Uriobkgms 1.=
Kd ud `mgisi mrtçfubi. Pndkr frntnfmom m su msnstkdtk pirquk di smoçm cnoujmr udm `mgnbnm ck furvms ckb FUfp fuyis pudtis ck tmdhkdfnm fid bm furvm ckb FU fid `irgm ck ‒[‗ tmgonîd `ukrid bis pudtis gçdngis kd fmcm furvm ckb FUfp. Kb msnstkdtk mbkhmom krm ngpisnobk cnoujmr ksm hrí`nfm. ¹M qunîd rkspmbcmrçm ustkc kd kstk ckomtk9 Mpiymr kb cnoujmdtk. Ks iovni quk hkigîtrnfmgkdtk VMF di pukck skr, figi gçdngi, ym quk ks tmdhkdtk m MF kd ud pudti fid udm pkdcnkdtk dkhmtnvm. Bm údnfm tmdhkdfnm sk pricufk, figi gçdngi, MF 1.?
Vupidhm quk udm kgprksm pricufk cis pricuftis cnstndtis, fuyms fmdtncmcks kstíd rkprkskdtmcms pir q= y q?. Uir bi hkdkrmb, pickgis rkprkskdtmr kb fisti titmb ck bm kgprksm fid FS (q=, q?). Kstm `udfnöd gukstrm kfidigçms ck mbfmdfk sn FS (q=, i) + FS (5, q?) FS (q=, q?) pmrm ticis bis dnvkbks ck pricuffnöd ck bis cis onkdks.
4
m. Kxpbnquk fid pmbmorms pir quî kstm `irgubmfnöd gmtkgítnfm ngpbnfm quk bis fistis skríd gís omjis kd kstm kgprksm quk `mornfm vmrnis pricuftis quk kd cis kgprksms quk sibi `mornfmd udi ck bis pricuftis pir skpmrmci. Uir ck`ndnfnöd bis fistis titmbks sid gkdirks fumdci mgois ]= y ]? sid pricufncis pir bm gnsgm kgprksm quk fumdci bis gnsgis dnvkbks ck smbncm sid pricufncis pir cn`krkdtks kgprksms \F (q=, 5) sngpbkgkdtk snhdn`nfm quk udm `nrgm pricufk sibmgkdtk q=W. o. Vn bis cis pricuftis `ukrmd, ck akfai, kb gnsgi onkd, kdtidfks picrçmgis ck`ndnr bm pricuffnöd titmb figi . Vupidhm quk, kd kstk fmsi, kb fisti prigkcni (7 FS/q) cnsgnduyk m gkcncm quk q mugkdtm. Ckgukstrk quk kstm kgprksm tmgonîd cns`rutm ck kfidigçms ck mbfmdfk fid bm ck`ndnfnöd prkskdtmcm mquç.
7
, 2 ,5/ , / 2 5,.
Vkm q 7 q= + q?, cidck tmdti bm q= y q? 4 5. Ckonci m quk / pir supuksti, Ckb Ckb gnsgi gici _ksugnkdci rkdcngnkdtis , bi quk ckgukstrm bms kfidigçms ck mbfmdfk.
, / 2 ,5. , 2 , 5 5,
1.6
Bis fmtkcrítnfis Vgnta y Jidks vmd m ksfrnonr ud bnori dukvi ck ndtricuffnöd m bm kfidigçm. Figi mutîdtnfis fnkdtç`nfis, amd cktkrgndmci quk bm `udfnöd ck pricuffnöd ckb bnori ks q7V=/?J=/?, Cidck q7 dúgkri ck píhndms ckb bnori tkrgndmci, V7 kb dúgkri ck airms-aigork kgpbkmcms pir Vgnta y J7 kb dúgkri ck airms kgpbkmcms pir Jidks. Vgnta fidsnckrm quk su trmomji vmbk $6 pir airm. Am ndvkrtnci >55 airms kd bm prkpmrmfnöd ckb prngkr oirrmcir. Jidks, fuyi trmomji sk vmbirm kd $=? pir airm, rkvnsmrí kb oirrmcir ck Vgnta pmrm tkrgndmr kb bnori. m. ¹Fuídtms airms tkdcrí quk kgpbkmr Jidks pmrm pricufnr ud bnori tkrgndmci quk tkdcrí =;5 píhndms9 ¹[di quk tkdcrí 6559 ¹[di quk tkdcrí 3;59
q7 >555.; j5.;7 65√j
͔
Vn q7 =;5, j7?; y j7=55 fumdci q7 655, y j7??; fumdci q73;5. o. ¹Fuíb ks kb fisti gmrhndmb ck bm píhndm dúgkri =;5 ckb bnori tkrgndmci9 ¹Ck bm gmdkrm 6559 ¹Ck bm 3;59 Fistis titmbks 76*>55+=?*j J7 q?/>55 FSF7 ?855+=?q?/>55
7 cFSF cq
7
q 7 q > 8s
q7 =;5 FG73 q7 655 FG71 q7 3;5 FG7=? 1.3
Vupidhmgis quk bm `udfnöd ckb pricuffnöd ck pripirfnidks `njms ck udm kgprksm kstí cktkrgndmcm pir q7 gçd (;l, =5b), ^ quk bms tmsms ck mbqunbkr ckb fmpntmb y kb trmomji kstíd cktkrgndmcms pir v7=, w76. FS7 vL + wB 7L +6B m. Fmbfubmr bms furvms ckb fisti prigkcni, kb gmrhndmb y kb titmb, m bi bmrhi pbmzi, ck bm kgprksm. ;l 7=5B, l 7 ?B FS 7 ?B + 6B 7;B Figi q7=5B, B7q/=5, pir bi quk
7 FGk 7 7 . 7 FG7 7 7 FS (q) 7;.
o. Vupidhi quk l kstí `nji kd =5 m firti pbmzi. Fmbfubk bms furvms ckb fisti prigkcni, kb gmrhndmb y kb titmb, m firti pbmzi, ck bm kgprksm. ¹Fuíb ks kb fisti gmrhndmb ck bm cîfngm udncmc9 ¹Ck bm dúgkri ;59 ¹Ck bm dúgkri =559 L 7 =5
q 7 gnd (;5, =5B)
2
Vn B ;. q7 =5B
FS7=5 + 6B 7 =5 +
FG 7
FGk 7 +
≫
Vn B ;. q7 ;5 FS7 =5 + 6B FGk7
+
FG ks nd`ndnti ym quk L kstm `nji y gmyir B di mugkdtm q. FG (q7=5)7 6/=5 (P skr rksubtmcis mdtkrnirks). FG (q7;5) 7 FG (q7=55) ks nd`ndnti. 1.;
[dm kgprksm quk pricufk omstidks ck aiflky tnkdk udm `udfnöd ck pricuffnöd cktkrgndmcm pir
√ .
q7 ?
M firti pbmzi, bm fmdtncmc ck kqunpi ck fmpntmb ck bm kgprksm ks `njm kd l7=55. Bm tmsm ck mbqunbkr ck l ks v7 $= y bm tmsm ckb smbmrni ck = ks w7$3. m. Fmbfubk bm furvm ckb fisti titmb, m firti pbmzi, ck bm kgprksm. Fmbfubk bm furvm ckb fisti prigkcni m firti pbmzi. Vn l 7 =55 q7 ?
√ . √ =55. √ √ 7 ? 7
q7 ?5
B7
FSF7 vL + wB7 =(=55)+3 FGkF7
7 =55+
7 +
o. ¹Fuíb ks bm `udfnöd ckb fisti gmrhndmb, m firti pbmzi, ck bm kgprksm9 ¹Fuíbks sid bis FSfp, bis FUfp y bis FGh sn bm kgprksm pricufk ?; omstidks9 ¹Fndfukdtm omstidks9 ¹Fnkd omstidks9 ¹Cisfnkdtis omstidks9 FGF7
Vn q7 ?;, FSF7 =55+ FGkF7
+ 7 3.?;
7 =50.?; FGF7
7 5.;5
Vn q7;5, FSF7 =55 + FGkF7
+ 7 ?.;5
Vn q7 =55, FSF7 =55 + FGkF7
+ 7?
Vn q7 ?55, FSF7 =55 + FGkF7
+ 7?.;5
7 =?;
FGF7
7 ?55
FGF7
7 = 7?
7;55
FGF7
73
f. Cnoujk bms furvms ck bis FUfp y bis FGhfp ck bm kgprksm. Ndcnquk bis pudtis fmbfubmcis kd kb ndfnsi mdtkrnir. Bis fistis
FGh
Gmrhndmbks 3
FU
Urigkcni ?
= 5.;
?;
;5
=;5
?55
q
c. ¹Bm furvm ck bis FGhfp cödck ndtkrskftm bm furvm ckb FUfp9 Kxpbnquk pirquk bm furvm ck bis FGhfp snkgprk ndtkrskftmrí bm furvm ck bis FUfp kd su pudti gís omji. Vnkgprk y fumdci kb fisti gmrhndmb ck pricufnr udm udncmc gís kstí pir ckomji ck bm furvm ck fisti prigkcni, bis fistis prigkcni fmkríd. Ckb gnsgi gici, sn kb fisti gmrhndmb ck pricufnr udm udncmc gís ks gmyir quk kb fisti prigkcni, kdtidfks bis fistis prigkcni skrí kb mugkdti. Uir bi tmdti, bm furvm ck VGF ckok ndtkrskftmr bm furvm VMF kd su pudti gís omji. Gnkdtrms bm FGF kstí pir ckomji ck bm FGkF. FGkF omjm ym quk bis FGk ckfrkfkd. Vngnbmrgkdtk, sn bm FGF kstí pir kdfngm ckb FGkF îstm ckok suonr ym quk FG gmyirks m bis FGk amfkd suonr bis FGk. Uir bi tmdti, bm FGF ckok firtmr m bm, FGkF kd kb pudti gçdngi.
Vupidhmgis mairm quk kb fmpntmb pmrm `mornfmr omstidks ck aiflky ks `nji kd l mb firti pbmzi.
̋
k. Fmbfubk kb fisti titmb ck bm kgprksm figi udm `udfnöd ck q, w, v y . q7 ?
̋ ̋
̋ ̋
.q? 7 3l
̋
B 7 q?/3
̋
FS7 v + wB7 v + wq?/3
`. Cmcis q, w y v, ¹bm kgprksm fögi ckok kbkhnr kb mfkrvi ck fmpntmb pmrm gndngnzmr kb fisti titmb9
7 v-wq /3̋ 7 5 ̋7 w v ?
5.;
?
-5.;
h. [tnbnzmdci bis rksubtmcis ckb ndfnsi `, fmbfubmr kb fisti titmb, m bmrhi pbmzi, ck bm `mornfmfnöd ck omstidks ck aiflky.
.∟. (/3̋) .. ... .. . .. ..
FS7 vL + wB7v .
+w .
.
7
7
7
q
a. Vupidhm quk w7 $3, v7 $=, y cnoujk bm furvm ckb fisti titmb m bi bmrhi pbmzi ck bm pricuffnöd ck omstidks ck aiflky. Ckgukstrk quk ks udm kdvibvkdtk ck bms furvms m firti pbmzi quk fmbfubö kd kb ndfnsi m tigmdci bis vmbirks ck , =55, ?55 y 355.
̋
Vn w7 3
v7=
FS7?q
̋ ̋ ̋
FSF ( 7 =55) 7 =55 + q?/=55< FSF 7 FS pmrm q 7 =55 FSF ( 7 ?55) 7 ?55 + q?/?55< FSF 7 FS pmrm q 7 ?55 FSF ( 7 355) 7 355 + q?/355< FSF 7 FS pmrm q 7 355 1.0
[d kgprksmrni mcqunkrk cis kgprksms quk pricufkd mrtk`mftis. Fmcm udm `mornfm pricuftis ncîdtnfis y tnkdk udm `udfnöd ck pricuffnöd cktkrgndmcm pir q7
n7 =,?.
Vnd kgomrhi, fmcm kgprksm tnkdk udm fmdtncmc cnstndtm ck kqunpi ck fmpntmb. Kd fidfrkti, bm kgprksm = tnkdk l= 7 ?;, y bm kgprksm ? tnkdk l? 7 =55, bms tmsms ck mbqunbkr ck l y b kstíd cktkrgndmcms pir w 7 v 7 $=. m. Vn kb kgprksmrni qunkrk gndngnzmr kb fisti titmb ck su pricuffnöd ck mrtk`mftis m firti pbmzi, ¹Fögi ckok msnhdmr bm pricuffnöd kdtrk bms cis kgprksms9 qtitmb 7 q= + q? q=7
?;
F=7 ?; + B= 7 ?; +
7 ;
/?;
q? 7 =5
F? 7 =5 +
/=55
+ Gnd. Fisti: B 7 ! ! 7 7 5 3 7 7 ? 7 5 =55 Ftitmb 7 F= + F? 7 =?; +
o. Cmci quk bm pricuffnöd am snci msnhdmcm ck `irgm öptngm kdtrk bms cis kgprksms, fmbfubk bms furvms ckb fisti gmrhndmb, kb prigkcni y kb titmb m firti pbmzi. ¹Fuíb ks kb fisti gmrhndmb ckb mrtk`mfti dúgkri =559 ¹Ckb mrtk`mfti dúgkri =?;9 ¹Ckb mrtk`mfti dúgkri ?559
3 7 7 =; 7 3;
Vustntuykdci kstm kxprksnöd kd F titmb, sk iotnkdk, bukhi ck udms fukdtms:
7 =?; 7 7 =55 7 ?=?;55 7 $=.05 =?; 7 $?.55 ?55 7 $6.?5 f. ¹Kb kgprksmrni fögi ckok msnhdmr bm pricuffnöd ck mrtk`mftis, m bmrhi pbmzi, kdtrk bms cis kgprksms9 Fmbfubk bms furvms ckb fisti prigkcni, kb gmrhndmb y kb titmb ck bm pricuffnöd m bmrhi pbmzi. BSF 7 L + B 7 ?q BMF 7 ? 7 BGF c. ¹Fögi rkspidckrçm mb ndfnsi mdtkrnir sn bms cis kgprksms gistrmrmd rkdcngnkdtis ckfrkfnkdtks m ksfmbm9 Vn amy rktirdis ckfrkfnkdtks m ksfmbm fid `udfnidks ck pricuffnöd ncîdtnfms, kdtidfks sk ckok rkpmrtnr bm pricuffnöd kdtrk mgoms pbmdtms kd pmrtks nhumbks. 1.8
Uickgis hkdkrmbnzmr bm `udfnöd ck pricuffnöd FKV pmrm pidckrmr bis `mftirks. Kd kb fmsi ck bis cis `mftirks, bm `udfnöd ks
7 , 7 m. ¹Fuíb ks bm `udfnöd ck fisti titmb pmrm udm kgprksm quk tnkdk kstm `udfnöd ck pricuffnöd9
7 = 7 =,
7 \⁃ W
o. Vn y pickgis ckgistrmr quk kstm `udfnöd ck pricuffnöd fidvkrhk kd bm `irgubm Fioo-Ciuhbms figi ¹Fuíb ks bm `udfnöd ckb fisti titmb pmrm kstm vkrsnöd pmrtnfubmr ck bm `udfnöd FKV9
7
↟ 5.
7 ∟∟ f. Bm pmrtk rkbmtnvm ckb fisti ckb trmomji kd udm `udfnöd ck pricuffnöd fid cis `mftirks kstí cktkrgndmcm pir Ckgukstrk quk kstm pmrtk ks fidstmdtk kd kb fmsi ck bm `udfnöd Fioo-Ciuhbms ckb ndfnsi o. ¹Bis pmrígktris figi m`kftmd bm pmrtk rkbmtnvm ckb trmomji9
/.
7 /
c. Fmbfubk bm `rmffnöd rkbmtnvm ckb fisti ckb trmomji pmrm bm `udfnöd hkdkrmb ckb FKV ndtricufncm gís mrrnom. ¹Bis fmgonis kd , figi fmgonmd ksm `rmffnöd9 ¹Bm kbmstnfncmc ck sustntufnöd, figi cktkrgndm bm cnrkffnöd ck kstk k`kfti9 ¹Kb tmgmýi ck bis pmrígktris fögi bm m`kftm9
, 7 \(⁃)W
Msç quk
7 ⁃∟
.
Bm pmrtnfnpmfnöd rkbmtnvm ck gmdi ck iorm ks udm frkfnkdtk kd `udfnöd ck o / m. Vn φ 4 = ck trmomji ck mffnidks sk gukvk kd bm gnsgm cnrkffnöd quk v / w. Vn φ 2 =, ck gmdi ck iorm rkbmtnvis ck mffnidks sk gukvk kd bm cnrkffnöd ipukstm m v / w. Ksti fidfukrcm fid bm ndtunfnöd ck fögi sustntufnöd ckok m`kftmr m bms mffnidks.
1.1
Bms kbmstnfncmcks prkfni pripni ckbm ckgmdcm ckpkdcnkdtk ck bis `mftirks fmpntmb y trmomji kstíd ck`ndncms figi
. y .
m. Fmbfubk kjkgpbi 1.?.
, 7 .
pmrm fmcm udm ck bms `udfnidks ck fistis quk fidtnkdk kb
Bms kbmstnfncmcks sk pukckd bkkr cnrkftmgkdtk ck bms `udfnidks ck ckgmdcm fidtndhkdtks kd kb Kjkgpbi 1.3. Umrm kb fmsi ck pripirfnidks `njms q sk gmdtnkdk fidstmdtk). Umrm bm Fioo ‖ Ciuhbms,
. 7 . 7 5
(ym
. 7 ∗ <. 7 . . 7 5.
. Mb pmrkfkr, kb FKV ck kstk `irgubmrni tnkdk
kbmstnfncmcks di fidstmdtks.
o. Ckgukstrk quk, kd hkdkrmb,
Ckonci m quk bms `udfnidks ck fisti sid aigihîdkms ck hrmci udi kd bis prkfnis ck kdtrmcm, bms `udfnidks ck ckgmdcm fidtndhkdtks sid aigihîdkms ck hrmci fkri kd bis prkfnis figi bi suhnkrk bm ndtunfnöd. [smdci kb tkirkgm ck Kubkr cm . Cnvncnkdci pir cm kb rksubtmci.
7 5
f. Ckgukstrk quk bms ckrnvmcms ck prkfnis fruzmcis ck bms `udfnidks ck ckgmdcm ckpkdcnkdtks sid nhumbks< ks ckfnr, ckgukstrk quk
. 7/.
ckgistrmr quk ckdtri ckb fisti titmb (
,
7 . [tnbnfk kstk akfai pmrm
cidck sid, rkspkftnvmgkdtk, bm pmrtk ckb trmomji FS) y ckb fmpntmb ckdtri ckb fisti titmb ( FS).
/
[tnbnfk kb tkirkgm ck ^iudh:
7 7 7 Mairm sk gubtnpbnfmd pir bm nzqunkrcm justi mb bmci ck ckrkfai c. [tnbnfk bis rksubtmcis ck bis ndfnsis o y f pmrm ckgistrmr quk
. . 7 5.
. . 7 5 . . 7 5
Gubtnpbnfmdci pir mffnidks kd bm pmrtk o ck rkdcngnkdtis Bm sustntufnöd ck bm pmrtk f rkdcngnkdtis
k. Ndtkrprktk bms cnstndtms rkbmfnidks ck kbmstnfncmc fid pmbmorms y kxpbnquk su ngpirtmdfnm hbiomb pmrm bm tkirçm hkdkrmb ck bm ckgmdcm ck `mftirks ck pricuffnöd. Sicis kstis rksubtmcis cmd fakquks ngpirtmdtks pmrm skr utnbnzmcis kd kb trmomji kgpçrnfi. 1.>
Vupidhmgis quk bm `udfnöd ck fisti titmb ck udm kgprksm kstí cktkrgndmcm pir
7 m. [tnbnfk kb bkgm ck Vakpamrc pmrm fmbfubmr bms `udfnidks ck ckgmdcm fid pricuffnöd fidstmdtk ck bis `mftirks b y l.
7 7 /
7 7 /
o. [tnbnfk bis rksubtmcis ckb mpmrtmci mdtkrnir pmrm fmbfubmr bm `udfnöd ck pricuffnöd suoymfkdtk ck q. Ckspkjmdci w/v ck B:
7 / 74 7
Vustntuykdci kd L: L7
/ 7 74
6/ / 6/ 7 / / / 7
Bm fubm ks udm `udfnid ck pricuffnid Fioo-Ciuhbms.
1.=5
Vupidhmgis quk bm `udfnöd ck fisti titmb ck udm kgprksm kstí cktkrgndmcm pir
7 (?√ ). m. utnbnfk kb bkgm ck Vakpamrc pmrm fmbfubmr bm `udfnöd ck ckgmdcm fid pricuffnöd fidstmdtk ckb `mftir,
.
7 7 ∟. 7 7 ∟. o. [tnbnfk bis rksubtmcis ckb ndfnsi mdtkrnir pmrm iotkdkr bm `udfnöd ck pricuffnöd suoymfkdtk ck
.
Ck udm pmrtk,
7 . , 7 . msn quk 7 = i ∟ ∟ 7 ∟ Bm `udfnöd ck pricuffnöd ks kdtidfks 7 ∟ ∟ ∟ f. figprukom kb rksubtmci utnbnzmdci bis rksubtmcis ckb kjkgpbi 1.? pmrm ckgistrmr quk bm `udfnöd ck fistis FKV, fumdci 7 5.;, 7 -= hkdkrm kstm `udfnöd ck fisti titmb.
Kstm ks udm `udfnöd ck pricuffnöd FKV fid 7 -=. Uir bi tmdti, 7 = / (= - ) 7 5,;. Figpmrmfnöd fid kb kjkgpbi 1.? gukstrm bm rkbmfnöd kdtrk bis pmrígktris ck bm `udfnöd ck pricuffnöd FKV y su `udfnöd ck fistk rkbmfnidmcm.
