8/12/2019 Unit & Dimension Theory_E
1/22
]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.
8/12/2019 Unit & Dimension Theory_E
2/22
]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
3/22
]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
4/22
]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
5/22
]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
8/12/2019 Unit & Dimension Theory_E
6/22
]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
8/22
]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
11/22
]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
(`) Kjkr ny okjs`ty 53
7
=K3
VKjkrny okjs`tyU 5 V=K3
U
3=K
3
75
UvdbuekV
UkjkrnyV5
1
337
B
_BE
5 E7B-7_3
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
B
_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
VHeU 5 VHkU VqvCU 5 VqKU
C
K5 VvU 5 B__7
(v` ) Nrgv`tgt`djgb hdrak Hn
53
3
r
Ne, Kbkatrdstgt`a hdrak H
k5
=?
7
3
3
r
q
3
3
r
Ne5
3
3
= r
q
?
7
VN=U 5
3
3
e
q5
3
3
e
)`t(5 GG3_3E3
(v```)
e
k5
CQ
KQ5
C
K5 VvU (hrde pgrt (v`)) 5 B__7
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
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
UHV5
U_GV
U_BEV77
377
5 E7B7_1G7
(v) Kbkatr`agb ]dtkjt gb (X) 9 Kbkatr`agb pdtkjt`gb kjkrny pkr uj`t algrnk X 5q
[
VXU 5UqV
U[V5
U_GV
U_BEV77
337
5 E7B3 _1G7
(v) Sks`stgjak (S) 9Hrde Dles bgw X 5 ` S
VXU 5 V`U VSU
VE7B3_1G7 U 5 VG7U VSU
VSU 5 E7 B3 _1G3
(v ) Agpga`tgjak(A) 9
A 5X
q VAU 5
UXV
UqV5
UG_BEV
U_GV7137
77
VAU 5 E7 B3 _?G3
(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
VE7B7_3U 5 VG7U VB7U VCU
VCU 5 E7BD_3G7
(v` ) Egnjkt a pkrekgc`b ty `j vgauue (=) 9
Hdrak /bkjntl cktwkkj twd w rks
H5
?
d
3
37
r
``
7
377
B
_BE5
U?V
UV D
3UBV
UGUVGV V
=U 5 E7B3_3G3
(`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
U_BEV 337
5 V U VB3U UBV
UFV
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
VBU 5UFV
UcV
VcU 5 VB7F7U
[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
-3F
-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
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