EH\ CMDSS 44 D@O CMDSS 4<
© <>4; Dmm rakjts rbsbrvbo. @h pdrt he tjas pulmacdtah` idy lb rbprhoucbo, sthrbo a` d rbtrabvdm systbi hr trd`siattbo a` d`y ehri hr ly d`y ibd`s, bmbctrh`ac, ibcjd`acdm, pjhthchpya`k, rbchroa`k, scd``a`k, wbl hr htjbrwasb watjhut tjb prahr pbriassah` he tjb dutjhr. Dmm tjb a`ehridtah` a` tjas b-lhhf as hltda`bo erhi tjb shurcbs lbmabvbo th lb trub d`o rbmadlmb. Jhwbvbr, tjb dutjhr ohbs`‑t tdfb d`y rbsph`salamaty ehr tjb dlshmutb dccurdcy he d`y a`ehridtah` pulmasjbo d`o tjb odidkbs hr mhss sueebrbo tjbrbuph`.
^ulmasjbo ly7 I. M. Ch`tdct7 B-idam AO7 rdgalum.id`kd.whrmoNkidam.chi
[H\OS E\HI _JB DR_JH\ _jas b-lhhf ch`tda`s sdipmb hlsbrvdtah` he dmm tjb bxpbraib`ts th lb oh`b a` cmdss ZA d`o cmdss ZAA. Lut tjb stuob`ts sjhumo `bvbr chpy tjb sdipmb rbdoa`ks kavb` a` tjas b-lhhf. _jb bxpbraib`ts sjhumo lb prhpbrmy oh`b a` scjhhm mdlhrdthry. Stuob`ts ds wbmm ds tbdcjbrs cd` usb tjas b-lhhf ds rbebrb`cb. _jb krdpjs drb `ht a`cmuobo a` tjas b-lhhf. _jas b-lhhf h`my ‚
‑
ch`tda` DAI ,
‚
‑
‚
‑
‚
‑
HLSB\UD_AH@ , CDMCRMD_AH@ , \BSRM_ . _jas ‚
‑
‚
‑
b-lhhf ohbs `ht ch`tda` ^\HCBOR\B , ^\BCDR_AH@ btc.
4. Rsb he Ubr`abr Cdmapbr.
-----
4-5
<. Rsb he Scrbw Kdukb.
-----
;-44
3. _h obtbria`b obtbria`b tjb rdoaus he curvdturb he d kavb` kavb` spjbracdm suredcb ly d spjbrhibtbr. spjbrhibtbr. -----
4<-42
?. _h obtbria`b tjb idss he twh oaeebrb`t hlgbcts usa`k d lbdi ldmd`cb. ----45-4; 2. _h ea`o tjb wbakjt he d kavb` lhoy usa`k pdrdmmbmhkrdi mdw he vbcthrs.
-----
4:-41
<
5. Rsa`k d saipmb pb`oumui, th pmht M-_ d`o M-_ krdpjs d`o jb`cb ea`o tjb beebctavb mb`ktj he sbch`os pb`oumui usa`k dpprhpradtb dpprhpradtb krdpj. ----<>-<< ;. _h stuoy tjb rbmdtah`sjap lbtwbb` ehrcb he maiata`k eractah` d`o `hridm rbdctah` d`o th ea`o tjb chbeeacab`t he maiata`k eractah` lbtwbb` d lmhcf d`o d jhrazh`tdm suredcb. ----<3-
:. _h ea`o tjb ohw`wdro ehrcb dmh`k d` a`cma`bo pmd`b, dcta`k h` d rhmmbr oub th krdvatdtah`dm pumm he tjb bdrtj d`o stuoy ats rbmdtah`sjap watj tjb d`kmb he a`cma`dtah`(κ) ly pmhtta`k d krdpj lbtwbb` ehrcb d`o sa` κ. ----<2-<5 1. _h obtbria`b tjb Qhu`k‑s ihoumus he bmdstacaty he kavb` warb ly usa`k Sbdrmb‑s dppdrdtus. ----<;-<1 4>. _h ea`o tjb spra`k ch`std`t he d jbmacdm spra`k erhi mhdo bxtb`sah` oadkrdi. ----3>-34
44. _h stuoy tjb vdradtah` a` vhmuib watj prbssurb ehr d sdipmb he dar dt ch`std`t tbipbrdturb ly pmhtta`k krdpjs lbtwbb` ^ d`o U, d`o lbtwbb` ^ d`o 4/U. ----3<-33 4<. _h obtbria`b tjb suredcb tb`sah` he wdtbr ly cdpammdry rasb ibtjho ----3?-35 43. _h obtbria`b tjb chbeeacab`t he vaschsaty he d kavb` vaschus maquao ly ibdsura`k tjb tbria`dm vbmhcaty he kavb` spjbracdm lhoy. ----3;-?> 4?. _h stuoy tjb rbmdtah`sjap lbtwbb` rhhi's tbipbrdturb he d jht lhoy d`o taib ly pmhtta`k d chhma`k curvb. ?4-?< 42. _h obtbria`b spbcaeac jbdt he d kavb` shmao ly ibtjho he iaxturbs. ----?3-?? 45. _h obtbria`b tjb spbcaeac jbdt he kavb` maquao ly
ibtjho he iaxturb.
-----
?2-?5
4;. a) _h stuoy tjb rbmdtah` lbtwbb` erbqub`cy d`o mb`ktj he d kavb` warb u`obr ch`std`t tb`sah` usa`k sh`hibtbr. aa) _h stuoy tjb rbmdtah` lbtwbb` tjb mb`ktj he kavb` warb d`o tb`sah` ehr ch`std`t erbqub`cy usa`k sh`hibtbr. ----?;-?1 4:. _h ea`o spbbo he shu`o a` dar dt rhhi tbipbrdturb usa`k d rbsh`d`cb tulb ly twh rbsh`d`cb phsatah`s. ----2>
41. _h obtbria`b rbsastd`cb pbr ci he d kavb` warb ly pmhtta`k d krdpj he phtb`tadm oaeebrb`cb vbrsus currb`t. ----2<-23 <>. _h ea`o rbsastd`cb he d kavb` warb usa`k ibtbr lraokb d`o jb`cb obtbria`b tjb rbsastavaty he ats idtbradm. idtbradm. 2?-22 <4. _h vbraey mdws he chila`dtah` he rbsastd`cb (Sbrabs & ^drdmmbm) usa`k ibtbr lraokb. ----25-2; <<. _h chipdrb b.i.e. he twh kavb` praidry cbmms usa`k phtb`tahibtbr. ----2:-21 <3. _h obtbria`b tjb a`tbr`dm rbsastd`cb he kavb` praidry cbmm usa`k d phtb`tahibt phtb`tahibtbr. br. ----5>-54
AA. Diibtbr he obsarbo rd`kb d`o vbraey tjb sdib. ----<5. _h ea`o tjb erbqub`cy he tjb DC ida`s watj d sh`hibtbr. -----
5?-55 5;-5:
<;. _h ea`o tjb vdmub he v ehr oaeebrb`t vdmubs he u a` cdsb he d ch`cdvb iarrhr d`o th ea`o tjb ehcdm mb`ktj. ----51-;< <:. _h ea`o tjb ehcdm mb`ktj he d ch`vbx iarrhr, usa`k d ch`vbx mb`s. ----;3-;? <1. _h ea`o tjb ehcdm mb`ktj he d ch`vbx mb`s ly pmhtta`k krdpjs lbtwbb` u d`o v hr lbtwbb` 4/u d`o 4/v. ----;2-;: 3>. _h ea`o tjb ehcdm mb`ktj he d ch`cdvb mb`s usa`k d ch`vbx mb`s. ----;1-:> 34. _h obtbria`b d`kmb he ia`aiui obvadtah` ehr d kavb` prasi ly pmhtta`k d krdpj lbtwbb` d`kmb he a`caob`cb d`o tjb d`kmb he obvadtah`. ----:4-:< 3<. _h obtbria`b tjb rberdctavb a`obx he d kmdss smdl usa`k trdvbmma`k iacrhschpb. ----:3-:? 33. _h ea`o tjb rberdctavb a`obx he d maquao ly usa`k ch`vbx mb`s d`o d pmd`b iarrhr. ----:2-:5 3?. _h ordw tjb A-U cjdrdctbrastac curvb he d p-` gu`ctah` a` ehrwdro lads d`o rbvbrsb lads. ----:;-1> 32. _h ordw tjb cjdrdctbrastac curvb he d Wb`br oahob d`o th obtbria`b ats rbvbrsb lrbdfohw` vhmtdkb. ----14-13
35. _h stuoy tjb cjdrdctbrastacs he d chiih` biattbr `-p-` hr p`-p trd`sasthr d`o th ea`o hut tjb vdmubs he currb`t d`o vhmtdkb kda`s. ----1?-1:
^D\_ A CMDSS ZA ^JQSACS
BZ^B\AIB@_
4
DAI Rsa`k Ubr`abr Cdmmapbrs ea`o
4.Oadibtbr he d sidmm spjbracdm/cyma`oracdm lhoy <.Oaib`sah` he d kavb` rbkumdr (sdy rbctd`kumdr) lhoy he f`hw` idss d`o jb`cb ea`o ats ob`saty. 3.A`tbr`dm oadibtbr d`o obptj he d kavb` cdmhraibtbr/lbdfbr cdmhraibtbr/lbdfbr d`o jb`cb ea`o ats vhmuib.
HLSB\UD_AH@S
4 ida` scdmb oavasah`
4. Ubr`abr Ch`std`t 9
`h. he vbr`abr scdmb oavasah` 4
9
4>
9 .4 ii 9 >.>4 >.>4 ci
<. Wbrh brrhr he Ubr`abr cdmapbrs 9 >> ci. 3. Wbrh chrrbctah` 9 >> ci.
_DLMBS
4.
OADB_B\ HE S^JB\B/CQMA@OB\ S^JB\B/CQMA@OB\
S. @h.
Ida` Scdmb \bdoa`k (I) (ci)
4. <. 3.
4.3 4.3 4.3
Ubr`abr Scdmb \bdoa`k ` Û U.C. Oa9 I + (` (ci) Û U.C.) Oavasah`s cha`caoa`k (`) 1 .>1 O49 4.31 : .>: O<9 4.3: 1 .>1 O39 4.31
Ibd` Oadibtbr 9 4.31 ci.
<.
OAIB@SAH@ HE _JB LMHCF Saob
Mb`ktj 4. M4 <. M< 3. M3 Lrbdotj 4. L4 <. L< 3. L3 Jbakjt 4. J4 <. J< 3. J3
Ida` Scdmb \bdoa`k (I) (ci)
Ubr`abr scdmb rbdoa`k Oavash` \bdoa`k
`Û U.C.
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5.2 5.2 5.2
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5.23 5.24 5.2?
Ibd` Mb`ktj, M 9 ;.?? ci. Ibd` Lrbdotj, L 9 2.<; ci.
Ibd` Jbakjt, J 9 5.23 ci.
CDMCRMD_AH@ Uhmuib 9 Mb`ktj (M) Û Lrbdotj (L) Û Jbakjt (J)
O9I+ (`Û U.C.)
9 (;.?? Û 2.<; Û 5.23) ci
3
3
9 <25.>3 ci Kavb` Idss 9 32 k.
Ob`saty 9
3.
Idss Uhmuib
3
9 >.435; k/ci
A@_B\@DM OADIB_B\ HE CDMH\AIB_B\/LBDFB\
S. @h.
4. <. 3.
Ida` Scdmb \bdoa`k (I) (ci)
2.; 2.; 2.;
Ubr`abr Scdmb \bdoa`k ` Û U.C. Oa9 I + (` (ci) Û U.C.) Oavasah`s cha`caoa`k (`) 2 >.>2 O49 2.;2 ; >.>; O<9 2.;; ? >.>? O39 2.;?
?.
OB^_J HE CDMH\AIB_B\/LBDFB\ CDMH\AIB_B\/LBDFB\
S. @h.
Ida` Scdmb \bdoa`k (I) (ci)
4. <. 3.
1.: 1.: 1.:
Ubr`abr Scdmb \bdoa`k ` Û U.C. Ja9 I + (` (ci) Û U.C.) Oavasah`s cha`caoa`k (`) 1 >.>1 J49 1.:1 5 >.>5 J<9 1.:5 : >.>: J39 1.::
Ibd` Oadibtbr 9 2.;2 ci.
Ibd` Jbakjt 9 1.:: ci. <
Uhmuib
o j
? 3.4? 2.;2 2.;2 1.:: ? <25.?<5 <25.?<5 ci
3
BZ^B\AIB@_
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DAI Rsa`k Scrbw Kdukb ea`o
4.Oadibtbr he d kavb` warb <._jacf`bss he d kavb` sjbbt. 3.A`tbr`dm oadibtbr d`o obptj he d kavb` cdmhraibtbr/lbdfbr cdmhraibtbr/lbdfbr d`o jb`cb ea`o ats vhmuib.
HLSB\UD_AH@S 4.^atcj he tjb scrbw kdukb 9 4 ii. ^atcj
<.Mbdst chu`t 9
9
Oavasah`s h` carcumdr scdmb
4 4>>
3. Wbrh brrhr 9 >> ci.
9 >.>4 ii 9 >.>>4 ci.
_DLMBS
4. OADIB_B\ HE _JB KAUB@ [A\B S. @h.
4.
Oadibtbr a` h`b oarbctah` ^atcj Carcumdr Scdmb Scdmb (I) (`4) (ii) > 5;
Oadibtbr a` pbrpb`oacumdr oarbctah` o4 9 I + ^atcj Carcumdr o4 9 I + `4 Û M.C. Scdmb Scdmb `< Û M.C. (I) (`<) (ii) >.>5; > 5> >.>5?
Hlsbrvbo Oadibtbr
o4 + o< Oa 9 >.>522
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_jb Oadibtbr he tjb kavb` warb 9 >.>52< ci.
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<. _JACF@BSS HE D KAUB@ SJBB_
S. @h.
Ida` Scdmb \bdoa`k (I) (ii)
Carcumdr Scdmb \bdoa`k Hlsbrvbo Udmub oavasah`s (`)
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M.C.
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4. <. 3.
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Ibd` _jacf`bss 9 9
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M.C.
3. _JACF@BSS HE _JB A\\BKRMD\ MDIA@D
S. @h.
4. <. 3. ?. 2. 5.
Ida` Scdmb \bdoa`k (I) > > > > > >
Carcumdr Scdmb \bdoa`k oavasah` (`) ;5 ;< ;2 ;: ;; ;2
` Û M.C. (ii) >.;5 >.;< >.;2 >.;: >.;; >.;2
Hlsbrvbo Udmub _a 9 I + ` Û M.C. (ii) >.;5 >.;< >.;2 >.;: >.;; >.;2
@uilbr he Chipmbtb Chipmbtb squdrbs watja` tjb lhu`odry he mdia`d 9 :1 @uilbr he jdme/ihrb jdme/ihrb tjd` jdme squdrbs squdrbs watja` tjb lhu`odry he mdia`d mdia`d 9 41 _htdm squdrbs 9 4>: <
<
Drbd he mdia`d 9 <; ci 9 <;>> ii .
Ibd` tjacf`bss he mdia`d 9 >.;5 ii.
Uhmuib he mdia`d 9 Suredcb drbd Û tjacf`bss <
9 <;>> ii Û >.;5 ii. 3
9 <>3:.2 ii 3
9 <.>3:2 ci
BZ^B\AIB@_
3
DAI _h obtbria`b tjb rdoaus he curvdturb he d kavb` spjbracdm suredcb ly d spjbrhibtbr.
EH\IRMDB RSBO \9
m
<
5j
+
j <
m 9 saob he bquamdtbrdm trad`kmb dt wjhsb vbrtacbs, tjb mbks he spjbrhibtbr mab. j 9 tjb oastd`cb ly wjacj tjb cb`trdm mbk jds th lb mhwbrbo hr rdasbo. \ 9 rdoaus he curvdturb he tjb kavb` spjbracdm suredcb.
HLSB\UD_AH@S ^atcj he tjb spjbrhibtbr 9 4 ii.
Mbdst chu`t he tjb spjbrhibtbr 9
patcj
Carcumdr scdmb oavasah`s 9 >.>4 ii 9 >.>>4 ci.
Oastd`cb lbtwbb` mbks he spjbrhibtbr m4 9 ?.? ci m< 9 ?.? ci m3 9?.? ci
Ibd` oastd`cb m 9
m4+m<+m3 3
9 ?.? ci.
_DLMBS 4. IBDSR\BIB@_ HE SDKA__D (j) (d) _ypb he suredcb7 ch`vbx S. @h.
\bdoa`k he Carcumdr scdmb (h` spjbracdm suredcb) (C4)
\bdoa`k he pmd`b kmdss smdl
@h. he rhtdtah`s (i)
Carcumdr scdmb rbdoa`k (C<)
Erdctah`dm rhtdtah` ` 9 |C4 – C C<|
j 9 patcj i + Mbdst chu`t ` (ci)
4. <. 3.
24 23 2<
4 4 4
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CDMCRMD_AH@ Ibd` vdmub he j 9 4.>;? ii 9 >.4>;? ci.
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m
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+
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9 3>.>;? ci.
Sulstatuta`k tjb vdmubs ehr m d`o j a` tjb dlhvb ehriumd d`o shmva`k, shmva`k, wb kbt \ 9 3>.>;? ci.
\BSRM_ _jb rdoaus he curvdturb he tjb kavb` spjbracdm suredcb 9 3>.>;? ci.
BZ^B\AIB@_
?
DAI _h obtbria`b tjb idss he twh oaeebrb`t hlgbcts usa`k d lbdi ldmd`cb.
HLSB\UD_AH@ 4. Idss he tjb lrdss spjbrb7
Std`odro idssbs 9 i 4 9 ? k. Erdctah`dm idssbs 9 i < 9 :.:
<. Idss he tjb arh` lhl7
Std`odro idssbs 9 i 4‑ 9 2 k. Erdctah`dm idssbs 9 i <‑ 9 23 ik. 9 >.>23 k.
_htdm idss 9 i ‑ 9 i4‑ + i<‑ 9 2.>23 k.
\BSRM_ Idss he tjb lrdss spjbrb 9 ?.:23 k.
BZ^B\AIB@_
2
DAI _h ea`o tjb wbakjt he d pdrdmmbmhkrdi mdw mdw he vbcthrs.
kavb`
lhoy
usa`k
EH\IRMDB RSBO \ 9
<
<
E4 +E< +
E4 d`o E< rbprbsb`t tjb ehrcbs bxbrtbo oub th wbakjt drrd`kbib`ts h` lhtj saobs.
HLSB\UD_AH@ Sbmbctbo scdmb ehr ordwa`k tjb pdrdmmbmhkrdi 9 2 k wt 9 4 ci. S. @h.
E4 (k wt)
E< (k wt)
Mb`ktj HD (ci)
Mb`ktj HL (ci)
Oadkh`dm HC (ci)
R`f`hw` wbakjt ([) (k wt)
4. <.
3> ?>
32 ?2
5 :
; 1
5.;2 5.;2
33.;2 33.;2
3.
2>
22
4>
44
5.;3
33.52
Ibd` vdmub he wbakjt he tjb lhoy, [ 9
[4
[
<
[
3
9 33.;< k wt.
3
BZ^B\AIB@_
5
DAI <
Rsa`k d saipmb pb`oumui, th pmht M-_ d`o M-_ krdpjs d`o jb`cb ea`o tjb beebctavb mb`ktj he sbch`os pb`oumui usa`k dpprhpradtb dpprhpradtb krdpj.
HLSB\UD_AH@S Ubr`abr Cdmmapbrs7 Ubr`abr Ch`std`t 9 >.>4 ci. Wbrh brrhr 9 >> ci. Jhhf mb`ktj he pb`oumui
9 4.; ci.
^B@ORMRI LHL S. @h.
Hlsbrvbo Oadibtbr (ci)
\doaus (ci)
4. <. 3.
<.< <.< <.<
4.4 4.4 4.4
Ibd` rdoaus 9 4.4> ci.
_DLMB EH\ _JB _AIB _DFB@ _H CHI^MB_B <> HSCAMMD_AH@S HSCAMMD_AH@S S. @h. Beebctavb ^b`oumui Mb`ktj (M) (ci) 4. <. 3. ?. 2. 5.
:> 1> 4>> 44> 4<> 43>
_aib _dfb` ehr <> hscammdtah` (t) (s) 35 3: ?> ?<.4 ?? ?2.:
t _9 s <>
_< (s<)
4.: 4.1 < <.4 <.< <.3
3.
CDMCRMD_AH@ 4. _jb M-_ krdpj as pmhttbo ly tdfa`k beebctavb mb`ktj he pb`oumui dmh`k x-dxas d`o taib pbraho he hscammdtah` dmh`k y-dxas. _jb krdpj chibs hut th lb d pdrdlhmd. <
<. _jb M-_ krdpj chibs hut lb d strdakjt ma`b. <
3. Orhp d pbrpb`oacumdr erhi d pha`t h` M-_ krdpj < wjbrb _ 9 ?, th ibbt x-dxas. _jb pha`t h` tjb x-dxas wjbrb tjb pbrpb`oacumdr erhi tjb krdpj edmms as tjb mb`ktj he tjb sbch`o‑s pb`oumui.
\BSRM_ Beebctavb mb`ktj he tjb sbch`o‑s pb`oumui as 4>> ci.
BZ^B\AIB@_
;
DAI _h stuoy tjb rbmdtah`sjap lbtwbb` ehrcb he maiata`k eractah` d`o `hridm rbdctah` d`o th ea`o tjb chbeeacab`t he maiata`k eractah` lbtwbb` d lmhcf d`o d jhrazh`tdm suredcb.
EH\IRMD RSBO Dcchroa`k th tjb mdw he maiata`k eractah`, Ehrcb he eractah`
Chbeeacab`t he maiata`k eractah` (ν) 9
@hridm rbdctah` rbdctah`
HLSB\UD_AH@S _dlmb 47 [bakjt he tjb lmhcf d`o ^d` Hlgbct Lmhcf
^d`
Hlsbrvbo [bakjt ( k wt)
4. <. 3. 4. <. 3.
35 35 35 3< 3< 3<
Ibd` wbakjt he tjb lmhcf 9 [ 4 9 35 k wt. Ibd` wbakjt he tjb ^d` 9 [ p 9 3<.2 k wt.
_dlmb <7 @hridm rbdctah` d`o maiata`k eractah` S. @h.
4. <. 3. ?.
[bakjt pmdcbo h` lmhcf ([<) (k wt.)
_htdm `hridm rbdctah`, @ 9 [4 + [< (k wt.)
[bakjts a` tjb ^d` 9 [
E 9 ([ p +[ )
(k wt.)
(k wt.)
<:1 352.3 ?2:.; 521.3
342 ?>4.3 ?1?.; 512.3
441.< 453.2 <>3.2 <:;.3
424.; 415 <35 341.:
E
Ibd` vdmub he ν 9 >.?;5;2
\BSRM_ _jb krdpj lbtwbb` maiata`k eractah` d`o `hridm rbdctah` as d ma`bdr krdpj wjacj aipmabs E ∙ @.
ν9
@
>.?:< >.;:: >.?;; >.?5>
BZ^B\AIB@_
:
DAI _h ea`o tjb ohw`wdro ehrcb dmh`k d` a`cma`bo pmd`b, dcta`k h` d rhmmbr oub th krdvatdtah`dm pumm he tjb bdrtj d`o stuoy ats rbmdtah`sjap watj tjb d`kmb he a`cma`dtah`(κ) ly pmhtta`k d krdpj lbtwbb` ehrcb d`o sa` κ.
HLSB\UD_AH@S
_dlmb 47 [bakjt he tjb ^d` d`o \hmmbr Hlgbct \hmmbr
^d`
Hlsbrvbo [bakjt ( k wt) 4. <. 3. 4. <. 3.
25 25 25 4<.2 4<.2 4<.2
Ibd` wbakjt he tjb lmhcf 9 [ \ 9 25 k wt. Ibd` wbakjt he tjb ^d` 9 [ ^ 9 4<.2k wt.
_dlmb <7 Ehr ehrcb ‚E‑ d`o D`kmb he A`cma`dtah` ‚κ‑ S. @h.
4. <. 3. ?. 2.
3>¾ ?>¾ ?2¾ 2>¾ 22¾
[bakjts a` pd` ehr E4+E< ihtah` E9^ E9^ E9 < Ohw`wdros Rpwdros + [ + [ (k (k (k wt.) ^4 ^< wt.) wt.) 4:.; 4<.5 34.< <2.4 <:.42 <5.2 <>.3 31 3<.: 32.1 3>.; <3.1 ?3.< 35.? 31.: 3? <;.3 ?5.2 31.: ?3.42 3;.1 <1.3 2>.? ?4.: ?5.4
Sa`
>.2 >.5? >.;4 >.;; >.:<
(k wt.)
E-[ Sa` (k wt.)
<: 32.:? 31.;5 ?3.4< ?2.1<
>.42 >.>5 >.>? >.>3 >.4:
[ Sa`
\BSRM_ 4. _jb ohw`wdro ehrcb E dcta`k dmh`k tjb a`cma`bo pmd`b h` rhmmbr h` dcchu`t he krdvatdtah`dm ehrcb he bdrtj as E 9 ik sa` κ.
<. _jb krdpj lbtwbb` E d`o sa` κ as ma`bdr d`o lbtwbb` E d`o κ as d sa`b curvb.
BZ^B\AIB@_
1
DAI _h obtbria`b tjb Qhu`k‑s ihoumus he bmdstacaty he kavb` warb ly usa`k Sbdrmb‑s dppdrdtus.
HLSB\UD_AH@S Mb`ktj he tjb bxpbraib`tdm warb, m 9 4>> ci.
Ehr rdoaus he warb7 ^atcj he scrbw kdukb 9 4 ii. Mbdst chu`t he tjb scrbw kdukb 9 >.>4 ii. Wbrh brrhr he scrbw kdukb 9 > ii.
_dlmb 47 Oadibtbr he [arb S. @h. 4. <. 3.
Oadibtbr Carcumdr Scdmb (C4) <> 41 <4
I.S.\. (S4) > > >
o4 9 S4 + (C4 M.C.) >.< >.41 >.<4
Ibd` oadibtbr 9 >.>< ci. Ibd` rdoaus 9 >.>4 ci.
[bakjt a` S. jd`kbr @h. (Fk wt.) 4. <. 3. ?. 2.
>.2 4.> 4.2 <.> <.2
Iacrhibtbr \bdoa`k Mhdoa`k R`mhdoa`k 4.5 <.:: 3.;4 ?.23
4.5 <.:: 3.;4 ?.23
Ibd` 4.5 <.:: 3.;4 ?.23
Ibd` bxtb`sah` ( m)
> >.:< 4.5? <.?; 3.<1
CDMCRMD_AH@ Q9
ikm <
ώr m
:
<
9 4.:1; Û 4> @/i
\BSRM_ Ehr tjb kavb` idtbradm he warb, Cdmcumdtbo Qhu`k‑s ihoumus 9
@/i
Udmub he Q erhi tjb tdlmb he ch`std`ts 9
∿
^brcb`tdkb brrhr 9
<
@/i
.
Sa`cb tjb E-m krdpj as d strdakjt ma`b, at dmsh prhvbs Jhhfb‑s Mdw.
<
BZ^B\AIB@_
4>
DAI _h ea`o tjb spra`k ch`std`t he d jbmacdm spra`k erhi mhdo bxtb`sah` oadkrdi.
HLSB\UD_AH@S _jb krdpj as pmhttbo watj mhdo h` x-dxas d`o scdmb rbdoa`k h` y-dxas.
S. @h.
Mhdo i (k wt.)
^ha`tbr \bdoa`k (ci)
4. <. 3. ?.
4>> <>> 3>> ?>>
5>.;? 54.<4 54.2? 54.1:
CDMCRMD_AH@ Ehrcb Ch`std`t he tjb spra`k, 1.: i @ E 4>>> f 4 m m i 4>>
1.: 4>>> 9 Smhpb he tjb curvb 44>> @ / i
9
> .: > ; 1 :
1.: 4>>>
@/i
4 4> >
9 >.;1? @/i
\BSRM_ _jb vdmub he spra`k ch`std`t ehr tjb kavb` jbmacdm spra`k 9 >.;1? @/i
44
BZ^B\AIB@_ DAI
_h stuoy tjb vdradtah` a` vhmuib watj prbssurb ehr d sdipmb he dar dt ch`std`t tbipbrdturb ly pmhtta`k krdpjs lbtwbb` ^ d`o U, d`o lbtwbb` ^ d`o 4/U.
HLSB\UD_AH@S \hhi tbipbrdturb
A`atadm rbdoa`k, _ 4 9 <:℃ Ea`dm rbdoa`k, _ < 9 <:℃
Ibd` rhhi tbipbrdturb, _ 9
_
_4
_ <
<
9 <:℃
Ldrhibtbr rbdoa`ks
A`atadm jbakjt he ibrcury, j 4 9 ;5 ci. Ea`dm jbakjt he ibrcury, j < 9 ;5 ci. Ibd` ldrhibtrac jbakjt he ibrcury J 9
j4
j<
<
9 ;5 ci.
_dlmb 47 Uhmuib d`o ^rbssurb Ibdsurbib`t S. @h.
4. <. 3. ?. 2.
Jk mbvbm Jk mbvbm a` tulb a` tulb D L 9 d (ci) 9 l (ci)
?3.< 2>.2 23.2 ?2 5>.2
^rbssurb jbdo j 9 l – l – d d (ci) +4:.; +4?.: > -4>.2 -45.2
B`cmhsbo dar Uhmuib prbssurb he tjb dar ^9J j B`cmhsbo (ci) U (iM) 1?.; 2.2 1>.: 2.: ;5 5.1 52.2 : 21.2 :.:
4/U
^U
>.4:< 2<>.:2 >.4;< 2<5.5? >.4?2 2.4<2 2.44? 2<3.5
D krdpj as pmhttbo lbtwbb` b`cmhsbo dar prbssurb (^) d`o vhmuib he b`cmhsbo dar (U), watj ^ dmh`k x-dxas d`o U dmh`k y-dxas. D`htjbr krdpj as pmhttbo lbtwbb` ^ d`o 4/U.
\BSRM_ 4. ^U as ch`std`t (watja` bxpbraib`tdm brrhr). <. _jb krdpj lbtwbb` ^ d`o 4/U as d strdakjt ma`b wjacj aipmabs tjdt ^ ∙ 4/U.
BZ^B\AIB@_
4<
DAI _h obtbria`b tjb suredcb tb`sah` he wdtbr ly cdpammdry rasb ibtjho
EH\IRMD RSBO _
rj k
<
[jbrb, _ 9 suredcb tb`sah` he wdtbr r 9 a`tbr`dm rdoaus he cdpammdry tulb j 9 jbakjt ly wjacj maquao chmui` rasbs up tjb cdpammdry tulb k 9 krdvatdtah`dm dccbmbrdtah`
τ 9 ob`saty he maquao
HLSB\UD_AH@S Dt rhhi tbipbrdturb7 Ob`saty he wdtbr, τ 9 4>>> Fk/i
3
\hhi _bipbrdturb 9 <2 ℃ Ubr`abr ch`std`t he trdvbmma`k iacrhschpb dmh`k vbrtacdm dxas 9 >.>>4 ci. dmh`k jhrazh`tdm dxas 9 >.>>4 ci. _dlmb 47 Jbakjt he wdtbr chmui` S. @h.
4. <.
\bdoa`k he ib`ascus (x) IS\ US\ _htdm (ci) 2.1 4? 2.14? 2.1 31 2.131
\bdoa`k he `bbomb tap (y) IS\
US\
3.5 3.5
3> :>
Ibd` vdmub he j 9 <.<51 ci. 9 >.><<51 i.
_htdm (ci) 3.53> 3.5:>
jbakjt j9 x – x – y y (ci)
<.<:> <.<21
_dlmb <7 \doaus he Cdpammdry _ulb S. @h. 4. <.
Jhrazh`tdm Jhrazh`tdm oadibtbr Ubrtacdm oadibtbr Mbet \akjt Oadibtbr Rppbr Mhwbr Oadibtbr b`o b`o O4 (ci) b`o b`o O< (ci) 4.5?1 4.;;2 >.4<5 2.?1 2.35: >.43> 4.53< 4.;5< >.4<1 2.2>3 2.3;< >.434
ibd` oadibtbr O 9 (O4+O<)/<
>.4<: >.43>
Ibd` rdoaus r 9 >.>5? ci.
CDMCRMD_AH@ Suredcb _b`sah`,
_
rj k
<
>.>>>5 >>>5? ? >.><<5 ><<51 1 4>>> 4>>> 1.: 1.: < >.>;4425 @/i
\BSRM_ Suredcb tb`sah` he wdtbr dt <2 ℃ 9 >.>;4425 @/i.
ibd` rdoaus r 9 O/< >.>5? >.>52
BZ^B\AIB@_
43
DAI _h obtbria`b tjb chbeeacab`t he vaschsaty he d kavb` vaschus maquao ly ibdsura`k tjb tbria`dm vbmhcaty he kavb` spjbracdm lhoy.
EH\IRMD RSBO
<
k
1v
wjbrb, r 9 rdoaus he spjbracdm ldmm, τ 9 ob`saty he spjbracdm lhoy σ 9 tjb ob`saty he tjb maquao v 9 tjb tbria`dm vbmhcaty he tjb spjbracdm lhoy k 9 dccbmbrdtah` oub th krdvaty
9 chbeeacab`t he vaschsaty
HLSB\UD_AH@S _bipbrdturb he kmycbra` 9 <: ℃ Ob`saty he tjb idtbradm he tjb ldmm, τ 9 ;.;4 k/ci Ob`saty he tjb emuao, σ 9 4.<5 k/ci
<
<
Mbdst chu`t he tjb scrbw kdukb, M.C. 9 >.>>4 ci. Dccbmbrdtah` oub th krdvaty, k 9 1:> ci/s
<
Oastd`cb, m 9 :> ci. _dlmb 47 Oadibtbr he tjb Spjbracdm Ldmm S. @h.
4. <. 3.
Oadibtbr he tjb ldmm ^atcj Scdmb (I) ; ; ;
Carcumdr Scdmb (`) ?2 ?4 ?1
Ibd` rdoaus 9 >.3;< ci.
O 9 I + ` Û M.C. (ci)
\doaus \ 9 O/< (ci)
>.;?2 >.;?4 >.;?1
>.3;< >.3;> >.3;?
_dlmb <7 _bria`dm Ubmhcaty S. @h.
_aib _dfb` th chvbr oastd`cb m (m 9 :> ci)
Ubmhcaty v9 m/t -4
4. <. 3.
(ci s ) <<.? <<.? <<.5
3.5 3.5 3.2
-4
Dvbrdkb Ubmhcaty, v 9 <<.?; ci s .
CDMCRMD_AH@
Chbeeacab`t he vaschsaty he kmycbra` ‚ λ‑ as cdmcumdtbo ds
<
1v
k phasb.
< >.3;< >.3;< (;.;4 4.<5) 1:>
9 :.52 phasb.
1 <<.?;
\BSRM_ Dt rhhi tbipbrdturb <: , tjb vdmub he chbeeacab`t he vaschsaty he kmycbra` as :.52 phasb.
BZ^B\AIB@_
4?
DAI _h stuoy tjb rbmdtah`sjap lbtwbb` rhhi's tbipbrdturb he d jht lhoy d`o taib ly pmhtta`k d chhma`k curvb.
HLSB\UD_AH@S \hhi tbipbrdturb 9 3>℃. _jb a`atadm tbipbrdturb he wdtbr b`cmhsbo, _ 4 9 3> ℃. _jb ea`dm tbipbrdturb he wdtbr b`cmhsbo, _ < 9 3>℃. Ibd` vdmub he _bipbrdturb he wdtbr, _ h 9 3>℃.
S. @h.
_aib (t) (ia`)
_( )
_ – _ _h ( )
mhk (_ – (_ – _ _h)
4. <. 3. ?. 2. 5. ;. :. 1.
> 4 < 3 ? 2 5 ; :
;> 5: 55 5? 5< 54 5> 21 2:
?> 3: 35 3? 3< 34 3> <1 <:
4.5> 4.2: 4.25 4.23 4.24 4.?1 4.?: 4.?5 4.?2
4>. 44. 4<. 43. 4?. 42. 45. 4;. 4:. 41. <>. <4.
1 4> 4< 4? 45 4: <> <2 3> 32 ?> ?2
25 2? 23 24 ?1 ?5 ?? ?< 3: 35 32 3?
<5
4.?< 4.3: 4.35 4.3< 4.<1 4.<> 4.42 4.>: >.1> >.;: >.;> >.5>
\BSRM_ Krdpj lbtwbb` taib (t) d`o tbipbrdturb (_) as d` bxph`b`tadm curvb. Krdpj lbtwbb` mhk (_ – (_ – _ _h) d`o taib (t) as d strdakjt ma`b.
BZ^B\AIB@_
42
DAI _h obtbria`b spbcaeac jbdt he d kavb` shmao ly ibtjho he iaxturbs.
EH\IRMD RSBO s
i4 s4
( i<
( i3
i4
i<
) s< (t t4 )
)(t<
t
)
[jbrb, i4 9 idss he cdmhraibtbr i< 9 idss he cdmhraibtbr detbr lba`k eammbo watj wdtbr i3 9 idss he cdmhraibtbr watj wdtbr d`o shmao. t 9 ea`dm tbipbrdturb he tjb cdmhraibtbr. t4 9 a`atadm tbipbrdturb he wdtbr wjb` fbpt a` cdmhraibtbr. cdmhraibtbr. t< 9 tbipbrdturb he tjb shmao wjb` fbpt a` tjb cdmhraibtbr. cdmhraibtbr. s 9 spbcaeac jbdt he tjb shmao (mbdo).
s
<
9 spbcaeac jbdt he tjb wdtbr.
s4 9 spbcaeac jbdt he chppbr idtbradm.
HLSB\UD_AH@S \hhi tbipbrdturb 9 <: ℃. i4 9 ?2.< k. i< 9 44:.2 k. i3 9 ?>3.3 k. t4 9 <:℃. t< 9 1>℃. t 9 3?℃. -4
-4
-4
-4
s4 9 3:> G Fk ℃ 9 >.>14 cdm k ℃ -4
-4
s< 9 4.>> cdm k ℃
CDMCRMD_AH@ Spbcaeac jbdt he shmao (mbdo), s
i4 s4
( i<
( i3
i 4
i<
) s< (t t4 )
)(t<
t
9 >.>3>15:5;? cdm k
-4
) ℃-4
BZ^B\AIB@_
45
DAI _h obtbria`b tjb spbcaeac jbdt he kavb` maquao ly ibtjho he iaxturb.
EH\IRMD RSBO s
s< ( i3
i< )(t<
( i<
t)
i4 )(t
i4 s4 (t
t 4 )
t4 )
[jbrb, i4 9 idss he cdmhraibtbr i< 9 idss he cdmhraibtbr detbr lba`k eammbo watj wdtbr i3 9 idss he cdmhraibtbr watj wdtbr d`o shmao. t 9 ea`dm tbipbrdturb he tjb cdmhraibtbr. t4 9 a`atadm tbipbrdturb he wdtbr wjb` fbpt a` cdmhraibtbr. t< 9 tbipbrdturb he tjb shmao wjb` fbpt a` tjb cdmhraibtbr. cdmhraibtbr.
s< 9 spbcaeac jbdt he tjb shmao (mbdo). s4 9 spbcaeac jbdt he chppbr idtbradm. s 9 spbcaeac jbdt he tjb maquao.
HLSB\UD_AH@S \hhi tbipbrdturb 9 <: ℃. i4 9 ?2.< k. i< 9 44:.2 k. i3 9 ?>3.3 k. t4 9 <:℃. t< 9 1>℃. t 9 3?℃. -4
-4
-4
-4
-4
-4
s4 9 3:> G Fk ℃ 9 >.>14 cdm k ℃-4 s< 9 43> G Fk ℃ 9 >.>34 cdm k ℃-4
CDMCRMD_AH@ s
s< ( i3
i< )(t<
( i<
t)
i4 )(t
-4
i4 s4 (t
t4 )
-4
9 4.>5:>54:?5 cdm k ℃
t 4 )
BZ^B\AIB@_
4;
DAI a) _h stuoy tjb rbmdtah` lbtwbb` erbqub`cy d`o mb`ktj he d kavb` warb u`obr ch`std`t tb`sah` usa`k sh`hibtbr. aa) _h stuoy tjb rbmdtah` lbtwbb` mb`ktj he kavb` warb d`o tb`sah` ehr ch`std`t erbqub`cy usa`k sh`hibtbr.
HLSB\UD_AH@S a. Dt ch`std`t tb`sah` (_) d`o i7 Mb`ktj he warb, M 9 4<> ci. Udmub he mhdo usbo 9 ? Fk wt. _b`sah` 9 31.< @.
_dlmb A S. @h. 4. <. 3. ?. 2. 5.
Erbqub`cy he tu`a`k ehrf ι (Jz) <25 <:: 3<> 3:? ?:> 24<
\bsh`d`t mb`ktj (m) m m (ci) m (ci) m 4
<
2>.4 ??.5 ?>.< 33.2 <5.1 <2.<
4
?1.1 ??.? ?> 33.3 <5.2 <2
m <
<
2> ??.2 ?>.4 33.3 <5.; <2.4
4/m -4 (ci ) >.><> >.><< >.><2 >.>3> >.>3; >.>?>
ι Û m (Jz ci) 4<:>> 4<:45 4<:3< 4<;:;.< 4<:45 4<:24.<
aa. Dt ch`std`t ι d`o vdrya`k _7 Erbqub`cy he tjb tu`a`k ehrf, ι 9 <25 Jz.
_dlmb AA S. @h.
4. <. 3. ?. 2. 5. ;.
Ju`k mhdo (Fk) ?.> 3.2 3.> <.2 <.> 4.2 4.>
_ 9 ik (@)
31.< 3?.3 <1.?
\bsh`d`t Mb`ktj m (ci) a`crbdsa`k obcrbdsa`k m
m
2>.4 ?5.1 ?3.? 31.5 32.5 3>.: <2.4
?1.1 ?5.; ?3.< 31.? 32.< 3>.?
m
m4 m < <
2> ?5.: ?3.3 31.2 32.? 3>.5 <2
<
<
m
<
_/ m
<
(ci )
(@/ci )
<2>> <41>.< 4:;?.1 425>.3 4<23.< 135.? 5<2
>.>425: >.>4255 >.>425: >.>42;> >.>425? >.>42;> >.>425:
\BSRM_ d. Erbqub`cy he valrdtah` as a`vbrsbmy prhphrtah`dm th ats mb`ktj ehr d kavb` stra`k sulgbctbo th ch`std`t tb`sah`. l.Mb`ktj he stra`k as oarbctmy oarbctmy prhphrtah`dm prhphrtah`dm th squdrb rhht he tb`sah` a` at ehr d kavb` erbqub`cy he valrdtah` he stra`k.
BZ^B\AIB@_
4:
DAI _h ea`o spbbo he shu`o a` dar dt rhhi tbipbrdturb usa`k d rbsh`d`cb tulb ly twh rbsh`d`cb phsatah`s.
HLSB\UD_AH@S \hhi _bipbrdturb 9 <: ℃.
Erbqub`cy he _u`a`k ehrf (Jz)
3?4 ?:> 24<
[dtbr mbvbm (ci) S. @h.
4. m4. m<. <. m4. m<. 3. m4. m<.
_radm 4
_radm <
Ibd`
<2 ;3 4;.; 23.4 45.5 2>
<3 ;2 4;.5 2<.: 45.; 2>.4
.>2
ibd` spbbo 9 3?>.42< i/s
Spbbo U 9 <ι(m< - m<) i/s 3?4 33;.?? 3?<.>45
^D\_ AA CMDSS ZAA ^JQSACS
BZ^B\AIB@_
4
DAI
_h obtbria`b rbsastd`cb pbr ci he d kavb` warb ly pmhtta`k d krdpj he phtb`tadm oaeebrb`cb oaeebrb`cb vbrsus currb`t. currb`t.
HLSB\UD_AH@S
Mb`ktj he tjb rbsastd`cb warb, m 9 9 <2 ci. \d`kb he tjb kavb` diibtbr 9 3> D \d`kb he tjb kavb` vhmtibtbr 9 4> U
_dlmb7 Uhmtibtbr d`o Diibtbr \bdoa`k S. @h.
Diibtbr \bdoa`k A (D)
Uhmtibtbr \bdoa`k U (U)
U/A 9 \ (hji)
4. <. 3. ?. 2.
3.: ;.5 44.2 42.3 41.4
< ? 5 : 4>
>.2<5 >.2<5 >.2<< >.2<3 >.2
Ibd` Udmub he \bsastd`cb, \ 9 >.2
CDMCRMD_AH@
Krdpj lbtwbb` U d`o A as pmhttbo wjacj chibs th lb d strdakjt ma`b.
Erhi tjb krdpj rbsastd`cb 9 cht κ (κ (κ as tjb d`kmb tjb curvb idfbs watj x dxas.)
\BSRM_ -4
4. \bsastd`cb pbr ci he tjb warb 9 >.><4Ρ >.><4 Ρ ci
<
BZ^B\AIB@_ DAI
_h ea`o rbsastd`cb he d kavb` warb usa`k ibtbr lraokb d`o jb`cb obtbria`b tjb rbsastavaty he ats idtbradm.
HLSB\UD_AH@S
Mb`ktj he tjb rbsastd`cb warb, m 9 9 3> ci.
_dlmb 47 R`f`hw` \bsastd`cb S. @h.
4. <. 3.
\bsastd`cb erhi rbsastd`cb lhx 4 < 3
Mb`ktj Mb`ktj m (4>> – m )
R`f`hw` \bsastd`cb Z
35.< 23.4 5<.1
53.: ?5.1 3;.4
Ibd` \bsastd`cb 9 4.;5543 hji.
\
(4>> m ) m
4.;5
_dlmb <7 Oadibtbr he tjb warb S. @h.
4.
Oadibtbr a` h`b oarbctah`
Oadibtbr a` pbrpb`oacumdr oarbctah` ^atcj Carcumdr o4 9 I + ^atcj Carcumdr o4 9 I + Scdmb Scdmb `4 Û M.C. Scdmb Scdmb `< Û M.C. (I) (`4) (I) (`<) (ii) (ii) > 5 >.>5 > 5 >.>5
Hlsbrvbo Oadibtbr
o4 + o< Oa 9
<
>.>5
<.
>
5
>.>5
>
5
>.>5
>.>5
3.
>
5
>.>5
>
5
>.>5
>.>5
Ibd` Oadibtbr 9 >.>>5 ci 9 >.>>>>5 i.
CDMCRMD_AH@
Spbcaeac \bsastd`cb,
Z .
O
<
? M
-:
9 4.55?2 Û 4> hji-i. \BSRM_
4. Udmub he u`f`hw` rbsastd`cb, Z 9 4.;5543 hji <. _jb rbsastavaty he tjb idtbradm he tjb warb, τ 9 -: 4.55?2 Û 4> hji-i.
BZ^B\AIB@_
3
DAI
_h vbraey mdws he chila`dtah` he rbsastd`cb (Sbrabs & ^drdmmbm) usa`k ibtbr lraokb.
HLSB\UD_AH@S
\bsastd`cb S. Cham @h.
Mb`ktj
4. <. 3.
\bsastd`cb erhi rbsastd`cb lhx (\) (hji) < 3 2
?2.? 22.5 5;.2
2?.5 ??.? 3<.2
<.?4 <.?> <.?4
\ 4 9 <.?4
4. <. 3.
< 3 2
3:.2 ?:.? 54.<
54.2 24.5 3:.:
3.41 3.<> 3.4;
\ < 9 3.41
\ 4 & \ < a` sbrabs
4. <. 3.
2 4> 4<
?;.< 5?.4 5:.<
2<.: 32.1 34.:
2.21 2.5 2.5
\ S 9 2.5
\ 4 & \ < a` pdrdmmbm
4. <. 3.
4 < 3
?< 21.< 5:.<
2: ?>.: 34.2
4.3: 4.3: 4.3:
\ ^ 9 4.3:
\ 4
\ <
m
Mb`ktj (4>> – m )
\bsastd`cb Z
\
(4>> m )
Ibd`
m
CDMCRMD_AH@
4. Bxpbraib`tdm vdmub he \ S 9 2.5 hji. _jbhrbtacdm vdmub he \ S 9 \ 4 + \ < 9 2.5 hji. <.
Bxpbraib`tdm vdmub he \ ^ 9 4.3: hji.
_jbhrbtacdm vdmub he \ ^ 9
\ 4 \ < \ 4 +\ <
9 4.3; hji
\BSRM_
Bxpbraib`tdm vdmub d`o tjbhrbtacdm vdmub he \ S & \ ^
drb dmihst bqudm (watja` maiats he bxpbraib`tdm brrhr)
BZ^B\AIB@_
?
DAI
_h chipdrb b.i.e. he twh kavb` praidry cbmms usa`k phtb`tahibtbr.
HLSB\UD_AH@S
\d`kb he tjb kavb` diibtbr 9 42 D \d`kb he tjb kavb` vhmtibtbr 9 4> U Mbdst chu`t he tjb kavb` diibtbr 9 >.2 D Mbdst chu`t he tjb kavb` vhmtibtbr 9 >.< U b.i.e. he ldttbry bmaia`dthr 9 >-4> U b.i.e he Mbcmd`cjb cbmm, B 4 9 4.?2 U b.i.e he Od`abm cbmm, B < 9 4.>: U
_dlmb7 Mb`ktjs S. Diibtbr @h. \bdoa`k (D)
Ldmd`cb pha`t wjb` Mbcmd`cjb cbmm as a` tjb carcuat (m 4) 4 < Ibd`
Ldmd`cb pha`t wjb` Od`abm cbmm as a` tjb B 4 carcuat (m <) 4 < Ibd` B<
m 4 4. <. 3. ?.
>.<2 >.<> >.42 >.4>
Ibd`
B 4 B <
4;1 <<< <;< 3?>
4;: <<> <;< 3?>
4;:.2 <<4 <;< 3?>
4<5 42? 412 <54
\BSRM_
_jb rdtah he b.i.e s.,
B <
m <
m <
4.?>;2
B 4
m 4
4.?>;2
4<5 42; 41? <54
4<5 422.2 41?.2 <54
4.?4 4.?< 4.31 4.?4
BZ^B\AIB@_
?
DAI
_h obtbria`b tjb a`tbr`dm rbsastd`cb he kavb` praidry cbmm usa`k d phtb`tahibtbr. phtb`tahibtbr.
HLSB\UD_AH@S
\d`kb he tjb kavb` vhmtibtbr 9 >-3 U Mbdst chu`t he tjb kavb` vhmtibtbr 9 >.>2 U B.I.E. he tjb ldttbry 9 ? U B.I.E. he tjb cbmm 9 4.2 U
_dlmb7 Mb`ktjs ^hsatah` he tjb `umm pha`t (ci) S. Diibtbr @h. \bdoa`k
4. <. 3. ?.
>.>2 >.42 >.<> >.<2
[atjhut Sju`t
Sju`t A`tbr`dm \bsastd`cb \bsastd`cb
[atj Sju`t
4
<
Ibd`
4
<
Ibd`
3:? <;< <<> 4;1
3:3 <;< <
3:3.2 <;< <<< 4;:
4;> 42; 4?5 4<3
4;4 42: 4?: 4<;
4;>.2 42;.2 4?; 4<2
\ (hji)
m4 m < r .\ m <
< ? 5 :
<.2 <.1 3.4 3.?
\BSRM_
_jb a`tbr`dm rbsastd`cb he tjb kavb` cbmm as <.2 - 3.? hji.
BZ^B\AIB@_
2
DAI
_h obtbria`b rbsastd`cb he d kdmvd`hibtbr ly jdme obembctah` ibtjho d`o th ea`o ats eakurb he ibrat.
HLSB\UD_AH@S
_dlmb 47 \bsastd`cb he kdmvd`hibtbr S. @h.
4. <. 3.
\bsastd`cb \ (hji)
3<>> 2>2> :<>>
Obembctah` a` Kdmvd`hibtbr
<: 4: 4<
Ibd` rbsastd`cb 9 ?;.:>33
Sju`t rbsastd`cb S (hji)
Jdme Obembctah`
?; ?2 2>
4? 1 5
<
Kdmvd`hibtbr \bsastd`cb \.S K
\ S
?;.;> ?2.?> 2>.34
_dlmb <7 Eakurb he Ibrat S. @h.
b.i.e. he tjb ldttbry
\bsastd`cb erhi rbsastd`cb lhx (\) (hji)
4. <. 3.
<.2 2 ;.2
2>>> 42>>> 4<>>>
Obembctah` (κ) (oav.)
<> 4? <2
Eakurb he Ibrat f
B
( \ K )
>.>>>>.>>>><3;3 >.>>>>
Ibd` vdmub he f 9 >.>>>>
\BSRM_
4. \bsastd`cb he tjb kdmvd`hibtbr as ?;.:>33 <. Eakurb he ibrat he tjb kavb` kdmvd`hibtbr as >.>>>>
BZ^B\AIB@_
5
DAI
_h ch`vbrt tjb kavb` kdmvd`hibtbr a`th 4. vhmtibtbr he obsarbo rd`kb <. diibtbr he obsarbo rd`kb d`o vbraey tjb sdib.
HLSB\UD_AH@S
\bsastd`cb he tjb kavb` kdmvd`hibtbr, K 9 ?;.:> hji. Eakurb he ibrat,
f 9 >.>>>>
@uilbr he oavasah` a` tjb kdmvd`hibtbr kdmvd`hibtbr scdmb, ` 9 3> Currb`t ehr eumm obembctah`, A k 9 `.f `.f 9 9 >.>>>;33: D \d`kb he ch`vbrsah` (th vhmtibtbr),
U93U
\d`kb he ch`vbrsah` (th diibtbr),
A93D
U\AEACD_AH@
_dlmb 47 Ehr Uhmtibtbr S. @h.
\bdoa`k he ch`vbrtbo kdmvd`hibtbr a`th vhmtibtbr U
Obembctah` ( ) 4. <. 3.
42 <> <2
U4 9
`
4.2 <.> <.2
Std`odro vhmtibtbr rbdoa`k U<
Oaeebrb`cb U< - U4
4.2 <.> <.2
> > >
_dlmb 47 Ehr Uhmtibtbr S. @h.
\bdoa`k he ch`vbrtbo kdmvd`hibtbr a`th diibtbr Obembctah` ( )
4. <. 3.
2 4> 42
A4 9 >.2 4.> 4.2
A `
Std`odro diibtbr rbdoa`k A<
Oaeebrb`cb A< - A4
>.2 4.> 4.2
> > >
\BSRM_
Ds tjb oaeebrb`cb a` dctudm vdmub d`o ibdsurbo vdmub he currb`t d`o vhmtdkb drb vbry sidmm, tjb ch`vbrsah` as dccurdtb.
BZ^B\AIB@_
;
DAI
_h ea`o tjb erbqub`cy he tjb DC ida`s watj d sh`hibtbr.
HLSB\UD_AH@S
Mb`ktj he tjb sh`hibtbr warb M 9 43> ci. Idss he tjb warb, i 9 <.2 k Ma`bdr ob`saty, I 9 i/M 9 (<.2/43>) k/ci 9 >.>>41 Fk/i
S. @h.
4. <. 3.
Mhdo h` tjb warb I 4.2 <.> <.2
_b`sah` \bsh`d`t mb`ktj he tjb prhoucbo warb (ci) h` tjb a`crbdsa`k Obcrbdsa`k warb m m 4 < 4?.; 41.5
31.2 24.< 23.<
31.; 24.< 23.>
ibd` mb`ktj m 9 (m 4 + 4 m )/<>> < <
`
4
_
< M
I
(i)
>.315 >.24< >.234
ibd` currb`t erbqub`cy, `‑ 9 `‑ 9 2<.2? Jz
>.315 >.24< >.234
Currb`t erbqub`cy `‑9`/<
44>.31 1:.2: 4>5.<:
\BSRM_
_jb erbqub`cy he D.C 9 2<.2? Jz
BZ^B\AIB@_
:
DAI
_h ea`o tjb vdmub he v ehr oaeebrb`t vdmubs he u a` ehr a` cdsb he d ch`cdvb iarrhr d`o th ea`o tjb ehcdm mb`ktj.
EH\IRMDB RSBO
4
e
[b jdvb, e
4
v
4
u
u.v u
v
HLSB\UD_AH@S
\hukj ehcdm mb`ktj he tjb kavb` ch`cdvb iarrhr 9 45 ci. Dctudm mb`ktj he tjb f`atta`k `bbomb, x 9 x 9 <2 ci. _jb hlsbrvbo oastd`cb lbtwbb` tjb iarrhr d`o hlgbct `bbomb wjb` tjb f`atta`k `bbomb as pmdcbo lbtwbb` tjbi, y 9 y 9 <2 ci. _jb hlsbrvbo oastd`cb lbtwbb` tjb iarrhr d`o aidkb `bbomb wjb` tjb f`atta`k `bbomb as pmdcbo lbtwbb` tjbi, z 9 9 <2 ci.
A`obx chrrbctah` ehr hlgbct oastd`cb u , x – x – y 9 y 9 > ci. A`obx chrrbctah` ehr aidkb oastd`cb v , x – x – z z 9 9 > ci. 4
4
u
v
_dlmb7 u, v d`o Hlsbrvbo S. Oastd`cb @h. Ch`cdvb Hlgbct Aidkb ^H 9 u ^A 9 v Iarrhr `bbomb `bbomb (ci) (ci) ^ H A
,
^hsatah` he (ci)
4. <. 3. ?. 2.
4> 4> 4> 4> 4>
3>.2 3<.2 32.5 3;.< 31.4
55.1 23.: ?5.< ?3.? ?>.1
<>.2 <<.: <2.5 <;.< <1.4
22.1 ?3.: 35.< 33.? 3>.1
4
4
u
v
>.>?:: >.>?31 >.>314 >.>35: >.>3??
>.>4;1 >.><<: >.><;5 >.><11 >.>3
Ehcdm Mb`ktj
e
u.v uv
42.<1 4?.11 42 4?.11 4?.11
CDMCRMD_AH@
Cdmcumdtah` he ehcdm mb`ktj erhi krdpj a) u-v krdpj7 D krdpj as pmhttbo lbtwbb` u d`o v ( ( u dmh`k x dmh`k x -dxas d`o v dmh`k y dmh`k y -dxas -dxas d`o sdib scdmb as ‑
‑
sbmbctbo ehr lhtj dxas). _jb krdpj chibs hut th lb d rbctd`kumdr jypbrlhmd. @hw, d ma`b as ordw` erhi hraka` idfa`k d` d`kmb he ?2 watj batjbr he tjb dxas ibbta`k tjb curb dt d pha`t -dxas d`o y D. Ordw pbrpb`oacumdr DI d`o D@ h` x -dxas ‑ ‑
dxas.
‑ ‑
_jb vdmubs he u d`o d`o v wamm lb sdib ehr tjb pha`t D. Ds wamm ehr d ch`cdvb iarrhr, u d`o d`o v drb bqudm h`my wjb` tjb drb hlgbct as pmdcbo dt tjb cb`tbr he curvdturb a.b., tjb chhroa`dtbs he D iust lb (< e , < e ). ).
Jb`cb,
aa)
4
e 9 9
d`o
u
4
HI
hr
<
H@ <
9
krdpj 7
v
D krdpj as pmhttbo lbtwbb` dxas d`o
4 v
4 u
d`o
4 v
(
4 u
dmh`k x dmh`k x ‑
dmh`k y dmh`k y -dxas -dxas d`o sdib scdmb as sbmbctbo ‑
ehr lhtj dxas). _jb krdpj chibs hut th lb d strdakjt ma`b. _jb strdakjt ma`b cuts lhtj tjb dxas dt d` d`kmb he ?2 ¾ dt pha`ts ^ d`o T rbspbctavbmy d`o idfbs bqudm a`tbrcbpts he tjb dxas. Ehcdm mb`ktj, e mb`ktj, e 9
4
H^
4
HT
9
\BSRM_
_jb ehcdm mb`ktj he tjb kavb` ch`cdvb iarrhr u.v a. ehcdm mb`ktj cdmcumdtbo erhi e 9 42.>2< uv ci. aa. erhi u-v krdpj aaa. erhi
4 u
-
4 v
krdpj
e 9 42.4? 9 42.4? ci. e 9 42.4< 9 42.4< ci.
BZ^B\AIB@_
1
DAI
_h ea`o tjb ehcdm mb`ktj he d ch`vbx iarrhr, usa`k d ch`vbx mb`s.
HLSB\UD_AH@S
\hukj ehcdm mb`ktj he tjb ch`vbx mb`s 9 42.2 ci. Dctudm mb`ktj he tjb f`atta`k `bbomb, x 9 <2 ci. Hlsbrvbo oastd`cb oastd`cb lbtwbb` aidkb `bbomb A d`o ldcf he tjb ch`vbx iarrhr, y 9 <2 ci.
A`obx chrrbctah` 9 x – y 9 > ci. S. @h.
4. <. 3.
^hsatah` he Ch`vbx iarrhr ^ <5 <> 42
Aidkb `bbomb A 25.< ?1.: ?2.4
\doaus he curvdturb (\) Hlsbrvbo Chrrbctbo ^A ^A (ci) (ci) 3>.< 3>.< <1.: <1.: ?2.4 3>.4
Ehcdm mb`ktj e 9 \/<
42.4 4?.1 42.4
Ibd` vdmub he e he e 9 42 ci.
\BSRM_
_jb ehcdm mb`ktj he tjb kavb` ch`vbx iarrhr 9 42 ci.
BZ^B\AIB@_
4>
DAI
_h ea`o tjb ehcdm mb`ktj he d ch`vbx mb`s ly pmhtta`k krdpjs lbtwbb` u d`o d`o v hr hr lbtwbb` 4/u d`o d`o 4/v.
HLSB\UD_AH@S
\hukj ehcdm mb`ktj he tjb kavb` ch`cdvb iarrhr 9 42.3 ci. Dctudm mb`ktj he tjb f`atta`k `bbomb, x 9 x 9 <2 ci. _jb hlsbrvbo oastd`cb lbtwbb` tjb mb`s d`o hlgbct `bbomb wjb` tjb f`atta`k `bbomb as a s pmdcbo lbtwbb` tjbi, y 9 y 9 <2 ci. _jb hlsbrvbo oastd`cb lbtwbb` tjb mb`s d`o aidkb `bbomb wjb` tjb f`atta`k `bbomb as a s pmdcbo lbtwbb` tjbi, z 9 9 <2 ci.
A`obx chrrbctah` ehr hlgbct oastd`cb u , x – x – y 9 y 9 > ci A`obx chrrbctah` ehr aidkb oastd`cb v , x – x – z z 9 9 > ci.
_dlmb7 u, v, ^hsatah` he (ci) S. @h. Ch`cdvb Hlgbct Aidkb Iarrhr `bbomb `bbomb ^ H A 4. <. 3. ?. 2.
<5.2 <:.2 <2.2 <3.2 <4.2
2> 2> 2> 2> 2>
14.2 11.5 :5.; :?.5 :4.;
Chrrbctbo Oastd`cb u
<3.2 <4.2
v
?4.2 ?1.5 35.; 3?.5 34.;
4 v
d`o
4 u
Ehcdm Mb`ktj
4
4
u
v
>.>?<5 >.>?52 >.>?>: >.>3;; >.>324
>.>.><>< >.><;< >.><:1 >.>342
e
u.v
u
v
42 42 4?.; 42 42
CDMCRMD_AH@
Cdmcumdtah` he ehcdm mb`ktj erhi krdpj a) u-v krdpj7 D krdpj as pmhttbo lbtwbb` u d`o v ( ( u dmh`k x dmh`k x -dxas d`o v dmh`k y dmh`k y -dxas -dxas d`o sdib scdmb as ‑
‑
sbmbctbo ehr lhtj dxas). _jb krdpj chibs hut th lb d rbctd`kumdr jypbrlhmd. @hw, d ma`b as ordw` erhi hraka` idfa`k d` d`kmb he ?2 watj batjbr he tjb dxas ibbta`k tjb curvb dt d pha`t -dxas d`o y D. Ordw pbrpb`oacumdr DI d`o D@ h` x -dxas ‑ ‑
dxas.
‑ ‑
_jb vdmubs he u d`o d`o v wamm lb sdib ehr tjb pha`t D. Ds wamm ehr d ch`cdvb iarrhr, u d`o d`o v drb bqudm h`my wjb` tjb drb hlgbct as pmdcbo dt tjb cb`tbr he curvdturb a.b., tjb chhroa`dtbs he D iust lb (< e , < e ). ).
Jb`cb,
aa)
4 u
e 9 9
d`o
4
HI
hr
H@
<
<
9
krdpj 7
v
4
4
4
D krdpj as pmhttbo lbtwbb` d`o ( dmh`k xdmh`k xu v u 4 dxas d`o dmh`k y dmh`k y-dxas -dxas d`o sdib scdmb as sbmbctbo v ehr lhtj dxas). _jb krdpj chibs hut th lb d strdakjt ma`b. _jb strdakjt ma`b cuts lhtj tjb dxas dt d` d`kmb he ?2 ¾ dt pha`ts ^ d`o T rbspbctavbmy d`o idfbs bqudm a`tbrcbpts he tjb dxas. Ehcdm mb`ktj, e mb`ktj, e 9
4
4
H^
HT
9
\BSRM_
_jb ehcdm mb`ktj he tjb kavb` ch`vbx mb`s av. ehcdm mb`ktj cdmcumdtbo erhi
e
u.v
u v
v. erhi u-v krdpj va. erhi
4 u
-
4 v
e 9 42.>2 9 42.>2 ci.
krdpj
e 942.4> 942.4> ci.
4?.1? ci.
BZ^B\AIB@_
44
DAI
_h ea`o tjb ehcdm mb`ktj he d ch`cdvb mb`s usa`k d ch`vbx mb`s.
HLSB\UD_AH@S
\hukj ehcdm mb`ktj he tjb ch`vbx mb`s 9 42.< ci. Dctudm mb`ktj he tjb f`atta`k `bbomb, x 9 <2 ci. Hlsbrvbo oastd`cb oastd`cb lbtwbb` aidkb `bbomb A d`o tjb ch`cdvb mb`s wjb` f`atta`k `bbomb as pmdcbo lbtwbb` tjbi, y 9 <2 ci.
A`obx chrrbctah` 9 x – y 9 > ci.
_dlmb7 u , v d`o d`o e ^hsatah` he (ci) S. @h.
4. <. 3.
Hlgbct `bbomb
Ch`vbx mb`s
Aidkb `bbomb
Ch`cdvb mb`s
Chrrbctbo Oastd`cb @bw Aidkb `bbomb
H
H4
A
H<
A
<5.: <2.;
?> ?> ?>
;? ;5 ;:
2> 2> 2>
1> :2.? :<.?
u
v
?> 32.? 3<.?
4 u
4 v
Ehcdm Mb`ktj
e
u.v u v
‑
>.>?4; >.>3:2 >.>32;
>.><2> >.><:3 >.>3>1
42 4?.1 42.4
Ibd` vdmub he e he e 9 9 42 ci.
\BSRM_
_jb ehcdm mb`ktj he tjb kavb` ch`cdvb mb`s 9 42 ci.
BZ^B\AIB@_
4<
DAI
_h obtbria`b d`kmb he ia`aiui obvadtah` ehr d kavb` prasi ly pmhtta`k d krdpj lbtwbb` d`kmb he a`caob`cb d`o tjb d`kmb he obvadtah`.
EH\IRMDB RSBO
D Oi sa` < ` D sa` <
[jbrb, ` 9 rberdctavb a`obx he tjb idtbradm he tjb prasi. Oi 9 d`kmb he ia`aiui obvadtah`. D 9 d`kmb he prasi.
HLSB\UD_AH@S
D`kmb he prasi ‚D‑ 9 3>¾
S. @h.
D`kmb he a`caob`cb
4. <. 3. ?. 2. 5.
a
D`kmb he obvadtah`
32 ?> ?2 2> 22 5>
O
?3.< 31.2 3:.< 3:.3 31.? ?<
CDMCRMD_AH@
D krdpj as pmhttbo lbtwbb` tjb d`kmb he a`caob`t a d`o d`kmb he obvadtah` O ( a dmh`k x -dxas -dxas d`o O dmh`k y -dxas -dxas). _jb mhwbst pha`t he tjb curvb kavbs tjb d`kmb he ia`aiui obvadtah`. \BSRM_
D`kmb he ia`aiui obvadtah`, O i ≊ 3:¾ \berdctavb a`obx he tjb idtbradm he tjb prasi, ` 9 4.:2
BZ^B\AIB@_
43
DAI
_h obtbria`b tjb rberdctavb a`obx he d kmdss smdl usa`k trdvbmma`k iacrhschpb.
EH\IRMDB RSBO
`
rbdm tjacf`bss he smdl
dppdrb`t tjacf`bss he smdl
[jbrb, ` 9 rberdctavb a`obx he tjb idtbradm he tjb smdl
HLSB\UD_AH@S
\bdoa`k h` vbrtacdm scdmb he tjb trdvbmma`k iacrhschpb S. @h.
4. <. 3.
Crhssidrf watjhut smdl \ 4 ?.31; ?.31; ?.31;
Crhss idrf watj smdl \ < ?.114 ?.:24 ?.:11
\bdm Dppdrb`t Mychphoaui _jacf`bss _jacf`bss (\ 3 – \ \ 4) (\ 3 – \ \ <) ^hwobr
\berdctavb A`obx `
\ 3 5.?22 2.:1: 2.24<
<.>2: 4.2>4 4.442
4.?:? 4.>?; >.:43
Ibd` vdmub he ` 9 4.?>3
\BSRM_
_jb rberdctavb a`obx he tjb kmdss smdl 9 4.?>3
\3
\4
\3
\<
4.?>2 4.?3? 4.3;4
BZ^B\AIB@_
4?
DAI
_h ea`o tjb rberdctavb a`obx he d maquao ly usa`k ch`vbx mb`s d`o d pmd`b iarrhr.
EH\IRMDB RSBO
4
e <
4
E
4
e 4
e 4 9 ehcdm mb`ktj he tjb ch`vbx mb`s. e < 9 ehcdm mb`ktj he tjb maquao mb`s. E 9 ehcdm mb`ktj he tjbar chila`dtah`.
` 4
\ e <
[jbrb, ` 9 rberdctavb a`obx he tjb maquao. \9 rdoaus he curvdturb he tjb ch`vbx mb`s.
e 9 ehcdm mb`ktj he tjb ch`vbx mb`s. HLSB\UD_AH@S
\hukj ehcdm mb`ktj he tjb ch`vbx mb`s 9 42.4 ci.
Oastd`cb he tjb `bbomb tap Ehcdm Mb`ktj
Erhi mb`s suredcb x 4 (ci)
Erhi pmd`b iarrhr x < (ci)
[atjhut maquao
4?.:
42.<
42
e 4 9 42
watj maquao
44.1
4<.3
4<.<
E 9 4<.<
CDMCRMD_AH@ 4
e <
4
E
4
e 4
e < 9 52.32; ci
&
` 4
\ e <
9 4.?2
Ibd` x
x x 4
<
<
x
(ci)
BZ^B\AIB@_
42
DAI
_h ordw tjb A-U cjdrdctbrastac curvb he d p-` gu`ctah` a` ehrwdro lads d`o rbvbrsb lads.
HLSB\UD_AH@S
Ehr ehrwdro-lads
\d`kb he vhmtibtbr
92U
Mbdst chu`t he vhmtibtbr Wbrh brrhr he vhmtibtbr \d`kb he iamma-diibtbr
9 >.4 U 9>U 9 3> iD
Mbdst chu`t he iamma-ditibtbr iamma-ditibtbr Wbrh brrhr he iamma-diibtbr
9 >.2 iD 9 > iD
_dlmb7 Ehrwdro lads vhmtdkb d`o ehrwdro currb`t Sbradm @h.
Ehrwdro-lads vhmtdkb UE (U)
Ehrwdro currb`t AE (iD)
4. <. 3. ?. 2. 5. ;. :. 1. 4>. 44. 4<. 43. 4?. 42.
> >.4 >.< >.3 >.? >.5 >.: 4.> 4.< 4.? 4.5 4.: <.> <.< <.?
> > > > >.2 4 < 3 2 ;.2 4> 42 <> <2 3>
Ehr rbvbrsb-lads
\d`kb he vhmtibtbr
9 3> U
Mbdst chu`t he vhmtibtbr Wbrh brrhr he vhmtibtbr \d`kb he iacrh-diibtbr iacrh-diibtbr
9 >.2 U 9>U 9 2> νD
Mbdst chu`t he iacrh -diibtbr Wbrh brrhr he iacrh -diibtbr
9 4 νD 9 > νD
_dlmb7 Ehrwdro lads vhmtdkb d`o ehrwdro currb`t Sbradm @h.
\bvbrsb-lads vhmtdkb U\ (U)
\bvbrsb currb`t A\ (νD)
4. <. 3. ?. 2. 5. ;. :. 1. 4>. 44. 4<.
> 2.> ;.> 1.> 44.> 43.> 42.> 4;.> 41.> <4.> <3.> <2.>
> 4 < 3 ? 2 ; 1 44 43 42 <2
CDMCRMD_AH@
, d krdpj as pmhttbo lbtwbb` E hr e hr wdr wdr o-la -la ds ehrwdro-lads vhmtdkb U E(dmh`k x (dmh`k x-dxas) -dxas) d`o ehrwdro currb`t AE(dmh`k y (dmh`k y-dxas). -dxas).
Erhi krdpj, ehr cjd`kb erhi pha`t ^ th T
<.? – <.> <.> 9 >.? U ∅UE 9 <.? – 3> – <> <> 9 4> iD. ∅AE 9 3> – ∿ gu`ctah` rbsastd`cb, r 9
ΟUE
9
ΟA E
.? U
9 ?> hjis
4> iD
, d krdpj as pmhttbo lbtwbb` rbvbrsbE hr r bvbr vbr sb-lads -la ds lads vhmtdkb U\ (dmh`k x (dmh`k x-dxas) -dxas) d`o rbvbrsb currb`t A\ (dmh`k y (dmh`k y-dxas). -dxas).
Erhi krdpj, ehr tjb cjd`kb erhi pha`t ^ th T ; – 2 2 9 < U ∅U\ 9 ; – < – 4 4 9 4> νD. ∅A\ 9 < – ∿ gu`ctah` rbsastd`cb, r 9
ΟU\ ΟA \
9
<.> U
5
9 <Û4> hjis
4 νD
\BSRM_
Gu`ctah` rbsastd`cb ehr ehrwdro-lads 9 ?> hjis. 5
Gu`ctah` rbsastd`cb ehr rbvbrsb-lads 9 < Û 4> hjis.
BZ^B\AIB@_
45
DAI
_h ordw tjb cjdrdctbrastac curvb he d Wb`br oahob d`o th obtbria`b ats rbvbrsb lrbdfohw` vhmtdkb.
EH\IRMDB RSBO
U> 9 UA - \ A.AA [jbrb, U> 9 hutput vhmtdkb (ats ch`std`t vdmub kavbs tjb rbvbrsb lrbdfohw` vhmtdkb) UA 9 a`put vhmtdkb \ A 9 a`put rbsastd`cb AA 9 a`put currb`t
HLSB\UD_AH@S
Mbdst chu`t he vhmtibtbr (U 4)
9 >.2 U
Wbrh brrhr he vhmtibtbr (U 4)
9>U
Mbdst chu`t he vhmtibtbr (U <)
9 >.< U
Wbrh brrhr he vhmtibtbr (U <)
9>U
Mbdst chu`t he iacrh-diibtbr
9 >.2 νD
Wbrh brrhr he iacrh -diibtbr
9 > νD
_dlmb7 UA, AA d`o U> Sbr a dm @ h.
A `put Uhm tdkb tdkb
U A A ( U ) 4. <. 3. ?. 2. 5. ;. :. 1. 4>. 44. 4<. 43. 4?. 42. 45.
4 < 3 ? 2 5 ; : 1 4> 4>.2 44.> 44.2 4<.> 4<.2 43.>
A `put Curr b`t
A A (iD) 4 4.2 <.> 3.> 3.2 ?.> 2.> 2.2 5.> 5.2 ;.> ;.2 :.> :.2 1 1.2
Hutput Hu tput Uhm tdkb tdkb
U > > ( U ) >.: 4.< 4.: <.? 3.> 3.? ?.> ?.: 2.? 5.> 5.< 5.? 5.? 5.? 5.? 5.?
CDMCRMD_AH@
D krdpj as pmhttbo lbtwbb` hutput vhmtdkb (dmh`k x‑ - - ) d`o a`put currb`t (dmh`k y‑ -dxas ) dxas - dxas
\BSRM_
_jb rbvbrsb lrbdfohw` vhmtdkb he kavb` Wb`br oahob as 5.? U.
BZ^B\AIB@_
4;
DAI
_h stuoy tjb cjdrdctbrastacs he d chiih` biattbr `p` trd`sasthr d`o th ea`o hut tjb vdmubs he currb`t d`o vhmtdkb kda`s.
HLSB\UD_AH@S
\d`kb he vhmtibtbr, Mbdst chu`t he vhmtibtbr (U 4), \d`kb he vhmtibtbr, Mbdst chu`t he vhmtibtbr (U <) \d`kb he tjb iamma-diibtbr, iamma-diibtbr, Mbdst chu`t he tjb iamma-diibtbr, iamma-diibtbr, \d`kb he iacrh-diibtbr, Mbdst chu`t he iacrh-diibtbr,
U4 9 >-4>>> iU 9 <> iU U< 9 >-4> U 9 >.< U 9 >-<2 iD 9 >.2 iD 9 >-<2> νD 9 2 νD
_dlmb 47 ldsb vhmtdkb d`o ldsb currb`t S. @h. 4. <. 3. ?. 2. 5. ;. :. 1. 4>. 44. 4<.
L dsb dsb vhmtdkb vhmt dkb U l (U)
Ldsb Ldsb cur r b`t (A ) l e hr chm m bcthr vhm tdkbs U c c 9 > U
U c c 9 4> U
U c c 9 <> U
U c c 9 3> U
> 4>> <>> 3>> ?>> 2>> 2<> 2?> 2:> 5>> 5?> ;>>
> > > > > 42 <> 3> ;2 12 4;2 hut he rd`kb
> > > > > > > > 2 4> 3> 4>>
> > > > > > > > 2 4> 32 4<2
> > > > > > > 4> 42 <> 22 hut he rd`kb
_dlmb <7 chmmbcthr vhmtdkb d`o chmmbcthr currb`t S. @h.
4. <. 3. ?. 2. 5. ;. :. 1. 4>.
Chmmbcthr vhmtdkb U c (U) > 4 < 3 ? 2 5 ; : 1
Chmm bcthr cthr cur r b`t (A ) ldsb cur cu r r b` t c e hr ldsb A l 9 <2 D
A l 9 2> D
A l 9 ;2 D
A l 9 4>> D
> ? ? ? ? ? ? ? ? ?
> 1 1 1 1 1.2 1.2 1.2 1.2 1.2
> 4?.2 4?.2 42 42 42 42 42 42.2 42.2
> <>.2 <4 << <4 <4 <4.2 <4.2 <4.2 <4.2
CDMCRMD_AH@
4. Cdmcumdtah` ehr a`put rbsastd`cb (\ A) d krdpj as pmhttbo lbtwbb` ldsb vhmtdkb U l (dmh`k x-dxas) d`o ldsb currb`t A l (dmh`k y-dxas).
_jb rbcaprhcdm he smhpb kavbs us tjb a`put rbsastd`cb. ∿ \ A
U l A l
9 2.5 hji.
<. Cdmcumdtah` ehr hutput rbsastd`cb (\ h) d krdpj as pmhttbo lbtwbb` chmmbcthr vhmtdkb U c (dmh`k x-dxas) d`o ldsb currb`t A l (dmh`k y-dxas) _jb rbcaprhcdm he tjb smhpb kavbs tjb vdmub he hutput rbsastd`cb.
∿
\h
U c A c
9 >.<2 hji.
3. Cdmcumdtah` ehr currb`t kda` ( ) d krdpj as pmhttbo lbtwbb` ldsb currb`t A l (dmh`k x-dxas) d`o chrrbsph`oa`k chmmbcthr currb`t A c (dmh`k y-dxas). _jb rbsumta`k krdpj as d strdakjt ma`b. _jb smhpb tjb krdpj kavbs tjb vdmub he currb`t kda`, μ
∿
A c A l
9 >.
?. Cdmcumdtah` ehr vhmtdkb kda` D v7
∿
Dv
\h \ A
9 >.4>1<
\BSRM_
Ehr tjb kavb` chiih` biattbr trd`sasthr,
Currb`t kda`, μ 9 >.
Uhmtdkb kda`, Dv 9 >.4>1<
