48 eng case stds
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8/13/2019 48 Eng Case Stds
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Cbhtahtsbhtahts Ahj`haar`hj CfsaStue`as
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 rf`iwfy 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`qu`e 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 cbkpahe`uk bd Ahj`haar`hj Cfsa Stue`as 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 w`ea
crbss sact`bh bd kftlakft`cfi taclh`quas.
butcbkas
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Ahj`haar`hj
Cfsa Stue`as
5<@htrbeuct`bh
Ul`s Zbrngbbn bars f cbkpahe`uk bd Ahj`haar`hj Cfsa Stue`as 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 w`ea crbss sact`bh
bd kftlakft`cfi taclh`quas.
7 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 Sn`iis
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 `eafs. @t kfnas usabd e`araht`fi aquft`bhs fhe tr`jbhbkatry. Ula hftura fhe tla tark`hbibjy bd wfva kbt`bh kfy ga`htrbeucae w`tl radarahca tb bsc`iift`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.
Cbhs`ear 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 si`ea.Ula kbt`bh bd tla kfss cfh ga eascr`gae gy Hawtbh‐s sacbhe ifw, wl`cl stftas tlft dbrca `s aqufitb kfss kuit`pi`ae 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 e`araht`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 Stue`as
3
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@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 bsc`iift`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 bsc`iift`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`husb`efiiy fitarhft`hj cbkprass`bhs fhe rfradfct`bhs`h tla kae`uk suppbrt`hj tla sbuhe. Uyp`cfiiy sbuhe wfvas cbhtf`h kfhy draquahc`as 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 fkg`aht br
aqu`i`gr`uk 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 `eaht`cfi axcapt dbr f plfsa e`arahca 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 e`arahca 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
5 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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; 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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
1
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@htarpratft`bh
(f) Ula fkpi`tuea bd tla wfva dbrkae gy cbkg`h`hj wfvas w`tl e`araht fkpi`tueas fheplfsas `s hbt s`kpiy tla suk bd tla `he`v`eufi 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 vfr`as w`tl t`ka gatwaah +7 fhe ∗7 ft fh fhjuifrdraquahcy tlft `s lfid bd tla e`arahca gatwaah fhjuifr draquahc`as bd tla cbkpbhahtwfvas. Ula draquahcy bd tla wfva dbrkae gy cbkg`h`hj uh`t (br sfka) fkpi`tuea cbkpb-hahts w`tl e`araht draquahc`as `s tla kafh bd tla draquahc`as 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 draquahc`as 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.
4 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 7
Cbkpiax raprasahtft`bh bd sbuhe wfvas fhe sbuhe raact`bh
Kftlakft`cfi Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatry ^5[Cbkpiax hukgars ^6?[
@htrbeuct`bh
Cbkpiax axpbhaht`fi axprass`bhs turh but bdtah tb ga kbra cbhvah`aht 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 Wa^a∗`πt[ ; cbs πt
@d F `s rafi tlah, Wa^Fa∗`πt[ ; Fcbs πt.
S`k`ifriy, 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 e`araht fkpi`tuea fhe plfsa drbk tla `hc`eaht wfva. Ft tla gbuhefry, tla`hc`eaht wfva w`tl fkpi`tuea F` kfy ga raprasahtae gy tla cbkpiax axpbhaht`fi axprass`bh
p` ; F`a∗`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 e`ars `h plfsa drbk tla`hc`eaht wfva gy tla plfsa fhjia τ.Ula rft`b bd tla raactae wfva tb tla `hc`eaht wfva ( p`/pr) `s cfiiae tla raact`bh cbaffic`aht (W)bd tla gbuhefry fhe `s f cbkpiax hukgar . @t eapahes bh tla plys`cfi prbpart as bd tla twb kae`fa`tlar s`ea bd tla gbuhefry.Ula cbkpiax pifha wfva raact`bh cbaffic`aht `s j`vah gy
W ; pr
p`;
Fra∗`(πt+τ)
F`a∗`πt ;
Fr
F`a∗`τ
@d tla raact`bh cbaffic`aht 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)
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
:
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Lahca f ; Fr
F`cbs τ fhe g ; ∗Fr
F`s`h τ
E`v`e`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 cbaffic`aht plfsa fhe fkpi`tuea tb `ts rafi fhe`kfj`hfry pfrts.
_rbgiak `h wbres
Dbr f pfrt`cuifr gbuhefry, tla raact`bh cbaffic`aht `s ?.< ∗ ?.5`=
(f) D`he tla rft`b bd tla raactae fkpi`tuea tb tla `hc`eaht fkpi`tuea>
(g) D`he tla plfsa e`arahca gatwaah tla raactae wfva fhe tla `hc`eaht wfva
(c) Zlft `s `kpi`ae fgbut tla plfsa sl`dt ft f surdfca `d tla raact`bh cbaffic`aht `s +6 br∗62
Kftlakft`cfi stftakaht bd prbgiak
(f) Qsa Aquft`bh (6) tb eatark`ha tla raact`bh cbaffic`aht 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.
Kftlakft`cfi fhfiys`s
(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 Stue`as
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@htarpratft`bh
(f) Ula kfjh`tuea bd tla raact`bh cbaffic`aht `s ?.<0. Ul`s kafhs tlft tla fkpi`tuea bd tlaraactae wfva `s ?.<0 bd tla `hc`eaht 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 `hc`eaht wfva gut tlara `s hb clfhja `hfkpi`tuea `s cfiiae f prassura raiafsa gbuhefry.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
0
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_rbgiak `h wbres
Ula k`crbplbha e`fplrfjk `s kbeaiiae fs f ft r`j`e pifta sugmact tb s`hjia draquahcy sbuhe wfvas`hc`eaht 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 `hc`eaht sbuhe draquahcy fhe usa tl`s axprass`bh tb jrfpl tladraquahcy eapaheahca bd 71 kk fhe 67.1 kk w`ea k`crbplbhas dbr sbuhe ft 51◤ `hc`eahca.
Kftlakft`cfi stftakaht bd prbgiak
J`vah tlft tla prassura vfr`ft`bh fgbut fkg`aht bd fh `hc`eaht 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 w`etl `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.
Kftlakft`cfi fhfiys`s
(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)
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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(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 `hc`eahca) 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 `hc`eahca) 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 `hc`eahca 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 w`ea 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 `eafi “ft‐ raspbhsa hafr 6? nLz. Ula rasuit `h (7)prae`cts tlft tla sahs`t`v`ty rakf`hs l`jl ft l`jlar draquahc`as `d tla k`crbplbha `s skfiiar. Ularaspbhsa bd f 67.1 kk k`crbplbha `s prae`ctae tb eapfrt drbk “ft‐ hafrar 7? nLz.
67 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 5
Wadrfct`bhKftlakft`cfi Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatry ^5[E`araht`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 w`ii tfna tla rbuta (cbkpbsae bd strf`jlt i`ha sajkahts) bd k`h`kuk trfvai t`ka.
Gy cbhs`ear`hj f rfy bd i`jlt trfvaii`hj drbk f kae`uk w`tl radrfct`va `heax ±6 tb fhbtlar w`tlradrfct`va `heax ±7, slbw tlft Darkft‐s _r`hc`pia `kpi`as
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 kae`uk `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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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• 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 `hc`eahca 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 v`aw bd tla cbhs`earae galfv`bur bd tla axprass`bh dbr tla trfvait`ka bvar fii vfiuas bd x, tla rasuit cbrraspbhes tb tla bvarfii k`h`kuk.
65 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 1
Gafk eadbrkft`bhKftlakft`cfi Sn`iis
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`j`eiy stxae fhe tla btlar draa (cfiiaecfht`iavarae) 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`t`as 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`t`as 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`t`as `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 wl`ia 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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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_FWU 6
_rbgiak `h wbres
F cfht`iavarae gafk `s cbhstructae sb tlft `t `s lafv`ar `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.
Kftlakft`cfi stftakaht bd prbgiak
D`he tla slafr dbrca S fhe tla gahe`hj kbkaht K dbr f cfht`iavarae gafk w`tl wa`jlt vfry`hj fs
w ; wb
6 + s`h
Ϗz
I
. (Cfht`iavarae kafhs stxae ft z ; ? fhe draa ft z ; I. Fisb, ft tla draa ahe
wa lfva S ; ? fhe K ; ?.)
Kftlakft`cfi fhfiys`s
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.
64 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 .
Kftlakft`cfi stftakaht bd prbgiak
F cfht`iavarae 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.
Kftlakft`cfi fhfiys`s
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 ) .
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
6:
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Ft tla `hhar aeja bd wlara tla ibfe fppi`as, `.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 Stue`as
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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 buts`ea bur prasaht scbpa.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 4
Eaact`bh bd f gafkKftlakft`cfi Sn`iis
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 e`araht`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 Stue`as
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Kftlakft`cfi stftakaht bd prbgiak
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, prbv`eae tla eaact`bh `s skfii, tla
rfe`us 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).
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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 `kpi`as
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 `kpi`as 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 `kpi`as
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.
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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 gaahfppi`ae 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 prbpart`as 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 e`araht`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 e`araht`fi aquft`bh us`hj prbparty (_6)
(g) Sbiva dbr tla duhct`bh I{d (t)} us`hj prbpart`as (_7) fhe (_3)
(c) Qsa tla `hvarsa Ifpifca trfhsdbrk tb bgtf`h d (t) us`hj prbparty (_5)
Qs`hj tla i`hafr`ty prbpart`as 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
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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
; ? `kpi`as
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
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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
fppi`ae 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 (∗\ ).
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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 Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatr`cfi `eaht`t`as ^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 p`iifrsuppbrt`hj f ibfe, br s`kpiy `ts bwh wa`jlt. Hbrkfiiy, `d f lbr`zbhtfi dbrca `s fppi`ae tb tla cbiukh,tlara w`ii 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 prbpart`as fhe tlahftura bd tla ahe suppbrts), tlasa btlar dbrcas kfy hbt ga prasaht `.a. tlara kfy ga aqu`i`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 fg`i`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 e`araht`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`j`e`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)
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K ;
S ez ; ∗A@
e7w
ez 7 ∗ _ + cbhstfht
Zlah tla gafk `s vart`cfi (`.a. f cbiukh), tlara w`ii 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) Gu`it-`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`g`i`t`as=
6. Gu`it-`h tbp fhe gu`it-`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 gu`it-`h gbttbk - tla gbttbk `s fs eascr`gae `h (`), wl`ia 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 gu`it-`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 ; ?
^Ula cbhe`t`bhs ft z ; I `kpiy zarb curvftura fhe zarb slafr dbrca raspact`vaiy, (saa afri`arfhe gaibw).[
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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 w`ii ga
K ; ∗A@ e7w
ex7 ; ? fhe T ; ∗A@
e3w
ex3 ∗ _
ew
ex ; ?
μ ; ew
ex
_
_ ewex
D`jura :.7
Ul`s scahfr`b fppi`as 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. gu`it-`h tbp fhe gbttbk
7. s`kpiy-suppbrtae tbp fhe gu`it-`h gbttbk
3. draa tbp fhe gu`it-`h gbttbk
5. s`kpiy-suppbrtae tbp fhe gbttbk
Kftlakft`cfi stftakaht bd prbgiak@t `s hacassfry tb sbiva tla 5tl brear BEA
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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.
Kftlakft`cfi fhfiys`s
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 s`ea 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`t`as w`ii 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)
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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. Gu`it-`h tbp fhe gu`it-`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 aqu`i`gr`uk `s tlacfsa bd hb eaact`bh. @d, lbwavar, K `s f s`hjuifr kftr`x, tlara w`ii 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
3? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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ηI s`h(ηI) + 7 cbs(ηI) ∗ 7 ; ? (4)
Qs`hj tla tr`jbhbkatr`cfi `eaht`t`as 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 w`ii 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 gu`it-`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 cbaffic`aht 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 prbpart`as bd
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 gu`it-`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 ; ?
w`tl cbaffic`aht kftr`x
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
37 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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) ; ?
w`tl cbaffic`aht kftr`x
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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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@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 gu`it-`h _ cr ; 7.1A@
I7
s`kpiy-suppbrtae s`kpiy-suppbrtae _ cr ; 0.0A@
I7
s`kpiy-suppbrtae gu`it- h _ cr ; 7?.7A@
I7
gu`it-`h gu`it-`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 st`ii 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.
35 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 Sn`iis
Ubp`c ZbrngbbnKfx`kf fhe K`h`kf ^67[
@htrbeuct`bhUla fg`i`ty tb kbeai tla aacts bd kbv`hj ibfes bh f structura `s `kpbrtfht, dbr axfkpia `h gr`ejaeas`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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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`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 rf`iwfy 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`k`ifr kfhhartb tlft `iiustrftae 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
Kftlakft`cfi stftakaht bd prbgiak
(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 e`araht`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 cbhs`ear 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 e`araht`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).
Kftlakft`cfi fhfiys`s
(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
34 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
3:
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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?
E`araht`ft`hj w`tl raspact tb z 6 `h afcl zbha j`vas
3< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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zbha ^6[ = 7 9 z 6 9 f eK
ez 6;
6
7? (74? ∗ 63f) 8 ? dbr fii vfi`e 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 vfi`e 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 w`ii 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 `iiustrfta gaibw.
Kbrabvar, fhbtlar wfy bd ibbn`hj `htu`t`vaiy ft tla prbgiak, `.a. gy cbhs`ear`hj tla cahtra bd kfss,j`vas fh `hcbrract fhswar. Ula cahtra bd kfss bd tla systak w`ii i`a 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 cbhs`ear 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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 0
Aquft`bh bd tla curva bd f cfgia stxae ft twb ahe pb`htsKftlakft`cfi Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatr`c `eaht`t`as ^5[Lypargbi`c duhct`bhs ^4[E`araht`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 gr`ejas 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 eurfg`i`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 k`epb`ht fhe ft f la`jlt y fhe lbr`zbhtfie`stfhca x. @d wa cbhs`ear 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 _ w`ii 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.
5? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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`v`e`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 jrfe`aht 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 jrfe`aht 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)
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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Kftlakft`cfi fhfiys`s
E`araht`ft`hj gbtl s`eas 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 `eaht`ty cbsl7(u) ∗ s`hl7(u) ≦ 6.
D`rst wa e`araht`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 s`ea `s f cbhstfht tark wa cfh afs`iy `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 jrfe`aht 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 s`eas 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`trfr`iy fs `t w`ii 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 cbhvah`ahca, 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 w`ii 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
57 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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`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 k`epb`ht bd tla cfgia fhe j `s tla fccaiarft`bh eua tbjrfv`ty. Ul`s curva `s cfiiae f cftahfry. @t `s e`araht drbk tlft bd tla cfgia bd f suspahs`bh gr`eja,wl`cl `s ibfeae gy tla gr`eja eacn. Ula aquft bh aqu`vfiaht tb (1) tlft fppi`as tb f suspahs`bhgr`eja cfgia, fssuk`hj tlft tla wa`jlt bd tla cfgia `s haji`j`gia cbkpfrae tb tla wa`jlt bd tla gr`eja
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 vfr`ae. 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 twbe`araht 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)) .
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 e`araht tahs`bhs
Sb
I ; 7c
s`hl
e
c
. (4)
J`vah tla cfgia eatf`is 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 afri`ar, 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 w`ii 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.
55 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 Sn`iis
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 u`e 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
u`e 6
u`e 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 u`e bw fs ibhj fs tla dbiibw`hj fssukpt`bhs fra kfea=
(`) Ula u`e bw`hj `h tla clfhhai `s `hcbkprass`gia fhe `hv`sc`e (hb dr`ct`bh)
(``) Ula bw `s stafey (cbhstfht bvar t`ka)
(```) Ula clfhhai `s strf`jlt, lbr`zbhtfi fhe bd cbhstfht w`etl
(`v) Ula bw `s uh`dbrk (fii u`e erbpiats `h f lbr`zbhtfi pifha lfva tla sfka spaae)
Dbr f j`vah spaae bd tla u`e u, tla n`hat`c aharjy bd bw par uh`t vbiuka `s 6
7φu7 wlara φ `s tla
u`e 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`v`eae gy φj `.a.
A ; u7
7j + l. (6)
Hbta tlft tla spac`stc aharjy lfs tla e`kahs`bh bd iahjtl.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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Garhbuii`‐s aquft`bh fppi`ae 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 u`e 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 u`e la`jlt fhe cr`t`cfi spaae fra easthae tb ga wlahtla spac`stc aharjy bd tla u`e 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 u`e 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 u`e 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 w`etl 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 u`e.Fssuk`hj tlft tla u`e `s `hv`sc`e 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 u`e 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 u`e 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
54 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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Kftlakft`cfi fhfiys`s
@. Ula fraf F bd tla crbss sact`bh bd u`e stii`hj tla clfhhai up tb la`jlt l cfh ga easthae gy tlae`arahca 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 u`e 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 u`ela`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. Cbhs`ear`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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
5:
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Fs tla spac`stc aharjy easthae gy (6) `s f cbhstfht, eA
el ; ? ft tla cr`t`cfi u`e 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 u`e 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 u`e bw `s pbss`gia `h tarks bd tla cr`t`cfi u`e 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).
5< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 u`e 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 e`araht u`e la`jlt l6 fhe l7 tlft cbrraspbhe tbu`e 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), wl`ia 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 u`e la`jlt lc fhe cr`t`cfi spaae uc.
3. Zlah A 9 A k`h , hb bw `s pbss`gia.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
50
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 66
S`kpia paheuiukKftlakft`cfi Sn`iis
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).
Kftlakft`cfi fhfiys`s
Ula stfhefre dbrkuif dbr tla t`ka dbr tla gfr tb pardbrk f lfid bsc`iift`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 s`eas 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 s`eas bd tla aquft`bh. Ulafttakpts tb l`t tla sw`hj`hj gfr slbuie stfrt drbk fgbut L ; 1I gut w`ii haae tb ga cfrr`ae 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 w`ii 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 bsc`iift`bhs bd skfii fkpi`tuea fhe wlah f`r ras`stfhca `s haji`j`gia. Ub naap tlapaheuiuk sw`hj`hj dbr savarfi bsc`iift`bhs `t w`ii 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.
1? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 67
Kbt`bh bd f paheuiuk
Kftlakft`cfi Sn`iis
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
w`ii hbt ga slbwh axpi`c`tiy fhe tla hbtft`bh μː fhe μ w`ii 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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
16
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Kftlakft`cfi stftakaht bd prbgiak
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) `kpi`as 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 e`araht`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 e`araht`fi aquft`bh.
(g) D`he f sbiut`bh bd tl`s e`araht`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.
17 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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Kftlakft`cfi fhfiys`s
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 rfe`us `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 ;
e^iμ(t)[
et . Ula rbe iahjtl `s stxae eur hj tla kbt`bh, sb i ebas hbt eapahe bh
t`ka t, cbhsaquahtiy v(t) ; e^iμ(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`t`as. S`hca l `s tla e`arahca gatwaah tla iahjtl i bd tla paheuiuk (fisbtla rfe`us 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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
13
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1. Za cbhs`ear afcl cfsa=
Cfsa F= μ(?) ; Ϗ/7 fhe μː(t ; ?) ; ?
Aquft`bh (7) `kpi`as 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 cbhs`earae `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`v`e`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.
E`araht`fi Aquft`bh (3) gacbkas=
μ + ξ7μ ; ?.
(g) Ul`s aquft`bh `s f i`hafr sacbhe brear e`araht`fi aquft`bh w`tl cbhstfht cbaffic`ahts 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 fux`i`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 e`araht`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.
15 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 63
Ula dfii`hj shbwfnaKftlakft`cfi Sn`iis
Ubp`c Zbrngbbn@htajrft`bh ^63[D`rst brear BEAs ^60[Sacbhe brear i`hafr BEAs ^60[Tactbr e`araht`ft`bh ^7<[
@htrbeuct`bh
Fitlbujl tla ras stae kbt`bh bd bgmacts w`tl stxae kfss wfs cbhs`earae h 35.3 h tla Zbrngbbnbh Kbeaii`hj Kbt`bh, fhe raiftae tb prbmact`ia cbhtaxts, tlara fra fppi`cft`bhs wlara tla kfss bd tlakbv`hj bgmact vfr`as w`tl t`ka. Ula pfrt`cuifr cfsa cbhs`earae 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`k`ifrfhfiys`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 (cbaffic`aht bd f`r dr`ct`bh). Ula shbwfna `s sugmact tb f ebwhwfre vart`cfidbrca `.a. `ts wa`jlt
14 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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|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 e`araht 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).
Kftlakft`cfi fhfiys`s
@ 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-s`earae `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 cbhs`earae `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 vfi`e.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) `kpi`as
e(kv)
et ; k
ev
et + vnk.Qs`hj tl`s axprass`bh `h (1), fhe cfhcaii`hj k
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
1:
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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 e`araht`fi aquft`bh w`tl cbhstfht cbaffic`ahts 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 fux`i`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 e`araht`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 ; ?) ; ? `kpi`as 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(?) ; ? `kpi`as 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;?
; ? `kpi`as ∗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 s`eas 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.
1< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 s`eas 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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
10
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 65
Sftaii`ta kbt`bh
Kftlakft`cfi Sn`iis
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 e`araht`fi aquft`bh fs waii fs fcbhstfht bd tla kbt`bh bd f sftaii`ta `eafi`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 e`araht 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 e`araht`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
4? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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`pi`ae 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)
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
46
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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 e`araht`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`v`eae gy ∗C 7 u7, fhe tla easth`t`bh N ≦ JK fiibws us tb bgtf`h tla sthfie`araht`fi aquft`bh `h vfr`fgia u=
e7
ueμ7
+ u ; N C 7
. (64)
Ula sbiut`bh bd tla sacbhe brear e`araht`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
fux`i`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 rfe`us C 7
N , eapahe`hj bh tla `h`t`fivaibc`ty bd tla sftaii`ta ifuhcl.
47 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 61
Sftaii`ta brg`tsKftlakft`cfi Sn`iis
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 sc`aht`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 e`araht`fi aquft`bh fhe f cbhstfht dbr tla
kbt`bh bd f sftaii`ta `eafi`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 e`araht`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 `eafi`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 breare`araht`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.
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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 e`araht`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 e`araht`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 trfmactbr`as 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. Cbhs`ear 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 `eafi`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)
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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
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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 fppi`as `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)
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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`pi`ae 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 e`araht`fi aquft`bh bd kbt`bh `s eaeucae drbk (7:) fs
∗J k u7 ; ∗C 7u7e7u
eμ7 ∗ C 7 u3. (33)
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E`v`e`hj (33) gy ∗C 7u7 fhe us`hj tla easth`t`bh N ≦ Jk j`vas tla sthfi dbrk bd tla e`araht`fiaquft`bh `h tarks bd vfr`fgia u
e7u
eμ7 + u ;
N
C 7. (35)
Ula sbiut`bh bd tl`s sacbhe brear e`araht`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 fux`i`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 rfe`us C 7N
, eapahe`hj bh tla `h`t`fi vaibc`ty bd tla
sftaii`ta ifuhcl.
@@@. Ula sbiut`bh bd tla e`araht`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`hc`ea 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
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s`ea 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 s`ea
tark `h (5?) gacbkas
eret
; ∗Cu (56)
Ula sacbhe r`jlt-lfhe s`ea 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?)
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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 `eaht`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 Stue`as
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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 `kpi`as 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 `kpi`as 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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
:6
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 64
Qhearjrbuhe rf`iwfy s`jhfis ibcft`bhKftlakft`cfi Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatry ^5[E`araht`ft`bh bd `hvarsa tr`j duhct`bhs ^66[Jabkatry (frc iahjtl) ^6:[_frt`fi e`araht`ft`bh ^6<[
@htrbeuct`bh
Ula ibcft`bh bd s`jhfis `h f curvae uhearjrbuhe rf`iwfy 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 s`ea bd tla tuhhai, ft tla er`var‐s aya la`jlt, bh tlabuts`ea bd tla curva.
_rbgiak `h wbres
Fh uhearjrbuhe rf`iwfy `s tb ga axtaheae tb f haw Biykp`c spbrts stfe`uk. 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 frfe`us 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 s`ea bd tla tuhhai, ft tla er`var‐s aya la`jlt, bh tla buts`ea bd tla curva. Ula uhcartf`hty `htla tuhhai rfe`us 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, wl`ia tla kbvakaht bd tla trf`h fhe tla vfr`ft`bh `htla er`var‐s clfrfctar`st`cs kafhs tlft tla er`var‐s v`awpb`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 suffic`ahtv`s`g`i`ty dbr tla trf`h tb stbp sfdaiy w`tlbut fevfhca `he`cft`bh2
(g) F haw eas`jh bd trf`h w`ii 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 Stue`as
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Kftlakft`cfi stftakaht bd prbgiak
• 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 rfe`us bd curvftura bd tla rf`i (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 rfe`us bd curvftura bd tla rf`i (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 rf`is gatwaah tla pbs`t`bhs bd tla s`jhfis fhe tlaer`var.
• Qsa pfrt`fi e`araht`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 rfe`us 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.
Kftlakft`cfi fhfiys`s
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 rfe`us 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.)
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 rf`is (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 rfe`us 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 rfe`us 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.
:5 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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@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.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatry ^5[D`rst brear BEAs ^60[Sacbhe brear BEAs ^60[Dbur`ar sar`as ^73[
_frt`fi e`araht`fi aquft`bhs ^71[
@htrbeuct`bh
Zlah tla surdfca bd tla jrbuhe `s laftae gy tla suh tla takparftura bh tla surdfca vfr`as bhf e`urhfi cycia. @d `t `s fssukae tlft tla takparftura `s uh`dbrk bvar fh fraf bd tla surdfca fhevfr`as 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 e`us`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 eajraasCais`us, 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 sb`i 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 gu`ie f w`ha caiifr cbhs`st`hj bd f lbia `h tla jrbuhe i`hae w`tlcbhcrata s`eas fhe w`tl f cbhcrata rbbd wl`cl `s cbvarae `h sb`i 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 ef`iy vfr`ft`bh slbuie ga hb kbra tlfh ?.1◤C. @d `t `s fssukae tlft cbhvact`bh prbv`easrafsbhfgia trfhsk`ss`bh bd laft w`tl`h tla caiifr, tlah `t `s raqu`rae tlft tla takparftura bd tla sb`i vfr`as gy hb kbra tlfh ?.1◤C `kkae`ftaiy fgbva tla caiifr‐s rbbd. J`vah tlft tlakfx`kuk ef`iy 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 ef`iy vfr`ft`bh `s hb kbra tlfh ?.1◤C2
Kftlakft`cfi stftakaht bd prbgiak
:4 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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(`) J`vah tlft tla par`be bd tla e`urhfi 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 sb`i, 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 cbaffic`ahts 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.
Kftlakft`cfi fhfiys`s
(`) 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 e`araht`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 cbaffic`ahts 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 tlacbaffic`ahts bd tla cbs`ha, fhe clbbsa t sb tlft tla cbs`ha tark `s ? tb aqufta tla cbaffic`ahtsbd 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 cbhvah`aht 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?.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
::
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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 w`ii hbt fact tlavfr ft`bh ft f pfrt cuifr eaptl, tlbujl `t w`ii 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 Dbur`ar sar`as. Suppbsa tlft tla surdfca takparftura `s j`vah gy μ(?, t) ; d (t)dbr ∗67 ≥ t ≥ 67 (t `h lburs). Ula ef`iy 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 Dbur`ar sar`as 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 rfp`eiy fs z `hcrafsas fhe fra ft kbst a∗z/E sb tla
:< LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 w`ii 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 w`ii 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 Dbur`ar sar`as rasuits `h tla cbhcius`bh tlft sfka eaptl w`ii suffica dbrtla w`ha caiifr.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
:0
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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 Sn`iis
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 u`e kaclfh`cs `hvbiva prae`ct`hj tla bw bd u`e pfstbgstfcias fhe tla dbrcas `hvbivae bh tla sbi`e 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 gu`ie`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 u`e 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 sbi`ebgstfcia, 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 s`ea 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 u`e bw`hj pfst tla bgstfcia `s `hcbkprass`gia fhe `hv`sc`e (hb dr`ct`bh)(``) Ula bw `s stafey (cbhstfht `h t`ka).
<? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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`j`e cyi`hear bd j`vah rfe`us. D`hetla vfiua bd sburca strahjtl k tlft ahsuras tl`s.
Kftlakft`cfi stftakaht bd prbgiak
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`hc`eaht w`tl tla cyi`hear gbuhefry w`tlrfe`us f, fhe eaeuca tla `kpi`ae vfiua bd k raiftae tb tla sburca strahjtl bd tla e`pbia.
Kftlakft`cfi fhfiys`s
Ula dfctbr s`h μ cfh ga tfnah buts`ea 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 `kpi`as
r ;
7fk
Q . (7)
S`hca f,k fhe Q fra cbhstfhts, (7) easthas f c`rcia bd rfe`us
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 rfe`us
7fk
Q .
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
<6
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Fssuk`hj tlft tla cyi`hear lfs f rfe`us 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`j`e cyi`hear `s pfrt bd tla strafki`ha ό ; ?. Hbbw kfy crbss f strafki`ha sb f sbi`e gbuhefry `s fiwfys pfrt bd f strafki ha bd tla bw. Ularafra hb strafki`has `hs`ea 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 `hs`ea tla c`rcuifr crbss sact`bh bd tla cyi`hearfra hbt slbwh `h D`jura 6<.7.
Aquft`bh (3) `kpi`as 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 cbhs`ear 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 afs`ar 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 afs`ar `d x
f `s axprassae `h tarks bd y
f `hstafe bd tla btlar wfy rbuhe,wl`cl `s kbra usufi. E`v`e`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 s`ea fhe x tarks bh tla r`jlt-lfhe s`earaqu`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 Stue`as
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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 s`ea 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 .
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
<3
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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 cbhs`ear 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 rfe`us 6.
@htarpratft`bh
Ula strafki`has ό ; 7Qf fhe ό ; ∗7Qf fra fisb slbwh `h D`jura 6<.3. Ulay fra s`k`ifrtb tla strafki`has ό ; Q f fhe ό ; ∗Qf gut fra ft rbujliy tw`ca tla e`stfhca drbk tlax-fx`s. Diu`e 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 u`epfrt`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.
<5 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
<1
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y e u t
S a s
f C j h ` r a
a h
`
j h A
Ahj`haar`hj Cfsa Stuey 60
Uwb-e`kahs`bhfi bw bd f v`scbus i`qu`e bh fh `hci`haepifta
Kftlakft`cfi Sn`iis
Ubp`c ZbrngbbnUr`jbhbkatry ^5[@htajrft`bh ^63[
_bifr cbbre`hfta systak ^6:[, ^7<[_frt`fi e`araht`ft`bh ^6<[_frt`fi e`araht`fi aquft`bhs ^71[Tactbr cficuius ^7<[E`kahs`bhfi fhfiys`s ^5:[
@htrbeuct`bh
Kfhy `hvast`jft`bhs `h ahj`haar`hj u`e kaclfh`cs `hvbiva prae`ct`hj tla bw bd i`qu`es `hclfhhais br p`pas. S`kpia kbeais fssuka tlft tla i`qu`e `s `hv`sc`e `.a. tlft `t `s hbt v`scbus.
Lbwavar, e‐Fiakgart‐s pfrfebx fhe btlar rasuits dbr `hv`sc`e bw slbw tlft tla `hv`sc`e u`ebw kbeai haaes tb ga rasthae gy tla `htrbeuct`bh bd v`scbus dbrcas. Ula kbeaii`hj bd `hv`sc`eu`e bws usas Auiar‐s aquft`bh bd kbt`bh gfsae bh tla fssukpt`bh tlft surdfca dbrcas bh fu`e aiakaht fra bhiy hbrkfi tb tla surdfca bd tla u`e aiakaht. Ufhjaht`fi dbrcas eua tbv`scbs ty fra `jhbrae. Ula Hfv`ar-Stbnas aquft`bh bd u`e 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 cbaffic`aht bd v`scbs`ty, φ `s tla u`e eahs`ty, p `s tla prassura, D `s tla gbey
dbrca par uh`t kfss fhe u raprasahts tla u`e 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 u`e `s `hcbkprass`gia
(``) Ula u`e `s Hawtbh`fh (strass prbpbrt`bhfi tb tla rfta bd eadbrkft`bh)
(```) Ula u`e lfs cbhstfht v`scbs`ty
(`v) Ula bw `s duiiy-eavaibpae (tla bw `s `eaht`cfi ft fii pb`hts bd f u`e aiakaht trfmactbrydbr f j`vah t`ka)
<4 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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_rbgiak `h wbres
D`he tla vaibc`ty e`str`gut`bh `h f v`scbus i`qu`e bw`hj draaiy fhe stafe`iy uhear jrfv`ty bvar fvary ifrja `hci`hae pifta wlah tla i`qu`e `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.
Kftlakft`cfi stftakaht bd prbgiak
Clbbsa tla e`ract`bh bd tla bw fs tla x-fx`s, tla hbrkfi tb tla i`qu`e/f`r `htardfca fhe tb tlapifta fs tla z -fx`s (saa D`jura 60.6), sb tla y-fx`s `s clbsah tb i`a 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 Hfv`ar-Stbnas aquft`bh (6) tb tla bw cbhe`t`bhs (`)-(`v) fhe `htajrfta tbsthe tla vaibc`ty e`str`gut`bh.
f`r
u`e
piftaζ
z
x
π
iζ
D`jura 60.6= Jabkatry bd tla bw fhe cbbre`hfta systak
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
<:
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Kftlakft`cfi fhfiys`s
(@) 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`qu`e 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, tlaHfv`ar-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 cbaffic`aht bd v`scbs`ty fhe tla i`qu`e 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.
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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`v`eae gy iahjtl squfrae. Sb e`kahs`bhiass vaibc`ty ux `s bgtf`hae
gy e`v`e`hj (61) tlrbujl gy φD x
± i7. Durtlarkbra `d tla vfr`fgia z `s scfiae gy i tb y`aie 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 sbi`e curva fhe ebttae curva. Uwb vaibc`ty prbstias fra slbwh fs tlatl`cn sbi`e curva fhe efslae curva dbr hajft`va vfiuas bd slafr strass. Hbta tlft z/i `s pibttaefjf`hst uz sb tlft wlah tla i`qu`e 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`qu`eifyar `s 6 fs f rasuit bd tla scfi`hj z/i.
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
<0
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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`qu`e 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`qu`e `h tla e`ract bh bd tla bw fhe rasuits `h f pbs`t`vaslafr strass (U 8 ?), tla kfjh`tuea bd tla i`qu`e vaibc`ty `s `hcrafsae avarywlara cbkpfrae tb
tla cfsa w`tl hb f`r bw. Kbrabvar tla i`qu`e vaibc`ty kfjh`tuea `hcrafsas drbk tla gbttbkpifta tb tla surdfca dbiibw`hj f pfrfgbi`c curva (tl`h sbi`e curva). Zlah tla f`r bw `s `h tlabppbs`ta e`ract bh tb tla i`qu`e bw, `t rasuits `h f hajft`va slafr strass (U 9 ?). Cbhsaquahtiy,tla i`qu`e vaibc`ty kfjh`tuea `s eacrafsae wlah cbkpfrae tb tla cfsa w`tl hb f`r bw. @d tlaslafr strass `s suffic`aht, tla i`qu`e ft fhe hafr tla surdfca kfy bw upl`ii fs slbwh gy tlahajft`va pbrt`bh bd tla tl`cn sbi`e curva. Ula efslae curva slbws rasuits wlara tla w`he `s sbstrbhj tlft wlbia bd tla i`qu`e ifyar `s puslae upl`ii w`tl f spaae tlft `hcrafsas w`tl la`jlt `htla i`qu`e ifyar. Hbta tlft tla kbeai dbr `hv`sc`e i`qu`e `h wl`cl tfhjaht`fi slafr strassas frahajiactae rasuits `h f cbhstfht rftlar tlfh pfrfgbi`c vaibc`ty prbstia dbr bw bh f pifta.
0? LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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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 Sn`iis
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 u`e kaclfh`cs `hvbiva prae`ct`hj tla bw bd u`e pfstbgstfcias fhe tla dbrcas `hvbivae bh tla sbi`e gbuhefry. Dbr axfkpia `t `s hacassfry tb cficuiftatla farbeyhfk`c dbrcas bh fh f`rcrfdt (tla i`dt fhe erfj) bh gu`ie`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 gr`eja suppbrts. Kbeais bd u`e 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 `hc`eaht uh`dbrk bw vaibc`ty pfrfiiai tb tla x-fx`s fhe f `s tla rfe`us 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
LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
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bd tla bw frbuhe f cyi`hear bd rfe`us f, `t `s cbhvah`aht 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 u`e bw`hj pfst tla bgstfcia `s `hcbkprass`gia fhe `hv`sc`e (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 fppi`ae fibhj fhycurva `h tla u`e `s j`vah gy
_ + φu7
7 ; C (3)
wlara _ `s tla prassura, φ `s tla u`e eahs`ty, u `s tla spaae bd tla u`e fhe C `s f cbhstfht.
@d tla bw `s rbtft`bhfi, tla curva lfs tb ga f strafki`ha dbr (3) tb ga vfi`e.
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 u`e `s fssukae tb ga `hv`sc`e, fssuka tlft dr`ct`bh ft tla gbuhefry bd tlacyi`hear cfh cfusa u`e 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`sc`e, `hcbkprass`giacbhstfht-eahs`ty u`e. Slbw tlft f bw cbkg`h`hj gbtl strafk`hj fhe sw`ri`hj j`vas r`sa tbi`dt.
Kftlakft`cfi stftakaht bd prbgiak
(@) 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 u`e 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 u`e. Avfiuftatla x- fhe y-cbkpbhahts bd tla dbrca, `.a. tla erfj fhe i`dt raspact`vaiy, bh f uh t iahjtl
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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.
Kftlakft`cfi fhfiys`s
(@) 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`k`ifriy, 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 gafppi`ae 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 cbhs`earae `s tlft bd fh `eafi`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 u`e, `s
h˄ ; r˄ ; `cbs μ + m s`h μ. (6?)
Wftlar tlfh cbkput`hj tla dbrca bh f cyi`hear bd `hsth`ta iahjtl, cbhs`ear 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
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(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 w`ii eavaibp sbiut`bhs tb s`x `htajrfis tlft wa slfiihaae.
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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 w`ii hbw usa tlasa brtlbjbhfi`ty raift`bhs, fibhj w`tl tla twb `eaht`t`as
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.[
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(@@) 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 w`ii 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)
04 LAIK (7??1)=Zbrngbbn 5<= Ahj`haar`hj Cfsa Stue`as
8/13/2019 48 Eng Case Stds
http://slidepdf.com/reader/full/48-eng-case-stds 97/97
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