Electromagnetic Theory For Complete Idiots (Electrical Engineering For Complete Idiots) by DAVID SMITH

 

ECEOTXAJBFKET@O TLEAXP GAX OAJ^CETE @D@AT_

hy Dbv`d _j`tl 

 

Bcc X`flts Xeserved. Ka pbrt ag tl`s puhc`obt`ak jby he repraduoed `k bky garj ar hy bky jebks, `kocud`kf sobkk`kf, plataoapy`kf, ar atlerw`se w`tlaut pr`ar wr`ttek perj`ss`ak ag tle oapyr`flt lacder lacder.. Oapy Oa pyr` r`fl fltt © 8>8> 8>8>

 

Atler haams `k tle _er`es9 Brdu`ka gar Oajpcete @d`ats D`f`tbc _`fkbc ^r ^raoess`kf aoess`kf gar Oajpcete @d`ats Oaktrac _ystejs gar Oajpcete @d`ats O`rou`t Bkbcys`s gar Oajpcete @d`ats Hbs`o Eceotrak`os gar Oajpcete @d`ats D`f`tbc Eceotrak`os gar Oajpcete @d`ats

 

Tbhce ag Oaktekts

2. WEOTAX BKBCP_@_ 8. WEOTAX OBCOUCU_ ?. OUXW@C@KEBX OAAXD@KBTE _P_TEJ_ 4. ECEOTXA_ ECEOTXA_ T  TB BT@O_ 5. ECEOTX@O ECEOTX@O ^ATEKT@BC ^ATEKT@BC 7. OAKDUOTA OAKDUOTAX_ X_ & D@ECEOTX@O_ 6. JBFKETA_TBT@O_ JBFKETA_TBT@O_ . JB\VECC‐_ JB\VECC‐_ ERUBT@AK_ ERUBT@AK_ B^^EKD@\ B^^EK D@\ XEGEXEKOE_ XEGEX EKOE_ OAKTBOT

 

^XEGBOE _`koe tle t`je Jbxwecc flrst presekted b uk`fled tleary ag  Eceotr`o`ty bkd Jbfket`sj, tle flecd ag Eceotrajbfket`sj lbs seek kujeraus bdvbkoejekts. Hut tle gukdbjektbcs lbve rejb`ked jare ar cess tle sbje tlrauflaut tl`s per`ad. Bs aur sebrol gar jare bdvbkoed oajjuk`obt`ak teolkacafy `kteks`fles, tle recevbkoe ag tl`s flecd ag  ekf`keer`kf `s akcy fraw`kf dby hy dby. dby. _a `t `s very `jpartbkt gar b studekt ag Eceotr`obc ar Oajjuk`obt`aks ekf`keer`kf ar b plys`os ektlus`bst ta he gbj`c`br w`tl tle gukdbjektbcs ag tl`s suhieot. Ve waucd he cy`kf `g we w e ocb`jed tl`s haam `s gar tatbc hef`kkers w`tl ka jbtl hbomfraukd, @t‐s kat. Tl`s suhieot `s b lebd sorbtoler. sorbtoler. @t `s kat ake ag tlase t lase suhieots yau obk obsubccy obsub ccy cebrk w`tlaut w`tlaut su﬏ o`ekt jbtlejbt`ob jbtlejbt`obcc prerequ`s`te prerequ`s`tes. s. Ve dauht `g bky haam obk debc w`tl tl`s suhieot w`tlaut ttle le lecp ag jbtl. @k tl`s haam we prav`de b oako`se `ktraduot`ak `kta hbs`os ag Eceotrajbfket`o tleary, w`tl jare ejplbs`s ak tle plekajekak tlejsecves bkd cess ak tle der`vbt`aks bkd atler jbtlejbt`obc bspeots. Lawever, vbr`aus vbr `aus essekt`bc jbtl oakoepts c`me veotar bkbcys`s bkd veotar obcoucus lbve heek `kocuded `kocuded w`tl su﬏ su﬏ o`ekt detb`c. detb`c. Tle der`vbt`ak der`vbt`aks s ag jbky `jpartbkt tlearejs `k tl`s suhieot lbs bcsa heek `kocuded. B hbs`o mkawcedfe ag Obcoucus bkd saje ^lys`os bre tle akcy prerequ`s`tes requ`red ta gaccaw tle tap`os d`soussed `k tle haam. Ve've tr`ed ta expcb`k tle vbr`aus gukdbjektbc oakoepts ag Eceotrajbfket`o tleary `k tle s`jpcest bkd tle jast `ktu`t`ve jbkker w`tlaut bk aver av er rec`bkoe ak jbtl. Bcsa, we lbve tr`ed ta oakkeot tle vbr`aus tap`os w`tl rebc c`ge s`tubt`aks wlerever pass`hce. Tl`s wby evek flrst t`jers obk cebrk tle hbs`os ag Eceotrajbfket`o tleary w`tl j`k`juj e﬎art. Lapeguccy tle studekts w`cc ekiay tl`s

 

d`﬎erekt bpprabol ta tl`s taufl suhieot. Tle vbr`aus oakoepts ag tle suhieot bre brrbkfed caf`obccy bkd expcb`ked `k b s`jpce rebder-gr`ekdcy cbkfubfe w`tl `ccustrbt`ve flfures.  Tl`s haam `s kat jebkt ta he b repcboejekt gar tlase stbkdbrd Eceotrajbfket`o tleary texthaams, rbtler tl`s haam slaucd he v`ewed bs bk `ktraduotary text gar hef`kkers ta oaje `k fr`ps w`tl bdvbkoed cevec tap`os oavered `k tlase haams. Tl`s haam w`cc lapeguccy serve bs `ksp`rbt`ak ta cebrk Eceotrajbfket`o tleary `k frebter deptls bkd f`ve b frebter bppreo`bt`ak gar tl`s plekajekak. Xebders bre wecoaje ta f`ve oakstruot`ve suffest`aks gar Xebders tle `jpravejekt ag tle haam. Ekiay tl`s wakderguc suhieot. Tlbkm yau!

 

2. WEOTAX BKBCP_@_ 2.2 _OBCBX_ BKD WEOTAX_ @k plys`os, qubkt`t`es obk he ocbss`fled `kta 8 jb`k obtefar`es= _obcbrs bkd Weotars. Weotars. Oertb`k qubkt`t`es suol bs jbss, t`je, tejperbture, vacuje, deks`ty eto. obk he bdequbtecy represekted iust hy b kujer`obc vbcue, bk bjaukt ar b jbfk`tude. _uol qubkt`t`es bre obcced _obcbrs. Gar exbjpce, `g tle jbss ag bk ahieot `s f`vek bs 8 m`cafrbjs, tle vbcue 8 dekates tle kujher ag t`jes tle uk`t m`cafrbj `s oaktb`ked `k tlbt ahieot. Ak tle atler lbkd, oertb`k atler qubkt`t`es suol bs garoe, vecao`ty, d`spcboejekt, booecerbt`ak eto. obkkat he bdequbtecy represekted hy b kujer`obc vbcue bcake. Tle akcy wby ta jebk`kfguccy represekt suol qubkt`t`es `s hy oaks`der`kf tlej bs lbv`kf b d`reot`ak `k bdd`t`ak ta tle t le jbfk`tude. _uol qubkt`t`es tlbt lbve b jbfk`tude bs wecc bs b d`reot`ak bre obcced Weotars. Gar Gar exbjpce, `g yau wbcm 8 j`ces kartl, tlek wbcm 8 j`ces sautl, yau bre hbom bt yaur ar`f`kbc pas`t`ak `.e. yaur ket d`spcboejekt `s zera. Kaw `kstebd lbd we iust jekt`aked tle jbfk`tude bkd `fkared tle d`reot`aks, tlere‐s ka wby ag tecc`kf yaur ket d`spcboejekt, yau oaucd lbve wbcmed 4 j`ces strb`flt ar 4 j`ces cegt ar wl`olever wby pass`hce. Weotars bre represekted hy b d`reoted c`ke sefjekt, hbs`obccy brraws. Tle cekftl ag tle c`ke sefjekt dekates tle jbfk`tude bkd tle brrawlebd `kd`obtes tle d`reot`ak ag tle veotar.. Gar exbjpce, b veotar graj pa`kt B ta bkatler pa`kt veotar H `s dekated hy

. ^a`kt B `s obcced tle tb`c ag tle veotar

 

bkd pa`kt H `s obcced tle lebd ag tle veotar veotar.. Tle jbfk`tude ag b veotar

`s dekated bs

.

Kate tlbt b veotar `s olbrboter`zed akcy hy `ts jbfk`tude bkd d`reot`ak, bkd kat hy `ts pas`t`ak `k spboe. Tl`s `jpc`es tlbt bs cakf bs tle jbfk`tude bkd tle d`reot`ak bre ukbctered, yau bre gree ta jave tle veotar braukd.

2.8 _OBCBX JUCT@^C@OBT@AK Vlek b veotar `s juct`pc`ed hy b pas`t`ve sobcbr qubkt`ty (b pas`t`ve kujher), `ts jbfk`tude fets juct`pc`ed hy tle

 

sobcbr qubkt`ty bkd tle d`reot`ak rejb`ks ukbctered. _`j`cbrcy, `g b veotar `s juct`pc`ed hy b kefbt`ve sobcbr qubkt`ty, `ts jbfk`tude fets juct`pc`ed hy tle sobcbr qubkt`ty bkd `ts d`reot`ak reverses.  Tle praduot ag b sobcbr x bkd veotar

`s dekated bs x 

.

2.? BDD@T@AK AG WEOTAX_ Weotar bdd`t`ak `sk‐t bs strb`fltgarwbrd bs bdd`kf sobcbr qubkt`t`es. @k obse ag veotars, tle d`reot`aks lbve ta he oaks`dered bs wecc. Oaks`der tle gaccaw`kf exbjpce ag b sac`d hcaom he`kf bppc`ed 2> Kewtaks ag garoe graj 8 d`reot`aks bs slawk `k tle flfure hecaw.

 

@k wl`ol d`reot`ak da yau tl`km tle hcaom h caom w`cc jave bs b resuct ag tlese 8 garoes1 Ag oaurse, bcakf tle datted c`ke. Kaw `jbf`ke tlbt tle hcaom `s he`kf pusled w`tl 8> Kewtaks ag garoe graj tle hattaj bkd 2> Kewtaks ag garoe graj tle s`de bs hegare. @k wl`ol d`reot`ak da yau tl`km tle hcaom w`cc jave kaw1 Oajjak sekse sbys tle hcaom w`cc jave `k b d`reot`ak sc`fltcy ta tle cegt ag tle datted c`ke. Vlbt we iust d`d w`tl tlese 8 exbjpces `s W Weotar eotar bdd`t`ak. Kaw cet‐s oaks`der b fekerbc obse wlere 8 veotars bre ar`ekted bt b oertb`k bkfce graj ake atler bs slawk hecaw hecaw..

 

 Ta flkd tle suj ar tle resuctbkt ag 8 veotars, repas`t`ak bky  Ta ake ag tle veotars suol tlbt tle tb`c ag tle ake veotar oa`ko`des w`tl tle lebd ag tle atler veotar, tlek tle c`ke sefjekt tlbt oajpcetes tle tr`bkfce dekates d ekates tle resuctbkt veotar.. Tl`s `s obcced tle tr`bkfce cbw ag veotar bdd`t`ak. @k veotar tle flfure bhave, we repas`t`aked veotar OD suol tlbt t lbt tle tb`c (O) ag veotar OD oa`ko`des w`tl tle lebd (H) ag veotar BH. Lekoe tle resuctbkt `s f`vek hy tle veotar BD. Kaw `g  yau try tle sbje, hut hy repas`t`ak`kf veotar BH `kstebd ag  OD, yau w`cc fet tle exbot sbje resuct, prav`kf tlbt veotar bdd`t`ak `s oajjutbt`ve.

Ve obk extekd tl`s `deb ta bky ka ag veotars bkd tl`s ruce `s mkawk bs tle ^acyfak cbw ag veotar bdd`t`ak.

 

2.4 UK@T WEOTAX Bcakf b pbrt`oucbr d`reot`ak tlere obk he `kflk`te ka. ag  pass`hce veotars w`tl bcc ag tlej d`﬎er`kf `k jbfk`tudes graj ake bkatler. bkatler. Hut `k bky d`reot`ak, tlere obk akcy he b s`kfce veotar ag uk`t jbfk`tude, suol b veotar `s obcced tle Uk`t veotar. veotar. Tle s`fk`flobkoe ag tle Uk`t veotar `s tlbt bcc veotars `k b speo`flo d`reot`ak bre sobced vers`aks ag tle uk`t veotar `k tlbt d`reot`ak. B uk`t veotar `k tle d`reot`ak ag b veotar obk he ahtb`ked bs,

, dekated hy

2.5 XE_ACUT@AK AG WEOTAX Ake h`f bdvbktbfe ag us`kf veotars `s tlbt tley obk he resacved `kta bky ka. ag oajpakekts. @k tle s`jpcest obse, b

 

veotar obk he resacved `kta 8 oajpakekt veotars v eotars cy`kf `k tle sbje pcbke. _lawk hecaw `s b veotar , `g we drbw `ts praieot`aks akta bky 8 perpekd`oucbr bxes `k `ts pcbke (gar oakvek`ekoe we lbve p`omed tle x bkd y bxes), we fet b set ag 8 kew veotars  bkd ag veotar

. Tlese 8 kew veotars bre obcced tle oajpakekts .

Ve obk tbme tl`s ake step gurtler bkd resacve b veotar `kta ? oajpakekts `k ? d`jeks`ak. Vlbt‐s tle bdvbktbfe bdvbkt bfe ag da`kf tl`s1 Gar ake, resacv`kf veotars `kta perpekd`oucbr per pekd`oucbr oajpakekts bccaws us ta express tlej `k terjs ag tle Obrtes`bk oaard`kbtes, wl`ol we bre gbj`c`br w`tl. Oaks`der b veotar

bs slawk `k tle flfure hecaw hecaw.. @g we

resacve tl`s veotar `kta 8 perpekd`oucbr veotars, we fet 8

 

oajpakekt veotars

bkd

. Weotar

`s bcakf tle z-

bx`s, sa we‐cc cebve `t bs suol. Hut veotar ak tle atler lbkd c`es `k tle x-y pcbke, sa we ta keed ta resacve `t gurtler.. Kaw `g we oaks`der gurtler

bs b sepbrbte veotar bkd

gurtler resacve `t, we fet `ts 8 oajpakekts

bkd

. Tl`s

wby we lbve resacved tle veotar

`kta ? jutubccy

perpekd`oucbr veotars , y bkd z bxes respeot`vecy.

wl`ol c`es bcakf tle x,

bkd

 

Bky veotar obk he resacved `kta oajpakekts bcakf tle oaard`kbte bxes `k tl`s jbkker. jbkker. Hy da`kf tl`s, we lbve jbkbfed ta fekerbc`ze veotars b c`ttce h`t. Vlbt `g we sajelaw `koucobte tle `deb ag uk`t veotars graj cbst seot`ak `kta tl`s wlace tl`kf, Obk we fekerbc`ze veotars evek gurtler1  Tle `deb `s tlbt tle uk`t veotar `s tle jast hbs`o veotar bcakf b d`reot`ak bkd every atler veotar bcakf tlbt sbje d`reot`ak `s iust b sobced vers`ak ag tlbt uk`t veotar veotar.. _a `g we dekate tle uk`t veotars bcakf x, y bkd z bxes bs z

 ,

x

 bkd

y

 respeot`vecy, tlek bky veotar B obk he expressed bs9

Jbtl haams usubccy dekate tle uk`t veotars bcakf tle ? bxes bs , bkd , hut s`koe tlese katbt`aks lbve atler jebk`kfs `k eceotr`obc ekf`keer`kf, tle usubc prbot`oe `s ta fa gar

 ,

x

 bkd

y

 .

z

Bs b oaksequekoe ag tl`s fekerbc`zbt`ak, we obk dekate b veotar s`jpcy hy us`kf ? ka‐s. Gar exbjpce, (5, 8, -4) deka de kate tes s ttle le veot veotar ar 5 dekates tle veotar

 + 8

x

y

 + 8

y

 - 4

 . _`j`cbrcy, (>, 2, 8)

z

 . @k tle cbter exbjpce tle x

z

 

oajpakekt `s bhsekt, jebk`kf tle veotar c`es `k tle y-z pcbke.  Tle jbfk`tude ag b veotar obk he ahtb`ked graj `ts oajpakekts bs9

 Tl`s resuct obk he ebs`cy ahtb`ked hy us`kf tle ^ytlbfarbs tlearej.

2.7 DAT ^XADUOT Weotars obk he juct`pc`ed `k 8 pass`hce wbys= tle Dat praduot bkd tle Orass praduot. Tle dat praduot ag 8 veotars resucts `k b sobcbr qubkt`ty bkd tle orass praduot ag 8 veotars resucts `k bkatler veotar. veotar. Gar tl`s rebsak, tle dat praduot bkd tle orass praduot bre bcsa mkawk bs tle sobcbr praduot bkd tle veotar praduot respeot`vecy.  Tle dat praduot hetweek 8 veotars `s dekated bs (rebd bs B dat H). Jbtlejbt`obccy, tle dat praduot `s deflked bs9

 

 Tle dat praduot obk he `ktu`t`vecy tlauflt ag bs b jebsure ag tle s`j`cbr`ty ag twa veotars ar law wecc tley warm tafetler w`tl ake bkatler bkatler.. Oaks`der aur –gar –garoes oes ak b hcaom‚ exbjpce bfb`k.

 

@k tle flrst flfure, tle garoes B bkd H bre `k tle sbje d`reot`ak bkd lekoe tley bre warm`kf tafetler ta jave tle hcaom. Tleregare, tle dat praduot `s jbx`juj `k tl`s obse. @k seoakd flfure, tle garoe H tle `s bppc`ed bt bk bkfce ta tletle garoe B, tl`s ahv`auscy `s kat hest jetlad ta jave tle hcaom. Tle garoe H `s kat oaktr`hut`kf ta t a tle jat`ak ag  tle hcaom bs juol bs `t d`d `k tle flrst obse, lekoe tle dat praduot `s b sjbccer vbcue, hut frebter tlbk zera. @k tle tl`rd flfure, tle garoe H `s bppc`ed artlafakbccy ta tle garoe B, sa `t daesk‐t oaktr`hute ta jav`kf tle hcaom (`k tle d`reot`ak ag  B) bt bcc, lekoe tle dat praduot lere `s zera. @k tle gaurtl exbjpce, tle garoe H `s bppc`ed `k appas`te d`reot`ak ta garoe B, wl`ol jebks garoe H `s kat akcy kat oaktr`hut`kf oaktr `hut`kf ta tle jat`ak ag tle hcaom, hut `t‐s botubccy kefbt`kf tle e﬎eot ag  garoe B. _a, tle dat praduot `k tl`s obse `s kefbt`ve.

^rapert`es ag Dat ^raduot9 Oajjutbt`ve praperty9 u. u.v v : v.u D`str`hut`ve praperty9 u.( v + w) : u.v + u.w

( u + v) .( w + z) : u.w + u.z + v.w + v.z m ( u.v) : ( mu mu)) .v : u.( mv mv))  , wlere m `s b sobcbr u.u :  

8

b x . b x : b y . b y : b z . b z : 2 b x . b y : b y . b z : b z . b x : >

 

Exbjpce9

2.6 OXA__ ^XADUOT  Tle seoakd wby ag juct`pcy`kf 8 veotars `s obcced tle Orass praduot. Bs jekt`aked ebrc`er, tle orass praduot ag twa veotars resucts `k bkatler veotar. veotar. Jbtlejbt`obccy, tle orass praduot `s deflked bs9

Deflk`kf tle orass praduot `sk‐t bs strb`fltgarwbrd bs w`tl tle dat praduot. F`vek 8 veotars, tle orass praduot ag tle twa veotars represekts tle breb ag tle pbrbccecafrbj garjed w`tl tlese 8 veotars tbmek t bmek bs bdiboekt s`des. Bkd tle

 

d`reot`ak ag tle orass praduot veotar `s karjbc ta pcbke ag  tl`s pbrbccecafrbj. Tle orass praduot obk he `ktu`t`vecy tlauflt ag bs b jebsure ag tle artlafakbc`ty ag 8 veotars. Ocaser tle bkfce hetweek tle veotars `s ta 0 > defrees, cbrfer tle breb ag tle pbrbccecafrbj garjed bkd oarrespakd`kfcy cbrfer tle orass praduot veotar.

Kaw tlere‐s bkatler sjbcc prahcej, f`vek 8 veotars B bkd H, tlere bre twa pass`hce d`reot`aks tlbt tle t le orass praduot veotar oaucd pa`kt ta, upwbrd bkd dawkwbrd. Tl`s `s wlere tle r`flt lbkd tlujh ruce oajes `k, wl`ol stbtes tlbt –@g  yau ourc yaur flkfers ag yaur r`flt lbkd `k suol b wby tlbt  tle `kdex flkfer pa`kts `k tle d`reot`ak ag veotar B bkd  j`ddce flkfer `k pa`kts `k tle d`reot`ak ag H, tlek tle  tlujh pa`kts tle d`reot`ak ag BxH‚   . veotar  . Kaw `g yau try tle sbje gar HxB, tle tlujh w`cc pa`kt `k tle dawkwbrd d`reot`ak `.e. HxB : -BxH.

 

^rapert`es ag Orass ^raduot9 u x v : -v x u D`str`hut`ve praperty9 u x ( v + w) : ( u x v) +

( u x w) m ( u x v) : ( mu mu)) x v : u x ( mv mv))  , wlere m `s b sobcbr

uxu: > b x x b x : b y x b y : b z x b z : > b x x b y : b z , b y x b z : b x , b z x b x : b y

 

b y x b x : - b z , b z x b y : - b x , b x x b z : - b y Exbjpce9

 

8. WEOTAX OBCOUCU_ 8.2 _OBCBX BKD WEOTAX G@ECD_ Vlbt `s b flecd1 B flecd `s b plys`obc qubkt`ty tlbt obk he speo`fled everywlere `k spboe bs b gukot`ak ag pas`t`ak (x, y bkd z oaard`kbtes). Bkd tlere bre hbs`obccy 8 types ag flecds9 _obcbr bkd veotar flecds. B sobcbr flecd (ar b sobcbr gukot`ak) bssao`btes b sobcbr vbcue ar b jbfk`tude ta every pa`kt `k spboe.

Exbjpce9  D`str`hut`ak ag tejperbture `k b raaj. @g   Tejperbture  T ejperbture `k b raaj raaj `s f`vek hy sobcbr flecd T: xy   z   , 8

tlek bt pa`kt (2,2,2) tle tejperbture `s 2 uk`t bkd bt

?

 

bkatler pa`kt (8,?,2), tle tejperbture `s 2< uk`ts bkd sa ak 

Kate tlbt `t `s kat keoessbry gar b sobcbr flecd ta lbve kak zera vbcues bt every pa`kt `k spboe. _`j`cbrcy, b veotar flecd (ar b veotar gukot`ak) `s b flecd `k wl`ol b veotar obk he bss`fked ta every pa`kt `k spboe.

Exbjpce9  Wecao`ty ag ﬈aw bt d`﬎erekt pa`kts `k b ﬈u`d. @g  vecao`ty ag ﬈aw `s f`vek hy W : xy  x + xy 8 z  y ‟ z ?  z , tlek bt pa`kt (2,2,2) tle vecao`ty `s dekated hy tle veotar   +  28 

 - 

y

 . Bt bkatler pa`kt (2,8,?), tle vecao`ty `s 8 

z

 ‟ 86 

y

 bkd sa ak.

z

 +

x

x

 

8.8 ^BXT@BC DEX@WBT@WE & TLE ∆ A^EXBTAX Gar s`kfce vbr`bhce gukot`aks, tle der`vbt`ve jebsures tle olbkfe `k tle gukot`ak vbcue w`tl respeot ta tle `kdepekdekt Tle der`vbt`ve pbrt`bc der`vbt`ve `s vbr`bhce tle equ`vbcekt ag vbr`bhce. tle ard`kbry gar juct` gukot`aks. @t `s hbs`obccy tle der`vbt`ve ag tle gukot`ak w`tl respeot ta bky ake ag `ts vbr`bhces w`tl tle atler vbr`bhces lecd oakstbkt. ^brt`bc der`vbt`ve ag b gukot`ak g w`tl respeot ta x, y & z bre dekated dekate d bs

,

&

respeot` respeot`vecy vecy..

Hy oaks`der`kf tle atler vbr`bhces bs oakstbkt, we bre essekt`bccy reduo`kf tle juct` vbr`bhce gukot`ak ta b s`kfce vbr`bhce gukot`ak. Bkd tleregare, tle pbrt`bc der`vbt`ve

 

jebsures tle olbkfe `k tle gukot`ak vbcue w`tl respeot ta bky ake vbr`bhce.

Us`kf pbrt`bc der`vbt`ves, we obk deflke b kew aperbtar obcced tle dec ar tle kbhcb aperbtar, dekated hy tle syjhac bs9   ∆. @k Obrtes`bk oaard`kbtes, tle dec aperbtar `s deflked bs9

∆ `sk‐t rebccy b veotar, `t `s rbtler b d`﬎erekt`bc aperbtar. aperbtar. Vlek bppc`ed ta b s`kfce vbr`bhce gukot`ak, `t `s tle sbje bs tle stbkdbrd der`vbt`ve. Hut gar b juct`-vbr`bhce gukot`ak, `t obk he bppc`ed `k ? d`﬎erekt garjs (frbd`ekt, d`verfekoe, ourc) bkd ebol ag tlese garjs lbs b d`﬎erekt s`fk`flobkoe.

8.? FXBD@EKT B juct` vbr`bhce gukot`ak lbs juct`pce der`vbt`ves bt every pa`kt, ake w`tl respeot ta every vbr`bhce `t depekds ak. Gar exbjpce, b gukot`ak g( x, y)  lbs 8 der`vbt`ves ∈g/∈x bkd ∈g/∈y bt every pa`kt (slawk `k tle flfure hecaw). Ebol ag tlese

 

der`vbt`ves oarrespakd ta tle rbte ag olbkfe ag tle gukot`ak w`tl respeot ta b pbrt`oucbr vbr`bhce. Tle resuctbkt ar tle suj ag tlese der`vbt`ves dekates tle jbx`juj rbte ag  olbkfe ag tle gukot`ak wlek bcc tle vbr`bhces bre oaks`dered bcc bt ake. Tlbt `s tle frbd`ekt. @k atler wards, tle Frbd`ekt ag b juct` vbr`bhce gukot`ak `s b veotar tlbt pa`kts `k tle d`reot`ak ag frebtest `korebse (steepest scape) ag tle gukot`ak bt b pa`kt. @t `s dekated hy syjhac ∆ g  . Tle frbd`ekt `s bkbcafaus ta tle scape gar s`kfce vbr`bhce gukot`aks.

 Tle frbd`ekt obk he obcoucbted bs9

 

Exbjpce9  g  g : 8x + yz

_a bt b pa`kt sby ^(5,-?, ^(5,-?, 0 ), tle frbd`ekt ar tle jbx scape `s `s deka de kate ted d hy tle tle veo veota tarr 8

 + 0

x

 - ?

y

 .

z

@k bkatler wby, tle frbd`ekt obk he tlauflt ag bs bk aperbtar tlbt oakverts b sobcbr flecd (sobcbr gukot`ak) `kta b veotar flecd (veotar gukot`ak).

^rapert`es ag Frbd`ekt9

 

8.? D@WEXFEKOE Ve sbw law tle dec aperbtar obk he bppc`ed ak b sobcbr gukot`ak `k tle garj ag tle frbd`ekt. Ak b veotar flecd tle dec aperbtar obk he bppc`ed `k 8 wbys, flrst ag wl`ol `s tle D`verfekoe. D`verfekoe `s katl`kf hut tle dat praduot hetweek tle dec aperbtar bkd b veotar flecd.

@ktu`t`vecy, tle D`verfekoe represekts tle autwbrd ﬈aw ag b veotar flecd graj bk `kflk`tes`jbc vacuje bt b f`vek pa`kt `k b veotar flecd. @k atler wards, d`verfekoe `s b jebsure ag  tle extekt ta wl`ol b pa`kt (wl`ol `s essekt`bccy b t`ky vacuje) helbves bs b sauroe ag tle veotar flecd.  Ta ukderstbkd tle oakoept ag d`verfekoe hetter, `jbf`ke b  Ta veotar flecd bs b ﬈u`d ﬈aw bs slawk hecaw. hecaw.

 

Kaw `g yau oaks`der b sjbcc spler`obc vacuje, tle d`﬎erekoe hetweek tle autwbrd ﬈aw bkd tle `kwbrd ﬈aw `.e. tle ket autwbrd ﬈aw f`ves tle d`verfekoe ag tle ﬈aw `k tle sjbcc vacuje. Gar exbjpce, bt pa`kt B, bcc tle flecd c`kes bre pa`kted bwby graj tle vacuje, wl`ol jebks pa`kt B `s bot`kf bs sauroe ag tle ﬈ux, tleregare tle d`verfekoe bt tlbt pa`kt `s b pas`t`ve vbcue. Bt pa`kt H, saje ag tle flecd c`kes bre ﬈aw`kf `kta tle vacuje bkd saje bre ﬈aw`kf aut ag tle t le vacuje, hut heobuse tlere bre jare autwbrd ﬈aw`kf flecd c`kes, tle ket autwbrd ﬈aw `s pas`t`ve, tleregare t leregare d`verfekoe `s pas`t`ve bt pa`kt H bs wecc (hut `t lbs cess jbfk`tude oajpbred ta pa`kt B). Bt pa`kt O, tlere bre equbc ka. ag flecd c`kes ﬈aw`kf `kta tle vacuje bs tlere bre flecd c`kes ﬈aw`kf aut ag tle vacuje. Lekoe tle d`verfekoe bt pa`kt O `s zera.  Tle bhave exbjpce `s akcy gar hetter ukderstbkd`kf ag tle oakoept, `k rebc`ty tle d`verfekoe lbs katl`kf ta da w`tl tle ka. ag flecd c`kes ekter`kf ar ex`t`kf tle vacuje, `t lbs jare ta da w`tl tle jbfk`tude & d`reot`ak ag tle flecd c`kes.

 

Exbjpce9   : x

x +

yz

y +

?xz

z

 Tle d`verfekoe aperbtar oakverts b veotar flecd `kta b sobcbr flecd.

 

^rapert`es ag D`verfekoe9

8.4 OUXC  Tle Ourc `s tle orass praduot hetweek tle dec aperbtar bkd b veotar flecd.

 Tle ourc ag b veotar flecd desor`hes tle ratbt`akbc tekdekoy ag b veotar flecd bt b pa`kt `k ?d spboe. Oaks`der b ﬈u`d ﬈aw bs slawk `k tle tl e flfure hecaw. hecaw. Kaw `g yau oaks`der b sjbcc spler`obc hbcc tlbt `s gree `s ta ratbte `k bky d`reot`ak, `t w`cc ratbte d`﬎erektcy depekd`kf ak `ts caobt`ak `k tle ﬈u`d. Tle flecd veotars bot`kf ak tle splere deterj`kes hatl d`reot`ak bkd tle speed bt wl`ol `t ratbtes. Jbfk`tude ag tle ourc

 

veotar dekates tle speed ag ratbt`ak bkd tle d`reot`ak ag tle ourc dekates tle bx`s ag ratbt`ak ag tle splere.

@t `s `jpartbkt ta kate tlbt ourc regers ta tle j`orasoap`o ratbt`ak ag tle hbcc bt b pa`kt `k tle t le veotar flecd (`.e. bs `g  trbkscbt`akbc jat`ak ag tle hbcc graj tle pa`kt `s restr`oted) bkd kat tle jborasoap`o o`roucbt`ak ag tle hbcc `k tle flecd, `g  bky.. @g yau tl`km bhaut tle jat`ak ag tle ebrtl braukd tle bky suk, tle ourc `s bkbcafaus ta tle ratbt`ak ag tle ebrtl bkd kat `ts revacut`ak braukd tle suk.

Exbjpce9   : x

x +

yz

y +

?xz

z

 

 Tle d`verfekoe aperbtar oakverts b veotar flecd `kta bkatler veotar flecd.

^rapert`es ag Ourc9

8.5 JAXE ^XA^EXT@E_

 

8.7 C@KE @KTEFXBC C`ke `ktefrbc ar tle pbtl `ktefrbc `s tle `ktefrbc ag b gukot`ak evbcubted bcakf b ourve. Gar s`kfce vbr`bhce gukot`aks, tle c`ke `ktefrbc obk akcy he evbcubted aver b strb`flt-c`ke pbtl (gar ahv`aus rebsaks) bs slawk.

 Tle c`ke `ktefrbc `sstr`ps evbcubted hy oaks`der`kf b hukol ag  sjbcc reotbkfucbr ag w`dtl dx bkd su`tbhce le`flt

 

(wl`ol w`cc he equbc ta tle vbcue ag tle gukot`ak bt tle pa`kt `.e. g(x)  ), sa bs ta flcc up tle breb. Tle breb ag bk `kd`v`dubc str`p `s tleregare g( x) dx (w`dtl x le`flt) bkd lekoe tle breb ag aur `kterest `s s`jpcy tle `ktefrbc (sujjbt`ak) ag  tlese sjbcc brebs bcakf tle pbtl. Gar juct`vbr`bhce gukot`aks, tle c`ke `ktefrbc obk he evbcubted bcakf bky 8- d`jeks`akbc pbtl, kat iust b strb`flt c`ke.

 Tle c`ke `ktefrbc gar juct`vbr`bhce gukot`aks `s evbcubted `k tle sbje jbkker bs w`tl s`kfce vbr`bhce v br`bhce gukot`aks, exoept tlbt `k tl`s obse, tle c`ke ecejekt (w`dtl ag tle reotbkfucbr str`p) `s ka cakfer dx ar dy , hut b gukot`ak ag dx bkd dy .

 

@k fekerbc, tle c`ke `ktefrbc ag b gukot`ak g bcakf b ourve O

obk he evbcubted bs9

 Ta sujjbr`ze, tle c`ke `ktefrbc ag b sobcbr gukot`ak `s  Ta katl`kf hut tle breb ag tle gekoe orebted hy b ourve pbtl bkd `ts praieot`ak ak tle gukot`ak.

 

Kaw cet‐s jave ak ta c`ke `ktefrbt`ak `ktefr bt`ak ag b veotar flecd. Tle c`ke `ktefrbc ag b veotar flecd obk he `kterpreted bs tle bjaukt ag warm tlbt b garoe flecd daes ak bk ahieot bs `t javes bcakf b ourve. Vlek yau try ta jave b hcaom bcakf b ourve O `k spboe bt oakstbkt speed tlraufl b garoe flecd, b garoe bcwbys bots ak tle hcaom, jbm`kf `t ebs`er ar lbrder ta jave tle hcaom depekd`kf ak tle t le d`reot`aks ag tle garoes bt b pa`kt. @g tle garoe bots appas`te ta tle d`reot`ak ag tle pbtl, tlek yau lbve ta da warm ta meep tle hcaom jav`kf. Ak tle atler lbkd, `g tle d`reot`ak ag tle garoe `s `k tle d`reot`ak ag tle pbtl, tlek yaur iah heoajes ebs`er heobuse yau're he`kf b`ded hy bk auts`de garoe.

Bs jekt`aked `k tle prev`aus olbpter, tle dat praduot `s b jebsure ag law wecc 8 veotars warm w`tl w `tl ebol atler, lekoe tle dat praduot hetweek tle veotar flecd bkd tle d`﬎erekt`bc ourve ecejekt `ktefrbted bcakf tle ekt`re ourve f`ves tle tatbc warm dake.

 

8.6 _UXGBOE @KTEFXBC Gar sobcbr gukot`aks, tle surgboe `ktefrbc `s katl`kf hut tle vacuje ekocased hy b surgboe bkd `ts praieot`ak ak tle gukot`ak. Tle s`jpcest obse `s wlek tle breb `s pcbkbr ar 8d`jeks`akbc bs slawk.

 

 Ta flkd tle surgboe `ktefrbc `k suol b obse, tle surgboe `s  Ta sc`oed us`kf pcbkes ag sjbcc tl`omkess e`tler bcakf tle x-bx`s ar tle y-bx`s (bcakf tle x-bx`s `k tle flfure). Tlbt wby we obk oakvert tle surgboe `kta b hukol ag ourves ar c`kes, bccaw`kf us ta use tle c`ke `ktefrbc.

 

 Tle sc`oed breb `s f`vek hy tle c`ke `ktefrbc, Lere dc :dx , s`koe tle c`kes bre strb`flt bkd pbrbccec ta tle x-bx`s. Ta Ta ahtb`k tle vacuje, bcc we keed ta da `s ta t a oajh`ke tlese brebs, wl`ol obk he dake hy `ktefrbt`kf tlej w`tl respeot ta dy . Tleregare, tle surgboe `ktefrbc `s f`vek hy hy99

 Tl`s equbt`ak `s obcced tle dauhce `ktefrbc. Hut kat bcc surgboes bre pcbkbr, saje surgboes jbyhe spler`obc ar oak`obc ar bkytl`kf ecse, `k suol b obse we waucd requ`re b jare fekerbc`zed jetlad ta flkd tle surgboe `ktefrbc tlbk tle dauhce `ktefrbc. Tle `deb lere `s ta d`v`de tle wlace surgboe `kta sjbcc d`﬎erekt`bc surgboes ebol ag breb d_ , tlek tle vacuje oarrespakd`kf ta ebol d`﬎erekt`bc breb `s tle vacuje ag tle Grekol Grekol gry slbped ecejekt (pbrbccecap`ped) bhave `t. Tle vacuje ag tl`s

 

ecejekt `s gd_ bkd lekoe tle vacuje oarrespakd`kf ta tle ekt`re surgboe `s s`jpcy tle `ktefrbc ag tl`s d`﬎erekt`bc vacuje.

 Tle surgboe `ktefrbc ag veotar flecd obk he `kterpreted bs tle bjaukt ag flecd ﬈aw`kf (tl`km ag `t bs ﬈u`d ﬈aw) tlraufl b surgboe per uk`t t`je. Vlek tle veotar flecd `s karjbc ta tle surgboe, tle ﬈aw w`cc he jbx`juj bkd wlek tle surgboe `s pbrbccec ta tle surgboe, tlere w`cc he ka ﬈aw  ﬈aw  .  _a, ta obcoucbte tle tatbc flecd ﬈aw`kf tlraufl tle surgboe, bcc we keed ta da `s ta bdd up tle oajpakekt ag tle veotar flecd tlbt bre perpekd`oucbr ta tle surgboe.

 

Lekoe tle surgboe `ktefrbc ag b veotar flecd `s f`vek hy9

8.< D@WEXFEKOE D@WEXFEKOE TL TLEAXEJ EAXEJ Booard`kf ta tle D`verfekoe Tlearej –Tle autwbrd ﬈ux  ag b veotar flecd tlraufl b ocased surgboe `s equbc ta  tle vacuje `ktefrbc ag tle d`verfekoe ag tle veotar  flecd aver tle ref`ak ekocased hy tlbt ocased surgboe‚  .

 

Jbtlejbt`obccy, tle D`verfekoe tlearej obk he wr`ttek bs9

@t `s bcsa mkawk bs Fbuss's tlearej ar Astrafrbdsmy's tlearej. Bctlaufl tle stbtejekt bkd tle jbtlejbt`obc garjucb jby caam oajpc`obted, tle `ktu`t`ak hel`kd tl`s tlearej `s pretty strb`fltgarwbrd. Oaks`der bky ? d`jeks`akbc ahieot pcboed b veotar v eotar flecd. Ve lbve fake w`tl b patbta slbped ahieot bs slawk `k tle flfure. Kaw cet‐s oaks`der tle r`flt lbkd s`de ag tle equbt`ak flrst. Ve lbve bcrebdy cebrkt cebrkt tlbt tle d`verfekoe ag b veotar flecd `s jebsure ag tle autwbrd fa`kf ar ttle le sauroe c`me helbv`ar bt b pa`kt `k tle flecd. Bs tl`s ahieot ukder aur oaks`derbt`ak `s pcboed b veotar flecd, every pa`kt (sjbcc vacuje) `ks`de tle ahieot w`cc lbve b d`verfekoe, e`tler pas`t`ve ar kefbt`ve ar zera. @k aur flfure, we lbve slawk tle d`verfekoe bt ? pa`kts B, H bkd O. Kaw `g we `ktefrbte (suj up) tle d`verfekoe bt bcc suol pa`kts tlrauflaut tle vacuje, wlbt we fet `s tle tatbc d`verfekoe ar tle tatbc sauroe c`me c`me kbture ag tle ahieot bs b wlace, wl`ol `s essekt`bccy juol flecd we ar`f`kbtes graj tle ahieot. Tl`s `s exbotcy tle law sbje qubkt`ty ahtb`k hy tbm`kf tle surgboe `ktefrbc ag tle flecd borass tle gucc surgboe.

 

 Tle D`verfekoe tlearej `k cbyjbk‐s terjs terjs obk he stbted bs –Tle tatbc autwbrd flecd ﬈aw`kf aut graj tle surgboe ag bk  ahieot `s tle equbc ta tle suj ag tle flecds ﬈aw`kf aut graj  every s`kfce pa`kt `ks`de tle vacuje ag tle ahieot  ‚.  ‚. Vlbt tl`s tlearej (bkd tle ake `k tle kext seot`ak) rebccy daes `s flkd b oarrecbt`ak hetweek tle sjbccer plekajekak tlbt aoours ak `ks`de ag bk ahieot ta t a tle cbrfer plekajekak ak tle auter per`plery per`plery.. Hut tl`s `s kat tle rebsak we ekf`keers use tle d`verfekoe tlearej, gar us tle d`verfekoe tlearej prav`des b taac gar oakvert`kf surgboe `ktefrbcs `ktefrb cs wl`ol bre bre agtek d`﬏ d`﬏ ouct ta oaj oajpute pute `kta bk eb ebs`er s`er vacuje `ktefrbc. Tl`s `s espeo`bccy useguc wlek we debc w`tl saje gbj`c`br slbpe ar `g tle d`verfekoe resucts `k b s`jpce gukot`ak. Bk `jpartbkt tl`kf ta he kated `s tlbt tle d`verfekoe tlearej `s akcy bppc`obhce ta ocased surgboes.

8.< _TAM _TAM E‐_ T TLEA LEAXEJ XEJ  Tle _tame‐s tlearej tlearej stbtes tlbt –tle c`ke `ktefrbc ag b  veotar flecd raukd tlbt b ocased pbtl `s equbc ta tle 

 

surgboe `ktefrbc ag tle ourc ag flecd aver bky surgboe  haukded hy tlbt ocased pbtl‚  .  . Jbtlejbt`obccy, tle _tame‐s tlearej obk he wr`ttek bs9

 Tle `deb hel`kd tle _tame‐s tlearej tlearej `s qu`te s`j`cbr ta tle d`verfekoe tlearej. Oaks`der b pcbkbr surgboe pcboed `k b veotar flecd, tlere w`cc he b ratbt`akbc e﬎eot bt every pa`kt ak tle surgboe wl`ol `s f`vek hy tle ourc ag tle flecd. Kaw `g  we `ktefrbte (suj up) tle ourc bt bcc pa`kts tlrauflaut tle surgboe, wlbt we fet `s tle tatbc ratbt`akbc e﬎eot bcakf tle per`plery due ta tle veotar flecd. Tl`s `s tle sbje bs tle c`ke `ktefrbc ag tle veotar flecd bcakf tle auter haukdbry haukdbry..  Tle rebsak wly tl`s `s true obk he hetter ukderstaad w`tl tle lecp ag tle flfure hecaw. hecaw. @g yau drap b hbcc kebr ta tle haukdbry ocase ta pa`kt B, tle hbcc w`cc ratbte `k ocaomw`se d`reot`ak due ta tle ourc bt tle pa`kt, f`vek `t‐s bot`kf `k `sacbt`ak. Hut `g yau oaks`der tle ourc bt bcc pa`kts ak tle surgboe, b hbcc drapped kebr pa`kt B obk ka cakfer meep sp`kk`kf bt B, tle ourc bt pa`kt H w`cc `kev`tbhcy pusl tle hbcc graj pa`kt B ta pa`kt H bkd tlek `t‐cc jave graj pa`kt H ta pa`kt O bkd sa ak. @k e﬎eot e﬎ eot tle hbcc drapped bt tle haukdbry w`cc akcy meep jav`kf bcakf tle haukdbry bkd kat tle `kker surgboe. Tle ourcs ak tle surgboe obkoec ebol atler aut, jbm`kf `t seej c`me tle veotar flecd `s iust bot`kf bcakf tle t le per`plery, kat tlrauflaut tle surgboe.

 

?. OUXW@C@KEBX OAAXD@KBTE _P_TEJ_ _a gbr `k tl`s haam, wl`ce deflk`kf veotars bkd atler stu﬎, we used obrtes`bk ar reotbkfucbr oaard`kbte systej. Vl`ce obrtes`bk oaard`kbte systej `s tle s`jpcest bkd tle jast papucbr oaard`kbte systej, tlese bre kat tle jast oakvek`ekt ta use wlek `t oajes ta prbot`obc bppc`obt`ak. @k tl`s olbpter we‐cc hr`e﬈y `ktraduoe 8 atler oaard`kbte systejs9 Oyc`kdr`obc bkd _pler`obc. Lawever, gar s`jpc`o`ty we‐cc jastcy st`om w`tl us`kf obrtes`bk oaard`kbtes tlrauflaut tl`s haam.

?.2 OPC@KDX@OBC OAAXD@KBTE _P_TEJ @k Oyc`kdr`obc oaard`kbte systej, b pa`kt `s spboe `s speo`fled `k terjs ag, tle d`st d`stbko bkoe e ag tl tle e pa`kt pa`kt graj graj tle tle z-bx`s z-bx`s ( ) tle bkfce b lbcg pcbke oaktb`k`kf tle pa`kt jbmes w`tl tle tle x- bx`s bx`s `k tle tle bkt`o bkt`ocao caomw` mw`se se d`re d`reot` ot`ak ak ( ).  Tl`s bkfce `s obcced tle bz`jutl. tle d`stbkoe ag tle pa`kt graj tle xy pcbke (z) (sbje bs `k obrtes`bk oaard`kbtes)

 

@k bky oaard`kbte systej b pa`kt obk he deflked d eflked hy tle `kterseot`ak ag ? artlafakbc surgboes. @k obse ag obrtes`bk oaard`kbtes, tlese surgboes were ? pcbkes. Gar oyc`kdr`obc oaard`kbte systej tlese surgboes bre b oyc`kder, b pcbke bkd b lbcg pcbke bs slawk hecaw.

 

 Ta oaver tle ekt`re spboe, tle rbd`us ag tle oyc`kder obk he  Ta vbr`ed graj > ta , tle bz`jutl bkfce obk he vbr`ed graj > ta 8 -

ta ta

bkd tle tle d`stbk d`stbkoe oe graj graj tle tle xy pcb pcbke ke obk obk he vbr vbr`ed `ed gra graj j .

 

Oyc`kdr`obc ta Obrtes`bk oaard`kbtes9

 

Weotars `k Oyc`kdr`obc oaard`kbtes9

?.8 _^LEX@OBC OAAXD@KBTE _P_TEJ @k _pler`obc oaard`kbte systej, b pa`kt `s spboe `s speo`fled `k terjs ag, tle d`stbkoe ag tle pa`kt graj tle ar`f`k (r) tle bkfce tle c`ke ia`k`kf tle pa`kt ta tle ar`f`k jbmes w`tl jbm w`tl tle z- bx bx`s `s ( ). tle bkfce b lbcg pcbke oaktb`k`kf tle pa`kt jbmes w`tl tle xx- bx`s bx`s `k tle tle bkt`o bkt`ocao caomw` mw`se se d`re d`reot` ot`ak ak ( (sbje bs tle bz`jutl bkfce `k oyc`kdr`obc oyc`kdr `obc oaard`kbtes)

)

 

@k spler`obc oaard`kbte pa`kt `s deflked hyslawk tle `kterseot`ak ag b splere,systej, b oake bbkd b lbcg pcbke bs hecaw.

 

 Ta  T a oaver tle ekt`re spboe, tle rbd`us ag tle splere (r) obk he vbr`ed vbr `ed graj graj > ta

, tle scbkt scbkt bkfce bkfce ag tle oake oake ( ) obk he

vbr`ed vbr `ed graj graj > ta

rbd`bk rbd`bks, s, bkd tle bz`jutl bz`jutl bkfce bkfce (

he vbr`e vbr`ed d ggra raj j > tta a 8 rbd` rbd`bk bks. s.

) obk

 

_pler`obc ta Obrtes`bk oaard`kbtes9

 

Weotars `k _pler`obc oaard`kbtes9

 

4. ECEOTXA_TBT@O_

4.2 OACAUJH‐_ O ACAUJH‐_ CBV CBV Hetweek 26 ak tle haukdbry. Tle Cbpcboe‐s equbt`ak lbs bk `kterest`kf haukdbry. praperty, tlbt `ts sacut`ak (we lbve bssujed 8 sacut`aks lere) obkkat lbve caobc jbx`jb ar j`k`jb bkd `ts extreje vbcues just aoour bt tle haukdbr`es. Fa`kf hy tl`s praperty tle jbx`juj bkd j`k`juj ag W ? bre hatl zera, wl`ol `jpc`es tlbt W? just he zera everywlere bkd kat iust tle haukdbr`es.  Tleregare, W2 : W8 everywlere bkd tl`s gbcs`fles aur bssujpt`ak tlbt tle Cbpcboe‐s equbt`ak lbs 8 d`﬎erekt sacut`aks. Tle uk`quekess tlearej bcsa bppc`es ta ^a`ssak‐s equbt`ak bkd tle praag `s `dekt`obc.

 

7. OAKDUOTAX_ & D@ECEOTX@O_ 7.2 OUXXEKT Ve lbve d`soussed `k cekftl bhaut stbt`o olbrfes bkd tle plekajekak bssao`bted w`tl tlej. Kaw we‐cc turk aur bttekt`ak ta olbrfes `k jat`ak. @k tl`s olbpter we‐cc tbme sc`flt detaur bkd study bhaut law d`﬎erekt jbter`bcs helbve `k bk Eceotr`o flecd. Olbrfes `k jat`ak oakst`tute bk eceotr`o ourrekt. Bk Eceotr`o Ourrekt Ourr ekt obk he deflked bs tle rbte ag ﬈aw ag olbrfe

(eceotraks). Jbtlejbt`obccy,  Tle uk`t ag ourrekt ourrekt `s Bjpere, kbjed bg bgter ter Grekol Grekol jbtlejbt`o`bk bkd plys`o`st Bkdrà-Jbr`e Bjpìre. Ake bjpere ag ourrekt represekts represekts ake oaucajh ag eceotr`obc olbrfe jav`kf pbst b speo`flo pa`kt `k ake seoakd. ^r`ar ta eceotr`o`ty he`kf `dekt`fled w`tl tle eceotrak, `t wbs tlauflt tlbt tle javejekt ag pas`t`ve olbrfes `s wlbt praduoed eceotr`o ourrekt. Gar tl`s rebsak, tle d`reot`ak ag  ourrekt ourr ekt wbs bssujed ta he `k tle d`reot`ak ag ﬈aw ag  pas`t`ve olbrfes. _uol b ourr ourrekt ekt `s terjed bs oakvekt`akbc ourrekt. ourr ekt. Cbter ak, bs `t wbs estbhc`sled tlbt tle javejekt ag eceotraks `s wlbt `s botubccy respaks`hce gar eceotr`o ourrekt. Tl`s ourrekt `s terjed bs Eceotrak ourrekt bkd `ts d`reot`ak bssujed ta he appas`te ta tle ﬈aw ag eceotraks. Gar bkbcys`s purpases `t `s ebs`er `jbf`ke ourrekt bs tle ﬈aw ag pas`t`ve olbrfes.

 

7.8 OAKDUOTAX_ Oakduotars lbve taks ag caasecy haukd eceotraks ar gree eceotraks `ks`de tlej bkd due ta tlerjbc ekerfy, tley meep jav`kf `k rbkdaj d`reot`ak. Ebol ag tlese sjbcc javejekts oaktr`hute ta bk Eceotr`o ourrekt. ourrekt. Hut tle ourrekts praduoed hy tlese gree eceotraks bre `k rbkdaj d`reot`ak (appas`te ta tle d`reot`ak ag tle`r jat`ak) bkd wlek we oaks`der tle oakduotar bs b wlace, tlese ourrekts obkoec ebol atler aut bkd tle ket ourrekt `s zera.

Vlek bk eceotr`o flecd `s bppc`ed, tlese gree eceotraks exper`ekoe b oaucajh`o garoe bkd bs b resuct tle eceotraks w`cc stbrt jav`kf `k tle d`reot`ak appas`te ta tle eceotr`o flecd. Hut tle eceotraks w`cc st`cc passess tle rbkdaj vecao`ty oajpakekt due ta tlerjbc ekerfy. Heobuse ag tl`s tle eceotraks wak‐t pbrbde `k b strb`flt c`ke c`me ake j`flt `jbf`ke, tle rbkdajkess iust deorebses bkd tley sart ag  jave `k b oajjak d`reot`ak.

 

Bs tle eceotraks p`om up speed, tley oacc`de w`tl bdiboekt btajs bkd tle wbccs ag tle oakduotar bkd bs b resuct tley rehaukd `k rbkdaj d`reot`ak. Tlek tle eceotraks akoe bfb`k p`om up speed bkd tl`s praoess repebts `tsecg `tsecg.. Tl`s praoess `s obcced dr`gt`kf bkd tle bverbfe vecao`ty bttb`ked hy tle eceotraks dur`kf tl`s praoess `s obcced tle dr`gt vecao`ty (vd ).

 

 Tle dr`gt vecao`ty vecao`ty `s d`reotcy prapart`akbc ta ta tle Eceotr`o flecd.

Vlere

`s obcced tle jah`c`ty ag eceotraks.

7.? OUXXEKT DEK_@TP  @k flecd tleary we bre jare `kterested `k plekajekak tlbt aoour bt pa`kt rbtler tlbk w`tl`k b cbrfe ref`ak. _a `t `s oakvek`ekt ta `ktraduoe b kew qubkt`ty obcced ourrekt deks`ty, dekated hy I. Tle ourrekt deks`ty bt b pa`kt `s deflked bs tle ourrekt tlbt pbsses tlraufl b uk`t surgboe

 

breb karjbc ta tle d`reot`ak ag ourrekt. @t `s b veotar qubkt`ty, bkd `ts uk`t `s B/j 8 . Jbtlejbt`obccy,

Bt tle pbrt`oce cevec, tle ourrekt deks`ty veotar `kd`obtes tle caobc dr`gt ag eceotraks bt b pa`kt `ks`de b jbter`bc bkd `ts d`reot`ak `s appas`te ta tle d`reot`ak ag ﬈aw ag eceotraks bt tle pa`kt. Kaw cet‐s der`ve tle recbt`ak hetweek tle ourrekt deks`ty bkd tle dr`gt vecao`ty. Oaks`der b oakduotar ag orass

 

seot`akbc breb B bs slawk `k tle flfure.

_uppase, `k tl`s oakduotar tle eceotraks bre bre jav`kf w`tl b dr`gt vecao`ty (v d ). Tleregare `k b sjbcc t`je grbje dt  ,  , tle eceotraks waucd lbve trbvecced b d`stbkoe c  :  : v d dt  .  . @k tl`s t`je per`ad, eceotraks `k tle vacuje f`vek hy W : B c : B v   waucd waucd lbve trbvecced tlraufl tle w`re. @g tlere bre k d dt  eceotraks per uk`t vacuje, tlere w`cc he kW eceotraks `k tle vacuje W , bkd tle tatbc olbrfe passessed hy tlese eceotraks w`cc he dR : k (  B v d dt  ) ( -e)  .

 

Lere tle qubkt`ty (-ke)  `s katl`kf hut tle gree eceotrak olbrfe deks olb deks`t `ty y ( e ) ag tle oakduotar. oakduotar. Tle kefbt`ve s`fk ocebrcy `kd`obtes tlbt tle ourrekt ﬈aw `s `k tle d`reot`ak appas`te ta tle ﬈aw ag eceotraks.

7.4 ALJ‐_ CBV Ve bre bcc gbj`c`br w`tl tle ocbss`o vers`ak ag Alj's cbw, W:@X, wlere W `s tle vactbfe, @ `s tle ourrekt bkd X `s tle res`stbkoe. @k flecd tleary we obk express tle alj‐s cbw `k b d`﬎erekt garj, `k terjs ag Ourrekt deks`ty I bkd Eceotr`o flecd `kteks`ty E. Oaks`derourrekt b oakduotar ag cekftl  bkd orass-seot`akbc orass-seot`akbc breb B obrry`kf @  tlraufl  tlraufl `t. c  bkd

 Tle oakstbkt `s obcced tle Oakduot`v`ty ag tle jbter`bc bkd `ts uk`t `s _`ejeks per jeter. Gar bk `satrap`o oakduot`kf

 

jed`uj, tle oakduot`v`ty `s tle sbje bcakf bcc d`reot`aks,

lekoe tl`s recbt`ak obk he fekerbc`zed bs,  Tl`s `s tle veotar garj garj ar tle pa`kt garj garj ag tle Alj's cbw. cbw.

7.5 OAKT@KU@TP ER UBT@AK UBT@AK Cbw ag oakservbt`ak ag olbrfe stbtes tlbt – Eceotr`o olbrfe  obk ke`tler he orebted kar he destrayed  ‚ bkd we mkaw tlbt Eceotr`o ourrekt `s katl`kf hut tle ﬈aw ag olbrfes. _a graj oajjak sekse we obk `kger tlbt bky eceotr`o ourrekt ourr ekt ﬈aw`kf aut graj b ocased surgboe just he equbc ta tle rbte ag deorebse ag tle olbrfe ekocased hy tl`s ocased surgboe. Jbtlejbt`obccy,

 

 Tl`s `s tle pa`kt garj garj ag tle Oakt`ku`ty equbt`ak. equbt`ak. Bpbrt graj bcc tle jbtlejbt`obc katbt`ak tl`s resuct `s b d`reot oaksequekoe ag tle cbw ag oakservbt`ak ag olbrfe bkd katl`kf ecse. Bs gbr bs tle kefbt`ve s`fk `s oakoerked, tlbt‐s heobuse `k tl`s obse we oaks`dered ourrekt ourrekt bs tle ﬈aw ag  pas`t`ve olbrfes. Lbd we oaks`dered eceotrak ﬈aw, tlek tle ourrekt ourr ekt ﬈aw`kf aut waucd he equbc ta tle rbte ag `korebse ag  eceotraks w`tl`k tle ocased surgboe, `k wl`ol obse tle kefbt`ve s`fk waucdk‐t he requ`red, hut tle equbt`ak w`cc rejb`k tle sbje bs tle olbrfe deks`ty w`cc he kefbt`ve.

7.7 D@ECEOTX@O_ Ukc`me oakduotars, d`eceotr`o jbter`bcs da kat passess gree eceotraks. Tle eceotraks presekt `k b d`eceotr`o bre strakfcy haukd ta tle btaj bkd lekoe tley obkkat pbrt`o`pbte `k

 

oakduot`ak. Hut wlek suhieot ta bk Eceotr`o flecd, tlese haukd eceotraks w`cc exper`ekoe b oaucajh`o garoe bkd tl`s w`cc sl`gt tle`r recbt`ve pas`t`ak. @k b pacbr`zed btaj, tle oekter ag tle eceotrak ocaud w`cc ka cakfer bc`fk w`tl tle oekter(slawk ag tle `k kuoceus (tle hecaw). pratak ocaud), sarts tle flfure Hut tlegarj`kf btaj bsbbd`pace wlaceag  w`cc st`cc rejb`k keutrbc bs tle ka ag eceotraks `s equbc ta tle ka ag prataks. p rataks.

@k olbpter 4, we jekt`aked tlbt tle Eceotr`o ﬈ux deks`ty(D) `s tlbt pbrt ag tle Eceotr`o flecd `kteks`ty(E) tlbt daesk‐t tbme tle e﬎eot ag tle jed`uj `kta oaks`derbt`ak. Tle j`ss`kf c`km `s obcced ^acbr ^acbr`zbt`ak(^). `zbt`ak(^). ^acbr`zbt`ak ^acbr`zbt`ak ag b d`eceotr`o `s deflked bs tle tatbc d`pace jajekt `kduoed per uk`t vacuje ag b d`eceotr`o jbter`bc. @t `s b jebsure ag tle bdd`t`akbc ﬈ux deks`ty br`s`kf due ta pacbr`zbt`ak ag b d`eceotr`o. Iust c`me Eceotr`o ﬈ux deks`ty D, `t `s bcsa b veotar qubkt`ty bkd lbs tle sbje uk`t Oaucajhs/j8 .

 

 Tle pacbr`zbt`ak ag b d`eceotr`o d`eceotr`o jbter`bc `s d`reotcy d`reotcy prapart`akbcc ta tle bppc`ed eceotr`o flecd. Tle twa bre recbted prapart`akb bs, Vlere > `s obcced tle eceotr`o susoept`h`c`ty ag tle d`eceotr`o. @t teccs us law prake ar susoept`hce b d`eceotr`o `s ta pacbr`zbt`ak `k tle presekoe ag bk eceotr`o flecd. Us`kf tle bhave recbt`ak, tle Jbfket`o ﬈ux deks`ty obk he expressed expressed bs,  Tle gbotar (2+ >  ) `s obcced tle d`eceotr`o oakstbkt ar tle recbt`ve perj`tt`v`ty perj`tt`v`ty ag b jed`uj w`tl respeot ta gree spboe.

7.6 HAUKDBXP OAKD@T@AK_ ^rbot`obccy, bk eceotr`o flecd daesk‐t bcwbys rejb`k oakflked ^rbot`obccy, ta gree spboe ar b jed`uj. Agtek `t lbs ta pbss graj ake jed`uj ta bkatler, `t ta oaucd grajd`eceotr`o, gree spboeartagraj b d`eceotr`o ar graj ake d`eceotr`o bkatler b oakduotar ta gree spboe eto. Bky wby `rrespeot`ve ag tle type ag jed`b, bk Eceotr`o flecd pbss`kf hetweek 8 d`﬎erekt jed`b sbt`sgy saje haukdbry oakd`t`aks bt tle `ktergboe. @k tl`s seot`ak we‐cc try ta estbhc`sl tle fekerbc haukdbry oakd`t`aks gar bk eceotr`o flecd bt tle `ktergboe hetweek 8 jed`b. Oaks`der Eceotr`o flecd pbss`kf graj ake jed`uj ta bkatler borass b haukdbry bs slawk hecaw.

 

Ve obk resacve tle Eceotr`o flecd `k hatl jed`ujs bcakf tbkfekt`bc bkd karjbc d`reot`ak. Tlek tle 8 haukdbry oakd`t`aks bre ahtb`ked hy oajpbr`kf tle tbkfekt`bc bkd tle karjbc oajpakekts ak hatl s`des w`tl ebol atler.

2. Haukdbry oakd`t`ak gar tbkfekt`bc Eceotr`o flecd  Ta ahtb`k tle haukdbry oakd`t`ak gar  Ta gar tle tbkfekt`bc oajpakekt ag tle Eceotr`o flecd, oaks`der b ocased oaktaur bhod bs slawk hecaw.

 

Ve mkaw tlbt stbt`o eceotr`o flecd `s oakservbt`ve `k kbture `.e. tle c`ke `ktefrbc ag tle eceotr`o flecd tbmek bcakf b ocased pbtl `s zera. Tleregare,

_a tle tbkfekt`bc oajpakekt ag tle Eceotr`o flecd rejb`ks ukbctered borass borass bk `ktergboe hetweek 8 jed`b. 8.

Haukdbry oakd`t`ak gar karjbc Eceotr`o flecd

 Ta ahtb`k tle haukdbry oakd`t`ak gar  Ta gar tle karjbc oajpakekt ag tle Eceotr`o flecd, oaks`der b fbuss`bk p`cchax (iust b ocased ?d surgboe) bs slawk hecaw.

 

Kaw we obk bppcy Fbuss‐s cbw ta tl`s Fbuss`bk p`cchax bs gaccaws,

 

Vlere Ϗ `s tle sleet olbrfe deks`ty bt tle `ktergboe. `ktergboe. Tl`s haukdbry oakd`t`ak `jpc`es tlbt tle Karjbc oajpakekt ag  tle Eceotr`o flecd heoajes d`soakt`kuaus bs `t pbsses hetweek 8 jed`b.

 

6. JBFKETA_TBT@O_ 6.2 JBFKET@O G@ECD _tbt`o olbrfes praduoe eceotr`o flecd braukd tlej, hut olbrfes `k jat`ak bcsa praduoes bkatler m`kd ag flecd obcced tle jbfket`o flecd. @k 2, Dbk`sl plys`o`st Lbks Olr`st`bk Arsted ahserved tlbt b jbfket`o flecd surraukds b ourrekt obrry`kf ahieot.  Tle jast gukdbjektbc `deb `k eceotrajbfket`sj eceotrajbfket`sj `s tlbt tlere `s jbfket`o flecd surraukd`kf every ourrekt obrry`kf ahieot. Tlese jbfket`o flecds tbme tle slbpe ag oakoektr`o r`kfs braukd b strb`flt w`re, obcced tle jbfket`o flecd c`kes. Cbrfer tle ourrekt ﬈aw`kf tlraufl w`re, jare tle ka. ag  jbfket`o flecd c`kes. Tlese c`kes bre kat rbkdaj, tley lbve b d`reot`ak bkd `t obk he deterj`ked hy us`kf tle X`flt lbkd tlujh ruce. @t faes c`me tl`s, `g yau pa`kt yaur tlujh `k  tle d`reot`ak ag tle ourrekt, tlek tle flkfers ourc `k tle 

 

d`reot`ak ag tle flecd c`kes  

_`j`cbrcy, wlek ourrekt ﬈aws tlraufl tlraufl b oa`c, b jbfket`o flecd `s fekerbted braukd `t, suol tlbt tle oa`c bots c`me b jbfket w`tl b kartl bkd sautl pacbr`ty. Tle pbtterk ag flecd c`kes `s bs slawk hecaw.

 

Kate tlbt tle flecd c`kes bre oakoektr`o `g yau oaks`der b t`ky part`ak ag tle oa`c, hut tlese flecd c`kes bdd bkd obkoec ebol atler f`v`kf us tl`s e﬎eot`ve pbtterk. Hy tle wby tlese sarts ag oa`cs bre obcced _aceka`ds.

6.8 JBFKET@O G@ECD @KTEK_@TP  & JBFKET@O GCU\ DEK_@TP   Iust c`me c`me law tle Eceotr`o flecd `kteks`ty `kteks`ty `s b jebsure ag  strekftl ag tle Eceotr`o flecd, tle Jbfket`o flecd @kteks`ty dekated bs L, `s b jebsure ag tle strekftl ag b Jbfket`o flecd. @t `s b veotar qubkt`ty bkd `ts uk`t `s Bjperes per jetre (B/j). Jbfket`o ﬈ux deks`ty (H) ak tle atler lbkd `s tle bkbcafaus ta Eceotr`o ﬈ux deks`ty (D) `k eceotrastbt`os. eceotrastbt`os. @t `s tle tatbc kujher ag jbfket`o c`kes ag garoe pbss`kf tlraufl b uk`t breb. @t `s bcsa b veotar qubkt`ty bkd `ts uk`t `s Veher per jetre squbre (Vh/j8 ) ar Tescb.

 

Hatl jbfket`o flecd `kteks`ty & jbfket`o ﬈ux deks`ty bre jebsures ag strekftl ag jbfket`o flecd, hut tle d`﬎erekoe `s tlbt tle Jbfket`o ﬈ux deks`ty depekds ak tle kbture ag  jed`uj.  Tle twa qubkt`t`es bre bre recbted ta ebol a atler tler bs, Vlere `s obcced tle perjebh`c`ty ag tle jed`uj. @t `s b jebsure ag tle res`stbkoe ag b jbter`bc bfb`kst tle garjbt`ak ag b jbfket`o flecd. Gar gree spboe, perjebh`c`ty `s dekated bs

 bkd lbs b vbcue ag 4 x 2> -6 L/j.

>

6.? H@AT _BWBXT CBV H`at- _bvbrt Cbw stbtes tlbt –tle jbfket`o flecd `kteks`ty  praduoed due ta b ourrekt ecejekt bt b pa`kt `s  prapart`akbc ta tle praduot ag tle ourrekt bkd tle  d`﬎erekt`bc cekftl, s`ke ag tle bkfce hetweek tle  ecejekt bkd tle c`ke ia`k`kf tle pa`kt ta tle ecejekt, bkd `kversecy prapart`akbc ta tle squbre ag tle  d`stbkoe hetweek tle pa`kt bkd tle ecejekt‚  .  .

 

Jbtlejbt`obccy, tle H`at _bvbrt cbw obk he expressed bs,

 Tle oakstbkt ag prapart`akbc`ty prapart`akbc`ty `s m : 2 /4ώ. @k veotar garj, tle H`at _bvbrt cbw obk he expressed expressed bs,

 

6.4 B^^C@OB B^^C @OBT@A T@AK K AG H@AT-_BW H@AT-_BWBXT BXT CBV CB V 2. @kflk`tecy Jbfket`o cakf flecd strb`flt `kteks`ty due ta oakduotar9 Oaks`der bk `kflk`tecy cakf strb`flt oakduotar tlbt obrr`es b ourrekt @ bs slawk `k tle flfure. Ve keed ta flkd tle Jbfket`o flecd `kteks`ty bt b pa`kt ^ bt b perpekd`oucbr d`stbkoe r graj tle oakduotar. Oaks`der b sjbcc ourrekt ecejekt dc  bt  bt b d`stbkoe c  graj  graj tle ar`f`k.

Booard`kf ta H`at-_bvbrt cbw, tle Eceotr`o flecd due ta tl`s Booard`kf ecejekt dc  bt  bt ^ `s,

 

8. Jbfket`o flecd `kteks`ty due ta b O`roucbr caap9 Oaks`der b o`roucbr caap ag rbd`us r obrry`kf b ourrekt @ bs slawk `k tle flfure.

 

z graj ecejekt  Tle b d`stbkoe tle oekter tle o`roucbr dc  , r`kf.pa`kt Kaw ^ `g `s webtoaks`der b ourrekt  ag , tlek booard`kf ta H`at-_bvbrt cbw tle Jbfket`o flecd `kteks`ty due

 

ta tl`s ecejekt bt pa`kt ^ `s,

 Tle Jbfket`o flecd @kteks`ty bt tle oekter ag tle O`roucbr O`roucbr r`kf

(z : >) `s f`vek hy,

6.5 BJ^EXE‐_ O@XOU@TBC CBV

 

Graj ake cbw ta tle kext. Tle bjpere‐s o`rou`tbc cbw `s tle jbfketastbt`os vers`ak ag tle Fbuss‐s cbw gar Eceotrastbt`os. Bjpere‐s o`rou`tbc cbw stbtes tlbt –tle c`ke `ktefrbc ag tle  Jbfket`o flecd `kteks`ty L bhaut bky ocased pbtl `s  equbc ta tle d`reot ourrekt ekocased hy tlbt pbtl‚  .  .

 Tl`s ocased pbtl obk he ag bky slbpe, o`roucbr, o`roucbr, squbre, ar bkytl`kf rbkdaj bs slawk `k tle flfure bhave. Jbtlejbt`obccy, tle Bjpere‐s cbw obk he expressed expressed bs,

@k terjs ag tle Jbfket`o flecd deks`ty H, tle bjpere‐s cbw obk he expressed bs,

 

Bkatler wby ta express tle Bjpere‐s cbw `s `k terjs ag tle ourrekt deks`ty I.

@k pa`kt garj, tle Bjpere‐s o`rou`tbc cbw obk he expressed

bs, (Der`ve us`kf _tames tlearej) @t `s `jpartbkt ta rejejher tlbt tle Bjpere‐s o`rou`tbc cbw `s akcy bppc`obhce ta stebdy jbfket`o flecds (`.e. tlase praduoed hy DO ourrekts). ourrekts). Tl`s cbw `s tle predeoessar ta ake ag tle Jbxwecc‐s equbt`aks.  Tle exbot Jbxwecc‐s Jbxwecc‐s equbt`ak oarrespakd`kf oarrespakd`kf ta tl`s cbw, wbs devecaped bgter Jbxwecc l`jsecg jad`fled tl`s cbw ta `kocude tle obse ag vbry`kf jbfket`o flecds (jare `k olbpter 2>).

6.7 B^^C@OBT@AK AG TLE BJ^EXE‐_ O@XOU@T O@XOU@TBC BC CB CBV V  Tle Bjpere‐s o`rou`tbc o`rou`tbc cbw prav`des prav`des bk ebs`er jetlad ta flkd tle Jbfket`o flecd `kteks`ty tlbk us`kf tle H`at-_bvbrt cbw `k obses wlere syjjetry `s presekt.

2. Eceotr`o flecd `kteks`ty due ta @kflk`tecy cakf strb`flt oakduotar9 Oaks`der bk `kflk`tecy cakf strb`flt oakduotar obrry`kf ourrekt @ . Ta flkd tle jbfket`o flecd `kteks`ty bt b pa`kt ^ bt

 

b d`stbkoe r graj tle oakduotar, oaks`der bk bjper`bk pbtl ekocas`kf tle oakduotar pbss`kf tlraufl tle pa`kt. n

Kaw `g we oaks`der b d`﬎erekt`bc cekftl dc  bt  bt pa`kt ^, tlek booard`kf ta Bjpere‐s o`rou`tbc cbw,

 

8. Eceotr`o flecd `kteks`ty due ta @kflk`te pcbke sleet9 Oaks`der bk `kflk`te pcbke sleet w`tl ourrekt ourrekt ﬈aw`kf bs slawk `k tle flfure. Tle surgboe ourrekt ourrekt deks`ty `s bssujed ta he Is . Kaw oaks`der bk bjper`bk caap ag w`dtl b bs slawk.

 

Booard`kf ta Bjpere‐s o`rou`tbc cbw,

 

Oajpbre tl`s ta tle equbt`ak E : ς/ 8εa we lbd gar tle eceotr`o flecd graj bk `kflk`te sleet ag olbrfe.

6.6 JBFKET@O JAKA^ACE_ @k eceotr`o`ty, tlere bre pas`t`ve olbrfes bkd kefbt`ve k efbt`ve olbrfes bkd tley obk rejb`k `sacbted graj ebol atler. Hut `k obse ag Jbfket`sj, tlere `s ka suol tl`kf bs pas`t`ve pace bkd b kefbt`ve pace. _ure, every jbfket lbs b Kartl bkd b sautl pace, hut tley bcwbys ex`st `k pb`rs, `.e. Jbfket`o jakapaces da kat ex`st, bt cebst we lbvek‐t gaukd bky t`cc dbte. Kat oakv`koed yet1 Try hrebm`kf b hbr jbfket `kta 8 p`eoes, ebol kew p`eoe w`cc helbve bs b sepbrbte jbfket w`tl `ts awk kartl bkd sautl paces. Pau obk da tl`s bs juol t`jes bs yau wbkt, hut tle resuct w`cc he tle sbje, yau w`cc sjbccer bkd sjbccer p`eoes ag jbfkets, hut kever b sepbrbte kartl p`eoe bkd b sepbrbte sautl p`eoe. Tlere bre saje tleary ﬈abt`kf braukd bhaut tl`s helbv`ar ag  jbfkets, hut exbot rebsak `s st`cc cbrfecy b jystery.

 

_a `k rebc`ty, tlere `s ka sauroe ar s`km `k obse ag jbfkets ukc`me eceotr`o flecds. Tle jbfket`o flecd c`kes jby seej c`me tley ar`f`kbte bt tle kartl pace bkd terj`kbte bt tle sautl pace, hut bcc tley bre da`kf `s ta o`roucbte hbom ak tlejsecves, garj`kf oakt`kuaus caaps.

 

Bk `jpartbkt `kgerekoe obk he jbde graj tl`s helbv`ar ag  jbfkets, tlbt –Tle jbfket`o ﬈ux tlraufl b ocased  surgboe `s bcwbys zera‚  .  . Tl`s resuct `s obcced tle Fbuss‐s cbw gar Jbfket`sj. Oakkeoted tle dats yet1 Tle rebsak`kf `s s`jpce. Heobuse jbfket`o paces dak‐t d ak‐t ex`st `k `sacbt`ak, bky flecd c`kes tlbt ekter b vacuje w`cc `kev`tbhcy ex`t graj tle appas`te s`de, sa bs ta oajpcete tle caap, tleregare tle ket ﬈ux w`cc he zera. zera.

Jbtlejbt`obccy, tl`s cbw obk he put bs,  Tl`s `s ake ag tle 4 Jbxwecc‐s Jbxwecc‐s equbt`aks.

 

@k Weotar garj, tl`s cbw obk he expressed bs,