Unit 1-Fluid Mechanics

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)1G5>>2)1)1

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

IJGFETH\FU

IJGFETH\FU

Of`fzak Ijgfeth|f

Of`fzak Ijgfeth|f 7 7 Ti c`iw+ v`dfzuta`d a`d Ti c`iw+ v`dfzuta`d a`d a~~kx tbf ma~~kx tbf mfauvzfmf`t il tfm~fzatvzf+fauvzfmf`t il tfm~fzatvzf+ ~zfuuvzf a`d ~bxuheak ~zi~fzthfu il lkvhd.

~zfuuvzf a`d ~bxuheak ~zi~fzthfu il lkvhd.

U~fehlhe Ijgfeth|fu

U~fehlhe Ijgfeth|fu 7 7 At tbf f`d At tbf f`d il tbf v`ht xiv il tbf v`ht xiv ubivkd jf ajkf ti ubivkd jf ajkf ti 77



 khut tbf tfm~fzatvzf ueakfu.khut tbf tfm~fzatvzf ueakfu.



 ei`|fzt tbf tfm~fzatvzf ti Efkuhvu "·ei`|fzt tbf tfm~fzatvzf ti Efkuhvu "· E E (+ Labzf`bfht "·(+ Labzf`bfht "· L  L (+(+ Za`ch`f "

Za`ch`f " Z Z(+ Cfk|h` "(+ Cfk|h` " C  C ( ueakfu.( ueakfu.



 dflh`f ^zfuuvzf "dflh`f ^zfuuvzf " ^ ^ ( + Atmiu~bfzhe ^zfuuvzf "( + Atmiu~bfzhe ^zfuuvzf " ^  ^ atmatm( + Oavof( + Oavof

^zfuuvzf "

^zfuuvzf " ^  ^ OO(( + Ajuikvtf ^zfuuvzf "+ Ajuikvtf ^zfuuvzf " ^  ^ AA( a`d \( a`d \aevvm aevvm "" ^  ^ || ((



 eakevkatf ~zfuuvzf oavof vuh`o tbf lizmvka oh|f`.eakevkatf ~zfuuvzf oavof vuh`o tbf lizmvka oh|f`.



 dhllfzf`thatf mauu df`uhtx "dhllfzf`thatf mauu df`uhtx " ρ ρ(+ u~fehlhe wfhobt "(+ u~fehlhe wfhobt " ψ ψ (+ u~fehlhe(+ u~fehlhe

oza|htx "

oza|htx " u u ( + u~fehlhe |ikvmf "( + u~fehlhe |ikvmf " || ( a`d |hueiuhtx.( a`d |hueiuhtx.



 eakevkatf lkvhd ~zi~fzthfu vuh`o tbf lizmvka oh|f`.eakevkatf lkvhd ~zi~fzthfu vuh`o tbf lizmvka oh|f`.

V@HT 1

V@HT 1

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1):G5>>2)1):

H@TZIDVETHI

H@TZIDVETHI@ TI

@ TI LKVHD MFEBA@HEU

LKVHD MFEBA@HEU

Lkvhd Mfeba`heu - H`tzidvethi`

Lkvhd Mfeba`heu - H`tzidvethi`

  Lkvhd Mfeba`heu

  Lkvhd Mfeba`heu hu a ufethi` il a~~khfd mfeba`heu+ ei`efz`fd whtb tbf utathe a`dhu a ufethi` il a~~khfd mfeba`heu+ ei`efz`fd whtb tbf utathe a`d dx`amheu il khqvhdu a`d oaufu.

dx`amheu il khqvhdu a`d oaufu.

C`iwkfdof il lkvhd mfeba`heu hu fuuf`thak liz tbf ebfmheak f`oh`ffz+ jfeavuf tbf C`iwkfdof il lkvhd mfeba`heu hu fuuf`thak liz tbf ebfmheak f`oh`ffz+ jfeavuf tbf magizhtx il ebfmheak ~ziefuuh`o i~fzathi`u azf ei`dvetfd fhtbfz ~azthakkx iz titakkx h` magizhtx il ebfmheak ~ziefuuh`o i~fzathi`u azf ei`dvetfd fhtbfz ~azthakkx iz titakkx h` tbf lkvhd ~bauf.

tbf lkvhd ~bauf.

Tbf ba`dkh`o il khqvhdu hu mveb uhm~kfz+ ebfa~fz+ a`d kfuu tzivjkfuimf tba` ba`dkh`o Tbf ba`dkh`o il khqvhdu hu mveb uhm~kfz+ ebfa~fz+ a`d kfuu tzivjkfuimf tba` ba`dkh`o uikhdu.

uikhdu.

F|f` h` ma`x i~fzathi`u a uikhd hu ba`dkfd h` a lh`fkx dh|hdfd utatf ui tbat ht utaxu h` F|f` h` ma`x i~fzathi`u a uikhd hu ba`dkfd h` a lh`fkx dh|hdfd utatf ui tbat ht utaxu h` uvu~f`uhi` h` a lkvhd.

uvu~f`uhi` h` a lkvhd.

Lkvhdu a`d tbfhz ^zi~fzthfu Lkvhdu a`d tbfhz ^zi~fzthfu

Lkvhdu Lkvhdu

H` f|fzxdax khlf+ wf zfeio`hf tbzff utatfu il mattfz7 uikhd+ khqvhd a`d oau. Aktbivob H` f|fzxdax khlf+ wf zfeio`hf tbzff utatfu il mattfz7 uikhd+ khqvhd a`d oau. Aktbivob dhllfzf`t h` ma`x zfu~fetu+ khqvhdu a`d oaufu ba|f a eimmi` ebazaetfzhuthe h` wbheb dhllfzf`t h` ma`x zfu~fetu+ khqvhdu a`d oaufu ba|f a eimmi` ebazaetfzhuthe h` wbheb tbfx dhllfz lzim uikhdu. Jitb azf lkvhdu+ jvt kaecu tbf ajhkhtx il uikhdu ti illfz a tbfx dhllfz lzim uikhdu. Jitb azf lkvhdu+ jvt kaecu tbf ajhkhtx il uikhdu ti illfz a ~fzma`f`t zfuhuta`ef ti a dflizmh`o lizef.

~fzma`f`t zfuhuta`ef ti a dflizmh`o lizef. A

A  lkvhd  lkvhd hu a uvjuta`ef wbheb dflizmu ei`th`vivukx v`dfz tbf aethi` il ubfazh`o lizefu+hu a uvjuta`ef wbheb dflizmu ei`th`vivukx v`dfz tbf aethi` il ubfazh`o lizefu+ biwf|fz umakk tbfx max jf. Ei`|fzufkx+ hl a lkvhd hu at zfut+ tbfzf ea` jf `i ubfazh`o biwf|fz umakk tbfx max jf. Ei`|fzufkx+ hl a lkvhd hu at zfut+ tbfzf ea` jf `i ubfazh`o lizefu aeth`o a`d+ tbfzflizf+ akk lizefu h` tbf lkvhd mvut jf ~fz~f`dhevkaz ti tbf ~ka`fu lizefu aeth`o a`d+ tbfzflizf+ akk lizefu h` tbf lkvhd mvut jf ~fz~f`dhevkaz ti tbf ~ka`fu v~i` wbheb tbfx aet.

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)5G5>>2)1)5

1.1

1.1 TFM^FZATVZF TFM^FZATVZF V@HTUV@HTU

Tfm~fzatvzf ueakfu azf dflh`fd jx tbf `vmfzheak |akvf auuho`fd ti a uta`dazd lhrfd Tfm~fzatvzf ueakfu azf dflh`fd jx tbf `vmfzheak |akvf auuho`fd ti a uta`dazd lhrfd ~ih`t. Jx h`tfz`athi`ak aozffmf`t tbf uta`dazd lhrfd ~ih`t hu tbf fauhkx zf~zidvehjkf ~ih`t. Jx h`tfz`athi`ak aozffmf`t tbf uta`dazd lhrfd ~ih`t hu tbf fauhkx zf~zidvehjkf tzh~kf ~ih`t il watfz. Tbfuf azf zf~zfuf`tfd jx tbf utatf il fqvhkhjzhvm jftwff` utfam+ tzh~kf ~ih`t il watfz. Tbfuf azf zf~zfuf`tfd jx tbf utatf il fqvhkhjzhvm jftwff` utfam+ hef a`d khqvhd watfz.

hef a`d khqvhd watfz.

H` tbhu v`ht wf kfaz` biw ti ei`|fzt tfm~fzatvzfu h`ti Efkuhvu+ Labzf`bfht Cfk|h` a`d H` tbhu v`ht wf kfaz` biw ti ei`|fzt tfm~fzatvzfu h`ti Efkuhvu+ Labzf`bfht Cfk|h` a`d Za`ch`f ueakfu.

Za`ch`f ueakfu.

Tbf Efkuhvu tfm~fzatvzf ueakf vufu tbf v`ht dfozff Efkuhvu "·

Tbf Efkuhvu tfm~fzatvzf ueakf vufu tbf v`ht dfozff Efkuhvu "· E E (+ wbheb bau tbf uamf(+ wbheb bau tbf uamf mao`htvdf au tbf Cfk|h`. Tbvu tbf tfm~fzatvzf dhllfzf`efu azf hdf`theak i` jitb ueakfu. mao`htvdf au tbf Cfk|h`. Tbvu tbf tfm~fzatvzf dhllfzf`efu azf hdf`theak i` jitb ueakfu. Biwf|fz+ tbf fzi ~ih`t i` tbf Efkuhvu ueakf hu ubhltfd ti :?5

Biwf|fz+ tbf fzi ~ih`t i` tbf Efkuhvu ueakf hu ubhltfd ti :?5 C  C + au ubiw` jx tbf+ au ubiw` jx tbf likkiwh`o zfkathi`ubh~ jftwff` tbf Efkuhvu tfm~fzatvzf a`d tbf Cfk|h` tfm~fzatvzf7 likkiwh`o zfkathi`ubh~ jftwff` tbf Efkuhvu tfm~fzatvzf a`d tbf Cfk|h` tfm~fzatvzf7

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¾¾

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Jx dflh`hthi`+ tbf Za`ch`f ueakf+ tbf v`ht il wbheb hu tbf dfozff Za`ch`f "

Jx dflh`hthi`+ tbf Za`ch`f ueakf+ tbf v`ht il wbheb hu tbf dfozff Za`ch`f " Z Z( ( huhu ~zi~izthi`ak ti tbf Cfk|h` tfm~fzatvzf aeeizdh`o ti

~zi~izthi`ak ti tbf Cfk|h` tfm~fzatvzf aeeizdh`o ti

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A dfozff il tbf uamf uhf au tbat i` tbf Za`ch`f ueakf hu vufd h` tbf Labzf`bfht ueakf+ A dfozff il tbf uamf uhf au tbat i` tbf Za`ch`f ueakf hu vufd h` tbf Labzf`bfht ueakf+ jvt tbf fzi ~ih`t hu ubhltfd aeeizdh`o ti tbf zfkathi`

jvt tbf fzi ~ih`t hu ubhltfd aeeizdh`o ti tbf zfkathi`

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 H@^VT 

 H@^VT 

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)4G5>>2)1)4

uvjuthtvth`o Fqu. "1( a`d ":( h`ti Fq. "5(+ ht likkiwu tbat uvjuthtvth`o Fqu. "1( a`d ":( h`ti Fq. "5(+ ht likkiwu tbat

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Fram~kf 1.1 Fram~kf 1.1 Ei`|fzt :>>·

Ei`|fzt :>>·E E titi C  C ..

Uikvthi` ti Fram~kf 1.1 Uikvthi` ti Fram~kf 1.1 :?5 :?5 # # ¾¾ = = E E   C   C  :?5 :?5 :>> :>>## = =  C   C  4?5 4?5 = = Fram~kf 1.: Fram~kf 1.: Ei`|fzt :6> ·

Ei`|fzt :6> ·E E ti ·ti · L  L 

Uikvthi` ti Fram~kf 1.: Uikvthi` ti Fram~kf 1.: E  E   L   L == ## ¾¾ ¾¾ 5:5: 11..22

" " ((

:6>:6> 2 2 .. 1 1 5: 5:## = =  L   L  ¾¾ = = 42:42: Fram~kf 1.5 Fram~kf 1.5 Ei`|fzt 5<6 ·

Ei`|fzt 5<6 · L  L titi Z Z

Uikvthi` ti Fram~kf 1.5 Uikvthi` ti Fram~kf 1.5  L   L   Z  Z ==4<>4<>##¾¾ 5<6 5<6 4<> 4<>## = =  Z  Z 2:6 2:6 = =

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)6G5>>2)1)6

Fram~kf 1.4 Fram~kf 1.4

Ei`|fzt :>> ·

Ei`|fzt :>> · L  L titi Z Z

Uikvthi` ti Fram~kf 1.4 Uikvthi` ti Fram~kf 1.4  L   L   Z  Z == 4<>4<>##¾¾ :>> :>> 4<> 4<>## = =  Z  Z <<> <<> = = Fram~kf 1.6 Fram~kf 1.6 Ei`|fzt 46> Ei`|fzt 46> Z Z titi C  C  Uikvthi` ti Fram~kf 1.6 Uikvthi` ti Fram~kf 1.6 2 2 .. 1 1  Z  Z  C   C == 2 2 .. 1 1 46> 46> = =  C   C  :6> :6> = = Fram~kf 1.< Fram~kf 1.< Ei`|fzt 41> Ei`|fzt 41> C  C ti ·ti · L  L  Uikvthi` ti Fram~kf 1.< Uikvthi` ti Fram~kf 1.<

" "

:?5:?5

((

5:5: 2 2 .. 1 1 √√ ## = = ¾¾ L  L   C  C 

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41>41> :?5:?5

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5:5: 2 2 .. 1 1 √√ ## = =  L   L  ¾¾ = = :?2:?2..<<

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)<G5>>2)1)<

AETH\HTX 1A

AETH\HTX 1A

TFUT XIVZ V@DFZUTA@DH@O

TFUT XIVZ V@DFZUTA@DH@OJFLIZF XIV EI@TH@VF _HTB TBF @FRTJFLIZF XIV EI@TH@VF _HTB TBF @FRT H@^VT…!

H@^VT…!

1.1

1.1 _zhtf _zhtf tbf tbf lizmvka lizmvka ti ti ei`|fzt ei`|fzt tbf tbf likkiwh`o likkiwh`o tfm~fzatvzf tfm~fzatvzf ueakfu.ueakfu.

1.:

1.: Uik|f Uik|f tbf tbf ~zijkfmu ~zijkfmu jfkiw7jfkiw7

h(

h( Ahz f`tfzh`o a wft uezvjjfz hu at 165 ·Ahz f`tfzh`o a wft uezvjjfz hu at 165 · E E . _bat hu tbf tfm~fzatvzf fr~zfuufd h`. _bat hu tbf tfm~fzatvzf fr~zfuufd h` dfozff Za`ch`f9

dfozff Za`ch`f9 hh(

hh( Tbf oau utzfam tfm~fzatvzf f`tfzh`o a lajzhe lhktfz hu 41> ·Tbf oau utzfam tfm~fzatvzf f`tfzh`o a lajzhe lhktfz hu 41> · L  L . _bat hu tbf. _bat hu tbf tfm~fzatvzf fr~zfuufd h` dfozff Cfk|h`9

tfm~fzatvzf fr~zfuufd h` dfozff Cfk|h`9

··E 

E ti

ti C 

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·· L 

 L ti

ti Z

 Z

 Z

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)?G5>>2)1)?

LFFDJAEC I@ AETH\HTX 1A

LFFDJAEC I@ AETH\HTX 1A

1.1 1.1 1.: 1.: h(

h( Ei`|fzt Ei`|fzt tfm~fzatvzf tfm~fzatvzf lzim lzim Efkuhvu Efkuhvu ti ti Labzf`bfht.Labzf`bfht. ·· L  L = 5: # 1.2·= 5: # 1.2·E E 

·· L  L = 5: # 1.2"165(= 5: # 1.2"165( = 5>? ·

= 5>? · L  L 

Ei`|fzt tfm~fzatvzf lzim Labzf`bfht ti Za`ch`f. Ei`|fzt tfm~fzatvzf lzim Labzf`bfht ti Za`ch`f.  Z  Z = = ·· L  L # 4<># 4<>  Z  Z = = 5>? 5>? # # 4<>4<> = ?<? = ?<? Z Z

··E 

E ti

ti C 

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ti Z

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ti C 

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^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)2G5>>2)1)2

hh(

hh( Ei`|fzt Ei`|fzt tfm~fzatvzf tfm~fzatvzf lzim lzim Labzf`bfht Labzf`bfht ti ti Za`ch`f.Za`ch`f.  Z  Z = = ·· L  L # 4<># 4<>  Z  Z = = 41> 41> # # 4<>4<> = 2?> = 2?> Z Z

Ei`|fzt tfm~fzatvzf lzim Za`ch`f ti Cfk|h`. Ei`|fzt tfm~fzatvzf lzim Za`ch`f ti Cfk|h`.

2 2 .. 1 1  Z  Z  C   C == 2 2 .. 1 1 2?> 2?> = =  C   C  425 425 = =

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)8G5>>2)1)8

1.:

1.: DFLH@HTHI@ DFLH@HTHI@ IL IL ^ZFUUVZF^ZFUUVZF

Lkvhd whkk frfzt a `izmak lizef i` a`x jiv`dazx ht hu h` ei`taet whtb. Uh`ef tbfuf Lkvhd whkk frfzt a `izmak lizef i` a`x jiv`dazx ht hu h` ei`taet whtb. Uh`ef tbfuf jiv`dazhfu max jf kazof a`d tbf lizef max dhllfz lzim ~kaef ti ~kaef ht hu ei`|f`hf`t jiv`dazhfu max jf kazof a`d tbf lizef max dhllfz lzim ~kaef ti ~kaef ht hu ei`|f`hf`t ti wizc 

ti wizc h` tfzmu il h` tfzmu il ~zfuuvzf+~zfuuvzf+ ~ ~+ wbheb hu tbf lizef ~fz v`ht + wbheb hu tbf lizef ~fz v`ht azfa.azfa.

V`htu 7 @fwti`‘u ~fz uqvazf mftzf+

V`htu 7 @fwti`‘u ~fz uqvazf mftzf+ @m @m-:-:++comcom-1-1 u u-:-:.. "Tbf uamf v`ht hu akui c`iw` au

"Tbf uamf v`ht hu akui c`iw` au ^aueak+ ^a ^aueak+ ^a h.fh.f 1 ^a = 1@m1 ^a = 1@m-:-:(( Akui lzfqvf`tkx vufd hu tbf aktfz`ath|f

Akui lzfqvf`tkx vufd hu tbf aktfz`ath|f UH UH v`ht tbf jaz+v`ht tbf jaz+ wbfzf wbfzf 1 jaz = 1>1 jaz = 1>66 @m @m-:-:

 H@^VT 

 H@^VT 

a~~khfd  a~~khfd  hu hu  lizef  lizef tbf tbf wbheb wbheb i|fz  i|fz   Azfa  Azfa  Lizef  Lizef  ~zfuuvzf  ~zfuuvzf==

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)1>G5>>2)1)1>

1.5

1.5 MFAUVZFMF@T MFAUVZFMF@T IL IL ^ZFUUVZF ^ZFUUVZF OAVOFUOAVOFU

Ajuikvtf ^zfuuvzf+ Oavof ^zfuuvzf+ a`d \aevvm Ajuikvtf ^zfuuvzf+ Oavof ^zfuuvzf+ a`d \aevvm

H` a zfohi` uveb au ivtfz u~aef+ wbheb hu |hztvakkx |ihd il oaufu+ tbf ~zfuuvzf hu H` a zfohi` uveb au ivtfz u~aef+ wbheb hu |hztvakkx |ihd il oaufu+ tbf ~zfuuvzf hu fuuf`thakkx fzi. Uveb a ei`dhthi` ea` jf a~~ziaebfd |fzx `fazkx h` a kajizatizx wbf` fuuf`thakkx fzi. Uveb a ei`dhthi` ea` jf a~~ziaebfd |fzx `fazkx h` a kajizatizx wbf` a |aevvm ~vm~ hu vufd ti f|aevatf a jittkf. Tbf ~zfuuvzf h` a |aevvm hu eakkfd a |aevvm ~vm~ hu vufd ti f|aevatf a jittkf. Tbf ~zfuuvzf h` a |aevvm hu eakkfd ajuikvtf fzi

ajuikvtf fzi+ a`d akk ~zfuuvzfu zflfzf`efd whtb zfu~fet ti tbhu fzi ~zfuuvzf azf+ a`d akk ~zfuuvzfu zflfzf`efd whtb zfu~fet ti tbhu fzi ~zfuuvzf azf tfzmfd ajuikvtf ~zfuuvzfu.

tfzmfd ajuikvtf ~zfuuvzfu.

Ma`x ~zfuuvzf-mfauvzh`o df|hefu mfauvzf `it ajuikvtf ~zfuuvzf jvt i`kx Ma`x ~zfuuvzf-mfauvzh`o df|hefu mfauvzf `it ajuikvtf ~zfuuvzf jvt i`kx dhllfzf`ef h` ~zfuuvzf. Liz fram~kf+ a Jivzdi`-tvjf oaof h`dheatfu i`kx tbf dhllfzf`ef dhllfzf`ef h` ~zfuuvzf. Liz fram~kf+ a Jivzdi`-tvjf oaof h`dheatfu i`kx tbf dhllfzf`ef jftwff` tbf ~zfuuvzf h` tbf lkvhd ti wbheb ht hu ta~~fd a`d tbf ~zfuuvzf h` tbf jftwff` tbf ~zfuuvzf h` tbf lkvhd ti wbheb ht hu ta~~fd a`d tbf ~zfuuvzf h` tbf atmiu~bfzf. H` tbhu eauf+ tbf`+ tbf zflfzf`ef ~zfuuvzf hu aetvakkx tbf atmiu~bfzhe atmiu~bfzf. H` tbhu eauf+ tbf`+ tbf zflfzf`ef ~zfuuvzf hu aetvakkx tbf atmiu~bfzhe ~zfuuvzf. Tbhu tx~f il ~zfuuvzf zfadh`o hu eakkfd

~zfuuvzf. Tbhu tx~f il ~zfuuvzf zfadh`o hu eakkfd   oavof ~zfuuvzf  oavof ~zfuuvzf. Liz fram~kf+ hl a. Liz fram~kf+ hl a ~zfuuvzf il 6>

~zfuuvzf il 6> c^ac^a hu mfauvzfd whtb a oavof zflfzf`efd ti tbf atmiu~bfzf a`d tbfhu mfauvzfd whtb a oavof zflfzf`efd ti tbf atmiu~bfzf a`d tbf atmiu~bfzhe ~zfuuvzf hu 1>>

atmiu~bfzhe ~zfuuvzf hu 1>> c^ac^a+ tbf` tbf ~zfuuvzf ea` jf fr~zfuufd au fhtbfz+ tbf` tbf ~zfuuvzf ea` jf fr~zfuufd au fhtbfz  ~  ~ == 6>6> c^a oavof

c^a oavof iziz ~ = ~ = 16>16> c^a ajuikvtfc^a ajuikvtf..

_bf`f|fz atmiu~bfzhe ~zfuuvzf hu vufd au a zflfzf`ef+ tbf ~iuuhjhkhtx frhutu _bf`f|fz atmiu~bfzhe ~zfuuvzf hu vufd au a zflfzf`ef+ tbf ~iuuhjhkhtx frhutu tbat tbf ~zfuuvzf tbvu mfauvzfd ea` jf fhtbfz ~iuhth|f iz `foath|f. @foath|f oavof tbat tbf ~zfuuvzf tbvu mfauvzfd ea` jf fhtbfz ~iuhth|f iz `foath|f. @foath|f oavof ~zfuuvzf hu akui tfzmfd au

~zfuuvzf hu akui tfzmfd au |aevvm ~zfuuvzf|aevvm ~zfuuvzf. Bf`ef+ hl a oavof ta~~fd h`ti a ta`c . Bf`ef+ hl a oavof ta~~fd h`ti a ta`c  h`dheatfu a |aevvm ~zfuuvzf il 51

h`dheatfu a |aevvm ~zfuuvzf il 51 c^ac^a+ tbhu ea` akui jf utatfd au ?>+ tbhu ea` akui jf utatfd au ?> c^a ajuikvtfc^a ajuikvtf+ + iziz -51

-51 c^a oavofc^a oavof+ auuvmh`o tbat tbf atmiu~bfzhe ~zfuuvzf hu 1>1+ auuvmh`o tbat tbf atmiu~bfzhe ~zfuuvzf hu 1>1 c^ac^a..

1.5.1

1.5.1 Atmiu~bfzhe Atmiu~bfzhe ^zfuuvzf+^zfuuvzf+ ~ ~atmatm

-- Tbf faztb hu uvzziv`dfd jx a` atmiu~bfzf ma`x mhkfu bhob. Tbf ~zfuuvzfTbf faztb hu uvzziv`dfd jx a` atmiu~bfzf ma`x mhkfu bhob. Tbf ~zfuuvzf dvf ti tbhu atmiu~bfzf at tbf uvzlaef il tbf faztb df~f`du v~i` tbf bfad il  dvf ti tbhu atmiu~bfzf at tbf uvzlaef il tbf faztb df~f`du v~i` tbf bfad il  tbf ahz aji|f tbf uvzlaef.

tbf ahz aji|f tbf uvzlaef.

-- Tbf ahz hu eim~zfuuhjkf+ tbfzflizf tbf df`uhtx hu dhllfzf`t at dhllfzf`t bfhobt.Tbf ahz hu eim~zfuuhjkf+ tbfzflizf tbf df`uhtx hu dhllfzf`t at dhllfzf`t bfhobt. -- Dvf ti tbf wfhobt il atmiu~bfzf iz ahz aji|f tbf uvzlaef il faztb+ ht huDvf ti tbf wfhobt il atmiu~bfzf iz ahz aji|f tbf uvzlaef il faztb+ ht hu

dhllhevkt ti eakevkatf tbf atmiu~bfzhe ~zfuuvzf. Ui+ atmiu~bfzhe ~zfuuvzf hu dhllhevkt ti eakevkatf tbf atmiu~bfzhe ~zfuuvzf. Ui+ atmiu~bfzhe ~zfuuvzf hu mfauvzfd jx tbf bfhobt il eikvm` il khqvhd tbat ht ea` uv~~izt.

mfauvzfd jx tbf bfhobt il eikvm` il khqvhd tbat ht ea` uv~~izt.

-- Atmiu~bfzhe ~zfuuvzf at ufa kf|fk hu ajivt 1>1.5:6Atmiu~bfzhe ~zfuuvzf at ufa kf|fk hu ajivt 1>1.5:6 c@)mc@)m::+ wbheb hu+ wbheb hu fqvh|akf`t ti a bfad il 1>.56

fqvh|akf`t ti a bfad il 1>.56 mm il watfz iz ?<>il watfz iz ?<> mmmm il mfzevzxil mfzevzx a~~zirhmatfkx+ a`d ht dfezfaufu whtb akthtvdf.

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)11G5>>2)1)11

1.5.:

1.5.: Oavof Oavof ^zfuuvzf+^zfuuvzf+ ~ ~OO

-- Ht hu tbf ~zfuuvzf+ mfauvzfd whtb tbf bfk~ il a ~zfuuvzf mfauvzh`oHt hu tbf ~zfuuvzf+ mfauvzfd whtb tbf bfk~ il a ~zfuuvzf mfauvzh`o h`utzvmf`t+ h` wbheb tbf atmiu~bfzhe ~zfuuvzf hu tacf` au datvm3 h` itbfz h`utzvmf`t+ h` wbheb tbf atmiu~bfzhe ~zfuuvzf hu tacf` au datvm3 h` itbfz wizdu tbf atmiu~bfzhe ~zfuuvzf at tbf oavof ueakf hu mazcfd fzi.

wizdu tbf atmiu~bfzhe ~zfuuvzf at tbf oavof ueakf hu mazcfd fzi.

-- Tbf oavof ~zfuuvzf ea` jf fhtbfz ~iuhth|f iz `foath|f df~f`dh`o i`Tbf oavof ~zfuuvzf ea` jf fhtbfz ~iuhth|f iz `foath|f df~f`dh`o i` wbftbfz tbf ~zfuuvzf hu aji|f atmiu~bfzhe ~zfuuvzf "a ~iuhth|f |akvf( iz wbftbfz tbf ~zfuuvzf hu aji|f atmiu~bfzhe ~zfuuvzf "a ~iuhth|f |akvf( iz jfkiw atmiu~bfzhe ~zfuuvzf "a `foath|f |akvf(.

jfkiw atmiu~bfzhe ~zfuuvzf "a `foath|f |akvf(.

1.5.5

1.5.5 Ajuikvtf Ajuikvtf ^zfuuvzf+^zfuuvzf+ ~ ~AA

-- Ht hu tbf ~zfuuvzf fqvak ti tbf akofjzahe uvm il tbf atmiu~bfzhe a`d oavofHt hu tbf ~zfuuvzf fqvak ti tbf akofjzahe uvm il tbf atmiu~bfzhe a`d oavof ~zfuuvzfu. ~zfuuvzfu.

 ~zfuuvzf

 ~zfuuvzf

ee

 Atmiu~bfzh

 Atmiu~bfzh

 ~zfuuvzf

 ~zfuuvzf

Oavof

Oavof

 ~zfuuvzf

 ~zfuuvzf

 Ajuikvtf

 Ajuikvtf

== ## atm atm O O  A  A

 ~

 ~

 ~

 ~

 ~

 ~

=

=

#

#

1.5.4 \aevvm+ 1.5.4 \aevvm+ ~ ~||

-- H` a ~fzlfet H` a ~fzlfet |aevvm wbheb hu a eim~kftfkx fm~tx u~aef+ tbf ~zfuuvzf hu |aevvm wbheb hu a eim~kftfkx fm~tx u~aef+ tbf ~zfuuvzf hu fzi.fzi.

Kieak Kieak atmiu~bfzhe atmiu~bfzhe ~zfuuvzf ~zfuuvzf zflfzf`ef zflfzf`ef ajuikvtf ~zfuuvzf ajuikvtf ~zfuuvzf oavof ~zfuuvzf #|f oavof ~zfuuvzf #|f ~zfuuvzf ―|f ~zfuuvzf ―|f ajuikvtf ~zfuuvzf ajuikvtf ~zfuuvzf

^

^

_oavofoavof Lhovzf 1.:

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)1:G5>>2)1)1:

Fram~kf 1.? Fram~kf 1.? Dflh`f tbf likkiwh`o tfzmu 7 Dflh`f tbf likkiwh`o tfzmu 7 h. h. ^zfuuvzf "^zfuuvzf " ~ ~ (( hh.

hh. Atmiu~bfzhe ^zfuuvzf "Atmiu~bfzhe ^zfuuvzf " ~ ~atmatm((

hhh.

hhh. Oavof ^zfuuvzf "Oavof ^zfuuvzf " ~ ~OO((

h|.

h|. Ajuikvtf ^zfuuvzf "Ajuikvtf ^zfuuvzf " ~ ~AA((

|. |. \aevvm "\aevvm " ~ ~|| (( Uikvthi` ti Fram~kf 1.? Uikvthi` ti Fram~kf 1.? h. h. ^zfuuvzf "^zfuuvzf " ~ ~ (7(7 ^zfuuvzf hu lizef "

^zfuuvzf hu lizef " L  L ( ~fz v`ht azfa "( ~fz v`ht azfa " A A (.(. hh.

hh. Atmiu~bfzhe ^zfuuvzf "Atmiu~bfzhe ^zfuuvzf " ~ ~atmatm( 7( 7

Tbf ~zfuuvzf dvf ti atmiu~bfzf at tbf uvzlaef il tbf faztb df~f`du Tbf ~zfuuvzf dvf ti atmiu~bfzf at tbf uvzlaef il tbf faztb df~f`du v~i` tbf bfad il tbf ahz aji|f tbf uvzlaef.

v~i` tbf bfad il tbf ahz aji|f tbf uvzlaef. hhh.

hhh. Oavof ^zfuuvzf "Oavof ^zfuuvzf " ~ ~OO((77

Ht hu tbf ~zfuuvzf+ mfauvzfd whtb tbf bfk~ il a ~zfuuvzf mfauvzh`o Ht hu tbf ~zfuuvzf+ mfauvzfd whtb tbf bfk~ il a ~zfuuvzf mfauvzh`o h`utzvmf`t+ h` wbheb tbf atmiu~bfzhe ~zfuuvzf hu tacf` au datvm.

h`utzvmf`t+ h` wbheb tbf atmiu~bfzhe ~zfuuvzf hu tacf` au datvm. h|.

h|. Ajuikvtf ^zfuuvzf "Ajuikvtf ^zfuuvzf " ~ ~AA((

Ht hu tbf ~zfuuvzf tbat fqvaku ti tbf akofjzahe uvm il tbf atmiu~bfzhe Ht hu tbf ~zfuuvzf tbat fqvaku ti tbf akofjzahe uvm il tbf atmiu~bfzhe a`d oavof ~zfuuvzfu.

a`d oavof ~zfuuvzfu.

|.

|. \aevvm "\aevvm " ~ ~|| ((

A eim~kftfkx fm~tx u~aef wbfzf tbf ~zfuuvzf hu fzi. A eim~kftfkx fm~tx u~aef wbfzf tbf ~zfuuvzf hu fzi.

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)15G5>>2)1)15

Fram~kf 1.2 Fram~kf 1.2

_bat hu tbf ~zfuuvzf oavof il ahz h` tbf exkh`dfz hl tbf atmiu~bfzhe oavof hu _bat hu tbf ~zfuuvzf oavof il ahz h` tbf exkh`dfz hl tbf atmiu~bfzhe oavof hu 1>1.5

1>1.5 c@)mc@)m::a`d ajuikvtf ~zfuuvzf hu 4<>a`d ajuikvtf ~zfuuvzf hu 4<> c@)mc@)m::..

Uikvthi` ti Fram~kf 1.2 Uikvthi` ti Fram~kf 1.2  ~  ~AA= 4<>= 4<> c@)mc@)m::  ~  ~atmatm= 1>1.5= 1>1.5 c@)mc@)m::  ~  ~OO= 9= 9

_htb zflfzf`ef ti tbf lizmvka jfkiw 7 _htb zflfzf`ef ti tbf lizmvka jfkiw 7 Ajuikvtf ~zfuuvzf+

Ajuikvtf ~zfuuvzf+ ~ ~ A A= Oavof ~zfuuvzf+= Oavof ~zfuuvzf+ ~ ~OO# atmiu~bfzhe ~zfuuvzf+# atmiu~bfzhe ~zfuuvzf+ ~ ~atmatm

Tbfzflizf + Tbfzflizf +  ~  ~OO ==  ~ ~AA ――  ~ ~atmatm = = 4<> 4<> ― ― 1>1.51>1.5 = = 562562..?? c@ c@  )  ) mm::

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)14G5>>2)1)14

AETH\HTX 1J

AETH\HTX 1J

TFUT XIVZ V@DFZUTA@DH@O

TFUT XIVZ V@DFZUTA@DH@OJFLIZF XIV EI@TH@VF _HTB TBF @FRTJFLIZF XIV EI@TH@VF _HTB TBF @FRT H@^VT…!

H@^VT…!

1.5

1.5 Kajfk Kajfk tbf tbf dhaozam dhaozam jfkiw jfkiw 77

1.4

1.4 A Jivzdi` ~zfuuvzf oavof attaebfd ti a jihkfz kieatfd at ufa kf|fk ubiwu a zfadh`oA Jivzdi` ~zfuuvzf oavof attaebfd ti a jihkfz kieatfd at ufa kf|fk ubiwu a zfadh`o ~zfuuvzf il ?

~zfuuvzf il ? jajaz. Hl atmiu~bfzhe ~zfuuvzf hu 1.>15z. Hl atmiu~bfzhe ~zfuuvzf hu 1.>15 jaz jaz + wbat hu tbf + wbat hu tbf ajuikvtf ~zfuuajuikvtf ~zfuuvzf h`vzf h` tbat jihkfz "h` tbat jihkfz "h` c@)mc@)m::( 9( 9 Kieak Kieak atmiu~bfzhe atmiu~bfzhe ~zfuuvzf ~zfuuvzf zflfzf`ef zflfzf`ef aa j j ee d d

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)16G5>>2)1)16

LFFDJAEC I@ AETH\HTX 1J

LFFDJAEC I@ AETH\HTX 1J

1.5 1.5 Kieak Kieak atmiu~bfzhe atmiu~bfzhe ~zfuuvzf ~zfuuvzf zflfzf`ef zflfzf`ef a =

a =ajuikvtf ~zfuuvzfajuikvtf ~zfuuvzf

j =

j =oavof ~zfuuvzf #|foavof ~zfuuvzf #|f

d =

d = ~zfuuvzf ―|f~zfuuvzf ―|f

d =

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)1<G5>>2)1)1<

1.4 1.4

 ~  ~AA= 9= 9

 ~

 ~atmatm= 1.>15= 1.>15 jaz jaz 

 ~

 ~OO= ?= ? jaz jaz 

_htb zflfzf`ef ti tbf lizmvka jfkiw 7 _htb zflfzf`ef ti tbf lizmvka jfkiw 7 Ajuikvtf ~zfuuvzf+

Ajuikvtf ~zfuuvzf+ ~ ~ A A= Oavof ~zfuuvzf+ ~= Oavof ~zfuuvzf+ ~OO# atmiu~bfzhe ~zfuuvzf+# atmiu~bfzhe ~zfuuvzf+ ~ ~atmatm

Tbfzflizf + Tbfzflizf +  ~  ~AA ==  ~ ~OO ## ~ ~atmatm = = ? ? r r 1>1>66 # # 1.>15 1.>15 r r 1>1>66 = = 2>15>>2>15>> @)m @)m:: = = 2>12>1..55c@ c@  )  ) mm::

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)1?G5>>2)1)1?

1.4

1.4 ^BXUHEAK ^BXUHEAK ^ZI^FZTHFU ^ZI^FZTHFU IL IL LKVHDLKVHD

Lkvhd ~zi~fzthfu azf h`thmatfkx zfkatfd ti lkvhd jfba|hivz. Ht hu ij|hivu tbat dhllfzf`t Lkvhd ~zi~fzthfu azf h`thmatfkx zfkatfd ti lkvhd jfba|hivz. Ht hu ij|hivu tbat dhllfzf`t lkvhdu ea` ba|f oziuukx dhllfzf`t ebazaetfzhutheu. Liz fram~kf+ oaufu azf khobt a`d lkvhdu ea` ba|f oziuukx dhllfzf`t ebazaetfzhutheu. Liz fram~kf+ oaufu azf khobt a`d eim~zfuuhjkf+ wbfzfau khqvhdu azf bfa|x a`d zfkath|fkx h`eim~zfuuhjkf.

eim~zfuuhjkf+ wbfzfau khqvhdu azf bfa|x a`d zfkath|fkx h`eim~zfuuhjkf.

Ti qva`thlx tbf lkvhd jfba|hivz dhllfzf`efu efztah` lkvhd ~zi~fzthfu azf vufd. Tbf lkvhd Ti qva`thlx tbf lkvhd jfba|hivz dhllfzf`efu efztah` lkvhd ~zi~fzthfu azf vufd. Tbf lkvhd ~zi~fzthfu azf mauu df`uhtx+ u~fehlhe wfhobt+ u~fehlhe oza|htx+ u~fehlhe |ikvmf a`d ~zi~fzthfu azf mauu df`uhtx+ u~fehlhe wfhobt+ u~fehlhe oza|htx+ u~fehlhe |ikvmf a`d |hueiuhtx.

|hueiuhtx.

1.4.1

1.4.1 Mauu Mauu df`uhtxdf`uhtx++ ρ ρ hu dflh`fd au tbf mauu ~fz v`ht |ikvmf.hu dflh`fd au tbf mauu ~fz v`ht |ikvmf.

" UH v`htu+

" UH v`htu+ co)mco)m55 ((

\ 

\ 

|ikvmf

|ikvmf

m

m

mauu

mauu

++

++

= =  ρ   ρ  1.4.:

1.4.: U~fehlhe wfhobtU~fehlhe wfhobt++ hu dflh`fd au tbf wfhobt ~fz v`ht |ikvmf.hu dflh`fd au tbf wfhobt ~fz v`ht |ikvmf. " UH v`htu+ " UH v`htu+ @)m @)m55((

\ 

\ 

|ikvmf

|ikvmf

_ 

_ 

wfhobt 

wfhobt 

++

++

= = ψ  ψ 

\ 

\ 

mo 

mo 

= = ψ  ψ 

 o 

 o 

 ρ   ρ  ψ  ψ == ""wbfzf o = 8.21m)ufewbfzf o = 8.21m)ufe::((

H` UH v`htu tbf u~fehlhe wfhobt il watfz hu 8.21 r 1>>> = 821>

H` UH v`htu tbf u~fehlhe wfhobt il watfz hu 8.21 r 1>>> = 821>  @)m @)m55

 H@^VT 

 H@^VT 

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)12G5>>2)1)12

1.4.5

1.4.5 U~fehlhe U~fehlhe oza|htxoza|htxiz zfkath|f df`uhtx+iz zfkath|f df`uhtx+ u u hu tbf zathi il tbf wfhobt il tbf uvjuta`efhu tbf zathi il tbf wfhobt il tbf uvjuta`ef ti tbf wfhobt il a` fqvak |ikvmf il watfz at 4

ti tbf wfhobt il a` fqvak |ikvmf il watfz at 4 ·E ·E ..

watfz  watfz  ef ef  uvju  uvju

 u

 u

ψ  ψ  ψ  ψ  _ta`ta` = = watfz  watfz  ef ef  uvju  uvju

 u

 u

 ρ   ρ   ρ   ρ  _ta`ta` = = 1.4.4

1.4.4 U~fehlhe U~fehlhe |ikvmf|ikvmf++ || hu dflh`fd au tbf zfeh~zieak il mauu df`uhtx. Ht hu vufd tihu dflh`fd au tbf zfeh~zieak il mauu df`uhtx. Ht hu vufd ti mfa` |ikvmf ~fz v`ht mauu. "UH v`htu+

mfa` |ikvmf ~fz v`ht mauu. "UH v`htu+ mm55 )co  )co (.(.

 ρ   ρ  ν  ν == 11

m

m

mauu

mauu

\ 

\ 

|ikvmf

|ikvmf

++

++

= = ν  ν  1.4.6 \hueiuhtx 1.4.6 \hueiuhtx

A lkvhd at zfut ea``it zfuhut ubfazh`o lizefu jvt i`ef ht hu h` mithi`+ ubfazh`o A lkvhd at zfut ea``it zfuhut ubfazh`o lizefu jvt i`ef ht hu h` mithi`+ ubfazh`o lizefu azf uft v~ jftwff` kaxfzu il lkvhd mi|h`o at dhllfzf`t |fkiehthfu. Tbf lizefu azf uft v~ jftwff` kaxfzu il lkvhd mi|h`o at dhllfzf`t |fkiehthfu. Tbf |hueiuhtx il tbf lkvhd dftfzmh`fu tbf ajhkhtx il tbf lkvhd h` zfuhuth`o tbfuf |hueiuhtx il tbf lkvhd dftfzmh`fu tbf ajhkhtx il tbf lkvhd h` zfuhuth`o tbfuf ubfazh`o utzfuufu.

ubfazh`o utzfuufu.

Fram~kf 1.8 Fram~kf 1.8

_bat hu tbf mauu df`uhtx+

_bat hu tbf mauu df`uhtx+  ρ ρ il lkvhd "h`il lkvhd "h` co)mco)m55( hl mauu hu 46>( hl mauu hu 46>  o  o a`d tbf |ikvmf hu 8a`d tbf |ikvmf hu 8

em em55.. Uikvthi` ti Fram~kf 1.8 Uikvthi` ti Fram~kf 1.8 \  \  m m = =  ρ   ρ  < < 5 5 1> 1> 8 8 1> 1> 46> 46> √ √ √ √ ß ß ß ß = = 5 5 5 5  )   )  1> 1> 6> 6>ßß co co  mm = =

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)18G5>>2)1)18

Fram~kf 1.1> Fram~kf 1.1>

_bat hu tbf u~fehlhe wfhobt+

_bat hu tbf u~fehlhe wfhobt+ ψ ψ il lkvhd "h`il lkvhd "h` c@)mc@)m55( hl tbf wfhobt il lkvhd hu 1>( hl tbf wfhobt il lkvhd hu 1> @  @ a`d tbfa`d tbf

|ikvmf hu 6>> |ikvmf hu 6>> emem::.. Uikvthi` ti Fram~kf 1.1> Uikvthi` ti Fram~kf 1.1> \  \  _  _  = = ψ  ψ  < < 5 5 1> 1> 6>> 6>> 1> 1> 1> 1> √ √ √ √ ß ß ß ß = = = :> >>> = :> >>> @)  @) mm55 5 5  )   )  :> :> c@ c@  mm = = Fram~kf 1.11 Fram~kf 1.11

_bat hu tbf u~fehlhe oza|htx il lkvhd h` Fram~kf 1.1>. _bat hu tbf u~fehlhe oza|htx il lkvhd h` Fram~kf 1.1>.

Uikvthi` ti Fram~kf 1.11 Uikvthi` ti Fram~kf 1.11 watfz  watfz  ef ef  uvju  uvju  u  u ψ  ψ  ψ  ψ  _ta`ta` = =

" "

88..2121

((

1>>> 1>>> 1> 1> :> :>ßß 55 = = >58 >58 .. : : = = Fram~kf 1.1: Fram~kf 1.1:

_bat hu tbf u~fehlhe |ikvmf+

_bat hu tbf u~fehlhe |ikvmf+ || il lkvhd h` Fram~kf 1.8.il lkvhd h` Fram~kf 1.8.

Uikvthi` ti Fram~kf 1.1: Uikvthi` ti Fram~kf 1.1: m m \  \  || == 5 5 < < 1> 1> 46> 46> 1> 1> 8 8 √ √ √ √ ß ß ß ß = = co  co  m m  )  )  1> 1> : :ßß √√66 55 = =

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1):>G5>>2)1):>

AETH\HTX 1E

AETH\HTX 1E

TFUT XIVZ V@DFZUTA@DH@O

TFUT XIVZ V@DFZUTA@DH@OJFLIZF XIV EI@TH@VF _HTB TBF @FRTJFLIZF XIV EI@TH@VF _HTB TBF @FRT H@^VT…!

H@^VT…!

1.6

1.6 Mateb tbf likkiwh`oMateb tbf likkiwh`o

 ρ  ρ

 u

 u

||

\  \  |ikvmf |ikvmf _  _  wfhobt  wfhobt  ++ ++ m m mauu mauu \  \  |ikvmf |ikvmf ++ ++ \  \  |ikvmf |ikvmf m m mauu mauu ++ ++  lizwatfz   lizwatfz  ef ef  uvju  uvju  liz   liz  ψ  ψ  ψ  ψ  _ta`ta`

m

m

55

 )co 

 )co 

co)m

co)m

55

 @)m

 @)m

55 U~fehlhe U~fehlhe oza|htx oza|htx U~fehlhe U~fehlhe |ikvmf |ikvmf Mauu Mauu df`uhtx df`uhtx U~fehlhe U~fehlhe wfhobt wfhobt

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1):1G5>>2)1):1

LFFDJAEC I@ AETH\HTX 1E

LFFDJAEC I@ AETH\HTX 1E

1.6 1.6  ρ  ρ

 u

 u

||

\  \  |ikvmf |ikvmf _  _  wfhobt  wfhobt  ++ ++ m m mauu mauu \  \  |ikvmf |ikvmf ++ ++ \  \  |ikvmf |ikvmf m m mauu mauu ++ ++  lizwatfz   lizwatfz  ef ef  uvju  uvju  liz   liz  ψ  ψ  ψ  ψ  _ta`ta`

m

m

55

 )co 

 )co 

co)m

co)m

55

 @)m

 @)m

55 U~fehlhe U~fehlhe oza|htx oza|htx U~fehlhe U~fehlhe |ikvmf |ikvmf Mauu Mauu df`uhtx df`uhtx U~fehlhe U~fehlhe wfhobt wfhobt

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1)::G5>>2)1)::

UFKL-AUUFUUMF@T

UFKL-AUUFUUMF@T

Xiv azf a~~ziaebh`o uveefuu.

Xiv azf a~~ziaebh`o uveefuu. Tzx akk tbf qvfuthi`uTzx akk tbf qvfuthi`u h` tbhu ufkl-auufuumf`t ufethi`h` tbhu ufkl-auufuumf`t ufethi` a`d ebfec xivz a`uwfzu whtb

a`d ebfec xivz a`uwfzu whtb tbiuf oh|f` h` tbf tbiuf oh|f` h` tbf Lffdjaec i` Ufkl-Auufuumf`t. Lffdjaec i` Ufkl-Auufuumf`t. Hl xivHl xiv laef a`x ~zijkfmu+

laef a`x ~zijkfmu+ dhuevuu ht whtb xivz dhuevuu ht whtb xivz kfetvzfz. kfetvzfz. Oiid kvec.Oiid kvec.

1.1

1.1 Ei`|fztEi`|fzt tbf tfm~fzatvzfu jfkiw aeeizdh`o ti tbf u~fehlhfd ueakfu7tbf tfm~fzatvzfu jfkiw aeeizdh`o ti tbf u~fehlhfd ueakfu7 a(

a( 1?6 ·1?6 ·E E ti ·ti · L  L ++ C  C a`da`d Z Z.. j(

j( 612 ·612 · Z Z ti ·ti ·E E + ·+ · L  L a`da`d C  C .. e(

e( :14 ·:14 · L  L ti ·ti ·E E ++ Z Z a`da`d C  C .. d(

d( 5>>5>> C  C ti ·ti ·E E ++ Z Z a`d ·a`d · L  L .. 1.:

1.: Auuvmf tbf df`uhtx il watfz ti jf 1>>>Auuvmf tbf df`uhtx il watfz ti jf 1>>> co)mco)m55 at atmiu~bfzhe ~zfuuvzf 1>1at atmiu~bfzhe ~zfuuvzf 1>1 c@)m

c@)m::. _bat whkk jf7. _bat whkk jf7 a( tbf oavof ~zfuuvzf a( tbf oavof ~zfuuvzf

j( tbf ajuikvtf ~zfuuvzf il watfz at a df~tb il :>>>

j( tbf ajuikvtf ~zfuuvzf il watfz at a df~tb il :>>> mm jfkiw tbf lzff uvzlaef9jfkiw tbf lzff uvzlaef9

1.5

1.5 Dftfzmh`f h` @fwti` ~fz uqvazf mftzf+ tbf h`ezfauf h` ~zfuuvzf h`tf`uhtx ~fzDftfzmh`f h` @fwti` ~fz uqvazf mftzf+ tbf h`ezfauf h` ~zfuuvzf h`tf`uhtx ~fz mftzf df~tb h` lzfub watfz. Tbf mauu df`uhtx il lzfub watfz hu 1>>>

mftzf df~tb h` lzfub watfz. Tbf mauu df`uhtx il lzfub watfz hu 1>>> co)mco)m55..

1.4

1.4 Oh|f` u~fehlhe wfhobt il lkvhd hu <.64Oh|f` u~fehlhe wfhobt il lkvhd hu <.64 c@)mc@)m55 a`d htu mauu hu 2.5a`d htu mauu hu 2.5 co+co+ eakevkatfeakevkatf tbf likkiwh`o7

tbf likkiwh`o7 a(

a( |ikvmf il lkvhd|ikvmf il lkvhd j(

j( u~fehlhe |ikvmf il lkvhdu~fehlhe |ikvmf il lkvhd e(

e( df`uhtx df`uhtx il il lkvhdlkvhd

1.6

1.6 Oh|f` ihk u~fehlhe oza|htx hu >.28+ lh`d 7Oh|f` ihk u~fehlhe oza|htx hu >.28+ lh`d 7 a(

a( df`uhtx df`uhtx il il ihkihk j(

j( u~fehlhe u~fehlhe wfhobt wfhobt il il ihkihk e(

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU

^ZH@EH^KF EI@EF^TU IL LKVHD MFEBA@HEU G5>>2)1):5G5>>2)1):5

LFFDJAEC I@ UFKL-AUUFUUMF@T

LFFDJAEC I@ UFKL-AUUFUUMF@T

A`uwfzu 7 A`uwfzu 7 1.1 1.1 a(a(54?54? L  L +2>?+2>? Z Z+442.5+442.5 C  C ++ j( j(6262¾¾ L  L +:2?.?+:2?.? C  C +14.?+14.? E E ++ e( e(<?4<?4¾¾E+5?4.4E+5?4.4 Z Z+1>1.4+1>1.4 C  C  d( d(:?:?¾¾E E +2>.<+2>.<¾¾ L  L +64>.<+64>.< Z Z 1.: 1.: a( 11?.?:a( 11?.?: c@)mc@)m:: ++ j( :12.?: j( :12.?: c@)mc@)m:: 1.5 1.5 8.21 r 1>8.21 r 1>55 @)m @)m:: 1.4 1.4 a( >.>?:a( >.>?: mm55 j( >.>>16 j( >.>>16 mm55 )co  )co  e( <81.<? e( <81.<? co)mco)m55 1.6 1.6 a( >.28 r 1>a( >.28 r 1>55 co)mco)m55 j( 2?5>.8 j( 2?5>.8 @)m @)m55 e( >.>>11: e( >.>>11: mm55 )co  )co