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]LPQ@AQ

"egj`slfuegrplys`as.`j" 7

[J@_ @EKJQ@DJ]LPQ@AGB W[GJ_@_@KQ 9

_lk qugjt`t`ks wl`al agj ck ekgsurko cy gj `jstruekjt gjo cy ekgjs dh wl`al wk agj oksar`ck tlk

bgws dh plys`as grk agbbko plys`agb qugjt`t`ks. _`bb abgss \ wk lgvk stuo`ko egjy plys`agb qugjt`t`ks

kn. bkjntl, vkbda`ty, gaakbkrgt`dj, hdrak, t`ek, prkssurk, egss, okjs`ty kta.

Hujogekjtgbdr

Cgs`a qugjt`t`ks

Okr`vkoWugjt`t`ks

QuppbkekjtgryWugjt`t`ks

]lys`agb qugjt`t`ks grk dh tlrkk typks

7. Hujogekjtgb Cgs`a Wugjt`t`ks 9

_lksk grk tlk kbkekjtgry qugjt`t`ks wl`al advkrs tlk kjt`rk spgj dh plys`as. Gjy dtlkr qugjt`t`ks agj ck okr` vko hrde tlks k. Gbb tlk cgs`a qugjt`t `ks grk aldskj sual tlgt tlky sldubo ck o`hhkrkjt, tlgt ekgjs `jokpkj-

okjt dh kgal dtlkr. (`.k., o`stgjak (o) , t`ek (t) gjo vkbda`ty (v) agjjdt ck aldskj gs cgs`a

qugjt`t`ks (ckagusk tlky grk rkbgtko gs X 5t

o). Gj @jtkrjgt`djgb Drngj`zgt`dj jgeko AN]E

9 Nkjkrgb Adjhkrkjak dj wk`nlt gjo Ekgsurks, aldsk skvkj plys`agb qugjt`t`ks gs cgs`a dr

hujogekjtgb.

Bkjntl(B)

_`ek(_)

Egss(E)

_kepkrgturk(F)

Kbkatr`agbaurrkjt

(G)

Bue`jdus@jtkjs`ty

(Ao)

Gedujtdh

Qucstgjak(edb)

_lksk grk tlk kbkekjtgry qugjt`t`ks (`j dur pbgjkt) tlgts wly aldskj gs cgs`a qugjt`t`ks.

@j hgat gjy skt dh `jokpkjokjt qugjt`t`ks agj ck aldskj gs cgs`a qugjt`t`ks cy wl`al gbb

dtlkr plys`agb qugjt`t`ks agj ck okr`vko.

`.k.,

Agj ck aldskj gs cgs`a qugjt`t`ks (dj sdek dtlkr pbgjkt, tlksk e`nlt gbsd ck usko gs

cgs`a qugjt`t`ks)

Cut (B)Bkjntl

(G)Grkg

(X)Xkbda`ty

agjjdt ck usko gs cgs`a qugjt`t`ks gs

Grkg 5 (Bkjntl)3 sd tlky grk jdt `jokpkjokjt.

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]LPQ@AQ

"egj`slfuegrplys`as.`j" 3

3. Okr`vko Wugjt`t`ks 9]lys`agb qugjt`t`ks wl`al agj ck kxprkssko `j tkres dh cgs`a qugjt`t`ks (E,B,_....) grk agbbko okr`vko

qugjt`t`ks.

`.k., Edekjtue ] 5 ev

5 (e)t`ek

jto`spbgakek5

_

EB5 E7 B7 _ 7

Lkrk V E7 B7 _ 7 U `s agbbko o`ekjs`djgb hdreubg dh edekjtue , gjo wk agj sgy tlgt edekjtue lgs

7 O`ekjs`dj `j E (egss)

7 O`ekjs`dj `j B (bkjntl)

gjo 7 O`ekjs`dj `j _ (t`ek)

_lk rkprkskjtgt`dj dh gjy qugjt`ty `j tkres dh cgs`a qugjt`t`ks (E,B,_....) `s agbbko o`ekjs`djgb hdr-

eubg gjo `j tlk rkprkskjtgt`dj, tlk pdwkrs dh tlk cgs`a qugjt`t`ks grk agbbko o`ekjs`djs.

1. Quppbkekjtgry qugjt`t`ks 9Cks`oks skvkj hujogekjtgb qugjt`t`ks twd suppbkekjtgry qugjt`t`ks

grk gbsd okh`jko. _lky grk

]bgjk gjnbk (_lk gjnbk cktwkkj twd b`jks) Qdb`o gjnbk

H@JO@JN O@EKJQ@DJQ DH XGS@D[Q ]LPQ@AGB W[GJ_@_@KQ 9 Lk`nlt, w`otl, rgo`us, o`spbgakekjt kta. grk g f`jo dh bkjntl. Qd wk agj sgy tlgt tlk`r o`ekjs`dj

`s VBU

VLk`nltU

VBU

V^`otlU

Vrgo`usU

Vo`spbgakekjtU

lkrk VLk`nltU agj ck rkgo gs O`ekjs`dj dh Lk`nlt

Grkg 5 Bkjntl ^`otlQd, o`ekjs`dj dh grkg `s VGrkgU 5 VBkjntlU V^ `otlU

5 VBU VBU

5 VB3U

Hdr a`rabk

Grkg 5r3

VGrkgU 5 VU Vr3U5 V7U VB3U

5 VB

3

ULkrk `s jdt g f`jo dh bkjntl dr egss dr t`ek sd sldubojt ghhkat tlk o`ekjs`dj dh Grkg.Lkjak `ts o`ekjs`dj sldubo ck 7 (E=B=_=) gjo wk agj sgy tlgt `t `s o`ekjs`djbkss. Hrde

s`e`bgr bdn`a wk agj sgy tlgt gbb tlk jueckrs grk o`ekjs`djbkss.

V3==U

VE B _ U 5 7= = =

V-7U

V1U

3

7 O`ekjs`djbkss

VXdbuekU 5 VBkjntlU V^`otlU VLk`nltU5 B B B 5 VB1U

Hdr splkrk

8/12/2019 Unit & Dimension Theory_E

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]LPQ@AQ

"egj`slfuegrplys`as.`j" 1

Xdbuek 51

?r1

VXdbuekU 5

1

?Vr1U

5 (7) VB1U 5 VB1U

Qd o`ekjs`dj dh vdbuek w`bb ck gbwgys VB1U wlktlkr `t `s vdbuek dh g aucd`o dr vdbuek dh

splkrk.

O ` e k j s ` d j d h g p l y s ` a g b q u g j t ` t y w ` b b c k s g e k , ` t o d k s j t o k p k j o d j w l ` a l h d r e u b g w k

g r k u s ` j n h d r t l g t q u g j t ` t y .

Okjs`ty 5vdbuek

egss

VOkjs`tyU 5UvdbuekV

UegssV5 1B

E5 VE7B 1U

Xkbda`ty (v) 5t`ek

jto`spbgakek

VvU 5Ut`ekV

UjtO`spbgakekV5

_

B5 VE=B7_7U

Gaakbkrgt`dj (g) 5ot

ov

VgU 5 5 3

7

B__

B_

Edekjtue (]) 5 evV]U 5 VEU VvU

5 VEU VB_7U

5 VE7B7_7U

Hdrak (H) 5 egVHU 5 VeU VgU

5 VEU VB_3U

5 VE7B7_3U

^drf dr Kjkrny 5 hdrak o`spbgakekjtV^drfU 5 VhdrakU Vo`spbgakekjtU

5 VE7

B7

_3

U VBU5 VE7B3_3U

]dwkr 5t`ek

wdrf

V]dwkrU 5Ut`ekV

UwdrfV5

_

_BE 337 5 VE7B3_ 1U

]rkssurk 5Grkg

Hdrak

V]rkssurkU 5 UGrkgV

UHdrakV

5 3

377

B

_BE

5 E7

B 7

_ 3

8/12/2019 Unit & Dimension Theory_E

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]LPQ@AQ

"egj`slfuegrplys`as.`j" ?

7. O`ekjs`djs dh gjnubgr qugjt`t`ks 9 Gjnbk ()

(Gjnubgr o`spbgakekjt) 5rgo`us

Gra

VU 5Urgo`usV

UGraV5

B

B5 VE=B=_=U (O`ekjs`djbkss)

Gjnubgr vkbda`ty () 5t

VU 5UtV

UV5

_

75 VE=B=_7U

Gjnubgr gaakbkrgt`dj () 5ot

o

VU 5UotVUoV 5

__BE 7== 5 VE=B=_ 3U

_drquk 5 Hdrak Gre bkjntlV_drqukU 5 VhdrakU Vgre bkjntlU

5 VE7B7_3U VBU 5 VE7B3_3U

3. O`ekjs`djs dh ]lys`agb Adjstgjts 9 Nrgv` tg t`djgb Adjstgjt 9

e7 e3rHn Hn

@h twd cdo`ks dh egss e7

gjo e3

grk pbgako gt r o`stgjak, cdtl hkkb nrgv`tgt`djgb gttrgat`dj

hdrak, wldsk vgbuk `s,

Nrgv`tgt`djgb hdrak Hn

5 337

r

eNe

wlkrk N `s g adjstgjt agbbko Nrgv`tgt`djgb adjstgjt.

VHnU 5

UrV

UeUVeUVNV3

37

VE7B7_ 3U 5UBV

UEUVEUVNV3

VNU 5 E 7 B1 _ 3

Qpka`h`a lkgt agpga`ty 9_d `jarkgsk tlk tkepkrgturk dh g cdoy cy _, Lkgt rkqu`rko `s W 5 es_

Lkrk s `s agbbko spka`h`a lkgt agpga`ty.

VWU 5 VeU VsU V_U

Lkrk W `s lkgt 9 G f`jo dh kjkrny sd VWU 5 E 7B3_ 3

VE

7

B

3

_

3

U 5 VEU VsU VFUVsU 5 VE=B3_3F7U

8/12/2019 Unit & Dimension Theory_E

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]LPQ@AQ

"egj`slfuegrplys`as.`j" ;

Ngs adjstgjt S 9Hdr gj `okgb ngs, rkbgt`dj cktwkkj prkssurk (])

Xgbuk (X) , _kepkrgturk (_) gjo edbks dh ngs (j) `s

]X 5 jS_ wlkrk S `s g adjstgjt, agbbko ngs adjstgjt.

V]U VXU 5 VjU VSU V_U ............ (7)

lkrk V]U VXU 5 UGrkgV

UHdrakV

VGrkg BkjntlU

5 VHdrakU VBkjntlU

5 VE7B7_3U VB7U 5 E7B3_3

Hrde kqugt`dj (7)

V]U VXU 5 VjU VSU V_U

VE7B3_3U 5 VedbU VSU VFU

VSU 5 VE7B3_3 edb7 F7U

Adkhh`a `kjt d h v`sads`ty 9r

r

hv

X

@h gjy splkr`agb cgbb dh rgo`us r edvks w`tl vkbda`ty v `j g v`sadus

b`qu`o, tlkj v`sadus hdrak gat`jn dj `t `s n`vkj cy

Hv5 8rv

Lkrk `s adkhh`a`kjt dh v`sads`ty

VHvU 5 V8U VU VrU VvU

E7B7_ 3 5 (7) VU VBU VB_ 7U

VU 5 E7B 7_ 7

]bgjaf s adjstgjt 9@h b`nlt dh hrkqukjay `s hgbb`jn , kjkrny dh g pldtdj `s n`vkj cy

K 5 l Lkrk l 5 ]bgjafs adjstgjt

VKU 5 VlU VU

5 hrkqukjay 5]kr`do_`ek

7 VU 5

U]kr`do_`ekV

75

_

7

sd E7B3_ 3 5 VlU V_ 7U

VlU 5 E7B3_ 7

1. Qdek spka`gb hkgturks dh o`ekjs`djs 9 Quppdsk `j gjy hdreubg, (B + ) tkre `s ade`jn (wlkrk B `s bkjntl). Gs bkjntl agj ck gooko

djby w`tl g bkjntl, sd sldubo gbsd ck g f`jo dh bkjntl.

Qd VU 5 VBU

Q`e`bgrby adjs`okr g tkre (H ) wlkrk H `s hdrak. G hdrak agj ck gooko/suctrgatko w`tl ghdrak djby gjo n`vk r`sks td g tl`ro hdrak. Qd sldubo ck g f`jo dh hdrak gjo `ts rksubt (H )

sldubo gbsd ck g f`jo dh hdrak.

H g tl`ro hdrakgjo `ts o`ekjs`dj

w bb gbsd ck E B _7 7 3

sldubo ck g f`jo dhhdrak V 5 E B _ 7 7 3

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]LPQ@AQ

"egj`slfuegrplys`as.`j" 8

Subk Jd. 7 9 D j k q u g j t ` t y a g j c k g o o k o / suctrgatko w ` t l g s ` e ` b g r q u g j t ` t y d j b y g j o n ` v k r ` s k t d t l k s ` e ` b g r q u g j t ` t y .

Kxgepbk 7. 3t

5 Hv + 3x

H`jo o`ekjs`djgb hdreubg hdr VU gjo VU ( lkrk t 5 t`ek, H 5 hdrak, v 5 vkbda`ty, x 5 o`stgjak)

Qdbut`dj 9 Q`jak o`ekjs`dj dh Hv 5 VHvU 5VE7B7_3U VB7_7U 5 VE7B3_ 1U ,

sd

3x

sldubo gbsd ck E7B3_ 1

UxV

UV3

5 E7 B3_ 1

VU 5 E7B?_ 1

gjo

3xHv w`bb gbsd lgvk o`ekjs`dj E7B3_ 1 , sd B.L.Q. sldubo gbsd lgvk tlk sgek

o`ekjs`dj E7B3_ 1

sdUtV

UV3

5 E7B3_ 1

VU 5 E7B3_ 7

Kxgepbk 3. Hdr j edbks dh ngs, Xgjokr wggbs kqugt`dj `s

3X

g] (X c) 5 jS_

H`jo tlk o`ekjs`djs dh g gjo c, wlkrk ] `s ngs prkssurk, X 5 vdbuek dh ngs _ 5 tkepkrg-

turk dh ngs

Qdbut`dj 9

QdUXV

UgV3 5 E

7B 7_ 3 Qd VcU 5 B1

3UBV

UgV 5 E

7 B7 _ 3

VgU 5 E7 B; _3

8/12/2019 Unit & Dimension Theory_E

7/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" >

Subk Jd. 3 9 Adjs`okr g tkre s`j()Lkrk `s o`ekjs`djbkss gjo s`j

sLypdtkjkdu

bgr]krpkjo`au`s gbsd o`ekjs`djbkss.

^lgtkvkr adeks `j s`j(......) `s o`ekjs`djbkss gjo kjt`rk Vs`j (.......)U `s gbsd o`ekj-s`djbkss.

s`j(- - -)

o`ekjs`djbkss

o`ekjs`djbkss

Q `e` bg rby 9ads(- - -)

o`ekjs`djbkss

o`ekjs`djbkss

tgj(- - -)

o`ekjs`djbkss

o`ekjs`djbkss

(- - -)

o`ekjs`djbksso`ekjs`djbkss

k

bdn (- - -)k

o`ekjs`djbkss

o`ekjs`djbkss

Kxgepbk 1. 53v

Hs`j (t) (lkrk v 5 vkbda`ty, H 5 hdrak, t 5 t`ek)

H`jo tlk o`ekjs`dj dh gjo

Qdbut`dj 9

Qd VU 5UvV

UHV3 5 377

377

U_BV

U_BEV

5 E7B 7 _=

Kxgepbk ?. 5 3

3Hv

bdnk

3v

3wlkrk H 5 hdrak , v 5 vkbda`ty

H`jo tlk o`ekjs`djs dh gjo .

8/12/2019 Unit & Dimension Theory_E

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]LPQ@AQ

"egj`slfuegrplys`as.`j" .

_`ek pkr`do dh g s`epbk pkjoubue agj ok pkjo dj

Qd wk agj sgy tlgt kxprkss`dj dh _ sldubo ck `j tl`s hdre

_ 5 (Qdek Jueckr) (e)g ()c(n)a

Kqugt`jn tlk o`ekjs`djs dh BLQ gjo SLQ,

E=B=_7 5 (7) VE7Ug VB7Uc VB7_3Ua

E=B=_7 5 Eg Bc+a _ 3a

Adepgr`jn tlk pdwkrs dh E,B gjo _,

nkt g 5 = , c + a 5 =, 3a 5 7

sd g 5 = , c 53

7, a 5

3

7

sd _ 5 (sdek Jue ckr) E= B7/3 n 7/3

_ 5 (Qdek Jueckr)n

_lk qugjt`ty Qdek jueckr agj ck hdujo kxpkr`ekjtgbby. Ekgsurk tlk bkjntl dh g pkjoubue

gjo dsa`bbgtk `t, h`jo `ts t`ek pkr`do cy stdpwgtal.

Quppdsk hdr 5 7e, wk nkt _ 5 3 ska. sd

3 5 (Qdek Jueckr)

8/12/2019 Unit & Dimension Theory_E

10/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 7=

Qd wk agj sgy tlgt h 5 (sdek Jueckr) ()g ()c (])a

_

75 (7) VBUg VEB1Uc VE7B7_3Ua

EB_7 5 Ec + a Bg 1c a _3a

adepgr`jn pdwkrs dh E, B, _

= 5 c + a

= 5 g 1c a7 5 3a

nkt g 5 7 , c 5 - 7/3 , a 5 7/3

Qd h 5 (sde k jue ckr) ]7

^k agj kxprkss gjy qugjt`ty `j tkres dh tlk n`vkj cgs`a qugjt`t`ks.Kxgepbk 0. @h vkbda ty (X), hdrak (H) gjo t`ek (_) grk aldskj gs hujogekjtgb qugjt`t`ks , kxprkss ( ) egss

gjo (``) kjkrny `j tkres dh X,H gjo _Qdbut`dj 9

Bkt E 5 (sdek Jueckr) (X)g (H)c (_) a

Kqugt`jn o`ekjs`djs dh cdtl tlk s`oks

E7B=_= 5 (7) VB7_7Ug VE7B7_ - 3 Uc V_7Ua

E7B=_= 5 Ec Bg + c _ g 3c + a

nkt g 5 7, c 5 7, a 5 7

E 5 (Qdek Jueckr) (X7 H7 _7) VEU 5 VX7 H7 _7U

Q`e`bgrby wk agj gbsd kxprkss kjkrny `j tkres dh X , H , _

Bkt VKU 5 Vsdek JueckrU VXUg VHU c V_U a

VEB3

_3

U 5 VEB_U VB_7

Ug

VEB_3

Uc

V_Ua

VE7B3_3U 5 VEc Bg + c _g 3c + aU

7 5 c6 3 5 g + c 6 3 5 g 3c + a

nkt g 57 6 c 5 7 6 a 5 7

K 5 (sdek Jueckr) X7H7_7 dr VKU 5 VX7UVH7UV_ 7U. _d h`jo dut uj`t dh g plys`agb qugjt`ty 9

Quppdsk wk wgjt td h`jo tlk uj`t dh hdrak. ^k lgvk stuo`ko tlgt tlk o`ekjs`dj dh hdrak `s

VHdrakU 5 VE7B7_3U

Gs uj`t dh E `s f`bdnrge (fn) , uj`t dh B `s ektkr (e) gjo uj`t dh _ `s skadjo (s) sd uj`t dh hdr ak agjck wr`ttkj gs (fn)7 (e) 7 (s) 3 5 fn e/s3 `j EFQ systke. @j ANQ systke, uj`t dh hdrak agj ck wr`ttkj

gs (n)7 (ae)7 (s)3 5 n ae/s3.

8/12/2019 Unit & Dimension Theory_E

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]LPQ@AQ

"egj`slfuegrplys`as.`j" 77

B@E@_G_@DJQ DH O@EKJQ@DJGB GJGBPQ@Q 9

Hrde O`ekjs`djgb gjgbys`s wk nkt _ 5 (Qdek Jueckr)n

sd tlk kxprkss`dj dh _ agj ck

_ 53 n

_ 5 n

s`j (.....)

dr dr

_ 5 ;=n

_ 5n

bdn (......)

dr dr

_ 5 3n

_ 5n

+ (t=)

O`ekjs`djgb gjgbys`s odksjt n`vk `jhdregt`dj gcdut tlk sdek Jueckr 9_lk o`ekjs`djgb adj-stgjt.

_l`s ektldo `s uskhub djby wlkj g plys`agb qugjt`ty okpkjos dj dtlkr qugjt`t`ks cy eubt`pb`-agt`dj gjo pdwkr rkbgt`djs.

(`.k., h 5 xg yc za)

@t hg`bs `h g plys`agb qugjt`ty okpkjos dj suj dr o`hhkrkjak dh twd qugjt`t`ks

(`.k.h 5 x + y z)

`.k., wk agjjdt nkt tlk rkbgt`dj

Q 5 u t +3

7gt3 hrde o`ekjs`djgb gjgbys`s.

_l`s ektldo w`bb jdt wdrf `h g qugjt`ty okpkjos dj gjdtlkr qugjt`ty gs s`jk dr ads`jk,bdngr`tle`a dr kxpdjkjt`gb rkbgt`dj. _lk ektldo wdrfs djby `h tlk okpkjokjak `s cy pdwkr

hujat`djs.

^k kqugtk tlk pdwkrs dh E,B gjo _ lkjak wk nkt djby tlrkk kqugt`djs. Qd wk agj lgvk djbytlrkk vgr`gcbk (djby tlrkk okpkjokjt qugjt`t ks)

Qd o`ekjs`djgb gjgbys`s w`bb wdrf djby `h tlk qugjt`ty okpkjos djby dj tlrkk pgrgektkrs, jdt

edrk tlgj tlgt.

Kxgepbk 7=. Agj ]rkssurk (]), okjs`ty () gjo vkbda`ty (v) ck tgfkj gs hujogekjtgb qugjt`t`ks 4

Qdbut`dj 9

], gjo v grk jdt `jokpkjokjt, tlky agj ck rkbgtko gs ] 5 v3 ,sd tlky agjjdt ck tgfkj gs

hujogekjtgb vgr`gcbks.

_d alkaf wlktlkr tlk ] , , gjo X grk okpkjokjt dr jdt, wk agj gbsd usk tlk hdbbdw`jn

egtlkegt`agb ektldo 9

V]U 5 VE7B-7_-3U

VU 5 VE7B-1 _=U

VXU 5 VE=B7_-7U

Alkaf tlk oktkre`jgjt dh tlk`r pdwkrs 9

8/12/2019 Unit & Dimension Theory_E

12/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 73

5 7 (1) (7)(7) 3 (7) 5 =,

Qd tlksk tlrkk tkres grk okpkjokjt.

O@EKJQ@DJQ CP QDEK Q_GJOGSO HDSE[BGK 9-

@j egjy agsks, o`ekjs`djs dh sdek stgjogro kxprkss`dj grk gsfko

k.n. h`jo tlk o`ekjs`dj dh (=

=)

hdr tl`s, wk agj h`jo o`ekjs`djs dh =

gjo =, gjo eubt`pby tlke, cut `t w`bb ck vkry bkjntly prdakss.

@jstkgo dh tl`s, wk sldubo iust skgral g hdreubg, wlkrk tl`s tkre (=

=) adeks.

@t adeks `j a 5==

7

(wlkrk a 5 spkko dh b`nlt)

=

=5 3a

7

V=

=U 5 3a

75 3)_/B(

75 B3 _3

Kxgepbk 77. H`jo tlk o`ekjs`djs dh

(`)=K3 (

= 5 pkre`tt`v`ty `j vgauue , K 5 kbkatr`a h`kbo)

(``)=

3

C

(C 5 Egnjkt`a h`kbo , =

5 egnjkt`a pkrekgc`b`ty)

(```)BA

7(B 5 @jouatgjak , A 5 Agpga`tgjak)

(`v) SA (S 5 Sks`stgjak , A 5 Agpga`tgjak)

(v)S

B(S 5 Sks`stgjak , B 5 @jouatgjak)

(v`)CK (K 5 Kbkatr`a h`kbo , C 5 Egnjkt`a h`kbo)

(v``) N=

(N 5 [j`vkrsgb Nrgv`tgt`djgb adjstgjt ,=

5 pkre`tt`v`ty `j vgauue )

(v```)e

k

(

k5 Kbkatr`agb hbux 6

e 5 Egnjkt`a hbux)

Qdbut`dj 9

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7

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3

75

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1

337

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8/12/2019 Unit & Dimension Theory_E

13/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 71

(``)=

3C

3

7

5 Egnjkt`a kjkrny okjs`ty

=

3C

3

75 VEgnjkt`a Kjkrny okjs`tyU

=

3C5

UvdbuekV

UkjkrnyV5

1

337

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_BE

5 E7B-7_3

(```)BA

75 gjnubgr hrkqukjay dh B A dsa`bbgt`dj

BA

75 VU 5

_

75 _7

(`v) SA 5 _`ek adjstgjt dh SA a`rau`t 5 g f`jo dh t`ek

VSAU 5 Vt`ekU 5 _7

(v)S

B5 _`ek adjstgjt dh B S a`rau`t

S

B5 V t ekU 5 _7

(v ) egnjkt`a hdrak He

5 qvC , kbkatr`a hdrak Hk

5 qK

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K5 VvU 5 B__7

(v` ) Nrgv`tgt`djgb hdrak Hn

53

3

r

Ne, Kbkatrdstgt`a hdrak H

k5

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7

3

3

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q

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q

?

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O`ekjs`djs dh qugjt ` t `ks rkbgtko td Kbkatrdegnjkt`a gjo Lkgt (djby hdr \@ @ gj o \@ @ @ s t u o k j t s ) (`) Algrnk (q) 9

^k fjdw tlgt kbkatr`agb aurrkjt ` 5ot

oq5

krvgb`jtt`eksegbb

hbdwkgrnalsegbbg

8/12/2019 Unit & Dimension Theory_E

14/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 7?

V ` U 5UotV

UoqV

VGU 5t

UqVVqU 5 V GG7 _7 U

(` ) ]kre`tt`v ty `j Xgauue (=) 9

Kbkatrdstgt a hdrak cktwkkj twd algrnks Hk5 337

r

qqf

5 =?

7

337

r

qq

VHkU 5 UVU?V

7

= 337

UrV

UqUVqV

E7 B7 _3 5U)V7(

7

= 3UBV

UG_UVG_V

V=

U 5 E7 B1 _?G3

(` `) Kbkatr a H`kbo (K) 9Kbkatr`agb hdrak pkr uj`t algrnk K 5q

H

VKU 5UqV

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377

5 E7B7_1G7

(v) Kbkatr`agb ]dtkjt gb (X) 9 Kbkatr`agb pdtkjt`gb kjkrny pkr uj`t algrnk X 5q

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VXU 5UqV

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337

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(v) Sks`stgjak (S) 9Hrde Dles bgw X 5 ` S

VXU 5 V`U VSU

VE7B3_1G7 U 5 VG7U VSU

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(v ) Agpga`tgjak(A) 9

A 5X

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77

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(v` ) Egnjkt a h`kbo (C) 9egnjkt`a hdrak dj g aurrkjt agrry`jn w`rk H

e5 ` CVH

eU 5 V`U VU VCU

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Hdrak /bkjntl cktwkkj twd w rks

H5

?

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3

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(`x) @jouatgjak (B) 9Egnjkt a pdtkjt gb kjkrny stdrko j gj `jouatdr [ 57/3 B `3

V[U 5 V7/3U VBU V`U3

VE7 B3 _3U 5 (7) VBU (G)3

V B U 5 E7B3_3G3

8/12/2019 Unit & Dimension Theory_E

15/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 7;

(x) _lkregb Adjouat v ty9

Sgtk dh lkgt hbdw tlrdunl g adjouatdrot

oW5 G

ox

o_

UotV

UoWV5 VfU VGU

UoxV

Uo_V

U_V

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5 V U VB3U UBV

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7

V U 5 E7 B7 _1 F7

(x ) Qtkhgjs Adjstgjt )( 9

@h g cbgaf cdoy lgs tkepkrgturk (_), tlkj Sgtk dh rgo`gt`dj kjkrny ke`ttkoot

oK5 G __?

UotV

UoKV5 UV VGU V_?U

U_V

U_BEV 337

5 UV VB3

U VF?

U

UV 5 VE7 Bd _1 F?U

(x ) ^`kjs Adjstgjt 9

^gvkbkjntl adrrkspdjo`jn td egx. spkatrgb jtkjs ty .e

5_

c(wlkrk _ 5 tkep. dh tlk cbgaf cdoy)

Ve

U 5U_V

UcV

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[J@_ 9 [j`t 9

Ekgsurkekjt dh gjy plys`agb qugjt`ty `s kxprkssko `j tkres dh gj `jtkrjgt`djgbby gaakptko

akrtg`j cgs`a stgjogro agbbko uj`t.

Q@ [j`ts 9@j 70>7 , gj `jtkrjgt`djgb Drngj`zgt`dj AN]E 9 (Nkjkrgb Adjhkrkjak dj wk`nlt gjo Ekgsurk)oka`oko tlk stgjogro uj`ts, wl`al grk `jtkrjgt`djgbby gaakptko. _lksk uj`ts grk agbbko Q@ uj`ts

(@jtkrjgt`djgb systke dh uj`ts)

8/12/2019 Unit & Dimension Theory_E

16/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 78

7. Q@ [j`ts dh Cgs`a Wugjt`t`ks 9Q[j`ts

Cgsk Wugjt`tyJgek Qyecdb Okh`j`t`dj

Bkjntl ektrk e_lk ektrk `s tlk bkjntl dh tlk pgtl trgvkbko cy b`nlt `jvgauue our`jn g t`ek `jtkrvgb dh 7/300, >03, ?;< dh g skadjo(70)

Kbkatr`a Aurrkjt gepkrk G

_lk gepkrk `s tlgt adjstgjt aurrkjt wl`al, `h eg`jtg`jko `jtwd strg`nlt pgrgbbkb adjouatdrs dh `jh`j`tk bkjntl, dh jknb`n`cbk

a`raubgr ardss-skat`dj, gjo pbgako 7 ektrk gpgrt `j vgauue,w`bb prdouak cktwkkj tlksk adjouatdrs g hdrak kqugb td 3 x

7=-> Jkwtdj pkr ektrk dh bkjntl. (70?1.78 dh tlk tlkredoyjge a

tkepkrgturk dh tlk tr`pbk pd`jt dh wgtkr. (708>)

Gedujt dhQucstgjak

edbk edb_lk edbk `s tlk gedujt dh sucstgjak dh g systke, wl`aladjtg`js gs egjy kbkekjtgry kjt`t`ks gs tlkrk grk gtdes `j=.=73 f`bdnrge dh agrcdj-73. (70>7)

Bue`jdus@jtkjs`ty agjokbg ao

_lk agjokbg `s tlk bue`jdus `jtkjs`ty, `j g n`vkj o`rkat`dj, dhg sdurak tlgt ke ts edjdalrdegt a rgo`gt`dj dh hrkqukjay

;?= x 7=73 lkrtz gjo tlgt lgs g rgo`gjt `jtkjs ty `j tlgto`rkat`dj dh 7/80).

3 . _wd suppbkekjtgry uj`ts wkrk gbsd okh`jko 9 ]bgjk gjnbk [j`t 5 rgo`gj (rgo) Qdb`o gjnbk [j`t 5 Qtkrgo`gj (sr)

1 . Dtlkr abgss`h `agt`dj 9@h g qugjt`ty `jvdbvks djby bkjntl, egss gjo

t`ek (qugjt`t`ks `j ekalgj`as), tlkj `ts uj`t

agj ck wr`ttkj `j EFQ, ANQ dr H]Q systke.

Hdr EFQ systke 9@j tl`s systke Bkjntl, egss gjo t`ek grk kxprkssko `j ektkr, fn gjo skadjo. rkspkat`vkby.

@t adeks ujokr Q@ systke.

Hdr ANQ systke 9@j tl`s systke ,Bkjntl, egss gjo t`ek grk kxprkssko `j ae, nrge gjo skadjo. rkspkat`vkby.

Hdr H]Q systke 9@j tl`s systke, bkjntl, egss gjo t`ek grk ekgsurko `j hddt, pdujo gjo skadjo. rkspkat`vkby.

8/12/2019 Unit & Dimension Theory_E

17/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 7>

?. Q@ uj`ts dh okr`vko Wugjt`t`ks 9 Xkbda`ty 5

Qd uj`t dh vkbda`ty w bb ck e/s

Gaakbkrgt`dj 5t`ek

vkbda`ty`jalgjnk5

s

s/e5 3s

e

Edekjtue 5 evsd uj`t dh edekjtue w`bb ck 5 (fn) (e/s) 5 fn e/s

Hdrak 5 eg[j`t w`bb ck 5 (fn) (e/s3) 5 fn e/s3 agbbko jkwtdj (J)

^drf 5 HQuj`t 5 (J) (e) 5 J e agbbko idubk (I)

]dwkr 5t`ek

wdrf

[j`t 5 I / s agbbko wgtt (^)

; . [j`ts dh sdek plys`agb Adjstgjts 9 [j`t dh [j`vkrsgb Nrgv`tgt`djgb Adjstgjt (N)

H 5 337

r

)e)(e(N 3s

efn 5 3e

)fn)(fn(N

sd uj`t dh N 5 3

1

sfn

e

[j`t dh spka`h`a lkgt agpga`ty (s) 9W 5 es_I 5 (fn) (Q) (F)

[j`t dh s 5 I / fn F

[j`t dh=

9

hdrak pkr uj`t bkjntl cktwkkj twd bdjn pgrgbbkb w`rks `s9

H5

?

=

3

37

r

``

e

J5

)7(

=

)e(

(G))G(3 [j`t dh = 5 3G

e.J

8. Q@ ]rkh`x 9Quppdsk o`stgjak cktwkkj fdtg td Ig`pur `s 1=== e. sd

o 5 1=== e 5 1 7=== e

f`bd(f)

5 1 fe (lkrk f `s tlk prkh`x usko hdr 7=== (7= 1))

Quppdsk tl`afjkss dh g w`rk `s =.=; e

o 5 =.=; e 5 ; 7= e-3

akjt`(a)

5 ; ae (lkrk a `s tlk prkh`x usko hdr (7=3))

Q`e`bgrby, tlk egnj`tuok dh plys`agb qugjt`t`ks vgry dvkr g w`ok rgjnk. Qd `j drokr td kxprkss tlk

vkry bgrnk egnj`tuok gs wkbb gs vkry segbb egnj`tuok edrk adepgatby, AN]E rkadeekjoko sdek

stgjogro prkh`xks hdr akrtg`j pdwkr dh 7=.

8/12/2019 Unit & Dimension Theory_E

18/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 71 pe (v) >.; jeQdbut`dj 9

(`) ;e 5 ; 7= 8e(``) 1 fe 5 1 7=1 e

(```) 3= ee 5 3= 7= 1e

(`v) >1 pe 5 >1 7=73 e

(v) >.; je 5>.; 7= 0 e

Kxgepbk 71. H 5 ; J adjvkrt `t `jtd ANQ systke.

Qdbut`dj 9

H 5 ; 3s

efn

5 (;) 3

1

s

)ae7==)(n7=(

5 ; 7= ; 3s

aen(`j ANQ systke).

_l`s uj`t ( 3s

aen) `s gbsd agbbko oyjk

Kxgepbk 7?. N 5 8.8> 7= 77 3

1

sfn

eadjvkrt `t `jtd ANQ systke.

Qdbut`dj 9 N 5 8.8> 7= 77

3

1

sfn

e

5 (8.8>7=77) 3

1

s)n7===(

)ae7==(5 8.8> 7= < 3

1

sn

ae

Kxgepbk 7;. 5 3 n/ae 1

adjvkrt `t `jtd EFQ systke.

Qdbut`dj 9

5 3 n/ae1

5 (3) 13-

1

e)(7=

fn7=

5 3 7=1 fn/e1

8/12/2019 Unit & Dimension Theory_E

19/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 70

Kxgepbk 78. X 5 0= fe / ldur

adjvkrt `t `jtd e/s.

Qdbut`dj 9

X 5 0= fe / ldur 5 (0=)skadjo)8=(8=

)e7===(

X 5 (0=)

18==

7===

s

e

X 5 0= 7 pe `jtde.Qdb.

Bkt > pe 5 (x) e , Jdw bkts adjvkrt cdtl BLQ & SLQ `jtd ektkr> (7= 73) e 5 (x) x 7= 8 e

nkt x 5 > 7= 8

Qd > pe 5 (>7= 8)e

Qdek Q@ uj`ts dh okr`vko qugjt`t`ks grk jgeko ghtkr tlk sa`kjt`st, wld lgs adjtr`cutko `j tlgt h`kbo gbdt.

8/12/2019 Unit & Dimension Theory_E

20/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 3=

.

Kbkatr`a ]dtkjt`gbKeh.

(X 5q

[)

vdbt X I / A Fn e3

/ s1G

8/12/2019 Unit & Dimension Theory_E

21/22

]LPQ@AQ

"egj`slfuegrplys`as.`j" 37

0. Qdek Q@ uj`ts kxprkssko `j tkres dh tlk spka`gb jgeks gjo gbsd `j tkres dhcgsk uj`t s9

Q@ [j`ts

]lys`agb Wugjt`ty @j tkres dh spka`gbjgeks

@j tkres dh cgsk uj`ts

_drquk ( 5 Hr) J e Fn e3 / s3

Oyjge`a X`sads`ty

(Hv5 Gor

ov)

]d`sk`ubbk (] ) dr ]g s Fn / e s

@epubsk (I 5 H t) J s Fn e / s

Edoubus dh kbgst`a`ty

(P 5strg`j

strkss)

J / e3

Fn / e s3

Qurhgak _kjs`dj Adjstgjt (_)

(_ 5

H)

J/e dr I/e3

Fn / s3

Qpka`h`a Lkgt agpga`ty (s)(W 5 es _)

I/fn F

(dbo uj`t sA.n

agb) e

3s

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-7

_lkregb adjouat`v`ty (F)

(ot

oW5 FG

or

o_)

^ / e F e fn s-1

F-7

Kbkatr`a h`kbo @jtkjs`ty K 5q

HX/e dr J/A e fn s

-1G

-7

Ngs adjstgjt (S) (]X 5 jS_) dredbgr Lkgt Agpga`ty

(A 5_E

W

)

I / F edb e3

fn s-3

F-7

edb-7

ALGJNK DH J[EKS@AGB XGB[K ^@_L _LK ALGJNK DH [J@_ 9Quppdsk wk lgvk

5 > a e nktwkektrks,`jtd`t

adjvkrtwk@h5 e

7==

>

wk agj sgy tlgt `h tlk uj`t `s `jarkgsko td 7== t`eks (ae e),

tlk juekr`agb vgbuk ckagek7==

7t`eks

7==

>>

Qd wk agj sgy

Juekr`agb vgbukuj`t

7

^k agj gbsd tkbb `t `j g hdregb wgy b`fk tlk hdbbdw`jn 9

8/12/2019 Unit & Dimension Theory_E

22/22

]LPQ@AQ

Egnj`tuok dh g plys`agb qugjt`ty 5 (@ts Juekr`agb vgbuk) (uj`t)

5 (j) (u)

Egnj`tuok dh g plys`agb qugjt`ty gbwgys rkeg`js adjstgjt ,`t

w`bb jdt algjnk `h wk kxprkss `t `j sdek dtlkr uj`t.

Qd

juekr`agb vgbukuj`t

7

Kxgepbk 7