kuk gahe`hj kbkaht dbr f kuit`pia structura 310 Aquft`bh bd tla curva bd f cfgia stxae ft twb ahe pb`hts 5?

6? Cr`t`cfi wftar la`jlt h fh bpah clfhhai 51

66 S`kpia paheuiuk 1?67 Kbt`bh bd f paheuiuk 1663 Ula dfii`hj shbwfna 1465 Sftaii`ta kbt`bh 4?61 Sftaii`ta brg`ts 4364 Qhearjrbuhe rfìwfy s`jhfi ibcft`bh :76: Laft cbheuct`bh tlrbujl f w`ha caiifr rbbd :46< Uwb-e`kahs`bhfi bw pfst f cyi`her`cfi bgstfcia <?60 Uwb-e`kahs`bhfi bw bd f v`scbus i`què bh fh `hci`hae pifta <47? Dbrca bh f cyi`hear eua tb f twb-e`kahs`bhfi strafk`hj fhe sw`ri`hj bw 06

Iafrh`hj

Ul`s Zbrngbbn bddars f cbkpaheùk bd Ahj`haar`hj Cfsa Stueàs fs fh fee`t`bhfi

tafcl`hj fhe iafrh`hj rasburca tb tlbsa `hciueae `h tla prav`bus wbrngbbns.

Ula Zbrngbbn `s `htaheae tb ra`hdbrca hbt`bhs bd kbeaii`hj us`hj fh `hcrafs`hjiy wèa

crbss sact`bh bd kftlakft`cfi taclh`quas.

butcbkas

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Ahj`haar`hj

Cfsa Stueàs

5<@htrbeuct`bh

Ul`s Zbrngbbn bars f cbkpaheùk bd Ahj`haar`hj Cfsa Stueàs fs fh fee`t`bhfi tafcl`hj fheiafrh`hj rasburca tb tlbsa `hciueae `h Zbrngbbns 6 tb 5:. Ula kftlakft`cfi tbp`cs fhe tla raiavfhtZbrngbbns fra i`stae ft tla gaj`hh`hj bd afcl Cfsa Stuey.

Ula Zbrngbbn `s `htaheae tb ra`hdbrca hbt`bhs bd kbeaii`hj us`hj fh `hcrafs`hjiy wèa crbss sact`bh

bd kftlakft`cfi taclh`quas.

7 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey 6

Fee`hj sbuhe wfvas

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Sacbhe brear BEAs ^60[

@htrbeuct`bh

Zfvas fhe kftlakft`cfi kbeais dbr wfvas fra `kpbrtfht `h kfhy fspacts bd ahj`haar`hj. Ul`s k`h`-

cfsa stuey `htrbeucas tlasa raprasahtft`bhs fhe `htrbeucas btlar wfva-raiftae èafs. @t kfnas usabd eàraht`fi aquft`bhs fhe tr`jbhbkatry. Ula hftura fhe tla tark`hbibjy bd wfva kbt`bh kfy ga`htrbeucae w`tl radarahca tb bscìift`bhs bd f kfss bh f spr`hj. F duhefkahtfi prbparty bd fhy i`hafrspr`hj wfs e`scbvarae gy Wbgart Lbbna `h tla 6:tl cahtury fhe `s nhbwh fs Lbbna‐s ifw. Lbbnaaxprassae l`s sthe`hj gy stft`hj tla pbwar bd fhy spr`hjy gbey `s `h tla sfka prbpbrt`bh w`tl tla

axtahs`bh. Qs`hj kbra kbearh tark`hbibjy, Lbbna‐s ifw kfy ga stftae fs `d fhy aifst`c kftar`fi `s

e`stbrtae, tla dbrca prbeucae `h tla kftar`fi `s prbpbrt`bhfi tb tla fkbuht bd e`stbrt`bh. Fh fijagrf`cstftakaht bd tla ifw, w`tl tla hbtft`bh D dbr tla dbrca, x dbr tla fkbuht tlft tla spr`hj `s axtaheaefhe N dbr tla cbhstfht bd prbpbrt`bhfi`ty, `s

D ; ∗Nx (6)

Ula hajft`va s`jh fr`sas gacfusa tla dbrca `s `h tla bppbs`ta e`ract`bh tb tla axtahs`bh.

Cbhsèar tla kbt`bh bd f kfss (K ) fttfclae tb f lbr`zbhtfi (kfssiass) spr`hj tla btlar ahe bd wl`cl `s cbhhactae tb f stxae pb`ht. Fdtar fh `h t`fi e`spifcakaht tla kfss kbvas bh f dr`ct`bhiasslbr`zbhtfi sièa.Ula kbt`bh bd tla kfss cfh ga eascr`gae gy Hawtbh‐s sacbhe ifw, wl`cl stftas tlft dbrca `s aqufitb kfss kuit`piàe gy fccaiarft`bh (f)=

D ; Kf

Fccaiarft`bh `s tla rfta bd clfhja bd vaibc`ty v, `.a. ev

et, fhe vaibc`ty `s tla rfta bd clfhja bd pbs`t`bh

x, `.a. exet

, sb f ; e7xet7

fhe

D ; K e7x

et7 (7)

Aquft`hj (6) fhe (7), tla aquft`bh eascr`g`hj tla kbt`bh bd tla kfss `s

K e7x

et7 ; ∗Nx

Ula jaharfi sbiut`bh tb tl`s sacbhe brear eàraht`fi aquft`bh (saa 60.5, sugsact bh 7) `s

x ; C cbs

N K

t

+ E s`h

N K

t

LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs

3




@d tla kfss stfrts drbk rast fhe `s raiafsae wlah tla e`spifcakaht drbk tla rast pbs`t`bh `s C , tlahx ; F ft t ; ?. Fisb xː ; ? ft t ; ?. Qsa bd a`tlar bd tlasa `h`t`fi cbhe`t`bhs kafhs tlft E ; ?, sb

x ; Fcbs

N

K t

Ul`s raprasahts bscìift`bh fgbut x ; ? w`tl fkpi`tuea F fhe fhjuifr draquahcy πh ;

(N/K )(tla fhjuifr hfturfi draquahcy bd tla kbt`bh). F cbkpiata cycia bd bsc iift`bh bccurs `h t`ka U sucl tlft U ; 7Ϗ

(N/K ) fhe U `s tla par`be bd tla bscìift`bh.

Ula hfturfi draquahcy bd tla rasuit`hj kbt`bh, d h, cfh ga eaeucae drbk πh ; 7Ϗd h. Ula rasuit`hjkbt`bh `s nhbwh fs s`kpia lfrkbh`c kbt`bh.

F`r kbiacuias galfva fs skfii kfssas bh spr`hjs fhe kbva tb fhe drb eur`hj tla pfssfja bd f sbuhewfva. Skfii fkpi`tuea sbuhe wfvas cbhs`st bd s`husbèfiiy fitarhft`hj cbkprass`bhs fhe rfradfct`bhs`h tla kaeùk suppbrt`hj tla sbuhe. Uyp`cfiiy sbuhe wfvas cbhtf`h kfhy draquahcàs s`kuitfhabusiy.Ft sbka stxae ibcft`bh, tla fssbc`ftae vfr`ft`bh bd f`r prassura p w`tl t`ka t fgbut tla fkgàht br

aquì`grùk prassura `h f s`hjia draquahcy (sbkat`kas cfiiae kbhbclrbkft`c) pifha sbuhe wfvacfh ga raprasahtae gy=

p ; Fcbs(πt + τ)

wlara F `s tla fkpi`tuea bd tla wfva, d ; π/7Ϗ `s `ts draquahcy fhe τ `s f stxae fhjia cfiiae tlaplfsa.S`hca s`ha fhe cbs`ha wfvas fra èaht`cfi axcapt dbr f plfsa eàrahca bd Ϗ/7 (wl`cl cfh ga `hciueaegy kbe`dy`hj tla plfsa fhjia τ), wa cfh fisb raprasaht f sbuhe wfva gy=

p ; Fs`h(πt + τ)

.

_rbgiak `h wbres

(f) D`he tla fkpi`tuea fhe plfsa rasuit`hj drbk tla cbkg`hft`bh bd twb sbuhe wfvas bd tlasfka draquahcy, tla strst bd wl`cl lfs tw`ca tla fkpi`tuea bd tla sacbhe fhe tla sacbhelfs f plfsa iafe bd 51◤.

(g) Zlft `s tla rasuit bd cbkg`h`hj twb uh`t fkpi`tuea sbuhe wfvas tlft lfva zarb plfsagut f draquahcy eàrahca bd ε/7Ϗ2

Kftlakft`cfi stftakaht bd prbgiak

(f) D`he axprass`bhs dbr F fhe τ sucl tlft

F s`h(πt + τ) ; 7 s`h(πt) + s`h(πt + Ϗ

5)

(g) D`he axprass`bhs dbr F fhe π sucl tlft

F s`h(πt) ; s`h(πt) + s`h^(π + ε )t[

Kftlakft`cfi fhfiys`s

(f) Qs`hj tla tr`jbhbkatr`c dbrkuif s`h(\ + P ) ≦ s`h \ cbs P + cbs \ s`h P , j`vasF s`h(πt + τ) ; Fs`h(πt)cbs τ + F cbs(πt)s`h τ





; 7 s`h(πt) + s`h(πt + Ϗ

5)

; 7 s`h(πt) + s`h(πt)∝

7+

cbs(πt)∝ 7

; (7 + 6∝ 7

)s`h(πt) + 6∝ 7

cbs(πt)

Ul`s kafhs

F cbs τ ; 7 + 6∝

7fhe F s`h τ ;

6∝ 7

Sb

F7 cbs7 τ + F7 s`h7 τ ; F7 ;

7 +

6∝ 7

7

+ 6

7 ; 1 + 7

∝ 7 ; :.<7< (tb 3 e.p.)

`.a. F ; 7.< (tb 6 e.p.)

Sugst`tut`bh `h a`tlar bd tla aquft`bhs `hvbiv`hj gbtl F fhe τ j`vas τ ; 65.43◤.

(g) Qs`hj tla tr`jbhbkatr`c raift`bhsl`p

s`h \ + s`h P ≦ 7s`h

\ + P

7

cbs

\ ∗ P

7

, w`tl \ ; πt fhe P ; (π + ε )t

j`vas

s`h(πt) + s`h^(π + ε )t[

; 7 s`h

πt + (π + ε )t

7

cbs

πt ∗ (π + ε )t

7

; 7 cbs

ε

7t

s`h

(π +

ε

7)t

Ul`s lfs tla dbrk F s`h(πt) `d

F ; 7 cbs

ε

7t

fhe π ; π +

ε

7 ;

π + (π + ε )

7

π raprasahts tla fhjuifr draquahcy bd tla cbkg`hae wfva.


1




@htarpratft`bh

(f) Ula fkpi`tuea bd tla wfva dbrkae gy cbkg`h`hj wfvas w`tl eàraht fkpi`tueas fheplfsas `s hbt s`kpiy tla suk bd tla `he`vèufi cbkpbhaht wfva fkpi`tueas (wl`cl wbuiega 7+6 ; 3). Ula plfsa bd tla cbkg`hae wfva `s iass tlfh tlft bd tla iafe`hj cbkpbhaht.

(g) Ula fkpi`tuea bd tla cbkg`hae wfva vfràs w`tl t`ka gatwaah +7 fhe ∗7 ft fh fhjuifrdraquahcy tlft `s lfid bd tla eàrahca gatwaah fhjuifr draquahcàs bd tla cbkpbhahtwfvas. Ula draquahcy bd tla wfva dbrkae gy cbkg`h`hj uh`t (br sfka) fkpi`tuea cbkpb-hahts w`tl eàraht draquahcàs `s tla kafh bd tla draquahcàs bd tla cbkpbhaht wfvas.F i`stahar wbuie lafr tla cbkg`hae wfva fs f sbuhe w`tl uctuft`hj vbiuka `.a. wbuielafr gafts gatwaah cbkpbhaht wfvas. Dbr axfkpia `d twb sbuhes w`tl draquahcàs bd 7?? Lz fhe 7??1 Lz fra pifyae tbjatlar tla rasuit`hj wfva lfs f draquahcy bd 7??7.1Lz fhe f vbiuka tlft uctuftas ft f draquahcy bd 1 Lz. Fitlbujl tla prassura wbuie gauctuft`hj ft f draquahcy bd 7.1 Lz, tla hajft`va pfrt bd tla uctuft`bh wbuie hbt galafre.





y e u t

S a s

f C j h ` r a

a h

`

j h A


Cbkpiax raprasahtft`bh bd sbuhe wfvas fhe sbuhe raact`bh

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Cbkpiax hukgars ^6?[

@htrbeuct`bh

Cbkpiax axpbhaht`fi axprass`bhs turh but bdtah tb ga kbra cbhvahàht tb lfheia tlfh tla tr`jbhb-

katr`c duhct`bhs (s`ha fhe cbs`ha) tb raprasaht sbuhe wfvas fhe tla`r `htarfct`bh w`tl surdfcas. Ulat`ka eapaheahca `h f s`hjia draquahcy sbuhe wfva kfy ga wr`ttah

a∗`πt ; cbs(πt) ∗ ` s`h(πt)

Sb Waâ∗`πt[ ; cbs πt

@d F `s rafi tlah, Wa^Fa∗`πt[ ; Fcbs πt.

S`kìfriy, us`hj a∗`(πt+τ) ; cbs(πt + τ) ∗ ` s`h(πt + τ) tlah

Wa^Fa∗`(πt+τ)[ ; Fcbs(πt + τ)

Ula fhjia τ `s cfiiae tla plfsa bd tla sbuhe wfva fhe eatark`has tla fkpi`tuea wlah t ; ?, `.a.F cbs τ. F pifha sbuhe wfva `s f wfva `h wl`cl tla wfva drbhts (cbhtburs bd aqufi plfsa) fra pifhafhe pfrfiiai. @d f pifha sbuhe wfva `s raactae ft tla gbuhefry gatwaah twb kae`f, tlah, `h jaharfi,tla raactae wfva lfs eàraht fkpi`tuea fhe plfsa drbk tla `hcèaht wfva. Ft tla gbuhefry, tla`hcèaht wfva w`tl fkpi`tuea F` kfy ga raprasahtae gy tla cbkpiax axpbhaht`fi axprass`bh

p` ; Fà∗`πt

Ula raactae wfva kfy ga raprasahtae gy tla cbkpiax axpbhaht`fi axprass`bh

pr ; Fra`(πt+τ).

Ula raactae wfva lfs tla sfka fhjuifr draquahcy π, fkpi`tuea Fr fhe eàrs `h plfsa drbk tla`hcèaht wfva gy tla plfsa fhjia τ.Ula rft`b bd tla raactae wfva tb tla `hcèaht wfva ( p`/pr) `s cfiiae tla raact`bh cbafficàht (W)bd tla gbuhefry fhe `s f cbkpiax hukgar . @t eapahes bh tla plys`cfi prbpart as bd tla twb kae`fa`tlar sèa bd tla gbuhefry.Ula cbkpiax pifha wfva raact`bh cbafficàht `s j`vah gy

W ; pr

p`;

Fra∗`(πt+τ)

Fà∗`πt ;

Fr

Fà∗`τ

@d tla raact`bh cbafficàht bd f gbuhefry `s wr`ttah fs f + `g, `.a. f `s tla rafi pfrt fhe g `s tla`kfj`hfry pfrt, tlah

f + `g ; Fr

F`

a∗`τ ; Fr

F`

(cbs τ ∗ ` s`h τ)


:




Lahca f ; Fr

F`cbs τ fhe g ; ∗Fr

F`s`h τ

E`vè`hj tla sacbhe bd tlasa axprass`bhs gy tla strst, j`vas

tfh τ ;

∗g

f

br frctfh(

∗g

f

) ; τ (6)

Fisb

|W| ;∝

f7 + g7 ;

F7

r

F7`

cbs7 τ + F7

r

F7`

s`h7 τ ; Fr

F`

(7)

Aquft`bhs (6) fhe (7) raifta tla cbkpiax raact`bh cbafficàht plfsa fhe fkpi`tuea tb `ts rafi fhe`kfj`hfry pfrts.

_rbgiak `h wbres

Dbr f pfrt`cuifr gbuhefry, tla raact`bh cbafficàht `s ?.< ∗ ?.5`=

(f) D`he tla rft`b bd tla raactae fkpi`tuea tb tla `hcèaht fkpi`tuea>

(g) D`he tla plfsa eàrahca gatwaah tla raactae wfva fhe tla `hcèaht wfva

(c) Zlft `s `kpiàe fgbut tla plfsa sl`dt ft f surdfca `d tla raact`bh cbafficàht `s +6 br∗62


(f) Qsa Aquft`bh (6) tb eatark`ha tla raact`bh cbafficàht fkpi`tuea>

(g) Qsa Aquft`bh (7) tb eatark`ha tla plfsa>

(c) Qsa Aquft`bhs (6) fhe (7) tb eatark`ha cbrraspbhe`hj plfsa clfhjas.


(f) |W| ; |?.< ∗ ?.5`| ;

(?.<)7 + (?.5)7 ;∝

?.< ≌ ?.<055

(g) τ ; frctfh(∗ g

f) ; frctfh(

?.5

?.<) ; frctfh(?.1)

Lahca τ

≌?.5434 rfe`fhs

≌74.1:◤

(c) @d |W| ; +6 tlah g ; ?. Sb τ ; frctfh(∗ g

f) ; frctfh(?) ; ?

@d |W| ; ∗6 ; a`Ϗ tlah τ ; Ϗ

< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




@htarpratft`bh

(f) Ula kfjh`tuea bd tla raact`bh cbafficàht `s ?.<0. Ul`s kafhs tlft tla fkpi`tuea bd tlaraactae wfva `s ?.<0 bd tla `hcèaht wfva.

(g) Ulara `s f plfsa clfhja bd 74.1:◤

ft tla gbuhefry.

(c) τ ; ? kafhs zarb plfsa clfhja ft tla gbuhefry. F gbuhefry ft wl`cl tlara `s hb clfhja`h fkpi`tuea br plfsa bh raact`bh `s cfiiae fcbust`cfiiy lfre.

τ ; Ϗ kafhs tlft tlara `s 6<?◤ plfsa clfhja ft tla gbuhefry. F gbuhefry ft wl`cltla raactae wfva `s 6<?◤ but bd plfsa w`tl tla `hcèaht wfva gut tlara `s hb clfhja `hfkpi`tuea `s cfiiae f prassura raiafsa gbuhefry.


0






_rbgiak `h wbres

Ula k`crbplbha e`fplrfjk `s kbeaiiae fs f ft r`jè pifta sugmact tb s`hjia draquahcy sbuhe wfvas`hcèaht ft f j`vah fhjia (saa D`jura 3.6). Ear`va fh axprass`bh dbr tla eapaheahca bd tla fvarfjasbuhe prassura bh tla pifta bh tla `hcèaht sbuhe draquahcy fhe usa tl`s axprass`bh tb jrfpl tladraquahcy eapaheahca bd 71 kk fhe 67.1 kk wèa k`crbplbhas dbr sbuhe ft 51◤ `hcèahca.


J`vah tlft tla prassura vfr`ft`bh fgbut fkgàht bd fh `hcèaht pifha sbuhe wfva trfvaii`hj `h tlapbs`t`va x-e`ract`bh `s raprasahtae gy

p ; Fcbs(πt ∗ nx) (6)

wlara F `s tla fkpi`tuea, n (; 7Ϗ/ξ) `s tla wfvahukgar cbrraspbhe`hj tb wfvaiahjtl ξ, π (; 7Ϗd wlara d `s tla draquahcy) `s tla fhjuifr draquahcy, x `s tla cbbre`hfta `h tla e`ract`bh bd trfvai, t `stla t`ka fhe tla pifta wètl `s I k,

(f) wr`ta ebwh tla prassura ft tla cahtra bd tla pifta, `d tla cahtra bd tla pifta `s tla br`j`h(x ; ?) fhe t ; ?.

(g) wr`ta ebwh fh axprass`bh dbr tla prassura bh tla pifta wlah t ; ?

(c) wr`ta ebwh tla raift`bhsl`p gatwaah , x fhe μ wlara `s tla e`stfhca drbk tla cahtrafhe lahca fh axprass`bh dbr tla prassura ft e`stfhca drbk tla cahtra,

(e) wr`ta ebwh fh axprass`bh dbr tla fvarfja prassura fs tla `htajrfi bd tla axprass`bh dbrprassura ft e`stfhca drbk tla cahtra w`tl raspact tb gatwaah tla i`k`ts bd ¶I/7 fhecfrry but tla `htajrft`bh,

(a) fssuka F ; 6 fhe sugst`tuta I ; 71 kk br 67.1 kk, μ ; 51◤ `h tla rasuit drbk (e)>fhe jrfpl tla rasuit dbr d gatwaah ?.7 fhe 6?? nLz.


(f) Drbk (6), dbr x ; ? fhe t ; ?, tlah p ; F. Sb tla prassura lfs f kfx`kuk ft tlacahtra bd tla pifta.

(g) Zlah t ; ?, tlah p ; Fcbs(nx)

(c) x ; s`h μ, lahca ft e`stfhca drbk tla cahtra

p ; Fcbs(n s`h μ)

(e) Qs`hj rasuits drbk 65.7 bh cficuift hj tla kafh vfiua bd f duhct`bh

pfv ; 6

I

+I/7

∗I/7F cbs(n s`h μ) e ;

F

I

6

s`h μ

s`h(n s`h μ)

+I/7

∗I/7

; F

nI s`h μ

s`h

nI s`h μ

7

∗ s`h

∗nI s`h μ

7

; 7FnI s`h μ

s`h

nI s`h μ7

(7)


66




(a) Qs`hj F ; 6, I ; ?.?71 k (br ?.?671 k), n ; 7Ϗ/ξ, c ; dξ, μ ; 51◤ fhe c ; 353 ks∗6 `h (7) j`vas

pfv ; 353

∝ 7

?.?71Ϗd s`h

?.?71Ϗd

353∝

7 br pfv ; 357

∝ 7

?.?671Ϗd s`h

?.?671Ϗd

353∝

7 D`jura 3.7 slbws tla rasuit`hj jrfpl.

@htarpratft`bh

Ula rasuit (7) kafhs tlft dbr μ ; 0?◤ (hbrkfi `hcèahca) tla fvarfja prassura `s zarb wlahavar I/ξ`s fh `htajar fs prae`ctae `h tla `htrbeuct`bh. Dbr μ ; 0?◤, (7) prae`cts tlft tla fvarfja prassura lfsf k`h`kuk wlahavar I/ξ `s fh `htajar. Dbr μ ; ? (jrfz`hj `hcèahca) br dbr hafr zarb draquahcyd ↗ ?) tlah usa bd tla fpprbx`kft`bh s`h(x) ∹ x dbr skfii x (rfe`fhs) `h (7) j`vas

i`knI s`hμ↗?

pfv(d ) ↗ F

Sb, dbr jrfz`hj `hcèahca br hafr zarb draquahcy, tla s`kpia k`crbplbha kbeai prae`cts tlft tlafvarfja prassura `s `heapaheaht bd draquahcy.

? 7? 5? 4? <? 6???.00001

?.00004

?.0000:

?.0000<

?.00000

6.?????

Draquahcy nLz

F v a r f j a _ r a s s u r a

71 kk

67.1 kk

D`jura 3.7= _rae`ctae draquahcy raspbhsa bd 71 kk fhe 67.1 kk wèa k`crbplbhas

Hbta tlft tla prae`ctae fvarfja prassura fhe lahca tla sahs`t`v`ty bd tla 71 kk k`crbplbha eapfrtsbh tla vary sahs`t`va scfia usae bhiy drbk tla èafi “ft‐ raspbhsa hafr 6? nLz. Ula rasuit `h (7)prae`cts tlft tla sahs`t`v`ty rakf`hs l`jl ft l`jlar draquahcàs `d tla k`crbplbha `s skfiiar. Ularaspbhsa bd f 67.1 kk k`crbplbha `s prae`ctae tb eapfrt drbk “ft‐ hafrar 7? nLz.





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Wadrfct`bhKftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Eàraht`ft bh ^66[

@htrbeuct`bh

@h Bpt`cs, Darkft‐s _r`hc`pia stftas tlft wlah crbss`hj f gbuhefry wlara tla radrfct`va `heax clfhjas,i`jlt wìi tfna tla rbuta (cbkpbsae bd strf`jlt i`ha sajkahts) bd k`h`kuk trfvai t`ka.

Gy cbhsèar`hj f rfy bd i`jlt trfvaii`hj drbk f kaeùk w`tl radrfct`va `heax ±6 tb fhbtlar w`tlradrfct`va `heax ±7, slbw tlft Darkft‐s _r`hc`pia `kpiàs

s`h μ6s`h μ7

; ±7

±6

wlara μ6 fhe μ7 fra tla fhjias tlft tla i`jlt kfnas w`tl tla hbrkfi.

y

xB

F

]

G

Z

±6

±7

μ6

μ7

(?,p)

(q,∗r)

p

xq ∗x

r

D`jura 5.6

_rbgiak `h wbres

Ula rfy `s trfvaii`hj drbk F(?, p) tb G(q, ∗r), (w`tl p, q , r 8 ?) w`tl radrfct`va `heax ±6 dbr y 8 ?fhe radrfct`va `heax ±7 dbr y 9 ?. D`he tla tbtfi trfvai t`ka bd tla rfy drbk F tb G (`h tarks bd x,tla lbr`zbhtfi cbbre`hfta bd pb`ht ] ) fhe tla sthe cbhe`t`bh dbr `t tb ga f k`h`kuk.

(@t kfy ga fssukae tlft tla spaae bd i`jlt `h f kaeùk `s j`vah gy v ; c

± wlara ± `s tla radrfct`va

`heax fhe c `s tla spaae bd i`jlt `h f vfcuuk.)

Kftlakft`cfi fhfiys`sSuppbsa tlft i`jlt trfvais drbk F tb G v`f ] (x, ?), tlah


63




• tla vaibc`ty bh stfja F] `s c/±6 fhe bh stfja ]G `s c/±7

• tla iahjtl bd stfja F] `s

x7 + p7 fhe bd stfja ]G `s

(q ∗ x)7 + r7 (gy _ytlfjbrfs)

Lahca tla trfvai t`ka bh stfja F] `s ±6

c x7 + p7 fhe bh stfja ]G `s

±7

c (q ∗ x)7 + r7

j`v`hj f tbtfi trfvai t`ka bd

U ; ±6

c

x7 + p7 +

±7

c

(q ∗ x)7 + r7

@h brear tb sthe tla vfiua bd x wl`cl k`h`k`sas tla trfvai t`ka U , sthe eU

ex fhe tla c`rcukstfhcas

uhear wl`cl eU

ex ; ?. D`rstiy

eU

ex ;

±6

c .

6

7(x7 + p7)∗

6

7 . 7x + ±7

c .

6

7(q ∗ x)7 + r7

∗ 6

7

∗7(q ∗ x)

; ±6

cx

x7 + p7∗ ±7

cq ∗ x

(q ∗ x)7 + r7

Hbw, s`hca ∯BF] ; μ6 fhe ∯ZG] ; μ7 tlah

s`h μ6 ; x x7 + p7

fhe s`h μ7 ; q ∗ x (q ∗ x)7 + r7

sb

eU

ex ;

±6

c s`h μ6 ∗ ±7

c s`h μ7

fhe eU

ex ; ? wlah

±6

c s`h μ6 ;

±7

c s`h μ7

br

s`h μ6s`h μ7

; ±7

±6

@htarpratft`bh

Ula rasuit, wl`cl kfy ga stftae tlft tla rft`b bd tla i`has bd tla fhjias bd `hcèahca fhe radrfct`bh

ft tla gbuhefry `s tla rft`b bd tla radrfct`ca `he`cas, `s nhbwh fs Shaii‐s ifw. @t `s pbss`gia tb slbwtlft

e7U

ex7 8 ? wlah

s`h μ6s`h μ7

; ±7

±6

(`h dfct e7U

ex7 8 ? dbr fii vfiuas bd x) fhe lahca tlft Shaii‐s ifw

raprasahts tla k`h`kuk rftlar tlfh tla kfx`kuk trfvai t`ka. Fitarhft`vaiy, tl`hn`hj fgbut tlaslfpa bd U ;U (x), `t `s pbss`gia tb slbw tlft fs x gacbkas vary ifrja tlah U gacbkas vary ifrja.Fisb, fs x gacbkas vary ifrja fhe hajft`va, U gacbkas vary ifrja (fhe pbs`t`va), kafh`hj tlft tlastft`bhfry pb`ht `h gatwaah tlasa twb axtrakas kust ga f k`h`kuk, `.a. Shaii‐s ifw ebas `heaaeraprasaht f k`h`kuk trfvai t`ka. @h vàw bd tla cbhsèarae galfv`bur bd tla axprass`bh dbr tla trfvait`ka bvar fii vfiuas bd x, tla rasuit cbrraspbhes tb tla bvarfii k`h`kuk.





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Gafk eadbrkft`bhKftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Easth ta `htajrfis ^63[D`rst brear BEAs ^60[

@htrbeuct`bh

Ula gafk `s f duhefkahtfi pfrt bd kbst structuras wa saa frbuhe us. @t kfy ga usae `h kfhywfys eapahe`hj fs lbw `ts ahes fra stxae. Bha ahe kfy ga r`jèiy stxae fhe tla btlar draa (cfiiaecfhtìavarae) br gbtl ahes kfy ga rast`hj bh suppbrts (cfiiae s`kpiy suppbrtae). Btlar cbkg`hft bhsfra pbss`gia. Ulara fra tlraa gfs`c qufht`tàs bd `htarast `h tla eadbrkft bh bd gafks= tla eaact`bh,tla slafr dbrca fhe tla gahe`hj kbkaht. Dbr f gafk wl`cl `s suppbrt`hj f e`str`gutae ibfe bd w (kafsurae `h H k∗6 fhe wl`cl kfy raprasaht tla said-wa`jlt bd tla gafk br kfy ga fh axtarhfiibfe), tla slafr dbrca `s eahbtae gy S fhe kafsurae `h H fhe tla gahe`hj kbkaht `s eahbtae gy K fhe kafsurae `h H k.

Ula qufht`tàs K , S fhe w fra raiftae gy

eK ez ; S (6)

fhe

eS

ez ; ∗w (7)

wlara z kafsuras tla pbs`t`bh fibhj tla gafk. @d bha bd tla qufht`tàs `s nhbwh, tla btlars cfh gacficuiftae drbk tla Aquft`bhs (6) fhe (7). @h wbres, tla slafr dbrca `s tla hajft`va bd tla ear`vft`va(w`tl raspact tb pbs`t`bh) bd tla gahe`hj kbkaht fhe tla ibfe `s tla ear`vft`va bd tla slafr dbrca.Fitarhft`vaiy, tla slafr dbrca `s tla hajft`va bd tla `htajrfi (w`tl raspact tb pbst`bh) bd tla ibfe fhetla gahe`hj kbkaht `s tla `htajrfi bd tla slafr dbrca. Ula hajft`va s`jh `h Aquft`bh (7) raacts tla

dfct tlft tla ibfe `s hbrkfiiy kafsurae pbs`t`vaiy `h tla ebwhwfre e`ract`bh wlìa f pbs`t`va slafrdbrca radars tb fh upwfre dbrca.

@d f gafk `s cfiiae “i jlt‐, tlah `ts said-wa`jlt cfh ga `jhbrae. @h kfhy cfsas tl`s `s f pardactiyrafsbhfgia fssukpt`bh gacfusa tla eadbrkft`bh cfusae gy btlar dbrcas `s dfr kbra s`jh`stcfht tlfhtlft cfusae gy said-wa`jlt.

@d f ibfe `s cfiiae “cbhcahtrftae‐ tlah `t kfy ga rajfreae fs fct`hj ft f s`hjia pb`ht. @h rafi`ty fiidbrcas fra e`str`gutae bvar sbka fraf avah tlbujl tla fraf k`jlt ga vary skfii.


61




_FWU 6

_rbgiak `h wbres

F cfhtìavarae gafk `s cbhstructae sb tlft `t `s lafvàr `h tla cahtra tlfh ft tla ahes. Kbeai tlavfr`ft`bh `h ibfe gy w ; wb 6 + s`h

Ϗz

I fhe tlah sthe tla slafr dbrca fhe gahe`hj kbkaht fs

duhct`bhs bd pbs`t`bh fibhj tla gafk.


D`he tla slafr dbrca S fhe tla gahe`hj kbkaht K dbr f cfhtìavarae gafk w`tl wa`jlt vfry`hj fs

w ; wb

6 + s`h

Ϗz

I

. (Cfhtìavarae kafhs stxae ft z ; ? fhe draa ft z ; I. Fisb, ft tla draa ahe

wa lfva S ; ? fhe K ; ?.)


Aquft`bh (7) j`vas

S ; ∗

wez ; ∗

wb

6 + s`h

Ϗz

I

ez ; ∗wb

z ∗ I

Ϗ cbs

Ϗz

I

+ C

Ft tla draa ahe (z ; I), S ; ? sb

? ; ∗wb

I ∗ I

Ϗ (∗6)

+ C wlara C `s f cbhstfht.

sb C ; wbI

Ϗ + wbI fhe S ; ∗wb

z ∗ I

Ϗ cbs

Ϗz

I

+ wb

I

Ϗ + wbI

Ul`s axprass`bh cfh ga sugst`tutae `htb Aquft`bh (6) tb j`va

K ;

Sez ;

∗wb

z ∗ I

Ϗ cbs

Ϗz

I

+ wb

I

Ϗ + wbI

ez

; ∗wbz 7

7 +

I7

Ϗ7 s`h

Ϗz

I + wb

I

Ϗz + wbIz + N wlara N `s f cbhstfht.

K ; ? wlah z ; I sb

? ; ∗wbI7

7 + wb

I7

Ϗ + wbI

7 + N sb N ; wbI7

7 ∗ wb

I7

Ϗ ∗ wbI

7

`.a.

K ; ∗wbz 77

+ I7

Ϗ7 s`h Ϗz

I + wb

IϏ

z + wbIz + wbI7

7 ∗ wb

I7

Ϗ ∗wbI

7

Jrfpls bd tla ibfe`hj, slafr dbrca fhe gahe`hj kbkahts fra j`vah `h D`jura 1.6 bh tla haxt pfja.





Ibfe (w)Slafr Dbrca (S )

Gahe`hj Kbkaht (K )_bs`t`bh (z) _bs`t`bh (z)

_bs`t`bh (z)

(f) (g)

(c)

w ;

w? s`h

Ϗz

I

I

I

I

D`jura 1.6= Ula ibfe`hj (f), slafr dbrca (g) fhe gahe`hj kbkahts (c) fs duhct`bhs bd pbs`t`bh z

_FWU 7

_rbgiak `h wbres

F i`jlt gafk suppbrts f ibfe cbhcahtrftae ft tla draa ahe. D`he tla slafr dbrca fhe gahe`hj kbkahtsfs duhct`bhs bd pbs`t`bh z .


F cfhtìavarae gafk suppbrts f cbhcahtrftae dbrca Z ft tla draa ahe. D`he tla slafr dbrca S fhegahe`hj kbkaht K fs duhct`bhs bd pbs`t`bh z fibhj tla gafk. Kbeai tla cbhcahtrftae dbrca w`tlf dbrca e`str`gutae bvar f skfii raj`bh bd tla gafk.


Ula dfct tlft tla ibfe `s cbhcahtrftae ft f s`hjia pb`ht `s hbt fh `hsurkbuhtfgia prbgiak. @t cfh gafpprbx`kftae gy f ibfe bd Z/ε sprafe bvar tla butar ε iahjtl bd tla gafk. Fs ε fpprbfclas zarb,tla s`tuft`bh ravarts tb tla bha eascr`gae `h tla axfkpia.

Aquft`bh (7) j`vas S ; ∗ wez . @h tla butar ε pfrt bd tla gafk, tl`s gacbkas

S ; ∗

Z

ε ez ; ∗Z

ε z + C.

Ft tla axtrak`ty `.a. ft z ; I , tla slafr dbrca `s zarb `.a.

? ; ∗Z

ε I + C j`v`hj C ;

ZI

ε .

Sb `h tla butar pfrt bd tla gafk

S ;

∗Z

ε

z + ZI

ε

; Z

ε

(I

∗z ) .


6:




Ft tla `hhar aeja bd wlara tla ibfe fppiàs, `.a. ft z ; I ∗ ε,

S ; Z

ε (I ∗ (I ∗ ε )) ;

Z

ε ε ; Z.

Dbr tla kf`h pfrt bd tla gafk, Aquft`bh (7) j`vas

S ; ∗

wez ; ∗

?ez ; C 6, wlara C 6 `s f cbhstfht.

@h brear dbr tla slafr dbrca tb ga cbht`hubus ft z ; I∗ ε, C 6 slbuie ga sat aqufi tb Z.

Sb

S ;

Z ? 9 z 9 I ∗ ε

Z

ε (I ∗ z ) I ∗ ε 9 z 9 I

Ula axprass`bh dbr S cfh hbw ga sugst`tutae `htb (6) tb j`va (dbr I ∗ ε 9 z 9 I )

K ;

Sez ;

Z

ε (I ∗ z ) ez ;

Z

ε

Iz ∗ z 7

7

+ C

Zlah z ; I , K ; ? sb

? ; Z

ε

I7 ∗ I7

7

+ C ;

Z

ε

I7

7 + C sb C ; ∗ZI7

7ε

fhe

K ; Z

ε

Iz ∗ z 7

7

∗ ZI7

7ε ; ∗Z

7ε (I ∗ z )7

Ft z ; I ∗ ε,

K ; ∗Z

7ε (I ∗ (I ∗ ε ))7 ; ∗Zε

7

Dbr tla pfrt bd tla gafk j`vah gy ? 9 z 9 I ∗ ε, (6) j`vas

K ;

Sez ;

Zez ; Zz + C 6

Ft z ; I ∗ ε , K kust aqufi ∗Zε

7 (tb aqufi tlft j`vah gy tla prav`bus cficuift bh) sb

Z (I ∗ ε ) + C 6 ; ∗Zε

7 `kpiy`hj C 6 ; ∗Zε

7 ∗ Z (I ∗ ε ) ; Zε

7 ∗ ZI

Lahca K ; Zz + Zε

7 ∗ ZI ; Z (z ∗ I +

ε

7) fhe sb bvar tla aht ra gafk=

K ;

Z (z ∗ I + ε

7) ? 9 z 9 I ∗ ε

∗Z

7ε (I ∗ z )7 I ∗ ε 9 z 9 I

6< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




Fs ε `s fssukae tb ga skfii `.a. cibsa tb zarb, tla pfrt ? 9 z 9 I ∗ ε gacbkas tla pfrt ? 9 z 9 I`.a. tla wlbia gafk fhe tlaradbra=

S ; Z fhe K ; Z (z ∗ I + ε

7)

∗1?

∗ 5?

∗ 3?

∗ 7?

∗ 6?

?.7 ?.5 ?.4 ?.< 6

∗?.7

∗?.5

∗?.4

∗?.<

∗6

?.04 ?.0: ?.0< ?.00∗?.??1

∗?.?6

∗?.?61

∗?.?7

∗?.?71

∗?.?3

?.7

?.5

?.4

?.<

6

?.7 ?.5 ?.4 ?.< 6

?.7 ?

.5 ?

.4 ?

.< 6

( )( )(g)(f)

(f)(f)(e)(c)

D`jura 1.7= Ula ibfe (f), slafr dbrca (g) fhe gahe`hj kbkaht (c) (pius kfjh`stae pbrt`bh (e))

@htarpratft`bh

@t `s wbrtl hbt`hj tlft f cbhcahtrftae ibfe bd Z cfusas f clfhja `h tla slafr dbrca bd Z .

Kftlakft`cfi cbkkaht

Lara tla cbhcahtrftae ibfe wfs kbeaiiae w`tl f e`str`gutae dbrca. Ul`s prasahts f daw kftlakft`cfiprbgiaks. Kftlakft`cfiiy `t k`jlt ga gattar tb kbeai tla cbhcahtrftae ibfe w`tl f eaitf duhct`bh.@h brear tb eb tl`s bha haaes f nhbwiaeja bd eaitf duhct`bhs wl`cl `s butsèa bur prasaht scbpa.


60




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Eaact`bh bd f gafkKftlakft`cfi Snìis

Ubp`c Zbrngbbn@htajrft`bh ^63[Dburtl brear BEAs ^60[Ifpifca trfhsdbrks ^7?[

@htrbeuct`bh

F uh`dbrkiy ibfeae gafk bd iahjtl I `s suppbrtae ft gbtl ahes fs slbwh `h D`jura 6. Ula eaact`bhy(x) `s f duhct`bh bd lbr`zbhtfi pbs`t`bh x fhe bgays tla bre`hfry eàraht`fi aquft`bh (BEA)

e5y(x)

ex5 ;

6

A@ q (x) (6)

wlara A `s Pbuhj‐s kbeuius, @ `s tla kbkaht bd `hart`f fhe q (x) `s tla ibfe par uh`t iahjtl ft pb`htx. Za fssuka `h tl`s prbgiak tlft q (x) ; q (f cbhstfht).

Ula gbuhefry cbhe`t`bhs fra

(f) hb eaact`bh ft x ; ? fhe x ; I

(g) hb curvftura bd tla gafk ft x ; ? fhe x ; I.

y(x)x

I

q Gafk

Ibfe

Jrbuhe y

x

D`jura 4.6= Ula eaactae gafk fhe pfrfkatars `hvbivae`h tla kftlakft`cfi dbrkuift`bh

_rbgiak `h wbresD`he tla eaact`bh bd f gafk, suppbrtae sb tlft tlft tlara `s hb eaact`bh fhe hb curvftura bd tlagafk ft `ts ahes, sugmact tb f uh`dbrkiy e`str`gutae ibfe, fs duhct`bh bd lbr`zbhtfi pbs`t`bh fibhj tlagafk.

7? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs





D`he tla aquft`bh bd tla curva y(x) fssukae gy tla gahe`hj gafk tlft sbivas tla BEA (6). Qsatla cbbre`hfta systak slbwh `h D`jura 4.6 wlara tla br`j`h `s ft tla iadt axtrak`ty bd tla gafk. @htl`s cbbre`hfta systak, tla gbuhefry cbhe`t`bhs

(`) tlft tlara `s hb eaact`bh ft x ; ?,

(``) tlara `s hb eaact`bh ft x ; I,

(```) tlft tlara `s hb curvftura bd tla gafk ft x ; ?,

(`v) tlft tlara `s hb curvftura bd tla gafk ft x ; I,

kfy ga wr`ttah

(`) y(?) ; ?

(``) y(I) ; ?

(```) e7y

ex7

x;?

(`v) e7y

ex7

x;I

; ?

Hbta tlft ey(x)

ex fhe

e7y(x)

ex7 fra raspact`vaiy tla sibpa fhe, prbvèae tla eaact`bh `s skfii, tla

rfeùs bd curvftura bd tla curva ft pb`ht (x, y).

Kftlakft`cfi fhfiys`sUlara fra twb katlbes bd prbcaae`hj=

@. @htajrft`bh bd tla BEA (ebha lara bh pfjas 77-73 fhe `h 60, pfjas 4:-4<).

@@. Ifpifca trfhsdbrk bd tla BEA (ebha lara bh pfjas 75-71 fhe `h 7?, pfjas 1?-17).


76




Sbiut`bh gy `htajrft`bh bd tla BEA

Sbiv`hj tla BEA gy tla katlbe bd `htajrft`bh raqu`ras f dbur-dbie `htajrft`bh bd Aquft`bh (6) fhetla usa bd tla gbuhefry cbhe`t`bhs (`) - (`v) tb eatark`ha tla cbhstfhts bd `htajrft`bh `hvbivae.

@htajrft`hj Aquft`bh (6) bhca

A@ e3y(x)

ex3 ; qx + F (7)

@htajrft`hj f sacbhe t`ka

A@ e7y(x)

ex7 ; qx7/7 + Fx + G (3)

@htajrft`hj f tl`re t`ka

A@ ey(x)

ex ; qx3/4 + Fx7/7 + Gx + C (5)

@htajrft`hj f dburtl t`kaA@y(x) ; qx5/75 + Fx3/4 + Gx7/7 + Cx + E. (1)

Ula gbuhefry cbhe`t`bhs (`)-(`v) ahfgia eatark`hft`bh bd tla cbhstfhts bd `htajrft`bh F, G, C,fhe E.

@h Aquft`bh (1), tla strst gbuhefry cbhe`t`bh y(?) ; ? j`vas

A@y(?) ; q Ú (?)5/75 + F Ú (?)3/4 + G Ú (?)7/7 + C Ú (?) + E ; ? wl`cl `kpiàs

E ; ? (4)

Qs`hj tla sacbhe gbuhefry cbhe`t`bh y(I) ; ? `h Aquft`bh (1) j`vas

A@y(I) ; qI5/75 + FI3/4 + GI7/7 + CI + E.

Lahca, s`hca E ; ? drbk (4)=

qI5/75 + FI3/4 + GI7/7 + CI ; ? (:)

Qs`hj tla tl`re gbuhefry cbhe`t`bh e7y

ex7

x;?

; ? `h Aquft`bh (3) j`vas

A@ e7y

ex7

x;?

; q Ú (?)7/7 + F Ú (?) + G

wl`cl `kpiàs tlftG ; ?. (<)

Qs`hj tla dburtl gbuhefry cbhe`t`bh e7y

ex7

x;I

; ? `h Aquft`bh (3) j`vas

A@ e7y

ex7

x;I

; qI7/7 + FI ; ?

wl`cl `kpiàs

F ; ∗qI/7 (0)

Ula axprass`bhs dbr F fhe G `h (<) fhe (0) fra sugst`tutae `h Aquft`bh (4) tb sthe tla ifst uhnhbwhcbhstfht C.





Lahca qI5/75 ∗ qI5/67 + CI ; ? sb

C ; qI3/75 (6?)

D`hfiiy, Aquft`bh (1) fhe tla vfiuas bd tla dbur cbhstfhts drbk (4), (<), (0), (6?) iafe tb tla sbiut`bh

y(x) ; 6A@ qx5/75 ∗ qIx3/67 + qI3x/75[ (66)

@htarpratft`bhUla prae`ctae eaact`bh `s zarb ft gbtl ahes wl`cl `s f clacn tlft tla ahe cbhe`t`bhs lfva gaahfppiàe cbrractiy. @t `s pbss`gia tb clacn tlft `t `s sykkatr`cfi fgbut tla cahtra bd tla gafk gy us`hjtla cbbre`hfta systak (\, P ) w`tl I/7 ∗ x ; \ fhe y ; P fhe var`dy`hj tlft tla eaact`bh P (\ )`s sykkatr`cfi fgbut tla vart`cfi fx`s `.a. P (\ ) ; P (∗\ ).

Sbiut`bh gy Ifpifca trfhsdbrk

Ula dbiibw`hj Ifpifca trfhsdbrk prbpartàs fra haaeae=

I

ehd (t)

eth

;

hn;6

sn∗6 eh∗nd

exh∗n

x;?

(_6)

I {6} ; 6/s (_7)

I {th} ; h!/sh+6 (_3)

I∗6

{I {d (t)

}}; d (t) (_5)

Ub sbiva f eàraht`fi aquft`bh `hvbiv`hj tla uhnhbwh duhct`bh d (t) us`hj Ifpifca trfhsdbrks tlaprbcaeura `s=

(f) Zr`ta tla Ifpifca trfhsdbrk bd tla eàraht`fi aquft`bh us`hj prbparty (_6)

(g) Sbiva dbr tla duhct`bh I{d (t)} us`hj prbpartàs (_7) fhe (_3)

(c) Qsa tla `hvarsa Ifpifca trfhsdbrk tb bgtf`h d (t) us`hj prbparty (_5)

Qs`hj tla i`hafr`ty prbpartàs bd tla Ifpifca trfhsdbrk, (6) gacbkas

Ie

5

yex5 (x)∗ I{ q A@ } ; ?.

Qs`hj (_7) fhe (_3)

s5I{y(x)} ∗5

n;6

sn∗6e5∗ny

ex5∗n

x;?

∗ q

A@

6

s ; ?. (7)

Ula dbur tarks bd tla suk fra5

n;6

sn∗6 e5∗ny

ex5∗n ; e3y

ex3

x;?

+ ee7y

ex7

x;?

+ s7ey

ex

x;?

+ s3y(?).

Ula gbuhefry cbhe`t`bhs j`va y(?) ; ? fhe e7y

ex7 ; ?, sb (7) gacbkas


73




s5I{y(x)} ∗ e3y

ex3

x;?

∗ s7ey

ex

x;?

∗ q

A@

6

s ; ?. (3)

Lara e3y

ex3

x;?

fhe ey

exx;?

fra uhnhbwh cbhstfhts , gut tlay cfh ga eatark`hae gy us`hj tla rakf`h`hj

twb gbuhefry cbhe`t`bhs y(I) ; ? fhe e7y

ex7

x;I

; ?.

Sbiv`hj dbr I{y(x)}, (3) iafes tb

I{y(x)} ; 6

s5e3y

ex3

x;?

+ 6

s7ey

ex

x;?

+ q

A@

6

s1.

Qs`hj tla i`hafr`ty bd tla Ifpifca trfhsdbrk, tla `hvarsa Ifpifca trfhsdbrk bd tl`s aquft`bh j`vas

I∗6

{I{y(x)}} ;

e3y

ex3x;?

Ú I∗6 6

s5

+

ey

exx;?

Ú I∗6 6

s7

+

q

A@ I∗6 6

s1

.

Lahca

y(x) ; e3y

ex3

x;?

Ú I∗6

3!

6

s5

/3! +

ey

ex

x;?

Ú I∗6

6

s7

+

q

A@ I∗6

5!

6

s1

/5!

Sb us`hj (_3)

y(x) ; e3y

ex3

x;?

Ú I∗6{I{x3}}/4 + ey

ex

x;?

Ú I∗6{I{x6}} + q

A@ I∗6{I{x5}}/75.

S`kpi`dy`hj gy kafhs bd (_5)

y(x) ; e3y

ex3

x;?

Ú x3/4 + ey

ex

x;?

Ú x + q

A@ x5/75. (5)

Ub usa tla gbuhefry cbhe`t`bh e7y

ex7

x;I

; ?, tfna tla sacbhe ear`vft`va bd (5), tb bgtf`h

e7y

ex7(x) ;

e3y

ex3

x;?

Ú x + q

7A@ x7.

Ula gbuhefry cbhe`t bh e7y

ex7

x;I

; ? `kpiàs

e3y

ex3

x;?

; ∗ q

7A@ I. (1)

Qs`hj tla ifst gbuhefry cbhe`t`bh y(I) ; ? w`tl (1) `h (5)

ey

exx;?

; qI3

75A@

(4)

D`hfiiy sugst`tut`hj (1) fhe (4) `h (5) j`vas





y(x) ; q

75A@ x5 ∗ qI

67A@ x3 +

qI3

75A@ x.

@htarpratft`bhUla prae`ctae eaact`bh `s zarb ft gbtl ahes wl`cl `s f clacn tlft tla ahe cbhe`t`bhs lfva gaah

fppiàe cbrractiy. @t `s pbss`gia tb clacn tlft `t `s sykkatr`cfi fgbut tla cahtra bd tla gafk gy us`hjtla cbbre`hfta systak (\, P ) w`tl I/7 ∗ x ; \ fhe y ; P fhe var`dy`hj tlft tla eaact`bh P (\ )`s sykkatr`cfi fgbut tla vart`cfi fx`s `.a. P (\ ) ; P (∗\ ).


71




y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey :

Gucni`hj bd cbiukhs

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatr`cfi èaht`tàs ^5[Kftr`cas fhe eatark`hfhts ^:[Kftr x sbiut`bh bd aquft`bhs ^<[@htajrft`bh ^63[Dburtl brear BEAs ^60[

@htrbeuct`bh

F vart`cfi gafk (nhbwh fs f cbiukh), `s sugmact tb f cbkprass`va dbrca - dbr `hstfhca, f pìifrsuppbrt`hj f ibfe, br s`kpiy `ts bwh wa`jlt. Hbrkfiiy, `d f lbr`zbhtfi dbrca `s fppiàe tb tla cbiukh,tlara wìi ga btlar dbrcas tb rastbra `t tb `ts hbh-eaactae pbs`t`bh dbr axfkpia tlrbujl tla ahesuppbrts. Lbwavar, uhear pfrt`cuifr c`rcukstfhcas (eapaheaht upbh tla cbiukh‐s prbpartàs fhe tlahftura bd tla ahe suppbrts), tlasa btlar dbrcas kfy hbt ga prasaht `.a. tlara kfy ga aquì`gr ukpbs`t`bhs btlar tlfh zarb eaact`bh.

z ; ?

z ; I

cbkprass`bh

D`jura :.6

Ula`r ax`stahca cfh ga e`sfstrbus dbr tla cbiukh. Fhy fee`t`bhfi lbr`zbhtfi ibfe cfh cfusa `t tbeaact durtlar, fhe tla cbiukh‐s fgì`ty tb suppbrt `ts ibfe `s `h ebugt. Qhear sucl cbhe t`bhs tlacbiukh kfy gucnia fhe avahtufiiy grafn, tlus gr`hj`hj ebwh `ts ibfe (a.j. f rbbd). Ula eaact`bhbd f gafk sft`sstas tla 5tl brear bre`hfry eàraht`fi aquft`bh (BEA)

A @ e5

wex5 ∗ H e

7

wex7 ; q

wlara

w `s tla eaact`bh fs f duhct`bh bd pbs`t`bh xA@ `s tla axurfi r`jè`tyH `s tla tahs`bh `h tla gafk fheq `s tla ibfe parpahe`cuifr tb tla gafk

@h jaharfi, us`hj tla easth`t`bh `h Ahj`haar`hj Cfsa Stuey 1 (Aquft`bh 7), tla slafr dbrca `s j`vah gy

S ;∗ q ez ;

∗A@

e3w

ez 3 ∗_ ew

ez + cbhstfht

Fisb, drbk Ahj`haar`hj Cfsa Stuey 1 (Aquft`bh 6)





K ;

S ez ; ∗A@

e7w

ez 7 ∗ _ + cbhstfht

Zlah tla gafk `s vart`cfi (`.a. f cbiukh), tlara wìi ga hb ibfe parpahe`cuifr tb tla gafk (q ; ?).Ulara `s fh fx`fi cbkprass`va dbrca, eahbtae _ , wlara _ ; ∗H `.a. tla tahs`bh `s hajft`va. Eahbt`hj

tla vart`cfi pbs`t`bh gy z w`tl w ; w(z ), tla aquft`bh gacbkas

A @ e5w

ez 5 + _

e7w

ez 7 ; ?

@d tl`s aquft`bh lfs bhiy zarb sbiut`bhs tlah gucni`hj cfhhbt bccur> lbwavar, `d fhy btlar sbiut`bhax`sts, tlah tla cbiukh `s i`fgia tb gucni`hj. Ula gucni`hj r`sn `s fisb `huahcae gy tla hftura bd tlatbp (z ; I) fhe gbttbk (z ; ?) suppbrts bd tla cbiukh. Ulara fra tlraa typas bd suppbrt=

(f) Guìt-`h= Ul`s kafhs hb iftarfi kbvakaht fhe hb rbtft`bh.

(g) S`kpiy-suppbrtae= Ul`s kafhs hb trfhsift`bh br curvftura gut rbtft`bh `s pbss`gia.

(c) Draa= Ul`s kafhs tlft tla suppbrt `s draa tb kbva (trfhsifta br rbtfta gut hb curvftura).

Hb iftarfi kbvakaht raqu`ras w ; ?.

Hb rbtft`bh raqu`ras ew

ez ; ?.

Hb curvftura raqu`ras e7w

ez 7 ; ?.

Ulara fra dbur kf`h pbss`gì`tàs=

6. Guìt-`h tbp fhe guìt-`h gbttbk - tla tbp fhe gbttbk bd tla cbiukh fra uhfgia tb kbvafhe hb rbtft`bh `s pbss`gia. Ula gbuhefry cbhe`t`bhs fra

(f) Ft z ; ?, w ; ? fhe ew

ez ; ?

(g) Ft z ; I, w ; ? fhe ew

ez ; ?

7. S`kpiy-suppbrtae tbp fhe guìt-`h gbttbk - tla gbttbk `s fs eascr`gae `h (`), wlìa tlatbp kfy rbtfta gut hb trfhsift`bh `s pbss`gia. Ula gbuhefry cbhe`t`bhs fra

(f) Ft z ; ?, w ; ? fhe

ew

ez ; ?

(g) Ft z ; I, w ; ? fhe e7w

ez 7 ; ? (hb kbvakaht fhe hb curvftura bd tla cbiukh)

3. Draa tbp fhe guìt-`h gbttbk - tla gbttbk `s fs eascr`gae `h (`) gut tla tbp `s draa tb kbva.

(f) Ft z ; ?, w ; ? fhe ew

ez ; ?

(g) Ft z ; I, e7w

ez 7 ; ? fhe ∗A @

e3w

ez 3 ∗ _

ew

ez ; ?

Ûla cbhe`t`bhs ft z ; I `kpiy zarb curvftura fhe zarb slafr dbrca raspact`vaiy, (saa afriàrfhe gaibw).[


7:




Eua tb tla prasahca bd fx`fi ibfe _ bh tla crbss sact`bh bd tla gafk, tla rasuitfht slafr dbrca(parpahe`cuifr tb tla fx`s bd tla gafk gadbra eadbrkft`bh) `s hb ibhjar s`kpiy q fhy kbra. _ kfnas f cbhtr`gut`bh fs waii. Ula rasuitfht slafr dbrca `s (saa D`jura :.7)

T ; q + _

ew

ex ; ∗A@

e3w

ex3 + _

ew

ex

Zlah bha cbkas tb prascr`ga gbuhefry cbhe`t`bhs, a.j., dbr f draa-ahe wlara slafr dbrca `s`hvbivae, `t `s v`tfiiy `kpbrtfht tlft tla rasuitfht slafr dbrca T `s akpibyae `hstafe bd V.

Ula draa-ahe gbuhefry cbhe`t`bhs wìi ga

K ; ∗A@ e7w

ex7 ; ? fhe T ; ∗A@

e3w

ex3 ∗ _

ew

ex ; ?

μ ; ew

ex

_

_ ewex

D`jura :.7

Ul`s scahfr`b fppiàs tb cbiukhs suppbrt`hj karaiy tla`r bwh wa`jlt.

5. S`kpiy-suppbrtae tbp fhe s`kpiy-suppbrtae gbttbk - ft afcl ahe bd tla cbiukh, rbtft`bh

`s pbss`gia, gut trfhsift`bh hbt. Ula gbuhefry cbhe`t`bhs fra

(f) Ft z ; ?, w ; ? fhe e7w

ez 7 ; ?

(g) Ft z ; I, w ; ? fhe e7w

ez 7 ; ?

@h afcl cfsa, tlara fra 7 gbuhefry cbhe t`bhs ft gbtl tla tbp fhe gbttbk bd tla cbiukh, `.a. 5gbuhefry cbhe`t`bhs tb ahfgia sbiut`bh bd f 5tl brear BEA.

_rbgiak `h wbres

Eatark`ha tla cbhe`t`bhs `h tla cbiukh (`.a. _ fs f duhct`bh bd A , @ fhe I) tlft ahfgia gucni`hj `htla dbur cfsas

6. guìt-`h tbp fhe gbttbk

7. s`kpiy-suppbrtae tbp fhe guìt-`h gbttbk

3. draa tbp fhe guìt-`h gbttbk

5. s`kpiy-suppbrtae tbp fhe gbttbk

Kftlakft`cfi stftakaht bd prbgiak@t `s hacassfry tb sbiva tla 5tl brear BEA





A @ e5w

ez 5 + _

e7w

ez 7 ; ? (6)

uhear tla dbur sats bd gbuhefry cbhe`t`bhs

(`) z ; ?

⇗w ; ? fhe

ew

ez

; ? > z ; I

⇗w ; ? fhe

ew

ez

; ?

(``) z ; ? ⇗ w ; ? fhe ew

ez ; ? > z ; I ⇗ w ; ? fhe

e7w

ez 7 ; ?

(```) z ; ? ⇗ w ; ? fhe ew

ez ; ? > z ; I ⇗ e7w

ez 7 ; ? fhe ∗A @

e3w

ez 3 ∗ _

ew

ez ; ?

(`v) z ; ? ⇗ w ; ? fhe e7w

ez 7 ; ? > z ; I ⇗ w ; ? fhe

e7w

ez 7 ; ?

wl`cl cbrraspbhe tb tla dbur sats bd ahe suppbrts eascr`gae, fhe, `h afcl cfsa sthe tla cbhe`t`bh bh_ sucl tlft sbiut`bhs btlar tlfh w ; ? fra pbss`gia.


Ula jaharfi sbiut`bh bd tla BEA `s

w(z ) ; wc(z ) + w p(z )

wlara wc(z ) `s tla cbkpiakahtfry duhct`bh fhe w p(z ) `s tla pfrt`cuifr sbiut`bh. Lara, w p ; ?s`hca tla r`jlt-lfhe sèa bd (6) `s zarb.

Ub sthe tla cbkpiakahtfry duhct`bh, strst sugst`tuta w ; aξz `htb (6) tb jat

A @ ξ5 + _ ξ7

aξz ; ?

tlah tla clfrfctar`st`c aquft`bh `s

A @ ξ5 + _ ξ7 ; ?

br

ξ7(ξ7 + η 7) ; ? wlara η ;

_

A @

w`tl rbbts

ξ6 ; ξ7 ; ?, ξ3, ξ5 ; ¶`η

sb tlft tla cbkpiakahtfry duhct`bhs fraF + Gz fhe C s`h(ηz ) + E cbs(ηz )

j`v`hj tla jaharfi sbiut`bh bd tla BEA

w(z ) ; F + Gz + C s`h(ηz ) + E cbs(ηz ) (7)

wlara F, G, C fhe E fra eatark`hae gy tla gbuhefry cbhe`t`bhs.

@h brear tb fppiy tla gbuhefry cbhe`t`bhs, tla dbiibw`hj qufht`tàs wìi fisb ga haaeae.

ew

ez ; G + Cη cbs(ηz ) ∗ Eη s`h(ηz ) (3)

e7w

ez 7 ; ∗η 7

C s`h(ηz ) + E cbs(ηz )

(5)


70




e3w

ez 3 ; η 3

E s`h(ηz ) ∗ C cbs(ηz )

sb tlft, us`hj tla ifst fhe strst bd tlasa fhe tla axprass`bh η 7 ; _

A@ , `t cfh ga slbwh tlft

A @ e3

wez 3

+ _ ewez

; Gη 7 A @ (1)

Hbw tfna afcl cfsa `h turh.

6. Guìt-`h tbp fhe guìt-`h gbttbk - fppiy`hj tla gbuhefry cbhe`t`bhs (`) tb Aquft`bhs (3)fhe (5) j`vas tla dbur i`hafr aquft`bhs

F + E ; ?G + ηC ; ?F + GI + C s`h(ηI) + E cbs(ηI) ; ?G + Cη cbs(ηI)

∗Eη s`h(ηI) ; ?

wl`cl fra wr`ttah `h kftr`x dbrk fs

K

FGC E

; ?

wlara K `s tla kftr`x

6 ? ? 6

? 6 η ?6 I s`h(ηI) cbs(ηI)? 6 η cbs(ηI) ∗η s`h(ηI)

Ciafriy F ; G ; C ; E ; ? `s f sbiut`bh tb (:) fhe `h tla cfsa wlara K `s fh `hvart`giakftr`x, tl`s `s tla bhiy sbiut`bh. Lbwavar, tl`s j`vas w ; ? sb tlft tla bhiy aquì`grùk `s tlacfsa bd hb eaact`bh. @d, lbwavar, K `s f s`hjuifr kftr`x, tlara wìi ga fh `hsth`ta hukgar bd sbiut`bhs tb (:). Ulus, gucni`hj kfy bccur wlah tla eatark`hfht bd K `s zarb. Axprass`hjs;s`h(ηI) fhe c;cbs(ηI)

eat K ; |K | ;

6 ? ? 6

? 6 η ?6 I s c? 6 ηc ∗ηs

;

6 η ?I s c6 ηc ∗ηs

∗

? 6 η 6 I s? 6 ηc

;

s cηc ∗ηs

∗ η

I c6 ∗ηs

+

6 s? ηc

∗ η

6 I? 6

; ∗ηs7 ∗ ηc7 + η 7Is + ηc + ηc ∗ η

; η (ηIs + 7c ∗ 7) s`hca s`h

7

(ηI) + cbs

7

(ηI) ; 6iafe`hj tb tla dbiibw`hj cbhe`t`bh dbr f hbh-tr`v`fi (hbh-zarb) sbiut`bh





ηI s`h(ηI) + 7 cbs(ηI) ∗ 7 ; ? (4)

Qs`hj tla tr`jbhbkatr`cfi èaht`tàs s`h7μ ≦ 7s`h μ cbs μ fhe 6 ∗ cbs7μ ≦ 7s`h7 μ Aquft`bh(4) cfh ga wr`ttah fs

5 s`h(ηI7

)ηI7

cbs(ηI7

) ∗ s`h(ηI7

)

; ?

sb tlft a`tlar

s`h(ηI

7 ) ; ? (:)

br

ηI

7

cbs(ηI

7

)

∗s`h(

ηI

7

) ; ? (<)

Sbiut`bhs dbr (:) fra ηI ; 7hϏ, h ; 6, 7, 3, . . . j`v`hj

_ ; 5h7Ϗ7A@

I7 h ; 6, 7, 3, . . . (0)

tla skfiiast bd wl`cl (h ; 6) `s 5Ϗ7 A@/I7 30.1A@/I7.

Aquft`bh (<) kfy ga axprassae fs ηI

7 cbt

ηI

7

; 6. Ula strst sbiut`bh bd (<) `s 5?.3<A@/I7

wl`cl `s ifrjar.

Fs tla ibfe _ `hcrafsas drbk zarb, `t `s tla strst vfiua ft wl`cl `t wìi gucnia wl`cl `s bd `htarast.Ul`s cr`t`cfi vfiua `s tlaradbra dbuhe drbk (0) w`tl h ; 6 tb ga

_ cr ; 5Ϗ7 A@

I7 30.1

A@

I7

7. S`kpiy-suppbrtae tbp fhe guìt-`h gbttbk - fppiy`hj tla gbuhefry cbhe`t`bhs (``) tb Aquf-t`bhs (3), (5) fhe (1) j`vas tla dbur i`hafr aquft`bhs

F + E ; ?G + ηC ; ?F + GI + C s`h(ηI) + E cbs(ηI) ; ?C s`h(ηI) + E cbs(ηI) ; ?

w`tl cbafficàht kftr`x

K ;

6 ? ? 6? 6 η ?6 I s`h(ηI) cbs(ηI)? ? s`h(ηI) cbs(ηI)

fhe eatark`hfht (hbt`hj tla axpfhs`bh fibhj tla tbp rbw fhe tla usa bd prbpartàs bd


36




eatark`hfhts `h tla iadt-lfhe 3 gy 3 eatark`hfht)

eat K ;

6 ? ? 6? 6 η ?6 I s`h(ηI) cbs(ηI)

? ? s`h(ηI) cbs(ηI)

;

6 η ?I s`h(ηI) cbs(ηI)? s`h(ηI) cbs(ηI)

∗

? 6 η 6 I s`h(ηI)? ? s`h(ηI)

;

6 η ?I ? ?? s`h(ηI) cbs(ηI)

∗ s`h(ηI)

? 66 I

; ∗ηI cbs(ηI) + s`h(ηI)

iafe`hj tb tla cbhe`t`bh dbr f hbh-zarb sbiut`bh

ηI ; tfh(ηI)

w`tl strst sbiut`bh ηI ; 5.50, sb tlft

_ cr ; (5.50)7A@

I7 7?.7

A@

I7

3. Draa tbp fhe guìt-`h gbttbk - fppiy`hj tla gbuhefry cbhe`t`bhs (```) tb Aquft`bhs (3), (5),(1) fhe (4) j`vas tla dbur i`hafr aquft`bhs

F + E ; ?G + ηC ; ?C s`h(ηI) + E cbs(ηI) ; ?G ; ?


K ;

6 ? ? 6? 6 η ?? ? s`h(ηI) cbs(ηI)? 6 ? ?

fhe eatark`hfht (dbuhe gy axpfhe`hj ebwh tla iadt cbiukh fhe tlah fibhj tla gbttbk rbw)

eat K ;

6 ? ? 6? 6 η ?? ? s`h(ηI) cbs(ηI)? 6 ? ?

;

6 η ?? s`h(ηI) cbs(ηI)6 ? ?

;

η ?s`h(ηI) cbs(ηI)

; η cbs(ηI)iafe`hj tb tla cbhe`t`bh dbr f hbh-zarb sbiut`bh





cbs(ηI) ; ?

w`tl jaharfi sbiut`bh

ηI ;

(7h

∗6)

7 Ϗ h ; 6, 7, 3, . . .

tla skfiiast sbiut`bh bd wl`cl `s dbr h ; 6, `.a.

_ cr ; Ϗ7

5

A@

I7 7.1

A@

I7

5. S`kpiy-suppbrtae tbp fhe s`kpiy-suppbrtae gbttbk - fppiy`hj tla gbuhefry cbhe`t`bhs(`v) tb Aquft`bhs (3) fhe (1) j`vas tla dbur i`hafr aquft`bhs

F + E ; ?

E ; ?F + GI + C s`h(ηI) + E cbs(ηI) ; ?C s`h(ηI) + E cbs(ηI) ; ?


K ;

6 ? ? 6? ? ? 66 I s`h(ηI) cbs(ηI)? ? s`h(ηI) cbs(ηI)

fhe eatark`hfht (dbuhe gy axpfhe`hj fibhj tla sacbhe rbw fhe tlah fibhj tla tbp rbw)

eat K ;

6 ? ? 6? ? ? 66 I s`h(ηI) cbs(ηI)? ? s`h(ηI) cbs(ηI)

;

6 ? ?6 I s`h(ηI)? ? s`h(ηI)

;

I s`h(ηI)? s`h(ηI)

; I s`h(ηI)

iafe`hj tb tla cbhe`t`bh dbr f hbh-zarb sbiut`bh

s`h(ηI) ; ?

w`tl jaharfi sbiut`bh

ηI ; hϏ h ; 6, 7, 3, . . .

tla skfiiast sbiut`bh bd wl`cl `s dbr h;6, `.a.

_ cr ; Ϗ7A@ I7

0.0A@ I7


33




@htarpratft`bh

F cbiukh `s sugmact tb gucni`hj `d tla cbkprass`va dbrca _ `s aqufi tb bha bd savarfi cr`t`cfi vfiuas.Dbr afcl typa bd ahe suppbrt, tla strst cr`t`cfi vfiua `s j`vah gaibw.

tbp suppbrt gbttbk suppbrt strst cr`t cfi ibfe

draa guìt-`h _ cr ; 7.1A@

I7

s`kpiy-suppbrtae s`kpiy-suppbrtae _ cr ; 0.0A@

I7

s`kpiy-suppbrtae guìt- h _ cr ; 7?.7A@

I7

guìt-`h guìt-`h _ cr ; 30.1A@

I7

Dbr f j`vah cbiukh, tla strst cr`t`cfi vfiua cfh ga `hcrafsae gy `hcrafs`hj tla vfiua bd tla rft`b A@ I7

(`h prfct ca, `hcrafs`hj A@ rftlar tlfh eacrafs`hj I fs tla cbiukh prasukfgiy stìi lfs tb rafcl tbtla rbbd atc). Lbwavar, f kbra aact`va strftajy kfy waii ga tb ahsura gu it-`h suppbrts dbr tlacbiukh ft afcl ahe wl`cl cfh `hcrafsa tla strst cr`t`cfi vfiua gy f dfctbr bd fpprbx`kftaiy 7, 5 br 64eapahe`hj bh tla br`j`hfi typa bd suppbrt usae.





y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey <

Kfx`kuk gahe`hj kbkaht dbr f kuit`pia structuraKftlakft`cfi Snìis

Ubp`c ZbrngbbnKfx`kf fhe K`h`kf ^67[

@htrbeuct`bhUla fgì`ty tb kbeai tla aacts bd kbv`hj ibfes bh f structura `s `kpbrtfht, dbr axfkpia `h grèjaeas`jh. Ahj`haar`hj Cfsa Stuey 1 lfs `htrbeucae tla easth`t`bh bd gahe`hj kbkaht fhe `ts raif-t`bhsl`p tb slafr dbrca. Ula gahe`hj kbkaht `s f kafsura bd tla taheahcy dbr f gafk tb gahefhe `s f duhct`bh bd gbtl tla pbs`t`bh bd tla ibfe fhe tla pbs`t`bh bd tla pb`ht wlara tla gahe`hjkbkaht `s ga`hj kafsurae.

@kfj`ha tlft f ibfe bd wa`jlt Z kbvas bh tb f gafk B_ bd iahjtl I fhe tlft tla gahe`hj kbkaht`s kafsurae ft pb`ht F, wl`cl `s f e`stfhca f drbk tla iadt ahe bd tla gafk (Saa D`jura <.6). Zlahtla ibfe `s ft f e`stfhca z drbk B, tla iadt ahe bd tla gafk, tlah tla gahe`hj kbkaht `s (prbpbrt`bhfitb)

K ; Z

I ∗ f

I z z 9 f

I ∗ z I

f z 8 f

(6)

wl`cl lfs ear`vft`va

eK

ez ; Z

I ∗ f

I (8 ?) z 9 f

∗ f

I (9 ?) z 8 f

F jrfpl bd tla vfr`ft`bh bd K w`tl pbs`t`bh z fibhj tla gafk `s slbwh `h D`jura <.6.

K

? f Iz -fx`s

F _B

D`jura <.6= Tfr`ft`bh bd gahe`hj kbkaht kafsurae ft f pfrt`cuifr pb`ht

w`tl pbs`t`bh bd f ibfe fibhj tla gafk

Ula gahe`hj kbkaht kafsura ft F stfrts drbk f vfiua bd zarb fs tla wa`jlt kbvas bhtb tla gafkft B, `hcrafsas ft f cbhstfht rfta fs tla wa`jlt fpprbfclas F, rafclas f kfx`kuk fs tla wa`jltpfssas bvar F fhe tlah eacrafsas i`hafriy tb zarb fs tla wa`jlt kbvas durtlar tb tla btlar ahe bd tla gafk ft _. Gy cbhvaht`bh, s`hca pbs`t`va K raprasahts f sfjj`hj gafk, `t `s slbwh ebwhwfres


31




`h D`jura <.6. Havartlaiass, tla gahe`hj kbkaht `s ft f kfx`kuk wlah tla wa`jlt pfssas bvar tlapb`ht wlara `t `s ga`hj kafsurae.

Bdtah, rftlar tlfh fct`hj ft f s`hjia pb`ht, f ibfe fcts ft savarfi pb`ht wl`cl rakf`h f stxae e`stfhcafpfrt a.j. f trucn, f parsbh cyci`hj, f tfgia stfhe`hj bh dbur iajs, br rfìwfy wfjbhs cbupiae tbjatlar.

D`he`hj tla pb`ht ft wl`cl tla kfx`kuk gahe`hj kbkaht bccurs cfh ga ebha `h f s`kìfr kfhhartb tlft ìiustrftae fgbva.

_rbgiak `h wbres

F “trf`h‐ cbhs`sts bd tlraa wfjbhs w`tl wa`jlts bd 4 nH, 7 nH fhe 1 nH raspact`vaiy. Afcl wfjbh`s sapfrftae drbk `ts ha`jlgbur gy f cbhstfht e`stfhca bd 6 k. @t kbvas bhtb fhe crbssas f gafkbd iahjtl 7? k. @d tla gahe`hj kbkaht `s kafsura ft f pb`ht 63 k drbk tla ahe wlara tla wa`jltspfss bhtb tla gafk, sthe tla pbs`t`bh bd tla ‐trf`h‐ tlft prbeucas tla ifrjast gahe`hj kbkaht.

Zlft kafsur`hj ibcft`bh wbuie cbrraspbhe tb tla kfx`kuk gahe`hj kbkaht fs afcl wfjbh pfssasbvar `t2


(f) Cbhstruct f duhct`bh

K ;3

`;6

Z

(I ∗ f)

I z z 9 f

(I ∗ z )

I f z 8 f

(7)

dbr tla tbtfi gahe`hj kbkaht kafsurae ft f e`stfhca f drbk tla bhsat bd tla ibfe, j`vah tlft

Z 6 ; 4, Z 7 ; 7, Z 3 ; 1, I ; 7? fhe f ; 63.

(g) Gy eàraht`ft`bh, sthe tla pb`ht ft wl`cl tl`s duhct`bh `s f kfx`kuk.

(c) Wapaft tla fhfiys`s gut dbr f jaharfi pb`ht f fhe cbhsèar tla gafk `h dbur zbhas=

zbha ^6[= 7 9 z 6 9 f (gadbra tla wa`jlts pfss f)zbha ^7[= f 9 z 6 9 f + 6 (bhiy tla 4 nH wa`jlt lfs pfssae f)zbha ^3[= f + 6 9 z 6 9 f + 7 (tla 4 nH fhe 7 nH wa`jlts lfva pfssae f)zbha ^5[= f + 7 9 z 6 9 7? (fii tlraa wa`jlts lfva pfssae f)

Cficuifta K `h afcl bd tla zbhas, fhe tlah, gy eàraht`ft`bh, sthe tla rfhja bd f dbr wl`cl f

kfx`kuk gahe`hj kbkaht `s prbeucae wlah tla strst wa`jlt (4 nH) pfssas bvar tla kafsurakahtpb`ht. Wapaft tl`s prbcass dbr tla sacbhe wa`jlt (7 nH) fhe tla tl`re wa`jlt (1 nH).


(f) Eahbta tla pbs`t`bh bd tla 4 nH wa`jlt gy z 6 fhe tla pbs`t`bhs bd tla 7 nH fhe 1 nH wa`jlts gyz 7 (; z 6 ∗ 6) fhe z 3 (; z 6 ∗ 7) raspact`vaiy.

Drbk (7) tla kbkaht eua tb wa`jlt ` `s

K ; Z

:

7? z z 9 63

63

7? (7? ∗ z ) z 8 63





sb tlft

K 6 ; 4

:

7? z 6 z 6 9 63

63

7? (7?

∗z 6) z

6 8 63

;

4 Ú :

7? z 6 z 6 9 63

4Ú

63

7? (7?

∗z 6) z

6 8 63

K 7 ; 7

:

7? z 7 z 7 9 63

63

7? (7? ∗ z 7) z 7 8 63

;

7 Ú :

7? (z 6 ∗ 6) z 6 9 65

7 Ú 63

7? (76 ∗ z 6) z 6 8 65

K 3 ; 1

:

7? z 3 z 3 9 63

63

7? (7? ∗ z 3) z 3 8 63

;

1 Ú :

7? (z 6 ∗ 7) z 6 9 61

1 Ú 63

7? (77 ∗ z 6) z 6 8 61

Z`tl fii tlraa wfjbhs bh tla gafk (`.a. 7 9 z 6 9 7?), tla tbtfi kbkaht `s

K ;

(4 Ú :

7? z 6) + (7 Ú :

7? (z 6 ∗ 6)) + (1 Ú :

7? (z 6 ∗ 7)) 7 9 z 6 9 63

(4 Ú 63

7? (7? ∗ z 6)) + (7 Ú :

7? (z 6 ∗ 6)) + (1 Ú :

7? (z 6 ∗ 7)) 63 9 z 6 9 65

(4 Ú 63

7? (7? ∗ z 6)) + (7 Ú 63

7? (76 ∗ z 6)) + (1 Ú :

7? (z 6 ∗ 7)) 65 9 z 6 9 61

(4 Ú 637?

(7? ∗ z 6)) + (7 Ú 637?

(76 ∗ z 6)) + (1 Ú 637?

(77 ∗ z 6)) 61 9 z 6 9 7?

;

06z 6 ∗ <5

7? 7 9 z 6 9 63

∗70z 6 + 65:4

7? 63 9 z 6 9 65

∗40z 6 + 7?34

7? 65 9 z 6 9 61

∗640z 6 + 31347?

61 9 z 6 9 7?

(g) Ulus

eK

ez 6;

06/7? (8 ?) 7 9 z 6 9 63

∗70/7? (9 ?) 63 9 z 6 9 65

∗40/7? (9 ?) 65 9 z 6 9 61

∗640/7? (9 ?) 61 9 z 6 9 7?

Ulus, K `s fh `hcrafs`hj duhct`bh bd z 6 dbr z 6 9 63, gut f eacrafs`hj duhct`bh dbr z 6 8 63 fhe sblfs f kfx`kuk ft z 6 ; 63.


3:




Ula rasuit`hj vfr`ft`bh bd K w`tl pbs`t`bh `s slbwh `h D`jura <.7 dbr tla par`be eur`hj wl`cl fiitlraa wfjbhs fra bh tla gafk.

kafsurakaht pb`ht

? 7 63 65 61 7?

K

D`jura <.7= Tfr`ft`bh bd K w`tl pbs`t`bh bd tla ‐trf`h‐ wlah fii bd `t `s bh tla gafk

(c) Fs gadbra, fh axprass`bh dbr tla gahe`hj kbkaht `h afcl zbha bd tla gafk cfh ga dbuhe.

K ;

4

(7?

∗f)

7? z 6 + 7

(7?

∗f)

7? (z 6 ∗ 6) + 1

(7?

∗f)

7? (z 6 ∗ 7) 7 9 z 6 9 f

4 (7? ∗ z 6)

7? f + 7

(7? ∗ f)

7? (z 6 ∗ 6) + 1

(7? ∗ f)

7? (z 6 ∗ 7) f 9 z 6 9 f + 6

4 (7? ∗ z 6)

7? f + 7

(76 ∗ z 6)

7? f + 1

(7? ∗ f)

7? (z 6 ∗ 7) f + 6 9 z 6 9 f + 7

4 (7? ∗ z 6)

7? f + 7

(76 ∗ z 6)

7? f + 1

(77 ∗ z 6)

7? f f + 7 9 z 6 9 7?

;

z 67? (74? ∗ 63f) +

6

7? (67f ∗ 75?) 7 9 z 6 9 f

z 67?

(65? ∗ 63f) + 6

7? (637f ∗ 75?) f 9 z 6 9 f + 6

z 67?

(6?? ∗ 63f) + 6

7? (6:7f ∗ 7??) f + 6 9 z 6 9 f + 7

∗63fz 67?

+ 7:7f

7? f + 7 9 z 6 9 7?

Eàraht`ft`hj w`tl raspact tb z 6 `h afcl zbha j`vas





zbha ^6[ = 7 9 z 6 9 f eK

ez 6;

6

7? (74? ∗ 63f) 8 ? dbr fii vfiè f

zbha ^7[ = f 9 z 6 9 f + 6 eK

ez 6;

6

7? (65? ∗ 63f)

8 ? dbr f 9 65?/639 ? dbr f 8 65?/63

(≌ 6?.<)

zbha ^3[ = f + 6 9 z 6 9 f + 7 eK ez 6

; 67?

(6?? ∗ 63f)

8 ? dbr f 9 6??/639 ? dbr f 8 6??/63

(≌ :.:)

zbha ^5[ = f + 7 9 z 6 9 7? eK

ez 6; ∗63f

7? 9 ? dbr fii vfiè f

F kfx`kuk vfiua bd K `s craftae `d tla ear`vft`va fgbva clfhjas drbk f pbs`t`va tb f hajft`va vfiua.

Cbkpfr`hj tla galfv`bur bd eK

ez 6`h femfcaht zbhas wìi slbw tla `huahca bd afcl bd tla wa`jlts

pfss`hj bvar f.

• Cbkpfr`hj tla strst fhe sacbhe zbhas (`.a. upbh pfssfja bd tla 4 nH wa`jlt), tla ear`vft`va

clfhjas drbk pbs`t`va tb hajft`va dbr f 8 6?.<.

• Lbwavar, `d f 9 6?.<, tla ear`vft`va rakf`hs pbs`t`va `h zbhas ^6[ fhe ^7[, gut gacbkas hajft`va`h zbha ^3[ (`.a. upbh pfssfja bd tla 7 nH wa`jlt) `d, fee`t`bhfiiy, f 8 :.:.

• @d f 9 :.:, tla ear`vft`va rakf`hs pbs`t`va `h zbhas ^6[, ^7[ fhe ^3[, gut gacbkas hajft`va `hzbha ^5[ (`.a. upbh pfssfja bd tla 1 nH wa`jlt).

@htarpratft`bh fhe cbkkaht@h tla pfrt bd tla prbgiak wlara f ; 63, tla kbkaht ft tla pb`ht bd kafsurakaht `s jraftast dbrz 6 ; 63, `.a. fs tla 4 nH wa`jlt pfssas bvar `t. @htu`t`vaiy, `t kfy saak bgv`bus tlft tla jraftast

gahe`hj kbkaht slbuie bccur ft tla `hstfht bd pfssfja bd tla ifrjast wa`jlt, gut tl`s `s hbt fiwfystla cfsa, fs wa ìiustrfta gaibw.

Kbrabvar, fhbtlar wfy bd ibbn`hj `htu`t`vaiy ft tla prbgiak, `.a. gy cbhsèar`hj tla cahtra bd kfss,j`vas fh `hcbrract fhswar. Ula cahtra bd kfss bd tla systak wìi ià gatwaah tla 4 nH fhe 7 nHwa`jlts. Zlah tla kfx`kuk gahe`hj kbkaht bccurs, tla cahtra bd kfss `s st ii fpprbfcl`hj tlapb`ht bd kafsurakaht.

Hbw cbhsèar tla rasuits drbk tla pfrt bd tla prbgiak `hvbiv`hj frg`trfry f.

• Dbr f 8 6?.< tla kfx`kuk gahe`hj kbkaht bccurs upbh pfssfja bd tla strst fhe ifrjastwa`jlt.

• Dbr :.: 9 f 9 6?.< tla kfx`kuk gahe`hj kbkaht bccurs upbh pfssfja bd tla skfiiastcbkpbhaht bd tla ibfe, gut tla bha wl`cl eatark`has wlatlar br hbt kbra tlfh lfid tla tbtfiwa`jlt lfs pfssae.

• Dbr f 9 :.: tla kfx`kuk gahe`hj kbkaht bccurs upbh pfssfja bd tla sthfi wa`jlt.


30




y e u t

S a s

f C j h ` r a

a h

`

j h A


Aquft`bh bd tla curva bd f cfgia stxae ft twb ahe pb`htsKftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatr`c èaht`tàs ^5[Lypargbi`c duhct`bhs ^4[Eàraht`ft`bh ^66[@htajrft`bh ^63[Iahjtl bd curva (`htajrft`bh) ^65[

@htrbeuct`bhCfgias fra cbkkbh cbkpbhahts bd structuras. Ibhj cfgias fra dbuhe `h suspahs`bh grèjas fhe dbrkbvarlafe pbwar i`has struhj gatwaah pyibhs. Ula slfpa fhe tla fssbc`ftae tahs`bh `h fh bvarlafepbwar cfgia `s `kpbrtfht, dbr axfkpia, wlah fssass`hj `ts eurfgì`ty fhe tla ciafrfhca `t raqu`ras.

_rbgiak `h wbresD`he f jaharfi aquft`bh eascr`g`hj tla curva bd f cfgia stxae ft twb ahepb`hts ft tla sfka la`jltfgbva tla jrbuhe. Ula i`ha eahs`ty bd tla cfgia `s f cbhstfht, φ, fhe tlara `s hb ibfe bh tla cfgiafpfrt drbk `ts bwh wa`jlt.

Kftlakft`cfi stftakaht bd tla prbgiakZa cfh erfw f e`fjrfk bd tla cfgia fs `h D`jura 0.6 wlara F fhe G fra tla ahe pb`hts.

B

sK

F G

_

x

y τ

D`jura 0.6. F cfgia stxae ft twb ahepb`hts ft tla sfka la`jlt fgbva tla jrbuhe.

Za lfva kfrnae tla tahs`bh, U , bparft`hj ft f pb`ht, _ , bh tla cfgia fibhj f tfhjaht tb tla cfgia.Ula pb`ht _ `s ft f e`stfhca s fibhj tla cfgia drbk `ts kèpb`ht fhe ft f la`jlt y fhe lbr`zbhtfie`stfhca x. @d wa cbhsèar f slbrt sact`bh bd tla cfgia, K_ , tlah tla tbtfi dbrca eua tb jrfv`tybparft`hj bh tl`s sact`bh `s sφj, bparft`hj vart`cfiiy ebwhwfres. Ula tahs`bh ft pb`ht _ wìi lfvaf vart`cfi cbkpbhaht U s`h(τ). Ula dbrcas bparft`hj `h tla lbr`zbhtfi e`ract`bh fra U ? bparft`hj fttla pb`ht K fhe U cbs(τ), tla lbr`zbhtfi cbkpbhaht bd tla tahs`bh, bparft`hj ft _ . Fs tla cfgia`s stft`c, tla dbrcas gbtl `h tla lbr`zbhtfi fhe vart`cfi e`ract`bhs kust suk tb ? `heapaheahtiy.





Sb wa lfva=

`h tla vart`cfi e`ract`bh sφj ; U s`h(τ)

fhe

`h tla lbr`zbhtfi e`ract`bh U ? ; U cbs(τ)Hbta tlft f cbhsaquahca bd tla cbhstfht lbr`zbhtfi tahs`bh `s tlft tla rasuitfht tahs`bh `s kfx`kukwlara tla vart`cfi cbkpbhaht `s kfx`kuk. E`vè`hj tla strst aquft`bh gy tla sacbhe wa jat fraift`bhsl`p dbr tfh(τ)

sφj

U ?; tfh(τ) (6)

wlara s `s tla e`stfhca fibhj tla cfgia drbk K fhe tfh(τ) `s tla jrfeàht bd tla tfhjaht tb tlacurva y ; d (x) ft tla pb`ht _ .

Ula jabkatr`cfi `htarpratft`bh bd tla ear`vft`va bd tla duhct`bh `s tlft `t `s aqufi tb tla jrfeàht bd tla tfhjaht tb tla curva, `.a.

ey

ex ; tfh(τ) (7)

Za dbuhe `h 65.5 tlft tla iahjtl bd f curva gatwaah pb`hts x ; f fhe x ; g `s j`vah gy

s ;

gf

6 +

ey

ex

7

ex (3)

Sugst`tut`hj dbr tfh(τ) drbk Aquft`bh (6) `htb Aquft`bh (7) wa jat

eyex ; sφjU ?

Sugst`tut`hj dbr s drbk Aquft`bh (3) w`tl f ; ? fhe g ; x wa jat=

ey

ex ;

φj

U ?

x

?

6 +

ey

ex

7

ex

φ `s f cbhstfht - tla i`ha eahs`ty bd tla cfgia, j `s tla fccaiarft`bh eua tb jrfv`ty (≌ 0.<6 k s∗7)fhe U ? `s f cbhstfht raprasaht`hj tla lbr`zbhtfi tahs`bh ft tla pb`ht K . Za cfh rapifca fii bd tlasa

us`hj f s`hjia cbhstfht, c, wlara c ; U ?

φj

. S`hca tahs`bh `s dbrca par uh t fraf, c lfs e`kahs`bhs bd

iahjtl.

Ula prbgiak tlah gacbkas, sthe f duhct`bh y ; d (x) wl`cl sft`sstas tla aquft`bh

ey

ex ;

6

c

x

?

6 +

ey

ex

7

ex

(5)


56





Eàraht`ft`hj gbtl sèas bd (5) w`tl raspact tb x wa jat=

e7y

ex7

; 6

c 6 + ey

ex

7

(1)

Za sbiva `t gy tla sugst`tut`bh ey

ex ; s`hl(u) fhe tlah us`hj tla èaht`ty cbsl7(u) ∗ s`hl7(u) ≦ 6.

D`rst wa eàraht`fta tl`s sugst`tut`bh w`tl raspact tb x tb bgtf`h e7y

ex7 ; cbsl(u)

eu

ex.

Sb Aquft`bh (1) gacbkas

cbsl(u)eu

ex ;

6

c

6 + (s`hl(u))7

Drbk cbsl7(u) ∗ s`hl7(u) ≦ 6 wa bgtf`h 6 + (s`hl(u))7 ; cbsl(u) sb

cbsl(u)euex

; 6c

cbsl(u)

sb eu

ex ;

6

c.

Fs tla r`jlt-lfhe sèa `s f cbhstfht tark wa cfh afsìy `htajrfta tl`s w`tl raspact tb x fhe wa sthetlft

u ;

6

c ex

sb u ;

x

c + C wlara C `s sbka cbhstfht.Za nhbw tlft s`hl(u) ;

ey

ex ; ? wlah x ; ? (tla jrfeàht bd tla tfhjaht `s ? ft tla pb`ht K wl`cl

`s wlara tla cfgia `s lbr`zbhtfi). Sb wlah x ; ?, `d s`hl(u) ; ?, tlah u ; ? j`v`hj C ; ?, sb

u ; x

c

Ufn`hj tla s`hl bd gbtl sèas bd tl`s axprass`bh fhe sugst`tut`hj ey

ex ; s`hl(u) wa jat=

ey

ex ; s`hl

x

c@htajrft`hj fjf`h w`tl raspact tb x wa sthe

y ;

s`hl

x

c

ex ; ccbsl

x

c

+ E.

Lara E `s sbka haw cbhstfht wl`cl wa cfh clbbsa frg`trfrìy fs `t wìi hbt clfhja tla slfpa bd tlacurva gut must tla la`jlt bd tla cfgia fgbva tla jrbuhe cbrraspbhe`hj tb tla skfiiast vfiua bd y.

Dbr cbhvahàhca, wa cbuie clbbsa E ; ? j`v`hj tla aquft`bh bd tla cfgia fs= y ; ccbslx

c

. S`hca

cbsl(?) ; 6, tl`s kafhs tlft tla la`jlt bd tla ibwast pb`ht bh tla cfgia `s c. @d tla ahes bd tla cfgia

fra ft ¶e, tla`r la`jlts wìi ga ft c cbsle

c, s`hca cbsl(∗z ) ; cbsl z .

Fitarhft`vaiy, `d wa stx tla la`jlt L bd tla ahe pb`hts bd tla cfgia fhe tlara `s f stxae e`stfhca 7egatwaah tlak (`.a. tla ahe pb`hts fra ft (¶e, L )), wa cbuie usa tla dfct tlft bha ahe bd tla cfgia





`s ft (e, L ) tb bgtf`h

L ; ccbsl

e

c

+ E

tb eatark`ha E, fhe lahca wr`ta

y ; ccbslx

c

+ L ∗ c cbsl

e

c

.

Ula la`jlt bd tla ibwast pb`ht bh tla cfgia `s j`vah gy tl`s aquft`bh w`tl x ; ?. S`hca cbsl(?) ; 6tl`s j`vas

y(?) ; c + L ∗ c cbsl

e

c

; L ∗ c

cbsl

e

c

∗ 6

.

@htarpratft`bh

Za lfva slbwh tlft tla aquft`bh bd tla cfgia lfhj`hj gatwaah twb pb`hts ft tla sfka la`jlt fhesugmact bhiy tb `ts bwh wa`jlt `s j`vah gy

y ; ccbslx

c

wlara tla br`j`h `s ft tla ibwast pb`ht bd tla cfgia, fhe c ; U ?φj

wlara φ `s tla cbhstfht i`ha eahs`ty

bd tla cfgia, U ? `s tla lbr`zbhtfi tahs`bh ft tla kèpb`ht bd tla cfgia fhe j `s tla fccaiarft`bh eua tbjrfv`ty. Ul`s curva `s cfiiae f cftahfry. @t `s eàraht drbk tlft bd tla cfgia bd f suspahs`bh grèja,wl`cl `s ibfeae gy tla grèja eacn. Ula aquft bh aqu`vfiaht tb (1) tlft fppiàs tb f suspahs`bhgrèja cfgia, fssuk`hj tlft tla wa`jlt bd tla cfgia `s haji`j`gia cbkpfrae tb tla wa`jlt bd tla grèja

eacn, `se7y

ex7 ;

6

c.

Ul`s lfs f sbiut`bh bd tla dbrk y ; x7

7c + C 6x+ C 7, wlara C 6 fhe C 7 fra cbhstfhts, `.a. f pfrfgbif.

Ula slfpa bd tla lfhj`hj cfgia eapahes upbh tla tahs`bh U ?. Gy kafhs bd “tahs`bhars‐ ft tla ahestl`s cfh ga vfràe. D`jura 0.7 slbws tla rasuits bd cficuift`bhs dbr cfgias kfea bd kftar`fi w`tleahs`ty 1??? nj k∗3 wlah struhj gatwaah twb kfsts 1? k l`jl fhe 6?? k fpfrt fhe sugmact tb twbeàraht tahs`bhs.

Ula ibwast pb`ht bh tla cfgia w`tl ibwar tahs`bh `s 73.55 k l`jl, wlarafs dbr tla l`jlar tahs`bh tlaibwast pb`ht `s 3:.50 k l`jl.

Ula iahjtl I bd tla lfhj`hj cfgia w`tl ahe pb`hts sapfrftae gy 7e, cfh ga eatark`hae gy raturh`hjtb Aquft`bh (3). Lahca

I ;

e∗e

6 +

s`hlx

c

7ex

;

e∗e

cbslx

c

ex ;

c s`hl

x

c

+e

∗e; c

s`hl

e

c

∗ s`hl

∗e

c

; 7c (s`hl (ec)) .


53




U ? ; 1?????? H k∗7

U ? ; 71????? H k∗7

∗4? ∗ 5? ∗ 7? ? 7? 5? 4?x

y(x)

7?

5?

4?

D`jura 0.7= Jrfpls bd tla slfpas bd lfhj`hj cfgias w`tl tla sfka stxae ahe pb`hts

(1? k l`jl fhe 6?? k fpfrt) gut eàraht tahs`bhs

Sb

I ; 7c

s`hl

e

c

. (4)

J`vah tla cfgia eatfìs fhe jabkatry usae dbr D`jura 0.7, tla iahjtls fra prae`ctae tb ga must bvar6?5 k dbr tla cfgia w`tl tla l`jlar tahs`bh fhe must uhear 6?0 k dbr tla cfgia w`tl tla ibwar tahs`bh.

Fs rakfrnae afriàr, tla kfx`kuk rasuitfht tahs`bh U bccurs ft tla ahes bd tla cfgia wlara tla

vart`cfi cbkpbhaht bd tla tahs`bh `s kfx`kuk. Ula vart`cfi cbkpbhaht bd tla tahs`bh `s j`vah gyT ; sφj. Sb `ts kfx`kuk vfiua wìi bccur wlara s ; I/7, `.a. us`hj (4),

T kfx ; φjc s`hl

e

c

; U ? s`hl

e

c

Lahca tla kfjh`tuea bd tla kfx`kuk rasuitfht tahs`bh `s j`vah gy

U 7? + T 7kfx ; U ?

6 + s`hl7

e

c

; U ? cbsl

e

c

J`vah tla kfx`kuk tahs`bh ft tla cfgia ahes, tl`s cfh ga usae tb cficuifta tla lbr`zbhtfi tahs`bh

fhe lahca tla vfiua bd c.





y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey 6?

Cr`t`cfi wftar la`jlt `h fh bpah clfhhaiKftlakft`cfi Snìis

Ubp`c ZbrngbbnEar`vft`va bd pbiyhbk`fis ^66[Kfx`kf fhe k`h`kf bd duhct`bhs ^67[@htajrft`bh ^63[

@htrbeuct`bh

Ula kbeaii`hj bd bpah clfhhai bws `h hfturfi strafks, frt`stc`fi cfhfis, `rr`jft`bh e`tclas fs waii fs`h pfrt`fiiy stiiae p`pas br tuhhais `s f cbkkbh prbgiak `h uè kaclfh`cs dbr lyerfui`c ahj`haars. Ftyp`cfi prbgiak k`jlt ga tb sthe tla la`jlt bd wftar bw`hj `h tla clfhhai dbr f j`vah bw rfta.

draa surdfca

clfhhai wfii

uè 6

uè 7

D`jura 6?.6= Tart`cfi crbss sact`bh tlrbujl fh bpah clfhhai

Garhbuii`‐s aquft`bh fibhj f strafki`ha, wl`cl `s fh axprass`bh bd aharjy cbhsarvft`bh, cfh ga usaetb kbeai uè bw fs ibhj fs tla dbiibw`hj fssukpt`bhs fra kfea=

(`) Ula uè bw`hj `h tla clfhhai `s `hcbkprass`gia fhe `hv`scè (hb dr`ct`bh)

(``) Ula bw `s stafey (cbhstfht bvar t`ka)

(```) Ula clfhhai `s strf`jlt, lbr`zbhtfi fhe bd cbhstfht wètl

(`v) Ula bw `s uh`dbrk (fii uè erbpiats `h f lbr`zbhtfi pifha lfva tla sfka spaae)

Dbr f j`vah spaae bd tla uè u, tla n`hat`c aharjy bd bw par uh`t vbiuka `s 6

7φu7 wlara φ `s tla

uè eahs ty. Ft f j`vah wftar la`jlt `h tla clfhhai (z ; l) tla pbtaht`fi aharjy par uh`t vbiukaraift`va tb tla ibwast pb`ht `h tla clfhhai (la`jlt z ; ?) `s φjl wlara j `s tla jrfv`tft`bhfi cbhstfht.Ula spac`stc aharjy duhct`bh A `s j`vah gy tla suk bd tla n`hat`c fhe pbtaht`fi aharjy par uh`tvbiuka e`vèae gy φj `.a.

A ; u7

7j + l. (6)

Hbta tlft tla spac`stc aharjy lfs tla e`kahs`bh bd iahjtl.


51




Garhbuii`‐s aquft`bh fppiàe fibhj f strafki`ha `h tla draa surdfca (saa D`jura 6?.6) `s j`vah gy

_ ?φj

+ u7

7j + l ; cbhstfht (7)

wlara _ ? `s tla ftkbsplar`c prassura fhe φ `s tla uè eahs`ty. Drbk (6) fhe (7)

_ ?φj

+ A ; cbhstfht (3)

Fs _ ?, φ fhe j fra cbhstfht, A `s cbhstfht fisb. Sb, `h fh bpah lbr`zbhtfi clfhhai, tla spac`stcaharjy `s f cbhstfht bd tla bw. Ula cr`t`cfi uè la`jlt fhe cr`t`cfi spaae fra easthae tb ga wlahtla spac`stc aharjy bd tla uè bw `s f k`h`kuk.

_rbgiak `h wbresZftar `s bw`hj `h f lbr`zbhtfi clfhhai w`tl crbss sact`bh tlft `s sykkatr`cfi fgbut tla cahtrfivart`cfi fx`s. Ula clfhhai crbss sact bh gbuhefry la`jlt kafsurae drbk tla ibwast pb`ht `h tlaclfhhai (tla trbujl) `s prbpbrt`bhfi tb tla dburtl pbwar bd tla lbr`zbhtfi e`stfhca drbk tla fx`s bd sykkatry.

(`) Axprass tla cr`t`cfi wftar la`jlt `h tarks bd tla cr`t`cfi spaae dbr f j`vah cbhstfht vbiuka bwrfta fhe eaeuca tla k`h`kuk spac`stc aharjy dbr wl`cl bw `s pbss`gia.

(``) Cficuifta tla vfiuas bd tla cr`t`cfi uè la`jlt fhe k`h`kuk spac`stc aharjy `d tla bw rfta `s 5k3 s∗6 fhe tla cbhstfht bd prbpbrt`bhfi`ty clfrfctar`s`hj tla clfhhai crbss sact`bhfi gbuhefry`s 3 k∗3.

(```) Gy pibtt`hj f jrfpl bd tla eapaheahca bd tla spac`stc aharjy bh tla wftar la`jlt eatark`ha tla

pbss`gia bws fhe var`dy tla cr`t`cfi uè la`jlt fhe k`h`kuk spac`stc aharjy rasuits bgtf`haefgbva.

Kftlakft`cfi stftakaht bd prbgiak@. Ula clfhhai prbstia `s clfrfctar`sae gy tla curva z ; fx5 `h f radarahca drfka wlara tla br`j`h B`s ft tla trbujl, fhe f `s f cbhstfht clfrfctar`s`hj tla wètl bd tla clfhhai prbstia (saa D`jura 6?.7).Dbr f j`vah wftar la`jlt l `h tla clfhhai, sthe fh axprass`bh dbr tla crbss sact`bhfi fraf bd uè.Fssuk`hj tlft tla uè `s `hv`scè fhe bw`hj ft cbhstfht spaae `h fhy lbr`zbhtfi pifha, eaeuca fhaxprass`bh dbr tla vbiuka bw rfta T `h tarks bd l fhe tla spaae bd tla uè u.

@@. Qsa Garhbuii`‐s aquft`bh fibhj f strafki`ha tb bgtf`h tla spac`stc aharjy duhct`bh fhe sthe `tsk`h`kuk dbr f j`vah vbiuka bw rfta T . Ft tla k`h`kuk bd tla spac`stc aharjy duhct bh, tla cr`t`cfiwftar la`jlt cfh ga axprassae `h tarks bd tla cr`t`cfi uè spaae.

z ; fx5

(x, z )

∗(l/f)6/5 B (l/f)6/5

x

l

z

D`jura 6?.7= Clfhhai prbstia fhe cbbre`hfta systak






@. Ula fraf F bd tla crbss sact`bh bd uè stii`hj tla clfhhai up tb la`jlt l cfh ga easthae gy tlaeàrahca gatwaah tla crbss sact`bh frafs F6 fhe F7, wlara F6 `s tla ractfhjuifr fraf gatwaahz ; l fhe tla x-fx`s dbr x `h tla rfhja ^∗(l/f)6/5, (l/f)6/5[ fhe F7 `s tla fraf gatwaah tla curvaz ; fx5 fhe tla x-fx`s dbr x `h tla rfhja ^

∗(l/f)6/5, (l/f)6/5[. S`hca tla fraf F6 `s ractfhjuifr,

F6 ; 7

l

f

6

5

l br

F6 ; 7 l1/5

f6/5. (5)

Gacfusa bd tla sykkatry bd tla curva fx5 w`tl raspact tb tla vart`cfi fx`s, tla fraf F7 `s tw`ca tlafraf uhearhaftl tla curva z ; fx5 dbr x `h tla rfhja ^∗?, (l/f)6/5[. Sb

F7 ; 7 (l/f)6/5

?

fx5ex. (1)

Ula `htajrfi (1) cfh ga avfiuftae fs F7 ; 7f

x1

1

(lf)

6/5

?

; 7f

6

1

l

f

1/5

∗ ?

sb tlft

F7 ; 7l1/5

1f6/5. (4)

Drbk (5) fhe (4) tla tbtfi fraf `s j`vah gy

F ; <l1/5

1f6/5. (:)

Ula bw rfta `s j`vah gy tla prbeuct bd tla crbss sact`bhfi fraf fhe tla uè spaae, `.a. T ; F u.Lahca us`hj (:)

T ; <l1/5

1f6/5 u (<)

Ula fssukpt`bh bd uh`dbrk bw (u `s cbhstfht), kafhs tlft T `s cbhstfht fisb dbr f j`vah uèla`jlt `h tla clfhhai.

@@. Hbta drbk (6) tlft savarfi pf`rs bd vfiuas (u, l) kfy iafe tb tla sfka cbhstfht vfiua bd A . @d l `s `hcrafsae, u haaes tb ga eacrafsae tb naap A cbhstfht. Cbhsèar`hj hbw tla eapaheahca bd A bh vfr`fgia l, sthe tla cr`t`cfi vfiua lc tlft cbrraspbhes tb f k`h`kuk `h A (l). Wafrrfhj`hj (<) tbaxprass u `h tarks bd T fhe sugst`tut`hj dbr u `h (6) rasuits `h

A ; 6

7j

1T f6/5

<l1/5

7

+ l

br

A ; 71T 7f6/7l∗1/7

67<j + l (0)

Ufn`hj tla ear`vft`va bd (0) w`tl raspact tb l fhe naap`hj `h k`he tlft T `s cbhstfht

eA

el ; ∗1

7 Ú 71T 7f6/7l∗:/7

67<j + 6


5:




Fs tla spac`stc aharjy easthae gy (6) `s f cbhstfht, eA

el ; ? ft tla cr`t`cfi uè la`jlt lc. Sb

671T 7f6/7l∗:/7c

714j ; 6

Sbiv`hj dbr lc j`vas

lc ;

671T 7f6/7

714j

7/:

(6?)

Sugst`tut`hj dbr T drbk (<) j`vas lc ;

671f6/7

714j

7/:

Ú

<l1/5c

1f6/5 uc

5/:

.

Fdtar s`kpi`stcft`bh, tla raift`bhsl`p gatwaah cr`t`cfi uè la`jlt lc fhe tla cr`t`cfi spaae uc cbrra-spbhe`hj tb k`h`kuk spac`stc aharjy gacbkas

lc ; 1u7

c

5j (66)

Sugst tut`hj tla vbiuka bw rfta T drbk (<) `h tla spac`stc aharjy Aquft`bh (0) j`vas tla k`h`kukspac`stc aharjy dbr wl`cl uè bw `s pbss`gia `h tarks bd tla cr`t`cfi uè iavai lc, `.a.

A k`h ;

71

<l1/5

c

1f6/5

7

f6/7l∗1/7c

67<j + lc

Sugst`tut`hj dbr uc drbk (66) j`vas

A k`h ;

71

<l

1/5c

1f6/5

75jlc

1 f6/7l

∗1/7c

67<j + lc

wl`cl s`kpi`stas tb

A k`h ; :lc

1 (67)

(``) @d tla bw rfta T ; 5 k3 s∗6 fhe tla cbhstfht, f, easth`hj tla clfhhai prbstia `s 3 k∗3, tlah

us`hj j ; 0.< k s∗7 fhe (6?) lc ;

671 Ú 5736/7

714 Ú 0.<

7/:

, br

lc ≌ 6.?6 k. (63)

Ula k`h`kuk aharjy kfy ga avfiuftae us`hj (67) fhe

A k`h ≌ 6.13 k. (65)

(```) Aquft`bh (0) kfy ga usae tb pibt tla spac`stc aharjy duhct`bh A (l) fhe tla jrfpl `s slbwh `hD`jura 6?.3. Hbta tlft tla k`h`kuk bd tla curva bccurs dbr vfiuas lc fhe A k`h tlft fra cbhs`stahtw`tl rasuits (63) fhe (65).





A (l) (katras)

A k`h

A 8 A k`h

A 9 A k`h

l6 lc

l7

l (katras)?

6

7

3

5

1

4

:

<

0

6?

? ?.1 6 6.1 7 7.1 3 3.1 5 5.1 1

D`jura 6?.3= Eapaheahca bd tla spac`stc aharjy bh tla uè la`jlt `h bpah clfhhai bw

@htarpratft`bhUla jrfpl (D`jura 6?.3) slbws tlft

6. Zlah A 8 A k`h, twb bws kfy bccur ft eàraht uè la`jlt l6 fhe l7 tlft cbrraspbhe tbuè spaae u6 fhe u7 raspact`vaiy. Ula bw clfrfctar sae gy tla vfiuas l6 fhe u6 `s bdtahhfkae ’slfiibw fhe dfst‛ (br suparcr`t`cfi), wlìa tla bw clfrfctar`sae gy tla vfiuas l7 fheu7 `s hfkae ’eaap fhe sibw‛ (br sugcr`t`cfi) fs l7 8 l6 fhe u7 9 u6.

7. Zlah A ; A k`h, bhiy bha bw kfy bccur dbr cr`t`cfi uè la`jlt lc fhe cr`t`cfi spaae uc.

3. Zlah A 9 A k`h , hb bw `s pbss`gia.


50




y e u t

S a s

f C j h ` r a

a h

`

j h A


S`kpia paheuiukKftlakft`cfi Snìis

Ubp`c ZbrngbbnFijagrf - rafrrfhj`hj dbrkuifa ^6[

@htrbeuct`bh

F s`kpia paheuiuk `s kfea gy stx`hj f slbrt katfi gfr lbr`zbhtfiiy ft tla ahe bd f str`hj bd iahjtlI. Sbka stueahts erbp f gfii gafr`hj drbk f vfr`fgia la`jlt L e`ractiy fgbva tla gfr ft tla `hstfht

wlah `t pfssas tlrbujl tla cahtra bd `ts sw`hj fhe try tb l`t tla gfr ft `ts haxt raturh tb tlft pbs`t`bh.Slbw tlft tlay slbuie stfrt tla`r fttakpt ft f la`jlt L wl`cl `s si`jltiy iass tlfh 1I. (L`ht= hajiactf`r ras`stfhca> ybu haae tb nhbw tla stfhefre dbrkuif dbr tla par`be`c t`ka bd f paheuiuk).


Ula stfhefre dbrkuif dbr tla t`ka dbr tla gfr tb pardbrk f lfid bscìift`bh (cahtra tb ahe fhe gfcn)`s j`vah gy

U ; Ϗ

I

j.

Dbr f pfrt`cia erbpp`hj drbk rast tla raift`bhsl`p gatwaah tla t`ka bd dfii t fhe tla e`stfhca \ dfiiah`s \ ; 6

7jt7. Za w`sl tb clbbsa \ sb tlft tla t`ka bd dfii aqufis tla lfid-par`be bd tla paheuiuk‐s

sw`hj. Ul`s iafes tb tla rasuit=

Ϗ

I

j ;

7\

j

Squfr`hj tla tarks bh gbtl sèas iafes tb tla rasuit tlft \ ; Ϗ7

7 I.

S`hca Ϗ7 `s hafriy 6? (fctufiiy fgbut 0.<404) `t dbiibws tlft tla la`jlt \ `s hafriy 1I. Hbta tlfttla vfiua bd j ebas hbt ahtar `htb tla rasuit, s`hca `t cfhcais drbk gbtl sèas bd tla aquft`bh. Ulafttakpts tb l`t tla sw`hj`hj gfr slbuie stfrt drbk fgbut L ; 1I gut wìi haae tb ga cfrràe butgy tr`fi fhe arrbr, s`hca tlara fra twb dafturas wl`cl kafh tlft bur cficuift`bh `s bhiy fpprbx`kfta.D`rst, tla aact bd f`r ras`stfhca wìi kfna tla t`ka bd dfii bd tla gfii gafr`hj f i`ttia ibhjar tlfh tlat`ka j`vah gy bur cficuift`bh. Sacbheiy, tla taxtgbbn aquft`bh dbr tla par`be`c t`ka bd f paheuiuk`s bhiy fccurfta dbr bscìift`bhs bd skfii fkpi`tuea fhe wlah f`r ras`stfhca `s haji`j`gia. Ub naap tlapaheuiuk sw`hj`hj dbr savarfi bscìift`bhs `t wìi prasukfgiy ga j`vah f df`riy ifrja `h`t`fi fkpi`tueabd sw`hj> dbr ifrja fkpi`tueas bd sw`hj tla par`be`c t`ka `s f i`ttia ibhjar tlfh tlft j`vah gy tlatrfe`t`bhfi aquft`bh.





y e u t

S a s

f C j h ` r a

a h

`

j h A


Kbt`bh bd f paheuiuk

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatr`c duhct`bhs ^5[Ear`vft`va bd f prbeuct fhe duhct`bh bd f duhct`bh ^66[Kfcifur`h axpfhs`bh ^64[Cbhtbur pibtt`hj dbr f duhct`bh bd twb vfr`fgias ^6<[I`hafr sacbhe brear BEAs ^60[

@htrbeuct`bh

F s`kpia paheuiuk cbhs`sts bd f ’pb`ht kfss‛ bd kfss k sw`hj`hj `h f vart`cfi pifha ft tla ahebd f tl`h rbe bd iahjtl i fhe haji`j`gia wa`jlt. Ula fhjia μ(t) `s kafsurae fht`-cibcnw`sa drbk tlavart`cfi fhe eahbtas tla e`spifcakaht fhjia wl`cl tla rbe kfnas w`tl tla vart`cfi (saa D`jura 67.6).Dr`ct`bhfi aacts bd fii n`hes fra hajiactae.

Ula pbtaht`fi aharjy bd f sw`hj`hj kfss, k, `s easthae gy A p ; kjl wlara l `s tla la`jlt bd tlakfss fgbva `ts rast pbs`t`bh (saa D`jura 67.6). Zlah tla systak `s `sbiftae drbk fhy axtarhfi dbrca,tla tbtfi aharjy A ; A n + A p bd tla systak `s cbhsarvae (`.a. `t `s cbhstfht) fhe cfh ga wr`ttah fs

f duhct`bh bd tla twb vfr`fgias μ(t) fhe eμ(t)

et . @h tla rakf`hear bd tl`s stuey tla t`ka eapaheahca

wìi hbt ga slbwh axpi`c`tiy fhe tla hbtft`bh μː fhe μ wìi ga usae dbr eμ

et fhe

e7μ

et7 .

μ(t)

k l

D`jura 67.6= _frfkatars easth`hj tla kbt`bh bd tla paheuiuk

_rbgiak `h wbresDbrkuifta f kbeai dbr tla kbt`bh bd tla paheuiuk kfss. Qsa tl`s tb eatark`ha tla kbt`bh wlah

(f) tla kfss `s raiafsae drbk rast wlah tla paheuiuk `s lbr`zbhtfi

(g) wlah tla kfss `s j`vah fh `h`t`fi `hstfhtfhabus fhjuifr vaibc`ty bd 3? rfe s∗6 wlah `h `ts ibwastpbs`t`bh

(c) wlah tla fhjia bd e`spifcakaht bd tla paheuiuk `s skfii.


16





6. Qsa fh aharjy katlbe. J`vah tlft A n ; 6

7kv7 raprasahts tla n`hat`c aharjy bd f kfss k `h

kbt`bh ft `hstfhtfhabus spaae v(t) (vfry`hj w`tl t`ka t) ear`va fh axprass`bh dbr tla rbtft`bhfi

n`hat`c aharjy axprassae `h tarks bd tla fhjuifr spaae ːμ (ear`vft`va bd tla `hstfhtfhabus fhjiaw`tl raspact tb t`ka).

7. Axprass tla pbtaht`fi aharjy `h tarks bd k, j, i, fhe cbs μ(t) wlara j `s tla jrfv`tft`bhficbhstfht.

3. Lahca slbw tlft tla tbtfi aharjy A , wl`cl `s f duhct`bh bd μ fhe μː, ft fhy `hstfht, `s j`vah gy

A ; 6

7 ki7 μː7 + kji(6 ∗ cbs μ(t)). (6)

5. Fssuk`hj tlft j ; 0.< k s∗7, i ; ?.?: k fhe us`hj tla vfr`fgia rfhjas

∗6? rfe 9 μ(t) 9 6? rfe fhe ∗3? rfe s∗6 9 eμ(t)et

9 3? rfe s∗6, pibt cbhtburs bd tla duhct`bh

6

ki7A ;

6

7μː7 +

j

i(6 ∗ cbs μ(t)) (7)

`h tarks bd tla vfr`fgias μ(t), μː(t).

1. S`hca k, i fhe A fra cbhstfht eur`hj tla kbt`bh, Aquft`bh (7) `kpiàs tlft tla `h`t`fi cbhe`t`bhs

μ(t ; ?) fhe μː(t ; ?) eatark`ha tla cbhstfht bd tla kbt`bh, A

ki7. Afcl cbhtbur i`ha raprasahts

f cbhstfht aharjy vfiua A dbr vfr`ft`bh `h tla cbbre`hftas μ(t) fhe eμ(t)

et . Ula vfiua bd A

ki7easthas afcl cbhtbur bd tla pibt fhe clfrfctar`zas tla kbt`bh bd tla paheuiuk. Qsa tla cbhtburpibt tb eascr`ga tla paheuiuk kbt`bh `h tla dbiibw`hj cfsas=

Cfsa F= μ(?) ; Ϗ/7 fhe eμ

et

t;?

; ?.

Cfsa G= wlah tla kfss `s `h tla ibwast vart`cfi pbs`t`bh fhe lfs fh fhjuifr spaae 3? rfe s∗6

ft t ; ?.

Cfsa C= μ `s skfii sucl tlft s`h μ ≌ μ.

4. S`hca `t `s fssukae tlft tla systak `s `sbiftae drbk fhy axtarhfi dbrca, tla tbtfi aharjy `s

cbhsarvae, `.a. A `s cbhstfht `h t`ka fhe eA/et ; ?. Ufn`hj tla t`ka ear`vft`va bd Aquft`bh(6), slbw tlft tla eàraht`fi aquft`bh bd tla kbt`bh `s=

μ + ξ7 s`h μ ; ? (3)

wlara ξ7 ; j/i.

:. (f) Watf`h`hj bhiy tla strst tark bd tla Kfcifur`h axpfhs`bh bd tla s`ha duhct`bh dbr skfiifrjukahts, fpprbx`kfta Aquft`bh (3) fs f i`hafr sacbhe brear eàraht`fi aquft`bh.

(g) D`he f sbiut`bh bd tl`s eàraht`fi aquft`bh.

(c) Lahca eaeuca tlft tla kbt`bh `s par`be`c fhe sthe fh axprass`bh dbr tla par`be `h tarksbd tla paheuiuk iahjtl.






6. Ula n`hat`c aharjy bd f kfss k kbv`hj ft f spaae v `s j`vah gy A n ; 6

7kv7 . Ula kfss kbvas

fibhj fh frc wlbsa rfeùs `s i tlrbujl f e`stfhca es(t) ; v(t)et eur`hj fh `hsth`tas`kfi t`ka

`htarvfi et. Cbhsaquahtiy, tla spaae bd tla kfss cfh ga axprassae ft t`ka t gy v(t) ; es(t)et

.

Bh tla btlar lfhe, `d tla fhjia bd rbtft`bh μ(t) `s axprassae `h rfe`fhs, s(t) ; iμ(t) sb tlft

v(t) ; es(t)

et ;

eîμ(t)[

et . Ula rbe iahjtl `s stxae eur hj tla kbt`bh, sb i ebas hbt eapahe bh

t`ka t, cbhsaquahtiy v(t) ; eîμ(t)[

et ; i

e^μ(t)[

et ; iμː. Wapifc`hj v(t) gy tl`s axprass`bh, tla

n`hat`c aharjy A n cfh ga axprassae fs A n ; 6

7ki7 μː7.

7. Ula jbfi `s tb bgtf`h A p `h tarks bd tla pfrfkatars k, j, i, fhe tla vfr`fgia cbs μ(t).Cbhsaquahtiy, tla la`jlt l bd tla kfss fgbva `ts pbs`t`bh ft rast, haaes tb ga axprassae `h

tarks bd tlasa qufht`tàs. S`hca l `s tla eàrahca gatwaah tla iahjtl i bd tla paheuiuk (fisbtla rfeùs bd tla c`rcuifr trfmactbry) fhe tla e`stfhca i cbs μ gatwaah tla fx`s bd rbtft`bh fhetla brtlbjbhfi prbmact`bh bd tla pb`ht kfss k bh tla vart`cfi, l ; i ∗ i cbs μ ; i(6 ∗ cbs μ).Wapifc`hj l gy tl`s axprass`bh, tla pbtaht`fi aharjy `s j`vah gy A p ; kji(6 ∗ cbs μ).

3. Ula tbtfi aharjy `s tla suk bd tla n`hat`c fhe pbtaht`fi aharjy A ; A n + A p tlaradbra

A ; 6

7ki7 μː7 + kji(6 ∗ cbs μ).

5. D`jura 67.7 slbws tla cbhtbur pibt bd A

ki7 fs f duhct`bh bd tla twb vfr`fgias μ fhe μː (

6<).

6??7??

3??

5??

1??

4??

μ(rfe)

(rfe s∗6)

∗3?

∗ 7?

∗6?

6?

7?

3?

?

∗4 ∗5 ∗7 7 5 4?

eμ

et

D`jura 67.7= Cbhtbur pibts bd tla rft`b dbr tla s`kpia paheuiuk


13




1. Za cbhsèar afcl cfsa=

Cfsa F= μ(?) ; Ϗ/7 fhe μː(t ; ?) ; ?

Aquft`bh (7) `kpiàs tlft A

ki7 ;

6

7μː7(t ; ?)+

j

i(6∗cbs μ(?)) `s f cbhstfht wlah tla

paheuiuk `s `sbiftae drbk axtarhfi dbrcas. Qs`hj μ(t ; ?) ; Ϗ/7 fhe μː(t ; ?) ; ?

j`vas A

ki7 ; j/i ≌ 65?(rfe s∗6)7. Ul`s cbhtbur bd cbhstfht aharjy A `s ibcftae `h

D`jura 67.7 gatwaah tla twb cibsae cbhtbur ifgaiae A

ki7 ; 6?? (rfe s∗6)7 fhe 7??

(rfe s∗6)7.

Cfsa G= @d tla kfss `s `h tla ibwast vart`cfi pbs`t`bh fhe lfs fh fhjuifr spaae 3? rfe s∗6 ftt ; ?, tlah μ(t ; ?) ; ? fhe μː(t ; ?) ; 3? rfe s∗6. Ula cbrraspbhe`hj cbhtbur dbr

cbhstfht aharjy `s j`vah gy A

ki7 ; (3?)7/7 ≌ 51? (rfe s∗6)7.

Cfsa C= Ula cfsa wlah μ `s skfii `s cbhsèarae `h 4 fhe : gaibw.

4. Ufn`hj tla t`ka ear`vft`va bd Aquft`bh (6) us`hj tla dfct tlft A, k, i, j fra cbhstfht `h t`ka,

j`vas ? ; 6

7ki7

e

et^μː[7 ∗ kji

e

et^cbs μ[. Qs`hj tla axprass`bh dbr f ear`vft`va bd f prbeuct (dbr

tla strst tark) fhe tla ear`vft`va bd f duhct`bh bd duhct`bh (dbr tla sacbhe tark) tl`s raift`bhgacbkas=

? ; ki7 μː^μ[ + kjiμː s`h μ.

Fdtar e`vè`hj gy ki7 μː fhe us`hj tla easth`t`bh ξ7 ; j/i, tl`s gacbkas

μ + ξ7 s`h μ ; ?,

fs raqu rae.

:. (f) Ula Kfcifur`h axpfhs`bh ( 64.1) dbr tla s`ha duhct`bh `s=

s`h μ ; μ ∗ μ3/3! + . . .

Dbr skfii fhjias, tla suk cfh ga truhcftae fdtar tla strst tark `.a. s`h μ ≌ μ.

Eàraht`fi Aquft`bh (3) gacbkas=

μ + ξ7μ ; ?.

(g) Ul`s aquft`bh `s f i`hafr sacbhe brear eàraht`fi aquft`bh w`tl cbhstfht cbafficàhts dbrwl`cl `t `s hacassfry tb sthe f cbkpiakahtfry duhct`bh μ(t) ( 60.3). Ula stfhefreprbcaeura tb sbiva sucl aquft`bhs `s tb saan tla rbbts bd tla fuxì`fry aquft`bh

(k7 + ξ7)akt ; ?.

Ul`s `s sft`sstae `d k7 + ξ7 ; ? br k7 ; ∗ξ7. Cbhsaquahtiy tla rbbts k fra cbkpiax `.a.k ; ¶`ξ.

Ula sbiut`bhs bd tla eàraht`fi aquft`bh cfh ga axprassae fs μ(t) ; Fa`ξt + Ga∗`ξt.Btlar wfys bd wr`t`hj tlasa sbiut`bhs fra

μ ; C s`h ξt + E cbs ξt br μ ; D cbs(ξt ∗ τ)wlara F, G,C,E,D fhe τ fra cbhstfhts tlft fra eatark`hae drbk tla `h`t`fi cbhe`t`bhs.







y e u t

S a s

f C j h ` r a

a h

`

j h A


Ula dfii`hj shbwfnaKftlakft`cfi Snìis

Ubp`c Zbrngbbn@htajrft`bh ^63[D`rst brear BEAs ^60[Sacbhe brear i`hafr BEAs ^60[Tactbr eàraht`ft`bh ^7<[

@htrbeuct`bh

Fitlbujl tla ras stae kbt`bh bd bgmacts w`tl stxae kfss wfs cbhsèarae h 35.3 h tla Zbrngbbnbh Kbeaii`hj Kbt`bh, fhe raiftae tb prbmactìa cbhtaxts, tlara fra fppi`cft`bhs wlara tla kfss bd tlakbv`hj bgmact vfràs w`tl t`ka. Ula pfrt`cuifr cfsa cbhsèarae lara `s tla dfii`hj shbwfna. Ul`s `sfh axfkpia drbk hftura gut tlara fra ahj`haar`hj cbhtaxts sucl fs `h dbbe prbcass`hj wlara s`kìfrfhfiys`s kfy ga raiavfht.

_rbgiak `h wbres

F shbwfna dfii`hj vart`cfiiy `s sugmactae tb f dr`ct`bhfi dbrca prbpbrt`bhfi tb `ts kfss fhe spaae.Kbrabvar, tla cbheahsft`bh bd wftar vfpbur `s `hcrafs`hj tla kfss bd tla shbwfna ft f rfta prb-

pbrt`bhfi tb `ts kfss. D`he tla t`ka eapaheahca bd tla shbwfna‐s vart`cfi pbs`t`bh fssuk`hj tlft `tstfrts drbk rast w`tl f kfss k?. Fisb sthe tla t`ka eapaheahca bd tla shbwfna‐s kfss.

z ; ?

z

w(t)

D d

D`jura 63.6= Dbrcas bh tla dfii`hj shbwfna fhe pfrfkatars `hvbivae

`h tla kftlakft`cfi dbrkuift`bh

Kftlakft`cfi stftakaht bd prbgiakUla shbwfna slbwh `h D`jura 63.6 `s uhearjb`hj f vart`cfi dr`ct`bhfi dbrca D d prbpbrt`bhfi tb tlakfss, k(t), fhe tb tla spaae bd tla fna, v(t), tlaradbra

|D d | ; ck(t)v(t) (6)

wlara c `s f cbhstfht (cbafficàht bd f`r dr`ct`bh). Ula shbwfna `s sugmact tb f ebwhwfre vart`cfidbrca `.a. `ts wa`jlt





|w(t)| ; k(t)j (7)

wlara j `s tla fccaiarft`bh eua tb jrfv`ty. Ula cbheahsft`bh bd wftar vfpbur `s `hcrafs`hj tla kfssk(t) bd tla shbwfna ft f rfta prbpbrt`bhfi tb tla kfss bd tla fna sucl tlft

ek(t)

et ; nk(t) (3)

wlara n `s f cbhstfht. Ula vart`cfi pbs`t bh z (t) bd tla fna w`tl raspact tb t`ka t `s sugmactae tbtla `h`t`fi cbhe`t bhs

(`) spaae v(t ; ?) ; ?(``) kfss k(t ; ?) ; k?

(```) z (t ; ?) ; ?.

@ Zr`ta tla bre`hfry eàraht fi aquft`bh (BEA) bd tla sacbhe brear `h z (t) bgtf`hae gy fppiy`hjtla jaharfi dbrk bd Hawtbh“s ifw

h

D h ; e(k(t)v(t))et (5)

wl`cl stftas tla aqufi`ty gatwaah tla suk bd fii axtarhfi dbrcas bh tla shbwfna fhe tla rfta bd clfhja bd tla shbwfna‐s kbkahtuk.

@@ Sbiva gy tla twb katlbes bd sbiv`hj tla rasuit`hj BEA=(f) Ula sacbhe brear BEA `h z (t) cfh ga sbivae us`hj tla rasuits drbk tla jaharfi tlabry.

(g) Ula sfka rasuit cfh ga ear`vae gy sbiv`hj tla strst brear BEA v(t) ; ez (t)

et gy e`ract `htajrft`bh.

@@@ D`he tla t`ka eapaheahca bd tla shbwfna kfss k(t) gy e`ract `htajrft`bh bd Aquft`bh (3).


@ BEA bd tla kbt`bh us`hj Hawtbh“s ifwUla wa`jlt w(t) fhe tla dr`ct`bhfi dbrca D d (saa 35.3) fra tla bhiy twb axtarhfi dbrcas cbh-sèarae `h tl`s stuey. Z`tl tla clb`ca bd vart`cfi fx`s z slbwh `h D`jura 63.6, usa bd Aquft`bhs (7)fhe (6) `h Hawtbh‐s ifw (Aquft`bh (5)) iafes tb

k(t)j ∗ ck(t)v(t) ; e(k(t)v(t))

et . (1)

(Ula t`ka-eapaheahca bd k fhe v `s cbhsèarae `kpi`c`t drbk hbw bh sb wa wr`ta k dbr k(t) fhe vdbr v(t).)

Ul`s `s gfsae bh tla lypbtlas`s tlft tla cbbre`hfta systak `s f radarahca drfka `h wl`cl (5) `s vfiè.Ula jbfi `s tb bgtf`h f BEA `h tarks bd tla duhct`bh z (t) bhiy, tlaradbra, tla ruia fgbut tla ear`vft`vabd f prbeuct bd duhct`bhs `s usae fs gbtl tla kfss fhe tla spaae eapahe bh t`ka,

e(kv)

et ; k

ev

et + v

ek

et .

Aquft`bh (3) `kpiàs

e(kv)

et ; k

ev

et + vnk.Qs`hj tl`s axprass`bh `h (1), fhe cfhcaii`hj k


1:




ev

et + ^n + c[v ; j. (4)

Wacfii`hj tlft v ; ez

et, Aquft`bh (4) gacbkas

e7

z et7

+ ^n + c[ez et

; j. (:)

@@(f) Sbiut`bh us`hj tla jaharfi tlabry dbr sacbhe brear BEAUl`s aquft`bh `s f i`hafr sacbhe brear eàraht`fi aquft`bh w`tl cbhstfht cbafficàhts dbr wl`cl `t `shacassfry tb sthe f cbkpiakahtfry duhct`bh z (t) ( 60.3). Ula stfhefre prbcaeura tb sbiva suclaquft`bhs `s tb saan tla rbbts bd tla fuxì`fry aquft`bh

k7 + ^n + c[k ; ?.

Ul`s `s sft`sstae `d k ; ∗^n+c[ br k ; ?. Cbhsaquahtiy, tla rbbts k fra rafi fhe tla cbkpiakahtfryduhct`bh bd tla eàraht`fi aquft`bh cfh ga axprassae fs

z (t) ; Fa∗^n+c[t + G, wlara F fhe G cbhstfhts

@t `s afsy tb clacn tlft f pfrt`cuifr sbiut`bh bd Aquft`bh (:) `s z (t) ; j

n + ct+ E fhe, gy sukkft`bh

bd tla twb sbiut`bhs (cbkpiakahtfry duhct`bh + pfrt`cuifr sbiut`bh) tla jaharfi sbiut`bh `s

z (t) ; Fa∗^n+c[t + j

n + ct + A (<)

wlara A `s f cbhstfht (; G + E).Ula `h`t`fi cbhe`t`bh (```) z (t ; ?) ; ? `kpiàs tlft Fa∗^n+c[Ú? + A ; ? sb A ; ∗F.Cbhsaquahtiy, (<) gacbkas

z (t) ; F{a∗^n+c[t ∗ 6} + jn + c

t. (0)

Ula `h`t`fi cbhe`t`bh (`) v(?) ; ? `kpiàs tlft ez

et

t;?

; ?.

Ula ear`vft`va bd (0) `s

ez

et ; ∗F^n + c[a∗^n+c[t +

j

n + c.

Ularadbra, ez

et

t;?

; ? `kpiàs ∗F^n + c[a∗^n+c[Ú? + j

n + c ; ? wl`cl j`vas F ;

j

(n + c)7. Aquft`bh

(0) gacbkas

z (t) ; j

(n + c)7 {a∗^n+c[t ∗ 6 + ^n + c[t}. (6?)

@@(g) Sbiut`bh us`hj fh `htajrft`bh bd tla strst brear BEA `h vvv(ttt)

Stfrt`hj drbk tla BEA (4) bha cfh wr`ta ev ; ^j ∗ ^n + c[v[et, fhe `htajrft`hj gbtl sèas bd tlaaqufi`ty us`hj tla `h`t`fi cbhe`t`bh (`) v(?) ; ? v

?

ev

j ∗ ^n + c[v ;

t?

et. (66)

Clfhj`hj vfr`fgias, tla axprass`bh

ev

j ∗ ^n + c[v cfh ga wr`ttah fs ∗ 6

n + c

eP

P wlara P ; j∗^n+c[v.Hbta tlft ft t ; ?, P ; j.





Ularadbra (66) gacbkas

∗ 6

n + c

P j

eP

P ;

t?

et. (67)

Wacfii drbk 66.7 tlft tla ear`vft`va bd ih|P

| `s j`vah fs 6/

|P

|, sb (67) gacbkas,

t ; ∗ 6

n + c(ih |P | ∗ ih j) ; ∗ 6

n + c ih

|P |j

Wapifc`hj tla vfr`fgia P `h tarks bd v(t),

t ; ∗ 6

n + c ih

|j ∗ ^n + c[v|j

.

Ufn`hj tla axpbhaht`fi bd gbtl sèas fhe sbiv`hj dbr v bha sthes tlft

v ; j

n + c{6 ∗ a∗t^n+c[}. (63)

Wacfii tla easth`t`bh v ; ez

et fhe tla `h`t`fi cbhe`t`bh (```) z (?) ; ?, tlah `htajrfta (63)=

z?

ez ; j

n + c

t?

{6 ∗ a∗^n+c[} et

wl`cl iafes tb

z (t) ; j

(n + c)7{a∗^n+c[t ∗ 6 + ^n + c[t}.

@@@. U`ka eapaheahca bd tla shbwfna kfss

Ula shbwfna kfss k(t) cfh ga bgtf`hae gy e`ract `htajrft`bh bd Aquft`bh (3), `.a.ek

et ; nk.

Lahca kk?

ek

k ; n

t?

et.

Qsa bd tla `h`t`fi cbhe`t`bh (``) k(?) ; k? j`vas

nt ; ih kk

k?

sb

k ; k?ant.

@htarpratft`bh

Ula shbwfna‐s spaae rafclas f tark`hfi vfiua v ; j

n + c fs cfh ga saah gy putt`hj

ev

et ; ? `h

Aquft`bh (4). Ula shbwfna‐s kfss `hcrafsas axpbhaht`fiiy gut wlah tla wa`jlt gacbkas cbuhtar-gfifhcae gy tla dr`ct`bhfi dbrca, tla tbtfi dbrca bh tla systak `s zarb fhe tla shbwfna dfiis ft tla

cbhstfht spaae v ; j

n + c.


10




y e u t

S a s

f C j h ` r a

a h

`

j h A


Sftaii`ta kbt`bh

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Aquft`bh bd cbh`cs `h pbifr cbbre`hftas ^6:[Sacbhe brear BEAs ^60[Tactbr cficuius ^7<[_bifr cbbre`hfta systak ^7<[

_rbgiak `h wbresFppiy Hawtbh‐s ifw `h f pbifr cbbre`hfta systak tb eaeuca tla eàraht`fi aquft`bh fs waii fs fcbhstfht bd tla kbt`bh bd f sftaii`ta èafi`sae fs f pb`ht kfss `h tla jrfv`tft`bhfi staie bd f pifhatw`tl cahtra bd kfss stxae ft tla br`j`h. Sbiva tla eàraht fi aquft`bh dbr tla trfmactbry bd tla sftaii`tafhe prbva Napiar‐s strst ifw tlft tla brg`t cfh ga wr`ttah `h tla dbrk bd tla aquft`bh bd f cbh`c.

Kftlakft`cfi stftakaht bd prbgiakUla jrfv`tft`bhfi dbrca bh f sftaii`ta bd kfss k (w`tl cahtra bd kfss ft _ ) c`rci`hj tla Afrtl bd kfssK (w`tl cahtra bd kfss ft B) eua tb tla jrfv`tft`bhfi staie `s j`vah gy

D ; ∗JkK

r7 r˄, (6)

wlara J `s tla jrfv`tft`bhfi cbhstfht, r `s tla e`stfhca gatwaah tla twb cahtra bd kfss, r˄ `s tla uh`tvactbr fibhj tla rfe`fi fx`s `h tla e`ract`bh bd `hcrafs`hj r. Hawtbh‐s ifw D ; kf tbjatlar w`tl tlajaharfi rasuit bd tla fccaiarft`bh f `h pbifr cbbre`hftas dbr tla kbt`bh bd f kftar`fi pb`ht `h f pifha(rasuit ear`vae `h 5:.3 _lys`cs Cfsa Stuey 66)

f ;

e7r

et7 ∗ r

eμ

et

7

r˄ +

7 er

et

eμ

et + r

e7μ

et7

μ˄, (7)

fiibw tb wr`ta tla aquft`bhs bd tla kbt`bh bd tla sftaii`ta. Napiar‐s strst ifw `s bgtf`hae gy astfgi`sl`hj

f raift`bhsl`p gatwaah vfr`fgias r fhe μ gy ai`k`hft`bh bd tla t`ka vfr`fgia but bd tla aquft`bhs bd tla kbt`bh. Ula sacbhe brear eàraht`fi aquft`bh bgtf`hae `s sbivae gy clfhj`hj vfr`fgia r tb 6/r.

_ r

D

k

μ`

m μ˄

r˄

K

B

D`jura 65.6= Jrfv`tft`bhfi dbrca fhe cbbre`hfta systaks





Kftlakft`cfi fhfiys`sFppiy`hj Hawtbh‐s ifw w`tl Aquft`bhs (6) fhe (7) iafes tb

∗JkK

r7 r˄ ; k

e7r

et7 ∗ r

eμ

et7

r˄ + k 7 er

et

eμ

et + r

e7μ

et7 μ˄, (3)

Ula gfs`s vactbrs (r˄, μ˄) lfva gaah clbsah tb ga `heapaheaht (hbh-cbii`hafr), tlaradbra (3) iafes tbtla twb aquft`bhs,

∗J K

r7 ;

e7r

et7 ∗ r

eμ

et

7

(5)

7 er

et

eμ

et + r

e7μ

et7 ; ?. (1)

hbt`hj tlft e

etr7

eμ

et ; 7r

er

et

eμ

et

+ r7 e7μ

et7, Aquft`bh (1) kuit`piàe gy r iafes tb

e

et

r7

eμ

et

; ?, (4)

tlaradbra,

r7eμ

et ≦ C (:)

`s f cbhstfht bd tla kbt`bh. Hbta tlft C `s bdtah easthae fs tla rft`b bd tla fhjuifr kbkahtuk I

fhe tla kfss k bd tla sftaii`ta fs C ; r7eμ

et ;

I

k.

Qs`hj (:), Aquft`bh (5) cfh ga wr`ttah

∗J K

r7 ;

e7r

et7 ∗ C 7

r3 . (<)

Za hbw w`sl tb ear`va tla aquft`bh bd tla brg`t `h tla dbrk bd f raift`bh gatwaah r fhe μ. Easth`hjtla vfr`fgia u ≦ 6/r, (<) gacbkas

∗J Ku7 ; e7r

et7 ∗ C 7u3. (0)

Ufn`hj tla t`ka ear`vft`va bd r fhe rapifc`hj r gy 6/u j`vas

er

et ;

e

et6

u

; ∗ 6

u7

eu

et . (6?)

Bha `s saan`hj tb ai`k`hfta tla t`ka vfr`fgia but bd (0), tlaradbra, fppiy`hj tla clf`h ruia bh tlat`ka ear`vft`va bd u j`vas

er

et ; ∗ 6

u7

eu

eμ

eμ

et. (66)

Aquft`bh (:) fiibws us tb wr`ta eμ

et ; Cu7 fhe (66) gacbkas

er

et ; ∗C

eu

eμ. (67)


46




Ufn`hj tla t`ka ear`vft`va bd (67) w`tl tla clf`h ruia iafes tb

e7r

et7 ; ∗C

e

et

eu

eμ

; ∗C

e

eμ

eu

eμ

eμ

et. (63)

Sugst`tut`hj dbr

eμ

et drbk Aquft`bh (:) `h (63) iafes tb

e7r

et7 ; ∗C 7u7e

7u

eμ7. (65)

Ula eàraht`fi aquft`bh bd tla kbt`bh `s eaeucae drbk (0) fs

∗J K u7 ; ∗C 7u7e7u

eμ7 ∗ C 7u3. (61)

Aquft`bh (61) cfh ga e`vèae gy ∗C 7 u7, fhe tla easth`t`bh N ≦ JK fiibws us tb bgtf`h tla sthfieàraht`fi aquft`bh `h vfr`fgia u=

e7

ueμ7

+ u ; N C 7

. (64)

Ula sbiut`bh bd tla sacbhe brear eàraht`fi aquft`bh (64) `s j`vah gy tla suk bd tla cbkpiakahtfry

duhct`bh fhe tla pfrt`cuifr `htajrfi N

C 7. Ula cbkpiakahtfry duhct`bh `s bgtf`hae drbk sbiv`hj tla

fuxì`fry aquft`bh n7 + 6 ; ? wl`cl iafes tb n ; ¶ `. Ularadbra, tla sbiut`bh bd (64) cfh ga wr`ttah

fs u(μ) ; F cbs μ + G s`h μ + N

C 7 br `h tla dbrk u(μ) ; E cbs(μ ∗ μ?) +

N

C 7. Ufn`hj but f dfctbr

N

C 7 fhe kfn`hj tla vfr`fgia r tla sugmact iafes tb

6

r ;

N

C 7

6 +

C 7

N E cbs(μ ∗ μ?)

fhe tla brg`t bd

tla sftaii`ta `h tarks bd tla pbifr fhjia μ `s axprassae fs tla aquft`bh bd f cbh`c `h pbifr cbbre`hftas

r ;

C 7

N

6 + C 7E

N cbs(μ ∗ μ?)

. (6:)

@htarpratft`bh

Ula pfrfkatar a ≦ C 7E

N `s tla accahtr`c`ty, tla pb`ht B `s bha bd tla dbc` bd tla cbh`c fhe tla

i`ha μ ; μ? `s tla dbcfi fx`s. Ahj`haar`hj Cfsa Stuey 61 wl`cl dbiibws haxt slbws tlft fii vfiuas bd accahtr`c`ty a fra pbss`gia fhe tlft tla trfmactbry cfh ga fh aii`psa (a 9 6), fh lypargbif (a 8 6),

axcapt`bhfiiy f pfrfgbif (a ; 6), br f c`rcia bd cahtra B fhe rfeùs C 7

N , eapahe`hj bh tla `h`t`fivaibc`ty bd tla sftaii`ta ifuhcl.





y e u t

S a s

f C j h ` r a

a h

`

j h A


Sftaii`ta brg`tsKftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Aquft`bh bd cbh`cs `h pbifr cbbre`hftas ^6:[Sacbhe brear BEAs ^60[Tactbr cficuius ^7<[_bifr cbbre`hfta systak ^7<[

@htrbeuct`bh

Ula usa bd sftaii`tas fs taiacbkkuh`cft`bh eav`cas, jibgfi pbs`t`bh`hj tbbis br dbr scàht`stc eftfjftlar`hj tb kbh`tbr tla Afrtl‐s ftkbsplara, bcafhs fhe surdfca `s `hcrafs`hj. Ahj`haars `h clfrjabd pifhh`hj tla eftf fcqu`s`t`bh fhe `hdbrkft`bh trfhsk`ss`bh tb tla Afrtl‐s surdfca haae tb cbkputatla brg`t bd tla sftaii`ta fccbre`hj tb tla ifws bd jrfv`tft`bh stftae gy Hawtbh. Ul`s cfsa stueyslbws tlft tla `h`t`fi vaibc`ty ft ifuhcl `s cruc`fi `h eatark`h`hj tla sftaii`ta brg`t.

_rbgiak `h wbres@. Qsa Hawtbh“s ifw `h pbifr cbbre`hftas tb eaeuca tla eàraht`fi aquft`bh fhe f cbhstfht dbr tla

kbt`bh bd f sftaii`ta èafi`sae fs f pb`ht bgmact `h tla jrfv`tft`bhfi staie bd tla Afrtl w`tl cahtra bd kfss stxae ft tla br`j`h.@@. Sbiva tla eàraht`fi aquft`bh dbr tla trfmactbry bd tla sftaii`ta fhe prbva Napiar‐s strst ifw tlfttla trfmactbry cfh ga wr`ttah `h tla dbrk bd tla aquft`bh bd f cbh`c.@@@. Slbw tlft tla trfmactbry bd tla sftaii`ta cfh ga aii`pt`c, lypargbi`c br pfrfgbi`c eapahe`hj bh tlakfjh`tuea bd tla ifuhcl vaibc`ty.

Kftlakft`cfi stftakaht bd prbgiak@. Ula jrfv`tft`bhfi dbrca eua tb tla Afrtl bd kfss k ft B bh f sftaii`ta èafi`sae fs f pb`ht bgmactbd kfss k w`tl cahtra bd kfss ft _ `s

D ; ∗Jkkr7 r˄, (6)

wlara J `s tla jrfv`tft`bhfi cbhstfht, r `s tla e`stfhca gatwaah tla twb cahtra bd kfss, r˄ `s tla uh`tvactbr fibhj tla rfe`fi fx`s `h tla e`ract`bh bd `hcrafs`hj r (saa D`jura 61.6). Ub usa Hawtbh‐s ifwD ; kf tb bgtf`h tla aquft`bhs bd tla kbt`bh bd tla sftaii`ta, tla strst stap `s tb sthe fh axprass`bhdbr tla fccaiarft`bh f `h pbifr cbbre`hftas dbr tla kbt`bh bd f pb`ht `h f pifha. Ula sacbhe breareàraht`fi aquft`bh bd kbt`bh `s bgtf`hae gy(f) astfgi`sl`hj f raift`bhsl`p gatwaah vfr`fgias r fhe μ(g) ai`k`hft`hj tla t`ka vfr`fgia drbk tla aquft`bhs bd tla kbt`bh(c) clfhj`hj vfr`fgia r tb u ≦ 6/r.


43




y˄

μ˄r˄k

_ rD

k μx˄

D`jura 61.6= Jrfv`tft`bhfi dbrca fhe cbbre`hfta systaks

@@. Ula sbiut`bh bd tla sacbhe brear eàraht`fi aquft`bh bd tla kbt`bh `s j`vah gy tla suk bd tlacbkpiakahtfry duhct`bh fhe tla pfrt cuifr `htajrfi. Napiar‐s strst ifw `s bgtf`hae gy astfgi`sl`hj fraift`bhsl`p gatwaah vfr`fgias r fhe μ `h tla dbrk bd tla aquft`bh bd f cbh`c.

@@@. Axprass`hj tla `h`t`fi vaibc`ty T ? bd tla sftaii`ta `h f pbifr cbbre`hfta systak (saa D`jura 61.7),

sthe tla `htajrft`bh cbhstfhts bd tla sbiut`bh u ≦ 6/r bd tla eàraht`fi aquft`bh bd tla kbt`bh `htarks bd tla rfe`fi pbs`t`bh bd tla sftaii`ta r? ft t ; ?, tla vaibc`ty kfjh`tuea T ? (ft t ; ?), tlaifuhcl fhjia ζ?, N ≦ Jk, fhe f cbhstfht bd tla kbt`bh easthae fs

C ≦ r7eμ

et. (7)

E`scuss tla pbss`gia trfmactbràs bd tla sftaii`ta `h tarks bd tla vfiuas bd tla accahtr`c`ty bd tla cbh`caquft`bh bgtf`hae, fhe slbw tlft tla trfmactbry typas eapahe sbiaiy bh tla `h`t`fi vaibc`ty kfjh`tuea.

y˄μ˄

r˄

x˄

μ ; ?

r?

T bμ

_ ?

T ?

T ?rζ?

D`jura 61.7= @h`t`fi vaibc`ty `h f pbifr cbbre`hfta systak ft t ; ?

Kftlakft`cfi fhfiys`s@. Ula aquft`bhs bd tla kbt`bh bd tla sftaii`ta fra j`vah gy Hawtbh‐s ifw D ; kf. Cbhsèar tlajaharfi pifhfr kbt`bh bd f pb`ht _ wlbsa pbs`t`bh `s j`vah `h pbifr cbbre`hftas. Axprass tla rfe`fifhe fhjuifr cbkpbhahts bd tla fccaiarft`bh f `h pbifr cbbre`hftas. Ula pb`ht _ raprasahts, `h tl`saxfkpia, tla cahtra bd kfss bd f sftaii`ta èafi`sae fs f pb`ht bgmact `h tla jrfv`tft`bhfi staie bd tlaAfrtl. Ula pbs`t`bh bd f pb`ht _ cfh ga easthae gy tla Cfrtas`fh cbbre`hftas (x, y) bd tla pbs`t`bhvactbr B_ ; x` +ym fs slbwh `h D`jura 61.3 wlara ` fhe m fra tla uh`t vactbrs fibhj tla cbbre`hftafxas. Ula Cfrtas`fh cbbre`hftas (x, y) cfh ga axprassae `h tarks bd tla pbifr cbbre`hftas (r, μ), `.a.

x ; r cbs μ (3)

fhey ; r s`h μ. (5)





μ

r _

vy

vμ

vx

v

vr

μ

`

m

μ˄

r˄

y

xB

D`jura 61.3= Cfrtas`fh fhe pbifr cbbre`hfta systak

Ula vaibc`ty bd _ `s v ; e

et(B_ ). Sb, s`hca B_ ; x` + ym, tla cbkpbhahts (vx, vy) bd vaibc`ty

v ; vx` + vy m `h tla drfka (`, m) fra ear`vae drbk tla t`ka ear`vft`vas bd (3) fhe (5) hbt`hj tla dfct

tlft e`

et ; ? fhe

em

et ; ? s`hca tla uh`t vactbrs fibhj Bx fhe By fra stxae w`tl t`ka. Lahca

vx ; ex

et ;

er

et cbs μ ∗ r

eμ

et s`h μ (1)

vy ; ey

et ;

er

et s`h μ + r

eμ

et cbs μ. (4)

Ula cbkpbhahts (fx, fy) bd fccaiarft`bh f ; fx`+ fy m `h tla drfka (`, m) fra bgtf`hae drbk tla t`kaear`vft`va bd v `.a. drbk tla t`ka ear`vft`vas bd (1) fhe (4). Lahca

fx ; e7x

et7 ;

e7r

et7 cbs μ ∗ 7

er

et

eμ

et s`h μ ∗ r

eμ

et7

cbs μ ∗ re7μ

et7 s`h μ (:)

fy ; e7y

et7 ;

e7r

et7 s`h μ + 7

er

et

eμ

et cbs μ ∗ r

eμ

et

7

s`h μ + re7μ

et7 cbs μ. (<)

Ula cbkpbhahts (vx, vy) bd vaibc`ty fhe (fx, fy) bd fccaiarft`bh fra axprassae `h tarks bd tla pbifrcbbre`hftas. S`hca tla vaibc`ty vactbr v `s tla sfka `h gbtl gfs`s sats,

vx` + vy m ; vrr˄ + vμμ˄. (0)

_rbmact`bhs bd tla gfs`s vactbr (`, m) bhtb tla gfs`s vactbrs (r˄, μ˄) iafes tb (saa D`jura 61.5)

` ; cbs μr˄

∗s`h μμ˄ (6?)

m ; s`h μr˄ + cbs μμ˄ (66)

μμ

`

mμ˄

r˄

D`jura 61.5= _rbmact`bhs bd tla Cfrtas`fh gfs`s sat bhtb tla pbifr gfs`s satAquft`bh (0) tbjatlar w`tl (6?) fhe (66) j`vas


41




vx(cbs μr˄ ∗ s`h μμ˄) + vy(s`h μr˄ + cbs μμ˄) ; vrr˄ + vμμ˄. (67)

Ula gfs`s vactbrs (r˄, μ˄) lfva gaah clbsah tb ga `heapaheaht, tlaradbra (67) iafes tb tla twb aquft`bhs=

vr ; vx cbs μ + vy s`h μ (63)

vμ ; ∗vx s`h μ + vy cbs μ. (65)

Qs`hj (1) fhe (4) `h (63) fhe (65) j`vas

vr ; er

et (61)

vμ ; r eμ

et. (64)

Ula sfka katlbe cfh ga usae dbr tla cbkpbhahts (fr, fμ) bd tla fccaiarft`bh. Ula aquft`bhf ; fx` + fy m ; frr˄ + fμμ˄ j`vas

fr ; fx cbs μ + fy s`h μ (6:)

fμ ; ∗fx s`h μ + fy cbs μ. (6<)

Qs`hj (1) fhe (4) `h (61) fhe (64) iafes tb tla rfe`fi fhe fhjuifr cbkpbhahts bd tla fccaiarft`bh f`h pbifr cbbre`hftas

fr ; e7r

et7 ∗ r

eμ

et

7

(60)

fμ ; 7eret

eμet

+ re7μet7

. (7?)

Ula fhjuifr vaibc`ty eμ

et fhe fccaiarft`bh

e7μ

et7 fra bdtah eahbtae gy π fhe ζ raspact`vaiy. Ula

cbkpbhaht bd vaibc`ty fibhj μ˄ `s rπ. Ula cbkpbhaht bd fccaiarft`bh fibhj r˄ `hciueas hbt bhiy

tla sb-cfiiae rfe`fi fccaiarft`bh e7r

et7 gut fisb ∗rπ7 wl`cl `s tla cahtr`patfi fccaiarft`bh br tla

fccaiarft`bh tbwfre tla br`j`h. Ula ∗rπ7 tark `s tla bhiy rfe`fi tark tlft fppiàs `h cfsas bd c`rcuifr

kbt`bh s`hca, dbr r cbhstfht, e7r

et7 ; ?. Ula fccaiarft`bh fibhj μ˄ `hciueas hbt bhiy tla tark rζ, gut

fisb tla Cbr`bi`s fccaiarft`bh 7 er

et π.

Sugst`tut`hj tla rfe`fi fhe fhjuifr cbkpbhahts bd tla fccaiarft`bh `h pbifr cbbre`hftas (60) fhe(7?) `htb tla Hawtbh‐s ifw, Aquft`bh (6) iafes tb

∗Jkk

r7 r˄ ; k

e7r

et7 ∗ r

eμ

et

7

r˄ + k

7 er

et

eμ

et + r

e7μ

et7

μ˄. (76)

Qs`hj fjf`h tla dfct tlft tla gfs`s vactbrs (r˄, μ˄) lfva gaah clbsah tb ga `heapaheaht kafhs tlfttla r˄ fhe μ˄ cbkpbhahts cfh ga aquftae, iafe`hj tb

∗Jk

r7 ;

e7r

et7 ∗ r

eμ

et

7

(77)





fhe

7 er

et

eμ

et + r

e7μ

et7 ; ?. (73)

Hbt`hj tlft

e

et

r

7eμ

et

; 7r

er

et

eμ

et + r

7e7μ

et7 , Aquft`bh (73) kuit`piàe gy r iafes tb

e

et

r7

eμ

et

; ?. (75)

tlaradbra,

r7eμ

et ≦ C (71)

`s f cbhstfht bd tla kbt`bh. Hbta tlft C `s bdtah easthae fs tla rft`b bd tla fhjuifr kbkahtuk I

fhe tla kfss k bd tla sftaii`ta fs C ; r7 eμ

et ;

I

k.

Qs`hj (71), Aquft`bh (77) cfh ga wr`ttah

∗Jk

r7 ;

e7r

et7 ∗ C 7

r3 . (74)

Za hbw w`sl tb ear`va tla aquft`bh bd tla trfmactbry uhear tla dbrk bd f raift`bh gatwaah r fhe μ.

Easth`hj tla vfr`fgia u ≦ 6

r, (74) gacbkas

∗J k u7 ; e7r

et7 ∗ C 7u3. (7:)

Ufn`hj tla t`ka ear`vft`va bd r fhe rapifc`hj r gy 6/u j`vas

er

et ;

e

et

6

u

; ∗ 6

u7

eu

et. (7<)

Ub ai`k`hfta tla t`ka vfr`fgia drbk (7<), fppiy tla clf`h ruia bh tla t`ka ear`vft`va bd u tb j`va

er

et ; ∗ 6

u7

eu

eμ

eμ

et. (70)

Drbk (71) fhe us`hj u ≦ 6

r, wa lfva

eμ

et ; Cu7, fhe sb (70) gacbkas

er

et ; ∗C

eu

eμ . (3?)

Ufn`hj tla t`ka ear`vft`va bd (3?) gy kafhs bd tla clf`h ruia iafes tb

e7r

et7 ; ∗C

e

et

eu

eμ

; ∗C

e

eμ

eu

eμ

eμ

et. (36)

Qs`hj (71) `h (36) iafes tb

e7r

et7 ; ∗C 7u7e

7u

eμ7. (37)

Ula eàraht`fi aquft`bh bd kbt`bh `s eaeucae drbk (7:) fs

∗J k u7 ; ∗C 7u7e7u

eμ7 ∗ C 7 u3. (33)


4:




E`vè`hj (33) gy ∗C 7u7 fhe us`hj tla easth`t`bh N ≦ Jk j`vas tla sthfi dbrk bd tla eàraht`fiaquft`bh `h tarks bd vfr`fgia u

e7u

eμ7 + u ;

N

C 7. (35)

Ula sbiut`bh bd tl`s sacbhe brear eàraht`fi aquft`bh `s tla suk bd tla cbkpiakahtfry duhct`bhfhe tla pfrt`cuifr `htajrfi

N

C 7. Ula cbkpiakahtfry duhct`bh `s bgtf`hae gy sbiv`hj tla fuxì`fry

aquft`bh n7 + 6 ; ? wl`cl j`vas n ; ¶ `. Ularadbra, tla sbiut`bh bd (35) cfh ga wr`ttah fs

u(μ) ; Fcbs μ + G s`h μ + N

C 7 br `h tla fitarhft`va dbrk u(μ) ; E cbs(μ ∗ μ?) +

N

C 7. Ufn`hj but f

dfctbr N

C 7 fhe ravart`hj tb vfr`fgia r iafes tb

6

r ;

N

C 7

6 +

C 7

N E cbs(μ ∗ μ?)

.

D`hfiiy, tla trfmactbry bd tla sftaii`ta `h tarks bd tla pbifr fhjia μ cfh ga axprassae fs tla aquft`bhbd f cbh`c `h pbifr cbbre`hftas

r ;

C 7

N

6 + C 7E

N cbs(μ ∗ μ?)

. (31)

Ula pfrfkatar a ≦ C 7

N `s tla accahtr`c`ty, tla pb`ht B `s bha bd tla dbc` bd tla cbh`c fhe tla i`ha

μ ; μ? `s tla dbcfi fx`s. Sbiut`bh tb Vuast`bh @@@, j`vah gaibw, slbws tlft fii vfiuas bd accahtr`c`tya fra pbss`gia fhe tlft tla trfmactbry cfh ga fh aii`psa (a 9 6), fh lypargbif (a 8 6), axcapt`bhfiiy

f pfrfgbif (a ; 6), br f c`rcia bd cahtra B fhe rfeùs C 7N

, eapahe`hj bh tla `h`t`fi vaibc`ty bd tla

sftaii`ta ifuhcl.

@@@. Ula sbiut`bh bd tla eàraht`fi Aquft`bh (35) `s clbsah `h tla dbrk

u(μ) ; Fcbs μ + G s`h μ + N

C 7 (34)

Ula cbhstfhts F, G fhe C fra eatark`hae drbk tla `h`t`fi cbhe`t`bhs bd tla ifuhcl. ft t ; ?. Ula`h`t`fi rfe`fi pbs`t`bh bd tla sftaii`ta, ft pb`ht _ ?(r?, μ?) br _ ?(u?, μ?), cfh ga easthae fs

r ; r? ft t ; ?. (3:)

Ula rfe`fi fx`s cfh ga clbsah tb cb`hcèa w`tl tla x-fx`s ft t ; ? (saa D`jura 7), fhe tlaradbra

μ ; μ? ft t ; ?. (3<)

Hbta tlft tla `h`t`fi cbhe`t`bh (3:) cfh ga axprassae fisb fs u ; u? ; 6/r? ft t ; ?. Ft t ; ?, tlasftaii`ta lfs vaibc`ty T ? j`vah gy (saa D`jura 7),

T ? ; T ?rr˄ + T ?μμ˄. (30)

Aquft`bhs (61) fhe (64) j`va tla cbkpbhahts bd tla vaibc`ty v `h tarks bd pbifr cbbre`hftas fs

v ;

er

et r˄ + r

eμ

et˄μ. (5?)

Aquft`bh (5?) cfh ga axprassae `h tarks bd tla vfr`fgia u. Qs`hj tla clf`h ruia, tla strst r`jlt-lfhe





sèa tark `h (5?) cfh ga wr`ttah er

et ;

er

eμ

eμ

et. Qs`hj (71), `.a.

eμ

et ;

C

r7, er

et ;

C

r7er

eμ fhe tla

easth`t`bh u ≦ 6/r iafes tb er

et ; Cu7e(u∗6)

eμ ; ∗C

eu

eμ. Easth`hj u ≦ eu

eμ, tla strst r`jlt-lfhe sèa

tark `h (5?) gacbkas

eret

; ∗Cu (56)

Ula sacbhe r`jlt-lfhe sèa tark `h (5?) `s r eμ

et tlft cfh ga axprassae fs r

eμ

et ; r

C

r7. Fdtar

s`kpi`stcft`bh

r eμ

et ; uC. (57)

Qs`hj (56) fhe (57), Aquft`bh (5?) gacbkas

v ;

∗Cur˄ + Cuμ˄. (53)

@t cfh ga eaeucae drbk D`jura 7 tlft T ?r ; T ? cbs ζ? fhe T ?μ ; T ? s`h ζ?. Ularadbra, (53) iafes tb

T ? cbs ζ? ; ∗Cu? (55)

fhe

T ? s`h ζ? ; Cu?. (51)

Qs`hj u? ; 6/r? `h (51) j`vas tla cbhstfht,

C ; r?T ? s`h ζ?, (54)

fhe `h (55) j`vas u? ; ∗T ? cbs ζ?

C wl`cl, fdtar us`hj (54), cfh ga wr ttah fs

u? ; ∗ cbs ζ?

r? s`h ζ?. (5:)

Qs`hj tla `h`t`fi cbhe`t`bhs (3:) fhe (3<) `h (34) j`vas

u(?) ; Fcbs(?) + G s`h(?) + N

C 7 br

u? ; F + N

C 7. (5<)

Ufn`hj tla strst ear`vft`va bd (34) w`tl raspact tb vfr`fgia μ iafes tb fh fee`t`bhfi aquft`bh dbr tlacbhstfht G. @heaae,

eu

eμ

μ;?

; ∗F s`h(?) + G cbs(?), sb

G ; u?. (50)

Drbk (5<), tla cbhstfht F cfh ga axprassae fs F ; u? ∗ N

C 7.

Sugst`tut`hj dbr C drbk (54) iafes tb F ; 6

r?∗ N

r7?T 7? s`h7 ζ?

.

Fdtar easth`hj κ? ≦ r?T 7?N

,

F ; 6

r?

6 ∗ 6

κ? s`h7 ζ?

. (1?)


40




Drbk (50) fhe (5:), tla cbhstfht G cfh ga axprassae fs

G ≦ ∗ cbs ζ?

r? s`h ζ?. (16)

Z`tl tla usa bd N

C 7 ;

N

r7?T 7? s`h7 ζ?

; 6

r?κ? s`h7 ζ?

fhe (1?) fhe (16), Aquft`bh (34) cfh ga

axprassae fs f duhct`bh bd tla pfrfkatars r?, T ?, ζ?, N fhe μ. Lahca,

u(μ) ; 6

r?

6 ∗ 6

κ? s`h7 ζ?

cbs μ ∗ cbs ζ?

r? s`h ζ?s`h μ +

6

r?κ? s`h7 ζ?

,

br, fdtar tfn`hj tla dfctbr 6

r?κ? s`h7 ζ?

but bd u(μ),

u(μ) ; 6

r?κ? s`h7 ζ?

6 + (ζ? s`h7 ζ? ∗ 6)cbs μ ∗ κ? s`h ζ? cbs ζ? s`h μ

. (17)

Ula rfe`fi pbs`t`bh bd tla sftaii`ta `s bgtf`hae gy rafrrfhj`hj (17) fs

r(μ) ; r?κ? s`h7 ζ?

6 + (κ? s`h7 ζ? ∗ 6)cbs μ ∗ κ? s`h ζ? cbs ζ? s`h μ. (13)

Ula eahbk`hftbr `h (13) `s bd tla dbrk A cbs μ+D s`h μ w`tl A ; κ? s`h7 ζ?6 fhe D ; ∗κ? s`h ζ? cbs ζ?.Ub racbvar tla usufi dbrk

r(μ) ; p

6 + a cbs(μ ∗ ϑ) (15)

bd tla aquft`bh bd f cbh`c w`tl accahtr`c`ty a fhe pfrfkatar p ; r?κ? s`h7 ζ? `h pbifr cbbre`hftas,tla tr`jbhbkatr`c èaht`ty a cbs(μ ∗τ) ; acbs μ cbs τ + a s`h μ s`h τ `s usae (saa 5.3 Nay _b`ht

6:, pfja 54).Cbkpfr`hj tl`s tb A cbs μ + D s`h μ fhe aquft`hj dfctbrs bd cbs μ fhe s`h μ, j`vas a cbs τ ; A fhea s`h τ ; D.

Squfr`hj tlasa fhe sukk`hj, j`vas a7 cbs7 τ + a7 s`h7 τ ; A 7 + D 7 br a7 ; A 7 + D 7. Ularadbra,

a7 ; (κ? s`h7 ζ? ∗ 6)7 + κ7? s`h7 ζ? cbs7 ζ?.

Fdtar axpfhe`hj tla strst tark fhe us`hj cbs7 ζ? ; 6 ∗ s`h7 ζ? `h tla sacbhe tark tla aquft`bh dbraccahtr`c`ty gacbkas

a7 ; 6 + κ7? s`h5 ζ? ∗ 7κ? s`h7 ζ? + κ7

? s`h7 ζ?(6 ∗ s`h7 ζ?).

Ula tarks `h s`h5 ζ? cfhcai tb iafva

a7 ∗ 6 ; κ?(κ? ∗ 7)s`h7 ζ?. (11)

Ula s`jh bd a7 ∗ 6 wl`cl eapahes bh tla vfiuas bd κ?, eatark`has tla trfmactbry typa bd tla sftaii`ta.@h (11), tla ifuhcl fhjia ζ? bccurs bhiy `h tla squfra bd tla s`ha duhct`bh fhe tl`s `s fiwfys pbs`t`va.Cbhsaquahtiy tla ifuhcl fhjia κ? lfs hb aact bh a`tlar tla s`jh bd a7 ∗ 6 br, tlaradbra, bh tlatrfmactbry typa.

:? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




Wacfii`hj jaharfi rasuits drbk 6:.6 fhe 6:.3 bh cbh`c sact`bhs=

6. `d κ? 9 7, a7 ∗ 6 9 ? fhe a7 9 6. Ularadbra a 9 6 fhe tla trfmactbry `s fh aii`pt`cfi brg`t.Hbta tlft a ; ? `s bgtf`hae dbr A ; D ; ? (br F ; G ; ?), fhe Aquft`bh (76) gacbkasr(μ) ; p. Ula trfmactbry bd tla sftaii`ta bccurs ft cbhstfht rfe`fi e`stfhca p ; r?κ? s`h7 ζ?

drbk tla cahtra bd tla pbifr cbbre`hfta systak (tla cahtra bd tla Afrtl), tlaradbra, tla brg`t`s c`rcuifr.

7. `d κ? ; 7, a7 ∗ 6 ; ? wl`cl `kpiàs a7 ; 6. Ularadbra, a ; 6 fhe tla trfmactbry `s pfrfgbi`c.

3. `d κ? 8 7, a7 ∗ 6 8 ? iafe`hj tb a7 8 6. Cbhsaquahtiy, a 8 6 fhe tla trfmactbry `s lypargbi`c.

@htarpratft`bh

S`hca κ? ≦ r?T 7?N

, tla tlraslbie vfiua κ? ; 7 tlft eatark`has tla trfmactbry typas `kpiàs f vaibc`ty

kfjh`tuea T ? ; 7N

r?. Ularadbra tla trfmactbry typas eapahe sbiaiy bh tla ifuhcl vaibc`ty kfjh`tuea

fhe

6. `d T ? 9

7N

r?, a 9 6, tla brg`t `s aii`pt`cfi,

7. `d T ? ;

7N

r?, a ; 6, tla trfmactbry `s pfrfgbi`c,

3. `d T ? 8

7N

W?, a 8 6, tla trfmactbry `s lypargbi`c.

Hbta tlft dbr fh `h`t`fi rfe`fi pbs`t`bh r? bd tla sftaii`ta (kafsurae drbk tla br`j`h bd tla pbifrcbbre`hfta systak ft tla cahtar bd tla Afrtl), tla tlraslbie kfjh`tuea bd ifuhcl vaibc`ty, `.a. tla

’ascfpa spaae‛, tlft park`ts tla sftaii`ta tb iafva tla jrfv`tft`bhfi staie bd tla Afrtl `s T asc? ;

7N

r?.

Fs ibhj fs tla ifuhcl vaibc`ty `s iass tlfh tla ascfpa spaae, tla sftaii`ta dbiibws fh aii`pt`cfi brg`tfrbuhe tla Afrtl fdtar ifuhcl.


:6




y e u t

S a s

f C j h ` r a

a h

`

j h A


Qhearjrbuhe rfìwfy s`jhfis ibcft`bhKftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Eàraht`ft`bh bd `hvarsa tr`j duhct`bhs ^66[Jabkatry (frc iahjtl) ^6:[_frt`fi eàraht`ft`bh ^6<[

@htrbeuct`bh

Ula ibcft`bh bd s`jhfis `h f curvae uhearjrbuhe rfìwfy tuhhai `s `kpbrtfht s`hca tla trf`h er`varhaaes tb ga fgia tb saa tlak `h t`ka tb stbp ft tlak sfdaiy. Fitarhft`vaiy, fh fevfhca `he`cft bhbd tla fpprbfcl`hj s`jhfis lfs tb ga j`vah. Ft f spaae bd 31 nk lr∗6 tla k`h`kuk e`stfhca fibhjtla trfcn raqu`rae tb j`va tla er`var t`ka tb rafct tb saa`hj tla s`jhfis `h t`ka dbr tla trf`h tb stbpfhe lfva f sfdaty kfrj`h, `s 57 k. Qsufiiy, tla trf`h er`var‐s ayas fra `h tla cahtra bd tla tuhhai.Uyp`cfiiy, tla s`jhfi `s ibcftae ?.71 k drbk tla sèa bd tla tuhhai, ft tla er`var‐s aya la`jlt, bh tlabutsèa bd tla curva.

_rbgiak `h wbres

Fh uhearjrbuhe rfìwfy `s tb ga axtaheae tb f haw Biykp`c spbrts stfeùk. Ula tuhhai lfs fe`fkatar bd 5 k fhe tla cahtra i`ha bd tla tuhhai `s curvae `h tla dbrk bd fh frc bd f c`rcia w`tl frfeùs bd 7?? k. Fssuka tlft tla kfx`kuk spaae bd tla trf`h `s 31 nk lr∗6, tlft tla trf`h er`var‐sayas fra `h tla cahtra bd tla tuhhai, gbtl vart`cfiiy fhe lbr`zbhtfiiy, fhe tlft tla s`jhfi `s ?.71 kdrbk tla sèa bd tla tuhhai, ft tla er`var‐s aya la`jlt, bh tla butsèa bd tla curva. Ula uhcartf`hty `htla tuhhai rfeùs ft fhy pb`ht `s ?.71 k. Kbrabvar tla fctufi pbs`t`bh bd tla s`jhfi kfy, `h prfct`ca,ga up tb ?.?7 k drbk `ts `htaheae pbs`t`bh, wlìa tla kbvakaht bd tla trf`h fhe tla vfr`ft`bh `htla er`var‐s clfrfctar`st`cs kafhs tlft tla er`var‐s vàwpb`ht kfy vfry gy ?.1 k drbk tla pbs`t`bhqubtae.

(f) Cficuifta tla pb`ht ft wl`cl tla er`var strst saas tla s`jhfi, fhe tla uhcartf`hty `h tlft cficu-

ift`bh cfusae gy tla vfr`ft`bhs `h tla s`jhfi pbs`t`bh fhe er`var‐s pbs`t`bh. @s tlara sufficàhtv`s`gì`ty dbr tla trf`h tb stbp sfdaiy w`tlbut fevfhca `he`cft`bh2

(g) F haw eas`jh bd trf`h wìi kbva tla er`var‐s pbs`t`bh ?.4 k tb tla iadt. Cfh tl`s eas`jh bd trf`hga `htrbeucae w`tlbut raqu`r`hj axtrf s`jhfis2

:7 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs





• D`jura 64.6 eap`cts tla s`tuft`bh

7?? k

60< k

7?6.:1 k

μ6μ7

C

E

A

S

D`jura 64.6

• Cficuifta tla fhjia μ6 sugtaheae ft tla v`rtufi cahtra bd tla rfeùs bd curvftura bd tla rfì (C )gy tla iahjtl drbk tla er`var (E) tb tla aeja bd tla tuhhai (A ) wlara tla i`ha bd s`jlt drbktla er`var tb tla s`jhfis tbuclas tla tuhhai.

• Cficuifta tla fhjia μ7 sugtaheae ft tla v`rtufi cahtra bd tla rfeùs bd curvftura bd tla rfì (C )gy tla iahjtl drbk tla s`jhfis (S ) tb tla aeja bd tla tuhhai (A ) wlara tla i`ha bd s`jlt drbktla er`var tb tla s`jhfis tbuclas tla tuhhai.

• Cficuifta tla axpactae e`stfhca fibhj tla rfìs gatwaah tla pbs`t`bhs bd tla s`jhfis fhe tlaer`var.

• Qsa pfrt`fi eàraht`ft`bh tb eaeuca tla vfr`ft`bh `h tlasa e`stfhcas rasuit`hj drbk vfr`ft`bhsbh s`jhfis fhe er`var pbs`t`bhs. Dbr axfkpia `d x eahbtas tla rfeùs bd tla tuhhai fhe y eahbtastla e`stfhca drbk tla aeja bd tla tuhhai fibhj tla i`ha bd s`jlt tb tla er`var, usa

∄μ6 ; ∇μ6∇x

∄x + ∇μ6∇y

∄y

• Lahca cficuifta tla pbss`gia rfhja bd e`stfhcas fibhj tla trfcn fhe cbkpfra tla rasuit`hj rfhjatb tla raqu`rae k`h`kuk e`stfhca.


Ula jabkatry bd tla s`tuft`bh tb ga kbeaiiae `s slbwh `h D`jura 64.6 fgbva wlara C eahbtas tlav`rtufi cahtra bd tla tuhhai curvftura.

(f) Ula i`ha bd s`jlt ES drbk tla er`var tb tla s`jhfi jrfzas tla tuhhai ft A , sb CA `s parpahe`cuifr

tb ES . Ula rfeùs bd tla cahtra i`ha bd tla tuhhai `s 7?? k, sb CA ; 60< k, CE ; 7?? kfhe CS ; 7?6.:1 k. Ularadbra tla fhjia μ6 ; cbs∗6

60<

7??

; ?.6561 rfe`fhs (tb 5 e.p.)


:3




fhe μ7 ; cbs∗6

60<

7?6.:1

; ?.6036 rfe`fhs (tb 5 e.p.). Ularadbra tla e`stfhca gatwaah er`var

fhe s`jhfi fibhj tla rfìs (cficuiftae fs tla e`stfhca fibhj tla cahtra i`ha) `s

7??(μ6 + μ7) ; 44.07 k

Za rbuhe tl`s ebwh tb 44 k dbr tla wbrst cfsa scahfr`b (fs ibwar `s iass sfda).

Za haae tb cficuifta tla uhcartf`hty `h tl`s e`stfhca. Iat x eahbta tla rfeùs bd tla tuhhaifhe y tla e`stfhca drbk tla aeja bd tla tuhhai tb tla er`var fibhj CE, `.a. 7 k. Ul`s kafhs

tlft CE ; x + y, sb tlft μ6 ; cbs∗6

x

x + y

.

Lahca ∇μ6

∇x ; ∗ 6

6 ∗ x

x + y7

Ú y

(x + y)7 ; ∗ y

(x + y)

7xy + y7(6)

Lahca ∇μ6

∇y ; ∗ 6

6 ∗

x

x + y

7Ú ∗x

(x + y)7 ;

x

(x + y)

7xy + y7(7)

Clfhjas bd ∄x `h x fhe ∄y `h y prbeuca f clfhja ∄μ6 ; ∇μ6∇x

∄x + ∇μ6∇y

∄y `h μ6.

S`hca x ; 60< fhe y ; 7 tlah

∄μ6 ; ∇μ6∇x

∄x + ∇μ6∇y

∄y ; ∗?.???3155∄x + ?.?31?0∄y (tb 5 s`j.stj.).

Ula uhcartf`hty `h tla rfeùs bd tla tuhhai (∄x) `s ?.71 k fhe tlft `h tla pbs`t`bh bd tla er`var(∄y) `s ?.1 k. Ulasa cbuie lfva a`tlar s`jh. Sb tla kfx`kuk arrbr `s wlah ∄x ; +?.71 fhe∄y ; ∗?.1, j`v`hj ∄μ6 ; ?.?6:45. Fs wa lfva wbrnae tb 5 s`jh`stcfht stjuras (fhe tlasa frabhiy fpprbx`kft`bhs) tla ifst stjura `s uhcartf`h.

Dbr tla uhcartf`hty `h μ7 (raiftae tb tla trfcn iahjtl gatwaah A fhe S ), x ; 60< fhe∄x ; ?.71 fs gadbra. Kbrabvar tla axprass`bhs (6) fhe (7) cfh ga usae dbr tla pfrt`fiear`vft`vas bd μ7 axcapt tlft hbw y ; 3.:1 fhe ∄y ; ?.?7. Sb

∄μ7 ; ∇μ7∇x

∄x + ∇μ7∇y

∄y ; ∗?.???5<?6∄x + ?.?7131∄y ; ?.???47:

Ula tbtfi uhcartf`hty `h tla e`stfhca fibhj tla trfcn cahtra-i`ha gatwaah E fhe S `s

7??(∄μ6 + ∄μ7) ; 3.4:15 k.

(g) Dbr tla rav`sae pbs`t`bh bd tla er`var wa lfva y ; 6.5 fhe x ; 60<. Ul`s j`vas μ6 ;

cbs∗6

60<

600.5

; ?.674: rfe`fhs, fhe ∄μ6 ; ?.?767. μ7 fhe ∄μ7 fra fs gadbra sb tla e`stfhca

fibhj tla trfcn `s 7??(?.360<) ; 43.04 k fhe tla uhcartf`hty `s 7??(?.?7764) ; 5.537 k.





@htarpratft`bh

(f) Fs wa lfva wbrnae tb dbur stjuras bhiy tla rasuit 3.4:40 k `s uhcartf`h, sb rbuhe upwfres tb3.< k br 5 k tb tla hafrast katra (`.a. `hcrafsa tla uhcartf`hty). Sb tla fctufi e`stfhca `sgatwaah 44

∗5 ; 47 k fhe 44 + 5 ; :? k. Gbtl fra jraftar tlfh tla k`h`kuk fccaptfgia

e`stfhca bd 57 k, sb hb fee`t`bhfi s`jhfi `s raqu`rae.

(g) Wbuhe`hj up tla rasuit bd 5.537 k fjf`h, tla rfhja `s (`h wlbia katras) gatwaah 10 k fhe40 k. Fjf`h tl`s `s jraftar tlfh tla k`h`kuk fccaptfgia e`stfhca, sb hb fee`t bhfi s`jhfi `sraqu`rae.


:1




y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey 6:

Laft cbheuct`bh tlrbujl f w`ha caiifr rbbd

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[D`rst brear BEAs ^60[Sacbhe brear BEAs ^60[Dburàr saràs ^73[

_frt`fi eàraht`fi aquft`bhs ^71[

@htrbeuct`bh

Zlah tla surdfca bd tla jrbuhe `s laftae gy tla suh tla takparftura bh tla surdfca vfràs bhf eùrhfi cycia. @d `t `s fssukae tlft tla takparftura `s uh`dbrk bvar fh fraf bd tla surdfca fhevfràs w`tl t`ka fhe eaptl fhe tla takparftura ft eaptl z fhe t`ka t `s eahbtae gy μ(z, t), `t`s nhbwh tlft μ(z, t) sft`sstas tla eùs`bh aquft`bh=

∇μ

∇t

; n∇ 7μ

∇z 7

(6)

wlara n `s f pbs`t`va cbhstfht `.a. tla cbheuct`v`ty bd tla jrbuhe, μ `s kafsurae `h eajraasCaisùs, z `s kafsurae `h katras (kafsurae w`tl tla pbs`t`va e`ract`bh ebwhwfres), t `s t`ka`h lburs, fhe tla vfiua bd n dbr sbì kfy ga tfnah tb ga 6.< Ú 6?∗3 k7 lr∗6. F s`kpia duhct`bhtlft raprasahts tla vfr`ft`bh bd takparftura w`tl t`ka ft tla surdfca `s

μ(?, t) ; μ? + F(?) cbs πt (7)

dbr sbka cbhstfhts μ?, F(?) fhe π.

_rbgiak `h wbres

F e`scarh`hj ahj`haar w`slas tb guìe f w`ha caiifr cbhs`st`hj bd f lbia `h tla jrbuhe i`hae w`tlcbhcrata sèas fhe w`tl f cbhcrata rbbd wl`cl `s cbvarae `h sbì tb fh fpprbpr`fta eaptl. Fccasstb tla w`ha caiifr `s tb ga fibhj f tuhhai drbk fh ax`st`hj caiifr. Ula raqu`rakaht `s tlft tlatakparftura `h tla w`ha caiifr slbuie hbt clfhja fpprac`fgiy bvar slbrt par`bes. Kbra prac`saiy,tla efìy vfr`ft`bh slbuie ga hb kbra tlfh ?.1◤C. @d `t `s fssukae tlft cbhvact`bh prbvèasrafsbhfgia trfhsk`ss`bh bd laft w`tl`h tla caiifr, tlah `t `s raqu`rae tlft tla takparftura bd tla sbì vfràs gy hb kbra tlfh ?.1◤C `kkae`ftaiy fgbva tla caiifr‐s rbbd. J`vah tlft tlakfx`kuk efìy vfr`ft`bh ft tla surdfca bd tla jrbuhe `s 5?◤C, lbw eaap slbuie tla rbbd bd tlaw`ha caiifr ga tb ahsura tlft tla efìy vfr`ft`bh `s hb kbra tlfh ?.1◤C2






(`) J`vah tlft tla par`be bd tla eùrhfi vfr`ft`bh bd tla takparftura ft tla surdfca eascr`gae`h Aquft`bh (7) `s 75 lburs (t `s kafsurae `h lburs), tlft tla kfx`kuk takparftura `sft t ; ? (hbbh) fhe tla kfx`kuk vfr`ft`bh `s 5?◤C, cficuifta π fhe F(?).

(``) Dbr tla takparftura ft eaptl z `h tla sbì, fssuka f sbiut`bh bd tla dbrk

μ(z, t) ; μ? + F(z )cbs(πt ∗ Cz ) (3)

wlara C `s f pbs`t`va cbhstfht fhe F(z ) eapahes bh z bhiy. Sugst tuta tla axprass`bh(3) `htb (6) fhe aqufta cbafficàhts bd s`ha fhe cbs`ha tarks tb sthe axprass`bhs dbr C fhelahca μ(z, t).

(```) Qsa tla rasuit`hj fkpi`tuea `h tla axprass`bh dbr μ(z, t) tb eaeuca tla vfiua bd z sucltlft tla fkpi`tuea `s iass tlfh ?.1.


(`) π ; 7Ϗ/75 ; Ϗ/67.

@d μ(?, t) ; μ? + F(?) cbs πt, tla kfx`kuk fhe k`h`kuk vfiuas fra μ? + F(?) fheμ?∗F(?). Sb dbr tla kfx`kuk vfr`ft`bh (wl`cl `s 7F(?)) tb ga 5?◤C, raqu`ras F(?) ; 7?.

(``) Dbr μ(z, t) ; μ? + F(z )cbs(πt ∗ Cz ) tb sft`sdy tla eàraht`fi aquft`bh (6), wa kustlfva tlft

∗F(z )w s`h(πt∗Cz ) ; n{F(z )cbs(πt∗Cz )+7F(z )C s`h(πt∗Cz )∗F(z )C 7 cbs(πt∗Cz )}.

Fs tla aquft`bh lbies dbr fii t fhe z ≡ ?, wa cfh aqufta tla cbafficàhts bd tla cbs`ha fhe s`ha

tarks (br aqu`vfiahtiy dbr fhy stxae z , wa cfh clbbsa t sb tlft tla s`ha tark `s ? tb aqufta tlacbafficàhts bd tla cbs`ha, fhe clbbsa t sb tlft tla cbs`ha tark `s ? tb aqufta tla cbafficàhtsbd tla s`ha). Ul`s j`vas f strst brear BEA fhe f sacbhe brear BEA=

∗F(z )π ; 7nCF(z ) (5)

? ; F(z ) ∗ F(z )C 7 (1)

Aquft`bh (5) wlah sbivae j`vas F(z ) ; F axp(∗ π

7nC z ) dbr sbka cbhstfht F fhe Aquft`bh

(1) j`vas

C 7 ;

π7nC

7, br C 7 ;

π7n

, br C ;

π7n

(s`hca C cfhhbt ga hajft`va!).

Sb

μ(z, t) ; μ? + Fa∗Cz cbs(πt ∗ Cz )

@t `s si`jltiy kbra cbhvahàht tb rawr`ta tl`s `s tla dbrk

μ(z, t) ; μ? + Fa∗z/E cbs(πt ∗ z

E)

wlara E ; 6

C ;

7n

π . Fs `h (`) wa lfva F ; 7?.


::




Qs`hj tla nhbwh vfiuas bd n fhe w, E ; ?.67 k (tb 7 s`jh`stcfht stjuras). Ula vfr`ft`bh `htakparftura w`tl eaptl z `s 5?a∗z/E . Ul`s `s iass tlfh ?.1 wlah z 8 E ih(<?) ; ?.174 k,br ?.13 k tb twb s`jh`stcfht stjuras. Sb tla rbbd bd tla w`ha caiifr slbuie ga ft f eaptl bd ftiafst 13 ck.

@htarpratft`bh fhe Cbkkaht

Ula fssukpt`bh tlft zarb t`ka cbrraspbhes tb tla l`jlast takparftura wìi hbt fact tlavfr ft`bh ft f pfrt cuifr eaptl, tlbujl `t wìi fact `ts t`k`hj. Lbwavar tla fssukpt`bh tlfttla takparftura ft tla surdfca `s j`vah gy f cbs`ha duhct`bh `s s`kpia gut hbt vary rafi`st`c. @t`s pbss`gia tb ear`va tla rasuit bd us`hj f kbra sbpl`st`cftae kbeai drbk tlft bd tla s`kpia bhagy kafhs bd Dburàr saràs. Suppbsa tlft tla surdfca takparftura `s j`vah gy μ(?, t) ; d (t)dbr ∗67 ≥ t ≥ 67 (t `h lburs). Ula efìy vfr ft`bh `s hbt dfr drbk f tr`fhjuifr wfva dbrk,j`vah gy F + G|t|, dbr fpprbpr`fta cbhstfhts F fhe G. Bh tla `htarvfi (∗67, 67) tla duhct`bh

d (t) ; 6?

3 (67 ∗ |t|) , lfs tla cbrract slfpa fhe fkpi`tuea. @d d `s fh avah duhct`bh `t cfh ga

raprasahtae gy f suk bd cbs`has, `.a.

d (t) ; 6

7f? +

∘h;6

fh cbs

hϏt

67

(4)

wlara

fh ; 6

67

67∗67

d (s)cbshϏs

67

es (h ; ?, 6, 7, 3, . . . )

Ul`s `s tla Dburàr saràs bh (

∗f, f) w`tl f ; 67.

Qs`hj (4),

d (t) ; 6?

3 (67 ∗ |t|) ; 7? +

∘h;6

64?

(7h ∗ 6)7Ϗ7 cbs

(7h ∗ 6)Ϗt

67

.

Z`tl π ; (7h ∗ 6) Ϗ

67, tla cbrraspbhe`hj sbiut`bh tb j`va F cbs

(7h ∗ 6)Ϗt

67 ft tla surdfca `s

μh(z, t) ; F axp∗z ∝

7h ∗ 6

E cbs(7h ∗ 6)Ϗt

67 ∗ z

∝ 7h ∗ 6

E

wlara E ;

75n

Ϗ 9 ?.67 k fs gadbra. Ula sbiut`bhs cbrraspbhe`hj tb tla tarks wl`cl suk

tb j`va 6?

3 (67 ∗ |t|) fra sukkae tb j`va

7? +∘h;6

64?

(7h ∗ 6)7Ϗ7axp

∗z

∝ 7h ∗ 6

E

cbs

(7h ∗ 6)Ϗt

67 ∗ z

∝ 7h ∗ 6

E?

Ul`s duhct`bh sft`sstas (6) fhe lfs tla vfiua 6?

3 (67 ∗ |t|) ft z ; ?. Hbta tlft tla l`jlastvfiuas bd tla axpbhaht`fi tarks dfii b rfpèiy fs z `hcrafsas fhe fra ft kbst a∗z/E sb tla

:< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




tbtfi vfr`ft`bh `h takparftura `s ft kbst 7∘h;6

64?

(7h ∗ 6)7Ϗ7a∗z/E .

Hbw putt`hj t ; 67 fhe z ; ? slbws tlft

7? ∗∘h;6

64?(7h ∗ 6)7Ϗ7

; μ(?, 67) ; ?, sb∘h;6

64?(7h ∗ 6)7Ϗ7

; 7?

Ul`s kafhs tlft tla kfx`kuk vfr`ft`bh `s 5?a∗z/E fs dbr tla s`kpia kbeai, sb tla eaptl bd ?.13 k eaeucae w`tl tla s`kpia kbeai wìi suffica. @d f gattar ast`kfta `s bgtf`hae gy `sbift`hjtla strst vfr`fgia tark fhe slbw`hj tlft fii tla btlar axpbhaht`fi tarks fra ft kbst a∗z

∝ 3/E `t

`s dbuhe tlft f vary si`jltiy skfiiar eaptl wìi suffica.

@htarpratft`bh

@h sukkfry, us`hj f kbra sbpl`st`cftae kbeai dbr tla surdfca takparftura vfr`ft`bh tlfh (7)fhe axprass`hj d fs f Dburàr saràs rasuits `h tla cbhcius`bh tlft sfka eaptl wìi suffica dbrtla w`ha caiifr.


:0




y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey 6<

Uwb-e`kahs`bhfi bw pfst f cyi`her`cfi bgstfcia

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnFijagrf - rafrrfhj`hj dbrkuifa ^6[Ur`jbhbkatry ^5[_bifr cbbre`hfta systak ^6:[, ^7<[

@htrbeuct`bh

Kfhy `hvast`jft`bhs `h ahj`haar`hj uè kaclfh`cs `hvbiva prae`ct`hj tla bw bd uè pfstbgstfcias fhe tla dbrcas `hvbivae bh tla sbiè gbuhefry. Dbr axfkpia `t `s hacassfry tb cficuiftatla farbeyhfk`c dbrcas bh fh f`rcrfdt (tla i`dt fhe erfj), tla w`he ibfe`hj bh guìe`hjs fhetla erfj bh fii typas bd ifhe val`cias. @h wftar `t `s hacassfry tb cficuifta tla lyerbeyhfk`cdbrcas bh gbfts br bh gr eja suppbrts. Kbeaii`hj uè bw pfst bgstfcias usas tla cbhcaptsbd strafki`has fhe strafk duhct`bhs (ό). Strafki`has fra i`has bd cbhstfht ό. F sat bd strafki`has `h f j`vah raj`bh bd spfca raprasahts tla vaibc`ty staie. Bh tla gbuhefry bd f sbièbgstfcia, tla strafk duhct`bh `s cbhstfht.

@h pbifr cbbre`hftas (r, μ), axfkpias bd strafk duhct`bhs usae `h kbeai`hj bw fra (f) Qr s`h μdbr f uh`dbrk bw pfrfiiai tb tla x-fx`s wlara Q `s tla kfjh`tuea bd tla bw vaibc`ty fhe(g) 7fk s`h μ/r dbr f e`pbia cbhs`st`hj bd f sburca fhe s`hn bd aqufi strahjtl 7Ϗk ft aqufie`stfhcas f a`tlar sèa bd tla cbbre`hfta br`j`h (slbwh w`tl Cfrtas`fh cbbre`hftas (∗f, ?) fhe(f, ?) raspact`vaiy `h D`jura 6<.6).

Q

x

y

∗ f f

r

μB

sburca s`hn

D`jura 6<.6= Jabkatry bd tla e`pbia (sburca fhe s`hn) fhe cbbre`hfta systak

Dbr kfhy prbgiaks `t `s fssukae tlft

(`) Ula uè bw`hj pfst tla bgstfcia `s `hcbkprass`gia fhe `hv`scè (hb dr`ct`bh)(``) Ula bw `s stafey (cbhstfht `h t`ka).

<? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




Ula pr`hc`pia bd suparpbs`t`bh (eua tb Wfhn`ha) ahfgias tla fee`t`bh bd tla strafk duhct`bhsbd supar`kpbsae vaibc`ty staies. Dbr axfkpia, tla suparpbs`t`bh bd f stafey uh dbrk bw fhetla bw eua tb f e`pbia `s raprasahtae gy tla suk bd tla raspact`va strafk duhct`bhs=

ό ; Qr s`h μ

∗7fk s`h μ/r.

_rbgiak `h wbres

Slbw tlft tla strafk duhct`bh rasuit`hj drbk tla suk bd tla strafk duhct`bhs dbr f uh`dbrkbw fhe dbr f e`pbia cfh ga usae tb raprasaht bw pfst f r`jè cyi`hear bd j`vah rfeùs. D`hetla vfiua bd sburca strahjtl k tlft ahsuras tl`s.


Cficuifta pb`hts bh strafki`has ό ; ? fhe ό ; ¶Qf gfsae bh tla cbkg`hae strafk duhct`bh

ό ; Qr s`h μ ∗ 7fk s`h μ/r.

Slbw tlft pfrt bd tla strafki`ha ό ; ? `s f c`rcia cb`hcèaht w`tl tla cyi`hear gbuhefry w`tlrfeùs f, fhe eaeuca tla `kpiàe vfiua bd k raiftae tb tla sburca strahjtl bd tla e`pbia.


Ula dfctbr s`h μ cfh ga tfnah butsèa f grfcnat `h tla cbkg`hae strafk duhct`bh ό sb tlft

ό ; s`h μ

Qr7 ∗ 7fk

r

. (6)

@d ό ; ?, tlah

s`h μ

Qr7 ∗ 7fk

r

; ?.

Dbr tl`s tb ga trua tlah a`tlar (`) s`h μ ; ? br (``) Q r7 ∗ 7fk

r ; ?.

(`) @d s`h μ ; ?, tlah μ ; ? br Ϗ.

(``) @d

Q r7

∗7fk

r ; ?, tlah Qr

7

∗ 7fk ; ? wl`cl `kpiàs

r ;

7fk

Q . (7)

S`hca f,k fhe Q fra cbhstfhts, (7) easthas f c`rcia bd rfeùs

7fk

Q .

Cbhsaquahtiy, tla strafki`ha ό ; ? lfs tlraa pfrts=

(@) pb`hts sft sdy`hj tla cbhe`t`bh μ ; Ϗ, `.a. fibhj tla hajft`va x-fx`s,(@@) pb`hts sft`sdy`hj tla cbhe`t`bh μ ; ?, `.a. tla pbs`t`va x-fx`s,

(@@@) pb`hts bh f c`rcia bd rfeùs

7fk

Q .


<6




Fssuk`hj tlft tla cyi`hear lfs f rfeùs f aqufi tb tla e`stfhca bd tla sburca br s`hn tark `htla e`pbia drbk tla cbbre`hfta br`j`h, `t `s pbss`gia tb clbbsa

f ; 7fk

Q (3)

Ul`s ahsuras tlft tla c`rcuifr gbuhefry bd tla r`jè cyi`hear `s pfrt bd tla strafki`ha ό ; ?. Hbbw kfy crbss f strafki`ha sb f sbiè gbuhefry `s fiwfys pfrt bd f strafki ha bd tla bw. Ularafra hb strafki`has `hsèa tla c`rcuifr crbss sact`bh bd tla cyi`hear, tlaradbra tla strafki`has dbr∗f ≥ x ≥ f fhe ∗f ≥ y ≥ f wl`cl wbuie ga `hsèa tla c`rcuifr crbss sact`bh bd tla cyi`hearfra hbt slbwh `h D`jura 6<.7.

Aquft`bh (3) `kpiàs tlft tla sburca strahjtl lfs f vfiua j`vah gy k ; fQ

7 .

Z`tl tl`s vfiua bd k, tla strafk duhct`bh easthae `h (6) gacbkas

ό ; Q s`h μr7

∗f7

r

. (5)

Hbw cbhsèar tla strafki`ha ό ; Qf. Drbk (5), tl`s kafhs tlft

Qf ; Q s`h μ

r7 ∗ f7

r

wl`cl cfh ga wr`ttah fs

f ; r s`h μ

6 ∗ f7

r7. (1)

@t `s afsàr tb pibt tla fssbc`ftae strafki`has `d (1) `s axprassae `h tarks bd Cfrtas`fh cbbre`hftas(x, y). Qs`hj y ; r s`h μ fhe r ;

x7 + y7 (saa D`jura 6<.6), Aquft`bh (1) j`vas

y

f ;

6

6 ∗ f7

x7 + y7

. (4)

Ul`s cfh ga wr`ttah fs y

f ;

x7 + y7

x7 + y7 ∗ f7 br

y

f

x7 + y7 ∗ f7

; x7 + y7.

@h tl`s cfsa tla fijagrf `s afsàr `d x

f `s axprassae `h tarks bd y

f `hstafe bd tla btlar wfy rbuhe,wl`cl `s kbra usufi. E`vè`hj tlrbujl gy fy iafes tb

x7

f7 +

y7

f7 ∗ 6 ;

x7

fy +

y

f.

Wafrrfhj`hj tb jat fii tla y tarks bh tla iadt-lfhe sèa fhe x tarks bh tla r`jlt-lfhe sèaraqu`ras tlraa staps=

Stap 6

yf

7 ∗ y

f ∗ 6 ;

xf

7 6y

f

∗ 6 ,

<7 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




Stap 7

y

f

7∗y

f

∗ 6

6

∗ y

f

Ú

y

f;

x

f7

.

Stap 3

x

f ; ¶

∗y

f

7+

y

f + 6

y

f ∗ 6

Ú

y

f

. (:)

Ula pius fhe k`hus s`jhs fiibw raprasahtft`bh bd tla wlbia strafki`ha dbr pbs`t`va fhe hajft`va

vfiuas bd x. Aquft`bh (:) `s plys`cfiiy kafh`hjdui dbr y

f ; 6 fhe fs ibhj fs tla squfra rbbt `s

rafi, `.a. dbr

∗y

f

7+

y

f + 6

y

f ∗ 6

Ú

y

f

≡ ? (<)

D`rst wa ibbn ft tla quferft`c tark

∗y

f

7+

y

f + 6, (0)

tb eatark`ha wlara `t clfhjas s`jh. Qs`hj tla quferft`c dbrkuif w`tl y

f fs tla vfr`fgia, tla

rbbts bd (0) fra 6 ¶ ∝

1

7 .

Sb ∗y

f

7+

y

f + 6 ≡ ? dbr

6 ∗ ∝ 1

7 ≡ y

f ≡ 6 +

∝ 1

7 , (saa D`jura 6<.7).

6 ∗ ∝ 1

7

6 + ∝ 17

x

yf

D`jura 6<.7

Ula strst tlraa i`has bd Ufgia 6 slbw tla s`jhs bd tla tarks bh tla iadt-lfhe sèa bd (<) fhe

fiibw us tb eaeuca tla rfhja bd vfiuas bd y

f sft`sdy`hj (<). Ula bvarfii s`jh `s slbwh `h tla ifst

cbiukh bd Ufgia 6. Sb `t cfh ga cbhciueae tlft Aquft`bh (<) `s sft`sstae dbr 6 ∗ ∝

1

7 ≥ y

f ≡ ?

fhe 6 ≥ yf ≥ 6 +

∝ 17 .


<3




Ufgia 6= S`jhs bd tla tarks `h `haqufi`ty (<)

y

f

y

f ∗ 6 ∗

y

f7

+ y

f + 6

∗y

f

7+

y

f + 6

y

f ∗6

Ú y

f∗∘ 9

y

f 9

6 ∗ ∝ 1

7 hajft`va hajft`va hajft`va hajft`va

6 ∗ ∝ 1

7 ≥ y

f ≥ ? hajft`va hajft`va pbs`t`va pbs`t`va

? 9 y

f 9 6 pbs`t`va hajft`va pbs`t`va hajft`va

6 ≥ yf ≥ 6 + ∝ 1

7 pbs`t`va pbs`t`va pbs`t`va pbs`t`va

6 +∝

1

7 9

y

f 9 +∘ pbs`t`va pbs`t`va hajft`va hajft`va

Ula scfiae cbbre`hftas, x

f fhe

y

f, j`va rasuits tlft fra `heapaheaht bd tla vfiuas bd f.

Hbw cbhsèar tla strafki`ha ό ;

∗Qf, Aquft`bh (5) iafes tb

∗Qf ; Q s`h μ

r7 ∗ f7

r

wl`cl, fdtar us`hj y ; r s`h μ fhe r ;

x7 + y7, fs dbr ό ; +Qf, cfh ga wr`ttah fs

y

f ;

∗6

6 ∗ f7

x7 + y7

. (6?)

Aquft bh (6?) `s s`kpiy tla hajft`va bd (4). Ularadbra, w`tlbut durtlar fijagrf`c abrt `t cfh gaeaeucae tlft tla strafki`ha ό ; ∗Qf `s sykkatr`cfi tb tla strafki`ha ό ; Qf w`tl raspact

tb tla y

f ; ? fx`s.

Ula strafki has ό ; ? fhe ό ; ¶Qf fra pibttae `h D`jura 6<.3. Fs f rasuit bd tla clb`ca bd scfiae Cfrtas`fh cbbre`hftas,

x

f fhe

y

f, tla cyi`hear lfs f rfeùs 6.

@htarpratft`bh

Ula strafki`has ό ; 7Qf fhe ό ; ∗7Qf fra fisb slbwh `h D`jura 6<.3. Ulay fra s`kìfrtb tla strafki`has ό ; Q f fhe ό ; ∗Qf gut fra ft rbujliy tw`ca tla e`stfhca drbk tlax-fx`s. Diuè pfrt cias fpprbfcl`hj tla cyi`hear fibhj pfrfiiai strf`jlt i`has fi`jhae w`tl tlax-fx`s fra eaactae frbuhe tla cyi`hear. Ula fkbuht bd tla`r eaact bh `hcrafsas fs tlastrafki`ha fpprbfclas tla x-fx`s. Ul`s `s cbhs`staht w`tl `htu`t bh fgbut tla kbt`bh bd uèpfrt`cias cbhstrf`hae tb bw frbuhe f cyi her`cfi bgstfcia. Ula strafki has fra fisb `h varyjbbe fjraakaht w`tl bgsarvft`bhs axcapt `kkae`ftaiy `h drbht bd tla cyi`hear fhe gal`he tlacyi`hear wlara tla fssukpt`bh bd stafey bw ebas hbt lbie.





7Qf

Qf?

∗3 ∗7 ∗6 ? 6 7 3

x/f

y/f x

y

Qf

7Qf

∗6

∗7

∗3

6

7

3

D`jura 6<.3= Strafki`has jaharftae gy cbhstfht vfiuas bd tla strafk duhct`bh


<1




y e u t

S a s

f C j h ` r a

a h

`

j h A


Uwb-e`kahs`bhfi bw bd f v`scbus i`què bh fh `hci`haepifta

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[@htajrft`bh ^63[

_bifr cbbre`hfta systak ^6:[, ^7<[_frt`fi eàraht`ft`bh ^6<[_frt`fi eàraht`fi aquft`bhs ^71[Tactbr cficuius ^7<[E`kahs`bhfi fhfiys`s ^5:[

@htrbeuct`bh

Kfhy `hvast`jft`bhs `h ahj`haar`hj uè kaclfh`cs `hvbiva prae`ct`hj tla bw bd i`quès `hclfhhais br p`pas. S`kpia kbeais fssuka tlft tla i`què `s `hv`scè `.a. tlft `t `s hbt v`scbus.

Lbwavar, e‐Fiakgart‐s pfrfebx fhe btlar rasuits dbr `hv`scè bw slbw tlft tla `hv`scè uèbw kbeai haaes tb ga rasthae gy tla `htrbeuct`bh bd v`scbus dbrcas. Ula kbeaii`hj bd `hv`scèuè bws usas Auiar‐s aquft`bh bd kbt`bh gfsae bh tla fssukpt`bh tlft surdfca dbrcas bh fuè aiakaht fra bhiy hbrkfi tb tla surdfca bd tla uè aiakaht. Ufhjaht`fi dbrcas eua tbv`scbs ty fra `jhbrae. Ula Hfvàr-Stbnas aquft`bh bd uè bw `s f jaharfi sft bh bd Auiar‐saquft`bh tb `hciuea v`scbus dbrcas. @h Cfrtas`fh cbbre`hftas tla aquft`bh cfh ga wr`ttah

φeu

et ; ∗∊ p + φD + ±∊7u (6)

wlara ± `s tla cbafficàht bd v`scbs`ty, φ `s tla uè eahs`ty, p `s tla prassura, D `s tla gbey

dbrca par uh`t kfss fhe u raprasahts tla uè vaibc`ty. Hbta tlft tla tbtfi t`ka ear`vft`va

bparftbr `s easthae fs e

et ≦ ∇

∇t + u.∊.

Dbr kfhy prbgiaks `t `s fssukae tlft

(`) Ula uè `s `hcbkprass`gia

(``) Ula uè `s Hawtbh`fh (strass prbpbrt`bhfi tb tla rfta bd eadbrkft`bh)

(```) Ula uè lfs cbhstfht v`scbs`ty

(`v) Ula bw `s duiiy-eavaibpae (tla bw `s èaht`cfi ft fii pb`hts bd f uè aiakaht trfmactbrydbr f j`vah t`ka)





_rbgiak `h wbres

D`he tla vaibc`ty e`str`gut`bh `h f v`scbus i`què bw`hj draaiy fhe stafeìy uhear jrfv`ty bvar fvary ifrja `hci`hae pifta wlah tla i`què `s sugmactae tb f slafr strass bd nhbwh vfiua fs f rasuitbd f`r gibwh fcrbss `ts surdfca. Snatcl f jrfpl bd tla vaibc`ty prbstia dbr pbs`t`va, hajft`va fhe

vfh`sl`hj slafr strass `h e`kahs`bhiass dbrk.


Clbbsa tla e`ract`bh bd tla bw fs tla x-fx`s, tla hbrkfi tb tla i`què/f`r `htardfca fhe tb tlapifta fs tla z -fx`s (saa D`jura 60.6), sb tla y-fx`s `s clbsah tb ià fibhj tla pifta. Fssukatlft tla pifta `s `hsth`taiy ifrja fibhj tla y-fx`s. Ula bw `s kbeaiiae fs twb-e`kahs`bhfi `.a.tlara `s zarb vaibc`ty `h tla y-e`ract`bh. Cbhsaquahtiy, easth`hj u ; ux` + uy m + uzn fs tlavaibc`ty vactbr `h tla brtlbhbrkfi gfs`s sat (`, m, n),

(`) uy ; ?

(``) tla ear`vft`vas w`tl raspact tb vfr`fgia y fra zarb.

Ula bw `s stafey, tlaradbra

(```) tla ear`vft`vas w`tl raspact tb t`ka fra zarb.

Ula bw `s duiiy eavaibpae `h tla e`ract`bh bd tla bw, tlaradbra

(`v) tla ear`vft`vas w`tl raspact tb vfr`fgia x fra zarb.

(@) Zr`ta ebwh tla gbuhefry cbhe`t bhs ft z ; ? fhe z ; i tlft kust ga sft`sstae gy uus`hj cbhe`t`bhs (`)-(`v).

(@@) Fppiy tla cbht`hu`ty aquft`bh ∊u ; ?.

(@@@) Fppiy tla Hfvàr-Stbnas aquft`bh (6) tb tla bw cbhe`t`bhs (`)-(`v) fhe `htajrfta tbsthe tla vaibc`ty e`str`gut`bh.

f`r

uè

piftaζ

z

x

π

iζ

D`jura 60.6= Jabkatry bd tla bw fhe cbbre`hfta systak


<:





(@) Ula gbuhefry cbhe`t`bh ft z ; ? `s tlft u ; ? fs tla pifta `s ft rast fhe `s fssukae tlfttlara `s hb si`p gatwaah tla i`què fhe tla pifta. Ul`s kafhs tlft ux

z;?

; ? fhe ; uzz;?

; ?.

Ula gbuhefry cbhe`t`bh ft z ; i raiftas tla j`vah slafr strass U jaharftae gy tla f`r bw tbtla vaibc`ty `.a.

U ; ±∇ux

∇z

z;i

. (7)

(@@) Ula cbht`hu`ty aquft`bh ∊u ; ? cfh ga wr`ttah `h Cfrtas`fh cbbre`hftas fs

∇ux

∇x +

∇uy

∇y +

∇uz

∇z ; ?. (3)

Cbhe`t`bhs (`) fhe (`v) s`kpi`dy (3) fs

∇uz

∇z ; ?. (5)

(@@@) Qs`hj tla easth`t`bhs bd ∊ fhe ∊7 `h Cfrtas`fh cbbre`hftas j`vah `h 7<.7, tlaHfvàr-Stbnas aquft`bh (6) cfh ga wr`ttah fs tla tlraa aquft`bhs

φ

∇ux

∇t + ux

∇ux

∇x

; ∗∇p

∇x + φD x + ±

∇ 7ux

∇x7 +

∇ 7ux

∇y7 +

∇ 7ux

∇z 7

(1)

φ∇uy

∇t

+ uy∇uy

∇y ;

∗

∇p

∇y

+ φD y + ±∇ 7uy

∇x7

+ ∇ 7uy

∇y7

+ ∇ 7uy

∇z 7 (4)

φ

∇uz

∇t + uz

∇uz

∇z

; ∗∇p

∇z + φD z + ±

∇ 7uz

∇x7 +

∇ 7uz

∇y7 +

∇ 7uz

∇z 7

(:)

cbrraspbhe`hj tb prbmact`bhs bh tla tlraa Cfrtas`fh fxas. Ula gbey dbrca `h tl`s prbgiak `seua tb jrfv`ty, tlaradbra, D x ; j s`h ζ, D y ; ? fhe D z ; j cbs ζ wlara ζ `s tla `hci`hft`bhfhjia bd tla pifta tb tla lbr`zbhtfi. Cbhe`t`bhs (`) tb (`v) ahfgia s`kpi`stcft bh bd (1) tb (:)raspact`vaiy fs

? ; φD x + ±∇ 7ux

∇z 7 (<)

? ; ? (0)

φuz∇uz

∇z ; ∗∇p

∇z + φD z + ±

∇ 7uz

∇z 7 . (6?)

Qs`hj (5), Aquft`bh (6?) gacbkas

? ; ∗∇p

∇z + φD z. (66)

Ula cbafficàht bd v`scbs`ty fhe tla i`què eahs`ty fra cbhstfht, fhe (<) cfh ga `htajrftae bvarvfr`fgia z tb sthe tla vaibc`ty prbstia.

@htajrft`bh drbk i tb z j`vas ±

zi

∇ 7ux

∇z 7 ez ; ∗φ

zi

D x ez.

<< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stueàs




Ula x-cbkpbhaht bd tla gbey dbrca D x ; j s`h ζ ebas hbt eapahe bh z fhe cfh ga tfnah butbd tla `htajrfi. Sb

± ∇ux

∇z z

i

; ∗φD x z

i

ez, br

±

∇ux

∇z ∗ ∇ux

∇z

z;i

; ∗φD x(z ∗ i). (67)

Qs`hj gbuhefry cbhe`t`bh (7), (67) kfy ga rawr`ttah fs

±

∇ux

∇z ∗ U

±

; ∗φD x(z ∗ i). (63)

Hbta tlft tla `htajrft`bh wfs clbsah drbk i tb z `h brear tb usa tla (7) ft tl`s stfja. @htajrft`hj

bha kbra t`ka bvar vfr`fgia z stfrt`hj drbk (63) iafes tb

±

z?

∇ux

∇z ∗ U

±

ez ; ∗φD x

z?

(z ∗ i) ez,

wlara tla `htajrft`bh gbuhes lfva gaah clbsah drbk ? tb z `h brear tb usa tla gbuhefrycbhe t`bh ft z ; ?. Fdtar avfiuft`hj tla `htajrfis

±

ux(z ) ∗ ux

z;?

∗ Uz ; ∗φD x

z 7

7 ∗ iz

. (65)

Ula gbuhefry cbhe`t`bh ft z ; ? jfva ux

z;?

; ? tlaradbra (65) iafes tb

ux(z ) ; ∗φD x7±

z 7 + U + φD xi

± z. (61)

Ula e`kahs`bhs bd ± fra KI∗6U∗6. (Saa 5:.6 dbr f e`scuss`bh bd e`kahs`bhfi fhfiys`s).

Sb tla e`kahs`bhs bd φD x

± fra KI∗3IU∗7/(KI∗6U∗6) ; U∗6I∗6 wl`cl `s tla sfka fs tla

e`kahs`bh bd vaibc`ty e`vèae gy iahjtl squfrae. Sb e`kahs`bhiass vaibc`ty ux `s bgtf`hae

gy e`vè`hj (61) tlrbujl gy φD x

± i7. Durtlarkbra `d tla vfr`fgia z `s scfiae gy i tb yàie fe`kahs`bhiass (iahjtl) vfr`fgia fisb tlah

ux

z

i

; ∗6

7

z

i

7+

U + φD xi

φD xi

z

i. (64)

Axfkpias bd vaibc`ty prbstias ux(z ) dbr pbs`t`va fhe zarb vfiuas bd slafr strass fra snatclae `hD`jura 60.7 fs tla tl`h sbiè curva fhe ebttae curva. Uwb vaibc`ty prbstias fra slbwh fs tlatl`cn sbiè curva fhe efslae curva dbr hajft`va vfiuas bd slafr strass. Hbta tlft z/i `s pibttaefjf`hst uz sb tlft wlah tla i`què ifyar `s supar`kpbsae bh tla vaibc`ty prbstia `t `s raprasahtaegy tla fraf gatwaah tla pf`r bd pfrfiiai tl`cn lbr`zbhtfi gifcn i`has. Ula tl`cnhass bd tla i`quèifyar `s 6 fs f rasuit bd tla scfi`hj z/i.


<0




z/

ux

z

U 8 ?

U ; ?U 9 ?

z

u

B

∗?.5

6

∗?.< ?.5 ?.<

?.1

D`jura 60.7= Eapaheahca bd vaibc`ty prbstia bh slafr strass

@htarpratft`bh

Zlah hb f`r bw `s prasaht, tla slafr strass vfh`slas (U ; ?) fhe tla i`què vaibc`ty kfjh`tuea`hcrafsas drbk tla gbttbk pifta tb tla surdfca dbiibw`hj f pfrfgbi`c curva fs axpactae (ebttaecurva). Zlah f`r gibws bvar tla i`què `h tla e`ract bh bd tla bw fhe rasuits `h f pbs`t`vaslafr strass (U 8 ?), tla kfjh`tuea bd tla i`què vaibc`ty `s `hcrafsae avarywlara cbkpfrae tb

tla cfsa w`tl hb f`r bw. Kbrabvar tla i`què vaibc`ty kfjh`tuea `hcrafsas drbk tla gbttbkpifta tb tla surdfca dbiibw`hj f pfrfgbi`c curva (tl`h sbiè curva). Zlah tla f`r bw `s `h tlabppbs`ta e`ract bh tb tla i`què bw, `t rasuits `h f hajft`va slafr strass (U 9 ?). Cbhsaquahtiy,tla i`què vaibc`ty kfjh`tuea `s eacrafsae wlah cbkpfrae tb tla cfsa w`tl hb f`r bw. @d tlaslafr strass `s sufficàht, tla i`què ft fhe hafr tla surdfca kfy bw uplìi fs slbwh gy tlahajft`va pbrt`bh bd tla tl`cn sbiè curva. Ula efslae curva slbws rasuits wlara tla w`he `s sbstrbhj tlft wlbia bd tla i`què ifyar `s puslae uplìi w`tl f spaae tlft `hcrafsas w`tl la`jlt `htla i`què ifyar. Hbta tlft tla kbeai dbr `hv`scè i`què `h wl`cl tfhjaht`fi slafr strassas frahajiactae rasuits `h f cbhstfht rftlar tlfh pfrfgbi`c vaibc`ty prbstia dbr bw bh f pifta.





y e u t

S a s

f C j h ` r a

a h

`

j h A

Ahj`haar`hj Cfsa Stuey 7?

Dbrca bh f cyi`hear eua tb f twb-e`kahs`bhfi strafk`hjfhe sw`ri`hj bw

Kftlakft`cfi Snìis

Ubp`c ZbrngbbnUr`jbhbkatry ^5[Ibjfr`tlk`c duhct`bhs ^4[@htajrft`bh ^63[

Brtlbjbhfi`ty raift`bhs bd tr`jbhbkatr`c duhct`bhs ^63[_bifr cbbre`hfta systak ^6:[, 7<[_frt`fi ear`vft`vas ^6<[Surdfca htajrfis ^70[

@htrbeuct`bh

Kfhy `hvast`jft`bhs `h ahj`haar`hj uè kaclfh`cs `hvbiva prae`ct`hj tla bw bd uè pfstbgstfcias fhe tla dbrcas `hvbivae bh tla sbiè gbuhefry. Dbr axfkpia `t `s hacassfry tb cficuiftatla farbeyhfk`c dbrcas bh fh f`rcrfdt (tla i`dt fhe erfj) bh guìe`hjs sugmact tb w`he ibfe`hjfhe bh fii typas bd ifhe val`cias. @h wftar `t `s hacassfry tb cficuifta tla lyerbeyhfk`c dbrcas

bh gbfts br bh grèja suppbrts. Kbeais bd uè bw pfst bgstfcias , `.a. strafk`hj bw usatla cbhcapts bd strafki`has fhe strafk duhct`bhs ό. Strafki has fra i`has bd cbhstfht ό.(Saa Ahj`haar`hj Cfsa Stuey 6< D`jura 6<.3.) F sat bd strafki has `h f j`vah raj`bh bd spfcaraprasahts tla vaibc`ty staie.

Drbk Ahj`haar`hj Cfsa Stuey 6< tla strafk duhct`bh

ό6 ; Q s`h μ

r7 ∗ f7

r

(6)

raprasahts tla bw `h tla x-y pifha pfst fh `hsth`ta cyi`hear w`tl `ts fx`s `h tla z -e`ract`bh (saaD`jura 7?.6) wlara (r,μ,z ) fra tla pbifr cbbre`hftas `h tla radarahca drfka (r˄, μ˄, n˄), Q `s tlakfjh`tuea bd tla `hcèaht uh`dbrk bw vaibc`ty pfrfiiai tb tla x-fx`s fhe f `s tla rfeùs bd tlacyi`hear.

F sw`ri`hj bw frbuhe f cyi`hear bd fx`s z cfh ga kbeaiiae gy tla strafk duhct`bh bd f i`havbrtax e`ractae fibhj tla z -fx`s. F i`ha vbrtax kbeais f twb-e`kahs`bhfi rbtft`bhfi kbt`bhwlara tla vbrt`c`ty `s cbhcahtrftae fibhj fh fx`s. Ula strafk duhct`bh ό7 dbr f i`ha vbrtax bd strahjtl ν `s

ό7 ; ν

7Ϗ ih r + cbhstfht. (7)

Ula cbrraspbhe`hj strafki`has fra c`rcias `h tla xy pifha w`tl cahtra ft tla br`j`h. Ula cbhstfht`h (7) `s frg`trfry fhe vfh`slas wlah tfn`hj tla ear`vft`vas tb bgtf`h tla vaibc`ty. @h tla cfsa


06




bd tla bw frbuhe f cyi`hear bd rfeùs f, `t `s cbhvahàht tb clbbsa tla cbhstfht fs ∗ν

7Ϗ ih f sb

tlft ό7 ; ν

7Ϗ ihr

f

. Z`tl tl`s clb`ca bd cbhstfht, tla gbuhefry bd tla cyi`hear cbrraspbhes

tb tla strafki`ha ό7 ; ?.

Dbr kfhy prbgiaks `t `s fssukae tlft(`) Ula uè bw`hj pfst tla bgstfcia `s `hcbkprass`gia fhe `hv`scè (hb dr`ct`bh)

(``) Ula bw `s stafey (cbhstfht `h t`ka)

@d tla bw `s `rrbtft`bhfi fhe hb gbey dbrca `s prasaht, Garhbuii`‐s aquft`bh fppiàe fibhj fhycurva `h tla uè `s j`vah gy

_ + φu7

7 ; C (3)

wlara _ `s tla prassura, φ `s tla uè eahs`ty, u `s tla spaae bd tla uè fhe C `s f cbhstfht.

@d tla bw `s rbtft`bhfi, tla curva lfs tb ga f strafki`ha dbr (3) tb ga vfiè.

μK

K

B x

y

Q μ

r

f

m

`

r˄μ˄

D`jura 7?.6= Jabkatry bd tla bw pfst tla cyi`hear fhe cbbre`hfta systaks

_rbgiak `h wbres

Fitlbujl tla uè `s fssukae tb ga `hv`scè, fssuka tlft dr`ct`bh ft tla gbuhefry bd tlacyi`hear cfh cfusa uè tb sw`ri frbuhe tla cyi`hear wlah `t rbtftas frbuhe `ts fx`s. D`he tlahat surdfca dbrca bh f cyi`hear `h f twb e`kahs`bhfi sw`ri`hj bw bd fh `hv`scè, `hcbkprass`giacbhstfht-eahs`ty uè. Slbw tlft f bw cbkg`h`hj gbtl strafk`hj fhe sw`ri`hj j`vas r`sa tbi`dt.


(@) Qs`hj tla strafk duhct`bh j`vah gy Aquft`bh (7), sthe tla vaibc`ty staie u ; urr˄ + uμμ˄ euatb tla i`ha vbrtax gy ear`v`hj tla cbkpbhahts ur fhe uμ `h f pbifr cbbre`hfta systak. QsaGarhbuii`‐s aquft`bh (3) tb eaeuca tla prassura e`str`gut`bh _ bh tla surdfca bd tla cyi`hear.Ula dbrca cbrraspbhe`hj tb prassura _ bh f surdfca aiakaht eS `s _eS. Ula tbtfi dbrca D bh fsurdfca `s bgtf`hae gy sukk`hj tla aact bd tla uè prassura _ bvar fii sucl surdfca aiakahts`.a.

D ; ∗ S

_ ˄h eS (5)

wlara h˄ `s tla uh`t hbrkfi tb tla surdfca aiakaht eS tlft `s pb`ht`hj `htb tla uè. Avfiuftatla x- fhe y-cbkpbhahts bd tla dbrca, `.a. tla erfj fhe i`dt raspact`vaiy, bh f uh t iahjtl





cyi`hear w`tl `ts fx`s fibhj tla z -fx`s fhe eaeuca tlft tla hat surdfca dbrca eua tb tla i`havbrtax `s zarb.

(@@) Qs`hj (`) tla vaibc ty staie bgtf`hae `h tla Ahj`haar hj Axfkpia `h 7<.3 dbr f bwstrafk`hj pfst f cyi`hear fhe (``) tla vaibc`ty staie ear`vae `h (@), fppiy Garhbuii`‐s Aquft`bh(3) tb eaeuca tla prassura e`str`gut`bh _ bh tla surdfca bd tla cyi`hear. D`he tla x fhe ycbkpbhahts bd tla dbrca bh f cyi`hear bd uh`t iahjtl fibhj tla z -fx`s eua tb strafk`hj bwfhe slbw tlft erfj `s zarb gut tlft tla i`dt `s hbt zarb.


(@) Ula cbkpbhahts bd tla vaibc`ty `h pbifr cbbre`hftas `h tarks bd tla strafk duhct`bh ό7 fra

ur ; 6

r

∇ό7

∇μ (1)

fhe

uμ ; ∗∇ό7

∇r . (4)

Qs`hj (7) `h (1) iafes tb ur ; 6

r

∇

∇μ

ν7Ϗ

ih r + cbhstfht

, tla tark `h tla grfcnats ebas hbt

eapahe bh tla vfr`fgia μ, tlaradbra tla ear`vft`va `s zarb fhe

ur ; ?. (:)

S`kìfriy, us`hj (7) `h (4), uμ ; ∗ ∇

∇r ν

7Ϗ ih r + cbhstfht

, fhe

∇

∇r (ih r) ;

6

r , sb

uμ ; ∗ ν

7Ϗr. (<)

Fppiy`hj Garhbuii`‐s aquft`bh (3) ft f pb`ht K (f, μK ) bh tla surdfca bd tla cyi`hear slbwh `hD`jura 7?.6 iafes tb

_ +φ∗ ν

7Ϗf

77

; C sb

_ ; C ∗ φν7

<Ϗ7f7. (0)

Hbta tlft tla surdfca bd tla cyi`hear `s f strafki`ha fhe tlft Garhbuii`‐s aquft`bh (3) cfh gafppiàe ft fii pb`hts bd tla surdfca. Ula jaharfi axprass`bh bd tla dbrca bvar tla wlbia surdfca`s j`vah gy Aquft`bh (5). Ula surdfca tb ga cbhsèarae `s tlft bd fh èafi`sae cyi`hear bd `hsth`taiahjtl fibhj tla z -fx`s. Cbhsaquahtiy, tla uh`t hbrkfi tb tla surdfca aiakaht eS , h˄, wl`cl `spb`ht`hj `htb tla uè, `s

h˄ ; r˄ ; `cbs μ + m s`h μ. (6?)

Wftlar tlfh cbkput`hj tla dbrca bh f cyi`hear bd `hsth`ta iahjtl, cbhsèar f uh`t iahjtl cyi`hearw`tl `ts fx`s fibhj tla z -fx`s. Ula surdfca aiakaht gacbkas eS ; feμÚ 6 fhe usa bd (5) w`tl


03




(6?) j`vas D ; ∗ 7Ϗ?

_ (` cbs μ + m s`h μ)f eμ, s`hca tla `htajrfi bvar tla wlbia cyi`her`cfi

surdfca S raqu`ras tlft tla vfr`fgia μ ruhs drbk ? tb 7Ϗ.

Ula dbrca cfh ga eacbkpbsae `htb x- fhe y-cbkpbhahts=

x-cbkpbhaht= D erfj ; ∗f

7Ϗ

?

_ cbs μ eμ (66)

fhe

y-cbkpbhaht= D i`dt ; ∗f

7Ϗ?

_ s`h μ eμ. (67)

Hbta tlft tla gfs`s vactbrs ` fhe m fra cbhstfht fhe cfh ga tfnah but bd tlasa `htajrfis. Qs`hjAquft`bhs (66) fhe (67) fhe tla axprass`bh dbr prassura j`vah gy (0),

D erfj ; ∗

C ∗ φν7

<Ϗ7f7

f

7Ϗ

?

cbs μ eμ (63)

fhe

D i`dt ; ∗

C ∗ φν7

<Ϗ7f7

f

7Ϗ?

s`h μ eμ (65)

(Fs tla prassura ebas hbt eapahe bh tla pbifr fhjia `t cfh ga tfnah but bd tla `htajrfi.) Ula`htajrfis cfh ga avfiuftae fs

7Ϗ?

cbs μ eμ ;

s`h μ7Ϗ?

; ? (61f)

fhe 7Ϗ?

s`h μ eμ ; ∗

cbs μ

7Ϗ?

; ?. (61g)

Qs`hj (61f) fhe (61g) `h (63) fhe (65), wa cbhciuea tlft D erfj ; ? fhe D i`dt ; ?.

Ularadbra, tla hat surdfca dbrca bh f cyi`hear `h f sw`ri`hj bw `s zarb. Hbta fisb tlft tla hat

surdfca dbrca bh f cyi`hear `h f strafk`hj bw `s zarb.

Ula haxt sact`bh slbws tlft f cbkg`hft`bh bd strafk`hj fhe sw`ri`hj bw j`vas r`sa tb f hbh-zarb surdfca dbrca gut gadbra prbcaae`hj wa wìi eavaibp sbiut`bhs tb s`x `htajrfis tlft wa slfiihaae.





Qsadui `htajrft`bh rasuits

@h 63.4 tlraa “brtlbjbhfi`ty raift`bhs‐ wara `htrbeucae (pfjas 17-15). Uwb bd tlasa cfhga sukkfr`sae fs dbiibws=

Dbr fhy `htajars k fhe h=

@ k,h ≦ 7Ϗ?

s`h kx s`h hx ex ;

? (k ; h)? (k ; h ; ?)Ϗ (k ; h ; ?)

N k,h ≦ 7Ϗ?

s`h kx cbs hx ex ; ? (`h fii cfsas)

Za wìi hbw usa tlasa brtlbjbhfi`ty raift`bhs, fibhj w`tl tla twb èaht`tàs

cbs7μ ≦ 6 ∗ 7s`h7 μ (sb s`h7 μ ≦ 6 ∗ cbs7μ7

)

s`h7μ ≦ 7s`h μ cbs μ (sb s`h μ cbs μ ≦ s`h7μ

7 )

tb avfiufta s`x `htajrfis haaeae `h dutura cficuift`bhs. 7Ϗ?

s`h μ eμ ; @ 6,? ; ? (64f)

7Ϗ?

cbs μ eμ ; N ?,6 ; ? (64g)

7Ϗ?

s`h μ cbs μ eμ ; N 6,6 ; ? (64c)

7Ϗ?

s`h7 μ eμ ;

7Ϗ?

s`h μ s`h μ eμ ; @ 6,6 ; Ϗ (64e)

7Ϗ

?

s`h7 μ cbs μ eμ ; 7Ϗ

?

s`h μ s`h μ cbs μ eμ ; 7Ϗ

?

s`h μs`h7μ

7 eμ ;

6

7@ 6,7 ; ? (64a)

7Ϗ?

s`h3 μ eμ ;

7Ϗ?

s`h μ(6 ∗ cbs7μ

7 ) ;

6

7

7Ϗ?

s`h μ eμ ∗ 6

7

7Ϗ?

s`h μ cbs7μ eμ

; 6

7N 6,? ∗ 6

7N 6,7 ; ? (64d)

^Hbta= btlar katlbes cbuie ga usae tb bgtf`h tlasa s`x rasuits bd cbursa.[


01




(@@) Ula cbkpbhahts bd tla vaibc`ty staie `h pbifr cbbre`hftas dbr f strafk`hj bw pfst fcyi`hear fra ear`vae drbk tla strafk duhct`bh fs

ustrafkr ; Q cbs μ

r7 ∗ f7

r7 (6:)

fhe

ustrafkμ ; ∗Q s`h μ

r7 + f7

r7

. (6<)

Cbkg`h`hj (6:) fhe (6<) w`tl rasuits (:) fhe (<) dbr tla vaibc`ty staie eua tb tla i`ha vbrtaxbgtf`hae `h pfrt (@) j`vas

ur ; Q cbs μ

r7 ∗ f7

r7

(60)

fhe

uμ ; ∗Q s`h μ

r7 + f7

r7

∗ ν

7Ϗr (7?)

Fccbre`hj tb Garhbuii`‐s aquft`bh (3), tla prassura ft f pb`ht K (f, μK ) bh tla surdfca bd tlacyi`hear slbwh `h D`jura 7?.6 `s

_ ; C ∗ φu7

7 (76)

wlara u7 ; u7r + u7μ ft r ; f.

Aquft`bhs (60) fhe (7?) w`tl r ; f j`va

u7 ; 5Q 7 s`h7 μ + ν7

5Ϗ7f7 +

7Qν

Ϗf s`h μ (77)

Cficuift`hj D D D erfj

Qs`hj Aquft`bhs (66), (76) fhe (77), tla erfj bh tla cyi`hear `s j`vah gy

D erfj ; ∗f 7Ϗ?

C ∗ φ

7

5Q 7 s`h7 μ + ν

7

5Ϗ7f7 + 7Qν

Ϗf s`h μ

cbs μ eμ.

Ula `htajrfi cfh ga eacbkpbsae fs tla suk bd tlraa tarks fhe tla erfj cfh ga wr`ttah fs

D erfj ; ∗f (@ 6 + @ 7 + @ 3) . (73)

Za wìi avfiufta afcl bd tla cbkpbhaht `htajrfis `h turh us`hj tla rasuits (64f) tb (64d).

@ 6 ≦

C ∗ φν7

<Ϗ7f7

7Ϗ?

cbs μ eμ ; ? (us`hj 64g)

@ 7 ≦ ∗5φQ 7 7Ϗ?

s`h7 μ cbs μ eμ ; ? (us`hj 64a)





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48 Eng Case Stds