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The theory and application of tensor analysis

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
THE THEORY A ® M U C iS X C * OF TSfSQt AHAtfSlE
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UMI Number: DP11961
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G eom etrical sn8. ces. . . . * * . . * . * • « . * .
^Iffieultles and preeaitfcions 1» th e tfefiniticaa o f
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m m m rn m m
A e l r t l f a e l m m memm. to h e m #»
deeper In s ig h t* and a ia lw a i la b o r*
w n riu g f o r ord er,. u n ity *
8 * attem pt® to p » « m M * e t t # re s u lts
o f in v e s tig a tio n # « n i to e s ta b lis h a eorraysK M taM e twtanen tb m . and
m ental processes.
la th e ease o f i& w s tiffc tle n s o f jfe y s ie a l phenomena
h© u t ilis e s ifce fooB or^it© re la tio n s h ip ® t e t n m
m ental pro#eases*
g « ® e tr i® a l re p re s e n ta tio n , and a lg e b ra ic f o w l M .
& alm ost say attem p t to © staM lsts a correspondence to t® #©a p h ysical
ptm m m m . and m ental processes, th e p rin c ip le # ' o f g e o a a trle a l rep res en ta­
tio n o f a lg e b ra ic forw w las a re tssed*
MJmmt i» # d t a t # ly * th e question
o f itiM»ther a fo rm u la tio n in. term s o f © « « # a l* ® expresses something ffeieh
c h a ra c te ris e s th© p h y s ic a l oaourrasMS* o r th e co ordin ate sy®taro a ris e s *
A, s u ffic ie n t e o n d itio n u fc fc ii, i f f u l f i l l e d , te u m t* * th a t a staterasnt ha#
a meaning a h ie h 1® is ft® p « fti.« t o f th®
4ta&M fc*r
iNw«#r®, ' f i r s t
i«
co ordin ate system i®
te n so r ana ly s is attem p ts to do.*
th e method o f
09®r£Am t»tt#. *ad then, shews h®# to obtain, re s u lts
wM|®li#, though expressed in te r s t o f o o o r iim te # * do n o t depend upon th e
elto ie # o f - t t * ® *
fhns te n so r a n a ly s is fre rld e ® a s u ita b le m athem atical
to o l f o r th e e s ta b lia b s e n t o f a correspondence between p h y s ic a l phenomena
and mental. jroa®#®®#., fewfc i t s a p p lic a tio n to d if f e r e n t ia l geossetry p ro v id es t t # method o f reasoning to he used*
th e o b je c tiv e s o f th is th e s is a re to p reseat aa e* o f th e fundam ental
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
p rin c ip le s o f ten so r a n a ly s is sad d if f e r e n t ! * ! geom etry, sad t o a p p ly
these p rin c ip le s to shows
C l) hjm v e c to r a n a ly s is is
a n a ly s is i ( f ) k m im $m &g@f » dynam ical
in te -rp r e ta tio n j (.3) )m * to
to tenso r
n a tio n # can fee' given a geom etrical
tfee q u a n titie s in Lfcgrenee'e *nj«s»
tio n s from C a rte s ia n rmftmmmm.' ee o rS iaetee to g e n e ra l eisrvi lin e a r re fe re n c e
co ordin ates * and tbcaoe to an
of congruence#! (4 ) tew to ap p ly
th e p rin c ip le s o f tr a n s fo fw a ti® ©f I*gf'®^l§e, # equations to th e a n a ly s is
o f r e la tin g e le c t r ic a l m achinery*
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I3U
tm s m
£ m .m m s
l l » ftsa&uses&al M e * f erasing th e basis o f th e ab so lu te d if f e r e n t ia l
c a lc u lu s , p o p u la rly ksewn m- ten so r a n a ly s is ,, is t h a t o f ©hanging
v&rlm htes o r co o rd iim te g *
JSstFly stwJy in ©om»S8tioK •with te n s o r a n a ly s is
m s esn tered s to a t an in m r itta t fts a ira ti© fo rm *
The g en eral th e o ry o f
th e q u a d ra tic d if f e r e n t ia l f w m was inaugurated by th e w r k o f Eleim nn
which -was read b e fo re the fh iS e s e fh ie a l P a c a lty o f th e U n iv e rs ity o f
# © e ttl» g « a la 1864*
W«rimm (19) •
M esw m *® » r k w » i j«bli«fe® d in h is "CNMNanelte
B efore th e d e ta il® ©f tie taiiitn *# work were known, th e m ia
lin e s o f th e to e o ry w ere developed by L ip s h its (1 *5 «*«*. C h ris to ff© ! ( 4 ) ,
both o f whom found the eoajpenmt# o f th e a f f in e connection and th e e « m »
•tore te n s o rj C h r is to ffs ! els© used c o m ria n t d iffe r e n tia t io n *
Rierasnn* a
work dem onstrated th a t th e E uclidean g e o m tfy ,. th e "h y p e rb o lic * ^ o ® » try
O f B o ly a i and Lobatehevsky* and h is " e llip t ic * geometry were s p e c ia l
oases o f a s » tr i« geometry based u p m th e In v a ria n t q u a d ra tic d if f e r e n t ia l
Com ds8 • g jj. dsl db^*
© is same oorariant d iffe r e n tia tio n and th e discovery of i t s iis*
port&nce m s da© to E ie o i*
His studies and those o f Levi-Civita were
set forth, la a ©«d«ms«d form A» a J o in t p ip e r ( 1 8 ) *
It m s R ic e ! who
introduced a ls o the m m o f subscript® m 4 superscripts to d is tin g u is h
between, two different l a m of tran sfor netx cm known as covariant and
contr®.variant •
Ihe invariant theory of tensor analysis underlies the m o d e m dif-
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fe r e n t ia l geometry « f in fia lte s iiB a l displaoew m fcs*
& t h is f ie ld m
n e o ***## #? I»® s»ri««t% swefe * • tb® mttistm ©onrteeii on., haw© ©#*&#*£ g re a t
Q
im port*®©#*
$ t» e # g e w e try serves as a unde o r rm a # » ia g . I t is considered
d e s ira b le t # in tro d u ce a t le a s t .« « » o f m® »ep# s d w a c e i ten so r and nonte n s o r In v a ria n t obi# e te and r e la tio n # t » geo aetry*
K e w rth e le s ® , th e
iaaart-aQ t th e o ry o f te n s e r a n a ly s is a x is ta as a .trsaaefa o f a n a ly s is ,,
cosftpletely d iffa n m tia ts d frotn g«*si«tfy#
ffi® aa®e te n s o r mm tntrodu eed by ito s ta im is t t # th e o ry o f g e n e ra l
r e la t iv it y i s th e y e a r I f IS (6 ) *
What wm» p re v io u s ly eal le d tt® ab so lu te
d if f e r e n t ia l #al©«Sw8 soon a fte n w rd s beem * p o p u la rly known as te n so r
a n a ly s is ,
lo th llu s te to (7 * f t a p t * l ) and S M iag iO tt (S , p* 4 4 ) have
w ritte n good aecoim ts o f th e ffe y s io a l »ijpiif't© «»® ® o f te n s o rs .
U n til th e works o f W syl (8 4 , 2 8 ), Sddtngton (§> p.* t U ) , Sehouten ( 2 0 ),
and C artea (S )p d : if f « r M t ia l g#«a»trjr had been m o stly lim ite d to th e
te m r ia n t th e o ry based ©a tts® ftaadaaental fw y fam ti# fo r a o f 'B U n a m *
was: Wsgrl sit# f i r s t g e n e ra lize d
It
id e a o f H i# e a r ly w orkers by In tr o ­
ducing gen eral fiaioM oe® o f tit® ooorditiKbe# as s p o w tri® a f fin e eatm ectto n s 1® p la c e o f tfe® C fh ria to ffe l apsis®!# end by lita itiis g p a r a lle l d ie plassnszlts -to. in fin ite s im a l d is ta n c e s *
id d im g to a, Sehouten, and C arten
hires fw « i« lls ® d Way!*® iw itio i to ia e lo i® asyamslgrie connection®! th e
ro s a ttin g gmmmt r y is o fte n © ailed non-liesaaanaa geom etry*
There are two d iffe r e n t %-pss of transforsmtion coefficient® Wbidh
®ay ooour in teaser theory*
The e ld e r tensor theory was Halted to ©e-
e r ils a t® trw ssfO V ttatioas where t mmm*
t® w i M
refereae® to th e differentials of coordinates*
be considered with
In t h is ease the trs»«-
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fo m m M m c o e ffic ie n ts mmm p a r t ia l d e r im t ivmm o f th e o o ordim t© -variab les
concerned, th e e o a iitte i& e for the ©ad«tws©# of © m e t d if f e r e n t ia ls or
th e te t# g p ro ttX i% o o m aitto o * being .m tjLsftw ft, «© th a t & lin e a r I w
i*
fo rm atio n o f th e co ordin ate d if f e r e n t ia ls , for sasKHpl*® «©«M bo
in te g r a te , to o b ta in Hi® co o rd in ate irarl&bS®®*
Sneli tri® s f® » & ti« tts a re
»®aetiffi«e © a ile d holcaaosaons tran sfo m & tio n © * .There is , k o m m r, o wore
fpaofraJt typ e ® f tr a n s fo rm tio n $& obtoh t t * im te fp j& id lity e s n litio a s fo r
an #*&#% i f f f e r m t f a l need n o t b© s a tis fie d *
la t h is m m . the o rd er o f
p a r t ia l d iffe r e n tia tio n . ooaaot bo iator<fe«a®NI aad th e s a w r e s a lt ©bt& tn ad f a a asyissaetrie aetaoatatlaa: ro o a lto *.
t o l tr a n s f © m atione «r® scree-
iia e s c a lle d oop^olipiopooo t r a a » f s r « a t i« s ,
An o p tio n a l view p o in t to
th a t o f tra n s fo rm a tio n o f »« ttt*4io lo w »«s param eters ecmsiobs of co n sid er­
in g th® t #««®r eeaqpaoHsot* d e fin ed w ith re s p e c t to d if f e r e n t ia ls o f a r c ,
*8 $ trwosfowsed from. one sets o f d iffe r e s tiitt # . o f a re to an oth er s e t,
w ith o u t r e fu ir in g th e esrtstea#® o f th e w d e rljrim g -v ariab les o f the d i f ­
fe r e n tia ls o f a re *
te n s o r a n a ly s is .
t h is la t t e r te n s e r th e o ry is knomi as ia t r ia a io
l a ItS d ,
paper on nThe Seometry o f llewaam iast
s** urns published ( B) j he gam an egseelleafc account o f th e in tr in s ic
I t should now be
m ria b le s and o f
th e c h ie f « * » # * » of
begin a ta e a tw w t e f
( c le a r th a t transform s t i on© o f co o rd in ate
S i a tenss o f thews m rla b lo c a re
i s I th e re fo re i t seems d e s ira b le to
a n a ly s t*. w ith th e fundam entals o f tran sfo rm s-
tin s of c o o rd in a te s *
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Xm.
Transform s t i mi m i e o e r# l» b e » :
A point (a group o f n m & m M real. saiBibers) la m «^i»Rsioi»t
m m itm M (a c entinuous
or so t of potato) may bo thought o f
as being refluent*# by any a la&«p«3%l*&& mrtahtoo x*, eta®** t takes
the « l » i 1 to a# r e in s t a t in g **» coordinates of tins p o in t*
HM M
othOMPwtoo th e
(U nless
»tll be considered m l . )
I f a- r e a l fu a o tlo a * qA o f th e w t a t l o o x * s a tis fy th e Jaeobifta r e ­
la tio n s h ip
( f.l)
th e fvnsM»tiea» a re said to be independent*
th e n * I f
(2*2)
a?1)
the a q u a n titie s 5 * «ar® an o th er s o t o f ©©ordinates o f th e n-sm n ifo M *
*h«& th e © e o riin a te s x*- o f any p o in t P a re s u b s titu te d 3a eq u atio n ( 2 * 2 ) ,
these e q u a tio n s g i-m- th e x * eoertiisabes o f f * - th e. © fu st ions ( f * f ) d e fin e
a t r » 0f o i « t i «
o f eoordin& tes o f th e a M a m ifo ld i I f the© * equations a re
I t s e a r fm e b t« **» o f th e ©©ordinates* th e y « ro m id to d e fin e an a ffin e
tr a n s fo r m a tio n # !
li n o * . th e equations ( 2* 2) a re independent* th e y can be solved
sim u ltan eo u sly f o r th e■# * l a term s o f th e a ? *
This o p eratic® y ie ld s the
a e m o tio n s
C t*s )
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which reprmma» t
2.
t r t m Wm at* to th e x 4 co o rd in ates*
groiiookor d eIt& s end tt®
I f th e
as
o f tt ® ''* 1
e fa a tiim
(2.3)
a re d iffe re n tia te d .
w ith re s p e c t to- m jm rtlm s lM * o o o rd im t® x ^ # th e re i s a tta in e d
<^Xk „
3 3^
<3x 01
* ~
.
|
-
<* • 1» . * . » ffl*
/<s *v
(2 *4 )
tt®**® Hi® in d ie o s k and J Jw tioat® p ir t t e n la r x 3- te rm s, sad th e repeated
©< index in d ic a te s a ®carnation one* *
i t assty
o< is known as a
changed a t w i l l t® aay o t te r l i t e r a l in d ex *
index?
T h is os© o f a
r e la t e d index to in d ic a te a s a n a tio n Is ka«m a® -tt® aw asetioa comrentio n | nnlesa otherw ise s ta te d i t w i l l be need on th e soeeeoding pages
w ith o u t f i* r t t » r » « a tiw »
Sin®# th e a 1 a re
tedep«ad®s%.»
(2 * 4 )
u n ity o r zero depending -upon wl»ifa®y i is @<psal
a re e ith e r equal t o
o r n o t equal to
j j th a t
1«
tk
dx
S * » l i f k * 3
°j
S
.;
<5^ <•©" i r
( ® *8 )
k /
j*
These a re c a lle d th e IT®»»©k#y d eltas? th e y may be g e n e ra lize d fo r k
su b scrip ts sad k s u p e rs e rifts as®, denoted by (2 2 , p * 3 )
i t i 2 **■#- i *
<5.
*
*
3 | 3 2 * •» llr *
< *•*>
I f t t # superscripts o f (2 *6 )® re a l l d if f e r e n t , and th® ® »b seriftis a re th e
same m t o f numbers, tt® ra in © o f tt® symbo 1 1® 1 or -1 a e e o rtla g to
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a s mmm
s a rip ts i a th©
«a © 4 4 p#r«ti.ta.ti«s is f©#iir* 4 to arrange the super-
\mm order as the sntsofijjpt#* in a l l other cases its
vmlae is s e re *
is a t i
ox
OF
3 #
5*
(2 ,7 )
i
fy s a e fe g w a tj'« m .a a t r t, « # a
I f k i s equation (2 *7 ) is t » M
canstftnt and J is allow ed to assume
\ *h^*
th e values 1 to m# it lin e a r aquations to
°x
'3P#»uI t ,
flies©
dm1
equations m y fee seised lay irajw-r*# rules tills yields
ad * a
d*a
3UH- dm
eafieefcer o f - V
?5r
. dm.
^ac
dm
*
,
.
(2 .8 )
I f i* and a in (2 *8 ) &r© allow ed to as st*® -sa e e e s s iv s ly a l l th e ir
p o ssib le values by le t t in g 0( assume a l l I t * possible values fo r each
value of k m til# following matrix eaa fee foras-d#
dy
*d x^
s«4r
dac
dxa
dx
dx®
< > x * j£
iF
m
i'iW
a
<>*2
i«%
Ox
dVi
dx
x
« a tr lx . is kaosm as th e fei
►
*.#» .**r
ay
d rn
ox
(t.s )
dx® 1
>
i«
H
dm
trs a s f© rm t io n m a trix * i t i s used
■is. th e d e fin itio n , -of a te n s o r.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S im ila r ly , ©<jta&tieaia (2 *8 ) s ig h t ha.'m hmm solved, fo r fe e im-mmm
ftm c tio a s
A <ra
JdJz-. «sd th e a f ttr ix
(7X
dwr
# * • *« *
7?
6x l
d ^ 1
a *a
*“ " if
\ w -w A -
O%
<7 X
a #
4*
<2 . 10)
dw 2
:ftte " g r o u p * p r o p e r t y
I t should fee noted ’fe e t th e
afctafiees o f equations
( l * i ) and (2 *1 0 ) belong t» m g ftw y f f e e t i « # fe e im tri# ® # end fee. © fe ra *
tlo®,# to w hich th e y a r e #«®f$e®tNl s a tis fy fe e fo llo w ia g fe a r c o n d itio n s *
( l ) th e product e n tity 0 e f two e n t it le * A 1 belcffltga to th e sasi©
g ro e p *
( f ) th e w iA fflfe o f s e y s ra l «!»»«»% * ©bey th e a s s o c ia tiv e law o f
Ao Jb C • 1,0 (Bo C) •
(A °B )o 0 , e tc *
(3 ) One elem ent o f fe e grewp is fe e ® « » it e l« m » tw so f e a t s u it im­
p lic a tio n by i t Imm.'mn seay
(4 ) £*sh e le m is t fee© *& *ta s # r# »
» e l« a p d «
#@ th a t fe e produet o f
®s. © leiaeitt «&d i t # la s e rs # is, fe e u n it e lt m a t *
Thus the whole th eo ry o f fgrowps 1# i^ w d ie te ly a p p lic a b le to fe e t r * « * »
forsafttlett a® trie© # | th is aslc*# p o s s ib le th e s u b d iv is io n o f eoKp&s-as prob»
le s s in to # « w » l s im p le r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C*
la m r la g ta
.
Ae©«rA3Lsg t # Vehlea ( 2 2 , p m 1 4 ) nwa. €&£eet o f any s o rt «fat@ti is n o t
changed by
o f eoortlln& tee is c a lle d m
in v a r ia n t* *
'is
s a y gw&st* a n y s e t o f p © to ts , # M sa y p o in t fu n c tio n ar©
*
in v a r ia n t * .
I f a p o in t fta a o tio a is ^ p re s e n te d by ^ (a 1) in. ea a 4 oo eriin & t®
system , miA by (pi*1 ) i» a a y a r* eoerdinabe sy® t«% th e p o in t ftasc tlo a i s
e a lle tt an a b e a lu b *.•© a la r* e ls s f ly m t»m r4«ab .*
In sae«® «iiag s e c tio n # & -taore g en erel ©Is-as o f l a « r l « i » # h a rin g
s e v e ra l e#®p©»afes i » eadb c o o rd in a te system * w i l l be considered.
6*
c o n tra .v a ria n t y e e te ra
± .9'
flas a € iff« f* * « itis ls €**• o f tfe# a o o o v d iario u r t i M w i 1, th e #
being fu a o tlo a e o f n ay m ■la ft« f© ft# « it m rto bJ#® at** a re given by the
equations
Jg* „ J jL M
n e g le e ttn g in fin ite s im a ls o f I i
1;
e r # r i« r *
(2 ,1 1 )
fin e # these e fu a tlo a e are
lin e a r in th e d if f e r e n t ia ls * a n y ti^ a e fe m fttie n o f ©©ordinate® is lo c a lly
lin e a r .
lb© a b s tra c t o b je c t isfaich is determ ined by equations (2 * 1 1 ), a M
ahoso va lu e is iaft«f«asleat o f th e re fe re n c e ooordiaate- system is © filled
a © o s tra m ria a t v e c to r # th e a A if f e r « t i a l s mm c a lle d it s
ts *
la general *■ th e a b s tra c t eh$e«t mkm® a eee^enanfea V^Cae^) tran sfo rm from
th e **■ system to th e
ayste® iy th e ru le
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* 48-
^ C x %} m - ~ ~
dx?
OM
« » !* # H »
mm
Va ( x i ) il
(2 *1 2 )
o f t3bo- tx m ft fa r m ttm m t s ix o f
( 2 * 0 ) * is d o fia o d t * to# * e« .% ¥ s *» fl® a t iMMrtserj tfe® rel sod fo s itie n - o f
tfe® io d e a M fie & tla to p » t s e r i j * ©» * ©asfsssBmi is uwwl to iE d ie st® tt »
eoatrsmrlarit -ois&iMriteir of the eas^poft&atw
7.
* ftp
CJo'WSS.riaitfc sestet*®
" * * •
S * W » * .« V
W ^ F % jir w l f f i | p ftS#
ft© n p a r t ia l € « rim tiir® # of ® s c a la r fs S a t ftra e iie & (pCx4) n4tb
re s p e c t to th e s m ris fe ls s s^Cx:*) s r# gpmm hy t& « r s le of p a r t ia l d i f *
fw re a tis tiO K
** T
^\ txrAi
«it*r» terms of hi
neglsotsd#
■%
^3J\
-nr
ai * «
“ * 1 vS
? E j*
©^ ®s*i#r than tfa# f i r s t la. thm ia H a lts s ira s l® ar©
Tbss® p a r t ia l d o ri'fm ti’®## r®yr#®«s*t the components o f the
gysdleat of cp,
whimh is determined toy squat loss (g »1S) , and
the abstract
iffttisli is tw^fmsAmak of tto®
w c to r j tfc® partial
system is oalled ® e©variant
sap® its ssK^easntos*
I n gam®vml0 the
stopfer*** otojost * sImms m #®»p»«st® %-Cn1) trmsfess from m » x % system
t®
srf
*
sysrtws toy tfe® m l #
i, -
, •
tlaCx1),
a-
( 2 * 14)
is dsftoei to to© ft « ® m r i « a t . w o t s r i ife# lowered peeltiosa ©f th #
id satifi® ® M .sai pestseripto ©ft a esapottest is ®«®d to la d te a t# th e cove, r ia n t
c h a ra c te r o f tb® eeatpaacat*
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I
* 10**
8*
O e fln itic p o f a gm m m H . %mmmt
The abstract lawrlttit ©bjwb who*® c«b§»o & m tm to a coordinate
svst^m at
&ra d&zi0&&d fer
< 2 *1 8 )
O'I Dg «p>#* 0S
where ettteh «f tie iastto#* talw .«©®a#«iT»ly the v a ta e # 1 to » and yield
m Crto) aompemm,t@p and m km ® lew of tsmsfdraattoa to
if* * ia * * • %r m
5t J a *'** 4
d r « 4 |L .
b,fea . . . b 6 £73#'
- 4 ^ g i ^ S y . , , J 5«5L *
(2 .*1 6 )
c ) # - ^—3.
i® d e fin e d to be a mixed te n se r o f order r ♦ a%
ffe® ten so r f is c o n tra -
m r la n t to r tofti©#® and e o e a ris a t to s to il# # ® ,. as in d ic a te d by th e
p o s itio n s -of' to© in d ice s*.
§«*
Sysamotrlo «pd. s ^ f ^ s M ^ y t e t« g # o ry
O r d in a r ily , form ula (2 *1 8 ) y ie ld * n ^ r * s ^ independent components.
Sojsetis&s, b o w s e r , a p o rtio n o f toes# components a re #$tt«X to ab so lu te
#*&&» *» d show a d e f t o it *
th e same m ria a e © e lm ra e ta r*
cases *
to two o r wore o f t h e ir tod ic e * o f
I t to «»®t?»a&ry to d is tin g u is h between two
(a ) a y a g M tris i (b ) asttospssaotri©.*
th e #y s»# h ri« ©as® .im plies th e © fin a lity o f a l l th e components deduced
from th e in d ice s o f syw eetiy by any perm utation o f thee® in d ic e s .
For
*X *s $ l*» . tli® fo u rth o rd e r s y *® e trie © on tr& r& riao t t « e # r f has ccs&ponents
©beytog hit# fo lle w to i re la tio n s h ip ® s-
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3lc1i
T J
m- f * " * * « t 3
#
(2 *1 7 )
■ jJJJk m ig sjfcii -m |
-#Mfe to r a lr iiig » to il® # # o f a n t ir# a more c a re fu l
is re q u ire d .
A chosen p erm u tatio n ,
such as 12S ***i% .««*%- he used mm m mtmwmrnm p e rs u t& tio n j a l l the per­
m utations deduced from %M« tmimwvm®
by «a s w k a u a to r o f
s * A s a ^ « o f to© in d ie s * a re -called even pertBwtotlo&Sf- a l l th e p srw u tatioaaa deduced, fr& m t h is » fe r® » « ® j^ s w ta tio e . by mm. odd number of ®x—
©bang®# o f two la d le s * a re sa l la d odd jm rtttttatisn#.*
Complete a n ti—
s p e w tty re q u ire s tti& t « « t» psrsm tati^ as mmaamrm th e sign o f the com­
ponents , and t h a t odd1 p e ra u t& tla n * m m w m th e slfg&f components w ito two
id e n tic a l In d ic e s a re ©e®a#ipwitJy a s m *
For m
f l e t th e fo u rth order
« a tt-s y n » tr i© © o n tra m rta a t -taaa o r T ta g components obeying tb s fo llo w ­
in g ru le s *
*
m ^
m m y ijlle m ^
(2 *1 8 )
*»' # * '#%©*
com binations o f ten so r# mate® p o ssib le the co n stru ctio n o f
fu r th e r ten so rs from g iv e n ts n # o r« f th e re fo r® i t is reasons t i e to expect
th ese p r to© ip le a o f © ® ® M »bioiis to be- im p o rtan t to a p p lic a tio n .
fh® mm o f to e te a s e rs o f to® same varian ce c h a ra c te r ie fa ro s d by
aid in g . eerxtaMipend-tog @ -«^#ae»ts*
I f 1 and 3 a re two tenso r# o f to # same
® r i« r p * C* "* a ) end th® seme v a ria i* © #feara.©ter# ifcea t h e ir aim Is
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give® b y
Til# t« i# o r c h a ra c te r
t h is operatica* t» e a s ily dem onstrated by a tr a * » .
fo n s a tlm o f th e oen^w&e&t# to a d iffe r e n t co o rd in ate system .
now co o rd in ate system at
a tsmi§mm$xk o f T 1# represented fe;
dm
'3 ,... i ~
I
d 'x " ' dm *'
<>*•
d *"" *?
v—i ®b
C t*s i)
a ;1
♦ ”*—*•■ *»■#
c^ari
16 '
I f t» t t #
j *. . i m
J,
*
iJ&j
• * • b„
dm *"1.nW
dm®'
•ni'arnrW'■imwim..
x " 'd m *
—
bn
daj
dX
J*
**•
#rvi ♦* t„«
. . . ««
t ( . . . at
n
bj . . . b
T h is co n fin es th e ten so r c h a ra c te r ©I
The
i t # e ^ Im lw a t .
o f a tons or east be o f f lie d to m y mixed te a s e r o r
I t is fern ed % s e tt la g a ®®jatr&twri«8it indeac equal to a
c o v a ria n t in dex #&& p #-rf«rai»g th e l&dlea&ed s w w rtis a ,.
duces th e order o f t t # te n s o r by tw o *
C o n tractio n r e ­
The te n s o r c h a ra c te r o f th is ©pey-
a t io n 1® e a s ily dew sw rtw it#d by s ta r tin g w ith t t # d e f in lM c a o f a p
» lx # 4 te n se r
ttz
j, a, - S :
\ —«Ai
,, d x
. fe.
dm
***
. . . b .*
A c o n tra c tio n w ith re«j»#ob to tw o tn iic e e o f -o fp e s it# yfcrimme ttayachey
fas. th en wade*
This o p e ra tio n y ie ld #
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
hence
_ii
■#*#i t j 2 *-** J„
» «.
j 2 » • • J*
<2«24)
a te n s o r o f ardor. &»*&•»&} t th e r e fo r * th e $ota9a«trfttlex» i« com pleted*
o f •too ten*© *** © f any oorioooo e im ra e te r li&s by d e fin itio n
I # • • 1*1^ ♦ • * lq ^ -yi( * * • 1 ^
«5| •» • JnJo»'"» J*
J t *■*.*- J„
• * ♦ iq
J0 •» * J s
(2 .2 5 )
where f is the .frodoet- teaser* &sd f and 0 are the
It is otvloos that the resulting teaser V is a teaser o f order equal to
the «w» o f the orders o f f end 1* .f M s r u le provides a sert&od o f form ing
tensors o f toiler order frets oootero or ton#o ra o f low er -order*
11* ffa# q u o tie n t law
A w ttlttlla e fe r fo r a is- eo«sid«r»d to- t o an
sevsr&X m ria h le s aui to t# nspre*«wteWl«# im gaasraX* by
l(i, »*# ImJ,
5^*' *--*#
1, * . * i yyj
5i ♦ * * 3 r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(t* » )
11* oosffioisnfcs of soo-h-iea tnmrleai form i» tl* rari&bles f4' # *** T4%
«at
# .**# 11^ farm » t®s®«r 'with & mrlsses elsam oter opposite to that
of th* profcet of tt* mriabJesf this smtoatwot !• t a r n as 1*® quotient
Saw*
Si»e« o q w tio s i <8.*) represent* am. t i w r i * * , i t has th e esse valu©
ia any ooordlaaat® eystisa*
A(i,
iMjt # ,*
a ro te n so r ern p rn m fcs# i t is o n ly asossasary to shew
th a t tli®
tan s o r*
In o rd er to show t h a t the oeeffloimts
tr a n s fe r * l a th # e«a® m y as 1 *# components o f a
I f the multilinear t& m M p r# # ® # * la th® variables at2* is equated
to th® multilinear fo r a •*£**•*•& in th# variables *** so th a t
y ~
a c i,
3 ) f 4, * .* * t 4 - f j ( * * .*
i.
iw
J, •»* Jr
(2 . 27)
^
k, ... Jcm
ItCit,
kml , *♦ # 1P) ¥ £| * » *
S» »»■» % *
*' * ' '
1,
te d i f th e new m rl&bMm.. mm- «*pre«s®d in isms o f th e old by m m m o f
th® usual trmsformtton formulas so that
A . # ' **.* f 4* W*
vi
« , » V * Z , - M f ' f 1'
3*1 ^—
/Av
(J x 4ii!
<)* *"
( 2. 28)
J iin
a S *.
than, by equating eosffl® iossts i t i s elear UShmk
Jt ** A
B pitttlesi
f -»♦»
aA
-Mwgggf
TATA
*** —
7iF
12. . KS/
is th® m m as th© law o f t m m f emotion o f th® components
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
**1
21*
©f a t#®#or wltli r e ® a t » m r i a s % «nd a ©owwfijmt
A 1® mtm ©ospaosb# o f a jatxwt h « o *«r
. Jt **■*■ 3,
Mz
«g«Xi »*,# t r d 'x *'1
z m *.
,
© f o rd e r (rtra ) * tits®-
d~S^'y' <)x^'
-
t h * #*i«te® tim ti«M .*
••
-
( t .so)
t h is q ® *tl*s it Item is
w to -fal as m g r tt o r is ii 6 l t e i o r ©har&et#r»
12*
o f .* tsBawr isfc© m?wsstri.© m i aa ti^ stB B O tri©
th*..
A l l t®»s©r« mam w ithw r ssagsotrio*: «»ii^® j® ® ® tr£e.* o r a © o a ttw ttio a
o f Hmr two w ith re s p o e t to a a j two la d le * * o f the saae m ria s e e o h a r*e t# r«
fh ia ©am b * d«B acai*t¥*t*i by w sM »i:
»•» *a * * » « . 1 / f a, &a * * * m„ A
!*|
-*.* bp
I 1Oj bg
*2 1
”&p '
-a © « 8 p *« a t o f & ttexmor -S* sj
* t ,.«* * M
fe| fcj * » . bp
C t.s i)
w ith r« # p # # t to two iadieo® a , ©ad
% * and by »siki»g
* ! * 2 *« * &n
%(
Ab,b,... * * f
A.
*b, b,... b,* \
* * * *n
b,... i
a eessfoa&enh o f a ten so r A^ m b l**# p i*i© trte w ith re s p e c t to th *
(2.32)
in d ie e s j
tfe#»- by ftid ia g th # r * s p t# tiv » *##!*■ *& % * i t fo llo w # th a t
# , ®2 # *» a *
®fe, b2»»* bP
1
, a, ftj
. fin_ 1 I «*®i a^ * . • i „ . ep®>2a t . . . ©J
bg *. * bP H I b4ba **» b? ' b, b2**» bf
b *g * • . a * .
>, b e * » . bp
a.
b, bg . . . bp]
§*1 a* * . * tkn
b, ba * • • bp
tbs® th.©
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(2 .S S )
II#- Srterior products
The eoia§s»sats of r a«4 tra«0 i©aal. vwetor« (r <») having the eotso
•vmelam-m ofesiftoter m m %m «qf»ag«# to form the following m a t r ix
( 2 , p. 4 1 )
1
f t * ft*, ft ft
b1 #
b®
#ft ft ft.ft;# ft «.ft ft ft,-ft #
h r *
(2 *8 4 )
ftftftft 1*
Fro®, this mtrlsc' * «oftp$#tely «otl-sy»»trie tensor* fcnerot ** th®
#*t#rior prodmet & f M m m m % er»# was. he forssed %•#
C l) §hs»«l*ig a gpftttf o f r »
»
l i
a(3r
*?**^
from omaag th e numeral®
.... 1 # t* *#.»-* »# with tar wlthemt r«j*titims
C l) F w a la g th # d»tersdxMaxt
JUfc.*
ba
b#
* * * * ft?
ra
r 0 . * •*.* r^f
fth lo h h&s fo r It® f i r s t # Ottawa th®
a eo lw w o f the ta b le j f o r
£ tft eeooad c o l ram. th # o@lw«» {3 # * te »
(8 )
th e d#t® rtdjm nt. 'thos formed mqmml to ft o OHiponertt o f
tli# tensor f 0 wher*
ma
,c<p
*
4#i
»##ft*ft•*m*ft*«'«
*ft**ftft*
* : f t ft f t .ft #
#
(2 # 1 8 )
ft* ft ft
The two well tatowa U » « r« » s o f d«t#rm :i»Kftt theory —* mm. later ftiemge o f
tiro rows (or e drams) of ft irt«rai«at .rovers#* th* fttgebr&lo s-itpt of the
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-ts *
mad a
w it h two
« Q m l rows (o r «#1m m *) i» n u l —
deam strm t# tk ® o o a p lo to ly m a t i^ y iw t r l® etaw m eter o f the e x te r io r
jred aieb *
.H » «actert#r f red a ct# a re #©gs®ti«s
m u ltim e te r # *
14*
f r m m t t ^ y w m h F i® % «*© r «
offl^pQeeot®*
.ifea#. *
a rw rfcrloto d number o f i n -
A rm&metkm o f
tit#
w «b® y o f la d le # # ema b« m d « |
la ikw® fengpioff i * accom plished*
q u a n titie s a re n o t tensors
tfeey
Bio reduced
do n e t t*R »s#© fi» as tenso rs j th ey
a re s a ile d pseudo-tensors (2 * p * # i ) *
There mm two |a rte @ ip tl type# # f p s e ^ ® » # « « © f**i iMBiwiy* tea® or
d e n s itie s and to&cer
i w if li f
't i n t i
.^ )a # a
tfce-se two- type# w i l l b® given *m m
fr r t'T ffc
-y ^& . • * . «
dafct-eyw w w trl# teaser® o f th # eaefcerier product typ e F it * (2 *3§)~j have
a l l eaqM B eets w ith e it h e r th # #«sa» ab so lu te vain® o r ® «ret th is fa c t
jmk»m p o s s ib le the f© r a * tlfl» e f th e two most iiqpertaist s p e c ia l type® o f
p#®ado-l»*is«p«,|, n®wBly#, s e a le r d « a # ltio » and -seftlar m p m It t e s *
IS *
S calar d e n s itie s
The ab so lu te vmlwe o f tis# -osdato&h eewpeaeist* © f an mWt. o rd e r
e o a f& e to ly
ta a e o r f o f th e e x te r io r preduet ty p e
can be represented by IITI *. afeor#' the p o o ltle w ra lu o o f S’ i« given by
S' ■»
fh i# q u a n tity
( 1 2 5 * .*n i# th e **efe«s»e® o rd e r)
(2 .3 6 )
S' « •» be shown to- possess a s in g le eorsponm t j b a t i t
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u a t# r g » * a
o f m im
tin® m fm rm n m m m « r# ®ti«&g*d*
This
shjtegt to m to ® # f K m d e r * ira » » fo rm tto & o f ®eordto& t*s w i l l now t»
d©faoastrat#<l*
Tfe® tra n s fe rs # tie ® . foswKtla o f to® oosgMSssnt* o f to® te n so r C 1#
gtoa® t$r
a lp
,3 se01
<(3 * * * 1?
a y
&fa « *#! M*
a*^
C t*S7)
Wm
HU*
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b *" *
(If
is *m O1
?®® permutation
o f 1 2 ..o n .)
(2 .5 6 )
CL,
( i f a n odd jn m a a t a t io n ) •
• •IT
»0 » ** 25.
(2 *3 9 )
¥fh©n thes® aw lu es &r® s u b s titu te d to ®§«stti«tt (£ # !§ } * th e fo llo w in g
r e s u lt is o b tain ed *
»
If* C
^
a y
is a»£» w ith. a ♦ sig a e r a « sig n
ffe* stm of the terras ♦ G.M.
ay
d ,1
fo llo w in g b e t t o r to * f « j » t a t t o a is swan or odUtf I t to to # saws «s tfee
...
.
i s f i a i t l m of to# tiotMttt&oat; o f
-d i?
a*'
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d x
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a .a r
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(2.4 1)
Itmmm- th e dem onstration is e e iijle te d * i
lie s a re
is c a lle d a s c a la r d e n s ity *
is th e sane m y as are seal& r d e n s itie s ,
■felt® o n ly d iffe r*© ® # fasts g t l s t Mm s t a r tla g «»ti»N»yw»fari© te n s e r is
ee& trasarlsaat i » I t s in d i#e ® f thw * a s e a le r « « p te ity is given lay
1
,
afeer® T> is th e s e a la r o a f*© Ity a a i A
<8 *4 *}
is Ife® in verse o f the determ inant
o f eq u atio n (2 *4 1 ) *
if*
.fim Har
A. te n s o r d e n g ity is d e fin e d to ¥ # tfcs p ro iw s t # f any ten so r and a
* e * 3 * r d e n s ity * f o r
x iJ ' * * *
* *?■ • * * t" 1
c t*4 s )
wfa®r* If is a seal& r d e n s ity and f f j 1 **'* f" ’!® a esnpoBest o f any te n so r,.
tK " n
is a is a s o r d e n s ity , and n o t a tree- te a s e r*
Sms© a u th o rs , how ever, d e fin e
te n s o rs I n mmmh a way th a t they sea i» e l® 4 « as a w e ig h tin g fa c to r a
s e a le r d e n s ity mr s a p a e ity ( f t , ft.* I f ) *
?t»# f a r w la o f tfa n ® fo rt*M o n o f IX .f’1*"**
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n «, ,* * , « m
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where
#. 4>^
p rfw e * th a t i \
18*
is n o t & tre e te a s e r*
te a s e r
A te a s e r ea p a c ify Is
td a ay te a s e r *
as th e p ro te s t © f & s c a la r c a p ac ity
If
. .
OJ
St -*♦* S„
Tv
4
fc * r* f
* '*
t , **.* b„
represents such s teaser ©apseify* its rule of transf©ns&tion is
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00
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dm3 *
a s *I
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<3ac n
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W ,
a» *
■
d * 3' -*-*»
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(3-k. 1
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1 ■* * * 1
i
where w .
a s r~si i +**■ im
l i * * * 5n
d x An^ : ii » •*
<8*48)
f
th is tra n s fo rm a tio n proves t h a t « is s e t s tre e te n s o r,
A» Im p o rtan t emseque&ee © f 141!® # * f l® i t ii® * « f te a s e r d e n s itie s &ad
te n s e r c a p a c itie s 1st
th e wp€#Mffe # f m te a s e r dessiiar *etd a te a s e r
c a p a c ity lie e s a trw e /tsgassg!1* . t& is * * » e a s ily he deinc®s tra fe d *
Th®
■extra fa s te r l a -seek a p ro te s t o f a te a s e r deasiigr ami a te a s e r c a p a c ity
|~Sq* (2 *4 4 ) *» d Iq * (2 ,4 ?)J whleh
fre e s e t th e preduet from. ’being *
ftW '.ts w s e r $m
T 't •
A AT* S'
« TT>
* tru e s e & la r*
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(8 *4 9 )
fare® wh-loli tli# te a s e r rnhmrmetmw m f th e p ro d w it fo llo w s *
r tty
wlusae elem ent as ft s e e la r
19,
The sssterier.
of a
weetore 8 jx ,-8 g t
* ,.* .*
i s g£*M» fey
8 |* a
«
•
* 8 ^ » (&v&n fefwteti ortfa
•
9 » (tire
In d ic e s )
» 8D » {«M p®s*wteatto»)
This (te%ei?s3jj«iit y ie ld s aa e low est 8
.*■** 8 jxP-
.# *
*
«
*
«
*
*
%
« # « . j» •
8aat*1
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(2 *6 0 )
*
©J* th e s e a lia r^ iq a a e itjr type,*
The
is tfee la ila ite s tia ftX v e liis * o f th # fe y p e r-p tra lle lftp lp s ii construeted
from tin® i a f t o i t e s f « 1 te e te rs *
The
fm e S * o f 8
S 't * A 6 ' t
is # r® » fey [ i f .
(2 .4 2 }^ j -
*
(8 *6 1 )
th is e q s a tle a is e q u iv a le n t to
4br*
* .* 4^* > ,
„
c)(x
<
**.* a?1)
£
»
(**8 8 )-
c k x * ,*** a ^ )
who**®
i« the functional detersiin&at * It should be noted
O\M. mm* M )
th a t no mmmm® Off » ! « • @*paei%* has boos in tro d u ced *
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Wm
ffee m a rt* * * -mlemmsk. mm m aa.tl-a y m a e trl 0 teas o r
.to e le a s s t © f * v r f * m
mm be » | « i » a l * i fey a m m w A o rd er e a t i*
s y a a a tri® t« a a # r o f th e © x te rte r p r« to e t type*.
I f two -rector® S j* aad
&g* form th e side® o f a mmtfmm- *%« a * a t#. tiw a tb * tons or iA ® m components
’•an*
1a ,* *
6s
*aCt *
*
S,3Ea
>,j*.
8gX
“2*
« 6 1-
S gS;a
(2 . 53)
%*a V
fee.® A vAliam ©cgual t@ trie © th e a re a e f tfe#
©I©j8©&t«
elem ent w i l l fee u»«& I s Stefe©*© Wkmmm Is & la t e r © ©©ties*
Bi.l® m r f& m
'
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All of lab®- f©ramie* -ettlah lia-r© bee® presented in Chapter II hm-m
merely re q u ire d tit# eslsfcem* of
«jerdi«®t» systw* ia m » -
diwnsio&sut. ja a a ifo M f th ey ha1®* * s yet -lees fiy®» no g@o«®tri«Al la t e r *
potation o r sig»ift©«»«©.*
of s?»ssoyati«R
Si# inbroteotiee ©f a few a d d itio n a l -poetulatea
repreeesbetioe -will .glee g®»»tr!©#A siipalffioam®© to
tl»*# f«nnnilftft$ this Intros tarfet©® farwisiw© tb# basis for a nairersol
gees*trieel lattgwag# #r Method of r«*#«ii»gf thereby direr# p h ysical
p r e t & w ©*» be eaq^reeeed la the same t e p * { « «
Jtaalogowi equation© of
miiiws ©riglse mm he *i«u*li*e«l eat imtertretwft om » mmmmi basis *•- the
p ro p e rtie s of the ■mri.ous type# of s u rw s , stirfa c e s * and spaces being
iso-sorphlo with #*ae ffcgN&esl
the a p p lic a tio n of th is universal
J*ap»agp to physical problemis result# ia & *.g#-«©tr£#fttl.«B® o f physics*
la th© a p p lic a tio n o f geswetrfoel aetfeede to p h y s ic a l -problem* it
1* usually neeessary to idealise mbegre by wftktsf ©erfeaia ei»pli-fyMg
« s # » i» j» tle » *
th e e x te n t o f the -osa»f%ion« is alm ost a» 1weeree measure
o f th® ©osrplexity o f the geometry r e t i r e d to re p re s e n t a g iw s problem}
therefore a « f .essampti** are usually troqsiirod for eoajflwt prehlewe*
Alm ost to s tin e ti v e ly the fb y « i© f« t will make th e neeeeetury assumption#
fo r the- appti.-eation of Saelldeae sethod# to a gteea problem* often s a tis ­
fa c to ry results ere Obtained*
there are «#»© pro-ble-ss#. h©M*rert ehleh
or© not a©e»r»t«ly eaqpressiM# ia. -eoeh a simplified fo r® *
I t then becomes
necessary to adopt on® of th© ator® geaeral* ead usually »©*•© complex
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
««so»
geom etries f o r ®i@
o f sueh problem s* ' f i m succeeding section©
e f th is p re s e n ta tio n o f tenser th e o ry mm c h ie fly eoaeerned w ith th e
geae**i8l n«ftn^l«sateniam tosd MXmmmimt geom etries wfele-h ia®l«A® most o th er
g *o g » b rl« * a s s p e c ia l cases*
Wmtom ta k in g up th ese ge«B etri#«# however*
i t is d e s ira b le to a lie ib wmm eomesptlea o f sfeat Is mmm* by space*
I»
Kinds o f fgaeeat
Os® o f th e am jor frobXeitts o f ffeileaeflig r tferewgfc th e ages has been
th a t o f d a tiijm te ia s what space la *
th e re jnmmr* to ha no such th ta g as
sp&ee, tyetk o n ly spaces, because th e re Is »#♦©©!atsA w ith tte# n o tio n o f
opuse so m m j d iffe r e n t M a as ( I S ) .
mod© o f sense per©®pt io n *
fi®yal»©X®^oally# apace Is 'o n ly a
HbmbJi a epeeo Car spaeesj is a y l m t t . spaeat
i t i# w am & m em m 'f& *0.
m A o f '« n o # rta i» d im e n s io n a lity *
Fo r
is pl& oes1m a publl-s
or p h y s ic a l space# p h y s ic a l spaa® is m m m tm l a b s tra c tio n o f p riv a te
space *
Biysto&X space is l « 9 p i » 8 i # f is o tr o p ic * th y ® # -4 its ^ i# io a a l.# and
Jfeelidoaa iaa i t s geowsrtfery# i t
th e $feysl«to-t*
is th e opae* u*#ft alm ost tn s M a e b iv e ly try
fM s spa®# is based «p®». th e o p e ra tio n a l n o ta tio n s o f
apaee^intervaX or
«&4 th e e t » a li% « f space in te rv a ls i
i t possesses these prep® rtl«® o f r ig id bodies which a re IsMepeadeat o f
t h e ir m a te ria l c o n te n t*
» 'tfe ® » a M o im »
Then th e re a re th e g e o w h rie a l ©paces o f th e
G eom etrical spaces a re o f g re a t ix ^ o rta n e e ia p h y s ic a l
th e o ry *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
«*3X*»
1*
G eoE etric& l spaces
i
tkm ps^iAmSL. wffmm
fm 1fe# 2 a » t
1#
to u n c e rta in ty »
tip ®
N evertheless , i t
has hm m p o s s ib le t # b u ild a th e o ry i» ableh th e -e#»©*pt« ( p o in ts , linos,
p la n e s , * $ » * } a re a b s tra c tio n s t r m
knowledge j b a t c e rta in
p o s tu la te s which appear © p e im tlo m lijF reasonable asset t® assumed*
This
th eo ry is geew ettyj tiw apae* it d e fin e s I * a g e cm etrical space.
The m tts n a tle la a dietdngaisfee® c le a r ly Iwtween. two types- o f
g e e a e trle a l #p»«#«*. s a n e ly * te e te r apaeeey a e tr i# spaces f t , p . I f ) ,
A l l © f th # Im po rtant g e e w t r i* * mm re p re s e n ta b le i » one or 'bo th o f the
above type® « f spaae*
tm th e f i r s t ty p e ®# apae© th e p a r tie a la r pestm lates necessary f o r
th e d e fin itifflft o f a -v m tm in tmmm e f a -lin e a r # f» a ti.« » in th e u n it
ve cto rs alo n g t i » mwmnUL axes' are a d m itte d *
s e c to r space o f b
dimensions t r i l l c o n ta in a »*©*&la*fe» m tm oa each o f which i® d e fin e d a
p a r tic u la r u n ity o r -wm m m iAg s tic k
th an a v e c to r ▼ can be represented
a t a :giv®*t p o in t in th e tWHtber space is. b«r»® o f it® components v 1 , v2 , , . ^
by th e •$ u a tl« k
r * V1
ea *
(3 ,1 )
ffe# ©an© pcw td lat© * A i d t f e r a l t th e d e fin itio n o f a v e c to r son**s t t t t t t e &sm b a s is o f a ffin e g m m tw y #
A ffin e .geeasetry is <sfearaeteri-'»#d
by the o p eratio n s M the th e o ry e f lin e a r- *$®ebi©n©t i t is a n o n -s te tric e l
g eo setry « 4 tb * -sym abrl© ©enaseetlas £s<i» ( 3 *1 l ) j ♦ . ®se space defin ed
by a f f in e geom etry is a liw s a r apa#® *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tm th e «pae* ® t a f f i a *
'veobor g«wB»»trie# a© re la tio n s h ip s between,
tii# aa lM es . a&asig the v s ri« » « sxa* m
m
i n t i th e re fo re such ta jio r te a t p h y s ic a l
^ « b i t i e s as, Ha* m p tltia t® o f ft w a t e r « i th® e a g le between two ve c to rs
«wm©fc be
* aw ry H a lt # * a p p lic a tio n to $ h y * ie * l th eo ry r e s a lts .
m o f t t » iy m *^ « r« a » th aeran
*# 2 «*
fo r o b lif »
i * 4 « *3
(s .g )
Ifts g th # mmi » 4f»e»si<m® i # la tra d a a e d y _tit is
f o s t a l f t ^ t f t o ffs e t* . iw lrM a # © # ft' r a t io tat*#*® . th e w a it vector® along
th© d iffe r e n t axes aw* .ask*® tb® veobor @r a f fin e » f see a m e tric a l space*
ftos geometry o f m i® a s tr lo apt#© is Immmm as » t r i ©
b« f l- f t e r *
gaoMSti-y*.
or a
g#o»#t#yf i t m y
g ^ m tjr y f I t 1# an in fin ite s im a l
fh » " It a s a r * g a e » tr y o f la e lM ,. th * *fcyp© rb© lle* geosarfcry o f
B o ly a i and
s a t Use * « ll.i.p tlo * ,g « ® » try o f tia s a n a are
s p e c ia l ea®#« o f a » tr lo
Q uadratic form Ji|;
* ($*Jf3f| ®fei©S» defines, t h * »trieal'fr©ptrtl#s
of tfa# •$*«* »#«i not be f w a it lw *
14#»aa h im s e lf .§©t»tad o a t that th is
q u a d ra tic f orm was to b# .reg®*##* mm m ffcy®i*ial la a & itg r* a ia *# it reveal®
lta#lf in ee n ta rifa g a l for#*® Ct#* f * PQ 2), for exam ple* a® t&e origin of
I® scnethnes c a ll# * * la ta r « tl*
gaaas&ry*
Th is r a d ic a l ©haage £ * th e ©wtlooJi open th # a e tr lo a l f r & jia r t i* *
o f space as ^ r ta la iJ ig to ®f*te§# t t s s l f * In a o p s s io jitly o f th e m atte r i t
c o n ta in ® *, 1® th© b a s is M m which fe w rfte ls has sin ce developed.
B in cteln *®
.general th e o ry o f r e l a t i v i t y Is haws* e» a asjg&ttv© Q uadratic -fo m i i t
Is an atte m p t to b rin g g ra v lta tis ® a a * # le a tiro « « a ® a *ti# ffee»«M»a. to th *
aceo iiat o f gaoswtsry ( 6 ) *
B efore th e in tro d u ctio n - o f IfoyX'** s p m e trle a l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
»s a> 4U *» U B i*a a f f in e
s e p a ra te ly *
tfass* twa classes of phenomena stood
W s y l** work mare e ie s s ly a p p ro a e i*# & complete u n ific a tio n
o f g r a v ita tio n m 4 ® l® © tt« « a p i# .ii» •gkmmmaa tin ® hod th e work o f
Woyl and others recognised th a t fo r «a ia tts lto o iia ftl psoaotyy to ho.
la * & • « ■ * * t w ith a a tu r * I t m o t to h ***d afon. th * fuB dasaat& l conception
o f la fi» tto a ta o l p a r a lle l d l*p 2 **m rti& # t e s t * * * o f th© f i n i t e d is p la c e *
a * * t o f ll* * * * B la a - geesBtffp’*
f h * l$*memiilam geom etry * *« *» to re ta in .*
p ro to t t y duo t o I t * * « e ld i* & * l © rtg ia ixk t h * theory o f ©tarfa c e s , an
elem ent o f f i n i t e geom etry*
fh© m s tri* tpaft&rKti* fo rm permit© compari­
so n , w ith resp ect to le n g th , n o t « %
« f two v e c to rs a t th e •« * » p o in t,,
h a t a ls o v e c to rs a t' d iffe r e n t point©.*
S at,, aeoordimg to W#yl (IS ,, p * 203} ,
®a t r u ly In f in it e # is a l goowetry * * » t reco gnise o n ly th # p r in c ip le o f th®
feaasferpftoe- o f a le n g th f r e e e s * p o in t to an o th er I n f i n i t e l y near t o
th * f ir s t * ®
M
i # t a t # » a t fo r M is th * assaa p tien th a t th e tra n s fe re n c e
o f le n g th f r o * one p o in t to «*©&&«? a t a f i n i t e d istan ce Is ln te g r& b le ,
#v«a as -the trftasf@ r«ne© o f d ire e tie n Is *m H L ateg r*fele *
g « o » t r y , i»
A t r u ly
o fta lo a f t § , p * 80S) eases in to b e in g ,
w hich,. when a p p lie d to 1fc® -w orld, w a it* * g v m e tta ti** and * l * * t r o « * g * * t i *
phaaerawoa to a th aoiy' in which **1 1 p h y s ic a l ^ p ia a titi* * hav® a * e « la g
Is w o rld geometry#*
The e fra e ia a r* o f . th is g*«st«h«y is given co nsidera­
tio n on *t»«**& ias$ page#*
•special cases o f th *
loth. W©yl*s t n i tiowaaaa*® g * © * * tr ie * a re
geom etry*
On **©<m&t o f t h e ir
p h y s io l a d g ty f le a s e * th ese m e tric a l geos© tries a re sow©times c a lle d "W atural"
p , 3 6 2 ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
«*
P lf f lg u lt ie e and precautions -,in th e d e fin itio n ® f te a s e r . d e r lm tim ©
A ll o f th e tsm ser
s f ifes B resions seet-lose h a w s ig »
M fle a n e e is itw n o n -M e tric a l m o to r or a f f in e spe©©*
The*®
a r e , i n g e n e ra l* e ie s e ly t i e i t© * p o in t o f th e spates* elth o o g k*. in th e
mmm o f C a rte s ia n refe r*® #© fm *a»©# a® has1*. re s u lt© ,fr« ® -s lid in g tfen
^ a a t it ie ® t e s a y p o in t o f tfe® ©pause*
l a n ffto e space w ith Cnrtosim a
r#f© reae© fram es thm m fe m tim o f d iffe re ra M a tie n . o f te n s e r
©as&aes m w r y sisspi® fm m t ,tsit© m rie v # easgmmutm m m s e p a ra te ly d if« *
fe r e r tia te d ms in ordinary- ©mleulus.*
B at whoa c u r v ilin e a r axes ar©
ta tr c d u c d . t h . traziB fonafiticm
jn ,. (8 .8 ) ] « - a c t
h o i a re fu a o iim s o f -posit im&$ fitis f a s t MkJ«« meemmmmry thm in h ro fa o tlo a
o f an m&dtttfitim l p«©t©3fct» in © ri# r t o d e fin © te a s e r d iffe r e n tia t io n *
A p a r t ia l i e r i m t i w
© f & im m ttm
is d e fin e d ms th© lim it o f th e
te©r««!»fe in- m is * a f the- fw oeidon fcstpsos two n s i^ tb o rln g point© A m to
© -ssii.11 tmmmmm in mm o f tty* ©ooM iwktos* w t t l t th s ©i&ers a re » i s *
ta ia e i- © «nst© nt# d ivid e d by th© Mmammmt o f the © oo rtiaat© a# th© i**«
o rm m n t is allo w ed to- ©pg&nesii w ro #
fa sh m lm mast d e fin e th e proee&urm
ia the- fsawmt&SB o f te a s e r d * r l m t t w s .
¥© define the ,p,rM«l t « r i m t i w o f a
the m l m
of the
teaser at two neither img points* t and P* * most t» ©o»j»r©i,
I» gmmm%0
the ©empeneate of a tmxmmr h a w different t ran aferine.t±on. coefficients at
ten- different points § tMMwtmm that the » 1 « fo r teas or addition and
«»%bm®bt«w IS net m l id for * w k a «ms#f therefor© the partial derivative
formed is th® ostwl mgr is wst * tensor*-
It becomes necessary to Intro*
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dne# a p o s tu la te o f ©expert sen. b e fo re ft # © m ri*n t d e riv a tiv e o p e ra to r*
ftfaiefe* ■when a p p lie d to ft te n s e r* fu *a l*b » « ft d e riv a tiv e which is ft tr a #
te n s e r, ©an be d e fin e d *
4 * . the..
>4i«glft«©aftitt end tlift affine comeotloitt
•® §rl | i i # fw £<*»■) h m shown t h a t I t 1# p o s s ib le to tatrodw s# th e
e o m ria n t i i ^ m t i w e ftgw ffttor
s fe e ify is g a ja e trie mm was previous"*
% th o u g h t necessary) th© e-hole* o f a la e trie *. fc a w w r , 4©es p erm it s ta p le
d e fin itio n o f ft © © variant d e r lm tlv e *
la o u te r to 4 « fto ® a ® » r i « i d e r im t lv e , fo llo w in g B r illo u in (2 , p .
who as®# « e y l*s .ia©tho& o f approach* i t is f i r s t necessary to tra n s p o rt &
teaser iftfiasi. at eae point F l e a neighboring point !**■», Starting with
ft v e c to r a. (x 1) wfeifth is d e fin ed a t P (x 1,
* . , * xK) , th e c o n d itio n
for its-paapwllol € ia p l« e *a » » t to F* (it1 * S i 1, •••• Xs +
fe® defined*
8 jt ° ) w i l l
For some partlo'ttlftr iw£ftr«»«# system th® mwert© sepality of
i
—
'Sis mtmpommtm v r m i m a t r and f * ©«» bo ad m itted as d e fin itio n w ith o u t
lo ss o f g e n e r a lity haoftuaft t h is
an o th er re fe re n c e systw a*
th is
w H l a # l i f t p - ha r e a liz e d in
is aam aed re a liz e d in th e
co o rd in ate s f » i « x^f, th e re fo r#
Ca*«)
The subscript © fttftaAs f o r o' d isp lfteo d ftwalwatia®.. ■ If" both a id s * o f thi»
mm transformed to map other ©©ordinate eyst®®
the result is
'fe a t
<3*45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
v.i i x k *
&
•artor® J^JL, ( xfc) , « |
s * k) - - ^ r - f * *
o*
(x^ ♦
terse® ©f
s»fc> -?r.
(5 *» )
6 x k) * ® « i th a t thoae 1a «aa f< a»R ti« i.
,J S .
utlU.ing U f U * ' ,
*
. i A ;—
,
t h ~ r « . >£<*k *
value of u*(x*“) aafl. th®
of v S ijx ^ ) at the point f *
«*pr..«4 te
of the partial derivatives
5 M ® oporatloa y ie ld s * sogleotiag th® higher
tsars®*
y
<3a
<3#r
( s .6 3
« taee
a^Cx^} » * ~ r r a (x ^ ) a
<3m
a
(3,7)
*
It follows that
ox
a A *)
_
d x ' dx a
<3**
J jL ~ m a& ) 9
3x
(S.8)
ft® sa%stlt»tt«» of ’
(3#,S} in eqpattess. (3,6) yield®
aiC x*- ♦ S x fc) • a^ (x^ ) ♦
aa
<3 *
6xk ) • u H x^) ♦
i, t.
vfaa**
S
(3 ,9 )
dx
<$* k - & S l - i z i
3 x" 3 x
<3 x 3 x a
a (x )
<3x a
m® c V *
(3*10)
4
m
3V
3 * 1 <3i
#
.»
►a
a* ** *u ■ « *d■ "w
-« »
*
3x^ 3x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3 .1 1 )
earn be mhmm t » fa# a. 8a&*4ft&»or' % n*attity> 1% is c e lle d th e a f fin e
ecgsmee'fefaft*
I t Is m m m & it h e re th a t to® o rd e r o f d iffa r w a tla tls n 1 *
to m fa s ria lf. th a t i s , th a t the mymmtr y
•
I* '
* r i.
i
is tit® s y iw & trio f a r t o f i ^ , ®*isfas.,.
< « •“ >
.
Ufa®**# r
ffe ia mssftaiptii® render®
» p*e&% s liH ^ llfio a tta a to. m a y fa m n ls t*., b a t i t i# n e t necessary*
Sepatiaw® (55*11) g iw to * oasiditions o f p a r a lle l #4#flae«awent o f ft
4
■weeter n | th a t i s , th ey g lv a th e e U w afelefe th e m o to r ft a m ln a te d a t
P assumes whan I t Is moved p a r a lle l t® i t s e l f to P **
Th® w r y p * r t io « la r co ordin ates in whioh tin t esi^po»«ifaii o f to®
m o to r 'n a re ©<|m l a t th e two fe to tft .are c a lle d g#©d#aie coordinates *
Titer© is «» in f in it y o f ipoAtsAo Mfsfemm fo r im finttsaslB ial d is p la o a a e a ts i
t o fo o t,, any a r b it r a r y 11®©*®“ tran afo raafalon o f ooordinatoo s tile h in vo lves
eoaastasat o o e ffio le is t* o f h tm s fo p m a tlo n w n in taist* th e © q u a lity o f too
ooagKtao&ta*.
Tbes© yart«**s © o«rdi«»t#a m y h a w d iffo rm a t M m t w f t » i
fast to o y a t * s t i l l g@#<§#si© ©©ordinates fo r s ftffl# la s t ly e a a ll Meffl&o«a®at»-<
S*
eo variant- fte rlw fc tf® .©f a w e te p
I t iss new jw *» lfa l* t o d a fla o a © o m ria a t d e riv a tiv e -to:let y ie ld s a
ten so r o f low er o rd e r*
fe e faro# inoroao® that* to • ooaponent o f a v e c to r
ft fatfaww® tip® nearby p o in ts ? and P ** is « q « l fa# th e s a l me o f t h * m o to r
a t th e sad p o in t f * K iw is th e w in ® o f th e m o to r a t th e I n i t i a l p o in t P
id. to to # os-d p©l»fa§ th a t i s , th e tru e im opm s*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* 5 x^) ■»
•gni *
* r*
A
a® 5 x k *
(3 ,1 3 )
If .Utis equation 1* divides, b j tl» tr»# «rl«ti«n Sxk* th
© eom
ri&nt
derim
tdT© is ©
btftlpfti.#
f y
” ®*k •
*
r
l
“m
<3 - M >
lb
-®©es»*l«ttt tsrimttir# of * ©
©
<
©
»
riant wetor is fsrw-i In ft sim
iliter
w
ay* T
h
«s
u
k
ia
- iiffotme® imth
©rwalt# is that instead of &mixmd
tensor -o
f tii® seco
n
d order ft ©
©
©
©
riaist t«si«©
r@
£th
e
- •©
©
©
&
&order i»
©
htftL«8df also different
ar© obtained*
d«rimtiw of th® ©©variant,©
©
©
to
rf
"“" I *
1#
1#k
fh# emrnrlmat
i# given %
~ ^ b * ® ik va •
dxk
ik
( S .15)
*
lb©-affine eo
an
o
o
tl-en
s £ or the o«itr«mriftst an
d ©em
ri«rt ©
«
.©
#
#
-
mn. be related -easily tg
r form
ing a extracted p
ro
d
ta
«
fc of the tw
o vectors
a
n
dh
jr using th® r«l®tl«n ia t th# o
o
m
riw
aA
- derivative of the ©
©
&
l&
r
tisas farwNI 2* tfe
®m m ft# tli# jjsrtis.1 deilmtive of th© ©
oalttrj- that is*
■ Jjgg ( A ^ ) - f
^
«*) %
* “~Ag: C ^ i ) •
♦ W4
# *5*
*
Tj_ ♦ ** ™ | - *
from eqaftttoa® (5*1S ©mi {& *!? } th© r e s u lt
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
**•**>
(s.I T
(7 7
* rL
1* ©Metawl*
“”) Tt *
( S
FSroB
*
T« ) - 1 7 Ti * tt± - 0
*
(s’ie)
($-*l§ it fellow* that
or
r t
• s \ |
(S#20)
•s*t«r® tli© a fftm # eosmeeti on fo r ®#m sri«at d f f f e r « r tti& ti on is th e n e g a tiv e
o f th a t f o r @ «atr»vafi»«t- d lffe w is ila tfo it*
th© ©gswfiesfc d e rlw M v e e «%©y th # same rule© o f prodeete* swss, and
repetitions as do erdtsKry deriretieesj therefor# these rules will not
he- repeated here#
6.
€ o m rla n t d e r im tiv e s o f .te n so rs
If
(w1 T j) I * used i » « tp « tie a . C s *I§ ) In s te a d o f a4
then th e
e$uet£s&
®*k
\ < )#
• —
sfle
^
<3**
/
e r *
i«fc
3
\
»® * .
I
* * 3)
r J
«i T
*k
is o b ta in e d * Frost th # r a le o f fo rm a t!©& o f te a s e rs
e le a r
J
®g« ( 2 . Kg)
i t is
th a t (a 4 r j ) ie m mixed te a s e r o f th e eeeemftorder ? th e re fo r#
^22 „
3E
d jL
♦ r-*. t t - r-*
•* J
»
t1
»
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(s.tt)
Th is isattkoi o f ftoraittg
of
ixo d fo r - t i» prodwet o f F
R*
d iffo is s n tifttlo a & m be gossra!*
o r fo r ft §ea® rai mixed tensor o f order
ffe® r a le f o r forrafttlosa o f th® © aw irlgait # « r i« M r « o f tfe® ten sor f
ma. be as&3*«*«6& thm#*
•fe.1
(^jg-JL
* r
ml
«f 4
, - —,
III
f ’""' *'**lv
T ^ - ” f * . . . . + r-0
lj« ..! t
ml " i J . . J t
m Jtib.m *■»«
»
* r
ji
*
**■* •
(8*25)
■T-*
T*
C o m rifta t i o r i m t i ’w©* © f .|wK^*MN*H*©r»
In or& er to d ftto rte is * tfc® R o w e l* fo r t l » eorftrisast d a riv a titre o f a
so & lar
or. oft^ftei'ty i t is © «o r# »i# st to s ta r t w ith e ith e r a
pro&aafe o f a « m J ftr
and a ®o«ifepa'ttftri«»t -re o to r, o r a s c a la r
^ p a o it f *»»<® a e o m r ia a t nsotor-*
x o f th e
ttj* p a r t ia l € # r im t it « w ith r e e p e a t to
d e n s ity
A* * A #
la f i r s t fors»d«
<8.*I4)
fb ls d a r i m t i r e is giT#Ji' %
"a? ”
*P
a ? * i 1 > ? j*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5 * * , }
t* .
I f i t i * assumed tit a t toe tr e e efew g* lor in a
placement is » m m 0
f a r * p a r a lle l d is ­
(S*M> gif#©
**^ k *
* r i:
(8 *2 6 )
ffee s u b s titu tio n o f tM ® - r e s u lt f a e g tto tiw i (3 *2 5 ) prods©#®
*»
**
la* *
(S*f?)
Slue# it is psratoaibl# to In te rS c a n g*th® i a « f ladle#*» this e^uatlm
is #<piml@ia% t©
( *£>&** mA r ®
a#
^ a**
|
.
(3 *2 8 )
**/
,'This .partial derivative is a scalar density because there is a
eontxaetlsn ,«f th# toil®#®*
ft® ##»p«»«rfe® la^ represent a contravariant
•vector j toerefor# to order that both side® of th# #%satlm ® be scalar
densities*.'the t#ra i» the parentheses w e t be a eewrlant rector density*
This resalt yield© th*. following. deftoitioa for th# comrieat derivative
of m ©ealar densityt
J jl^ rn
* AT ^
*
(5*29)
»l»i,*ari,y.# for a s c a la r -estpeextp*
J — .. # '
* fc
|t| *W. ^ *: A *
jjgpT "*
—
^
& ^L*i*rM
fc© *
(3 *3 0 )
J tfte r th e e o m r iw t 4 # r l* * ti® » » o f a e a la r d e n s itie s and o f s e ttle r
e e p e e ltto e bare fee** d e fin e d th e e e w a ris a t d e rim tiv ® # o f te n se r d e n s itie s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* 4 IN *
«®d &£ b@a»©r
® a*iS y fo llo w fcgr th© o » s to * r y p ro ta e i m l® o f
d iffe r o n tia tie s i aa*l
d e ri v a t iw
asd €-3*30)*
Th© e o w rta n t
of * t e « # r <t«ssslty T is g t« m %
T d
a ^ r ATi j . . . k
()
1
•u -.|w ip
ft
■
*
.*
* |I_ »
ml.
*
t
T~i©
ax
iJ,
***©©*d
*©*!** ai*©
©t© **#
%
w ™ ^sl
&&&#>• .«®i
„
„■»>•. »o
r i j . . A * ****
^11
^Z x
m, • # *
*
Ta! * ’ *e
ah**#,®
>&b # # *0
*e :2
i j * . *k
*
<3*31)
* « ...*
wh.«r« th® © ®%®r the :T is m«ed to iffitie a b # th© t ensor-deris i t y © lm m «t«r
© f th # component *
A s iw tia r jN ia u lt i« ©bted&ed fo r a t « » © r c a p a c ity *
Sfo® « a ly d i f ­
fe re n c e between. th® d a rtiM tti1?® © f a testaor d e n s ity aad teas#** ca p a c ity
is th a t th e la s t t®*m * r ®
mu*
«»r^ t ^ i j * ! jc *
■^
eq uation (3 *3 1 ) i© rep laced by
th e C o w r l§i# T r^ p a w w m t*. t®na««r c a p a c ity c h a ra c te r
o f H » OOia|K®l®l3t*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8«
A b a o la t® d e r im t lT #
f t * # » # to r
§ « © t t i » s . to # c o o r d in a te s ©# * 110M I# p o in t I s a »pee# o f r dltns»«*
© le a s. is expressed * # * fm m ttm
o f some poraisetor t *
T e s te r « i s earrtsw t alo n g w ith l b * f © l» t# i t stay fee
to # t r e e a h a a g *
o f t b * e.m feoejsto o f u aft® * tv I *
p te p a lle l ile fta e e s e a t dta© t o - a ®h<mg® i » t *
I f a e o n tr & T a r ia n t
to d e te rm in e
s s fe je e te i t o a
I t f o llo w s fro m e q u a tio n
C8 * 13) t h a t
fh#s© expressions re js ’e e e a t th # eeaptaente o f th # ab so lu te or ia t r ia a i#
d e rim tiT ® o f th e « M t« r « *
9-»' ":.geo4»M, & lilie s
% # w a t e r « © f eq u atio n (3 * 3 t) m ig h t fee gH«M as th # T e lo c ity © f
th© m obile p o in t f th a t i« # i t m ight fee » p p # *w to d fey th # eomponentg
(3*89)
la t h i s e a se th e afem tlitto o r i a t r i a s i # ie r im tlir © w o u ld g ira th© t r u e
a b s o lu te
o f th # w M I e f e i n t | th a t %»m
would fee th # tr u e a c c e le ra tio n *
fear th # m otion to occur a t constant
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(:S*56)
* o.
f h i*
m sm all *© p » « b * f a •s tr a ig h t Uaaw> th e g rad u al
a. i»»A »gta...llB »-» ■
She c o n d itio n s , f o r ia fls tt# * fe tta l p a r a lle l b ra a s fo rb o f ten so rs have
boon give® *
:§ aa those tra a s p o rt* bo extend©# to* fla .lt© di® ts»e® st
S h i*
•«ja«8ttoe eaa be «a*w©ro# $u ib e © s a lly f o r s c a la r f » « s t it i# s by s ta r tin g
w ith th© valu es o f * s c a la r q> a t & p o in t f and a t a »#srby p o in t P:f„ f o l ­
low ing i r i l l o t t i a ( 2 , p * i f ) , by * p *« ify i& g what ffe e to r* determ ine th©
efcang# !m tfe© v * i« « o f cp fo r a p a r a lle l dfspXaesm m t fro ® P to P% tat#
a fte rw a rd s by oxtossSliag th * d i» fi» « « s t*ijt to f i n i t e in t e r v a ls , a c r ite r io n
M
h© ob tain© #*
I f 8 3^ a re th© mmtffmmmtm o f th© v e c to r 6 * » PP%
th e v a ria tio n , o f <p w i l l be p m p & i^ ie iia l t o th ese ecsapomentsj th a t i s .
St? * %
(3.$#)
where f & 1# a 'component o f a o o v a r im t v e c to r whloh serve® a® a propor­
t io n a lit y fa c to r *
I f I i s t p o in t a t * f is t t ©
approach©# alo n g ®mm paifc ?f*A Q *
Has -m l«# e f f * 1 Q m ast be equal t©
tfe© value o f cp a t t p in * ife© ©hasg© in <f>j th is
by a llo ttin g
ils ta a © **, i t can be
valwo mm.
6 * i a o r a t io n (5 *8 § ) to deer® **© t o d *
be expressed
and% in te g ra tin g
th e r o s u ltia g ©aspressioii alo ng the path W * M $ th * r e s u lt 1 * th a t
f t dat *
PP*iQ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8.87)
-IS
I f ©tgtastion (S»S?) is «a* iaab#{gr#feX# r e ls iio s i* the® I t a s s t b# tru # f o r
any # tb # r path TB5U
For th # fs fe f i t
<s»ss)
It
is c le a r f e a t th e It s # in te g r a l
(8*89')
most -vanish f o r
I t «mw» be shown t h a t fo r t h l# in te g ra l
t© be s # r# fe a t fjj. must be fe e g ra d ie n t o f th e s e a ls r fu n c tio n cp *
t M # dem onstration I t 1 *
ll#
fo r
t© mm S toke*# Th##r«&*
Sfofee*g theorem
S to ke*» tIwor#s* i« a w r y asssftal theorem by mmma® o f whleh * 11b#
ia t e p ^ t l around a closed o m to a r of' a e u rf*# # is #aqa****#& 1® t#f® « o f
as ta ls g r s l o v e r tb s « » rf# « # *
A- e ® m r i« l re c to r fu n c tio n w ith components
■tend i t # f i r s t d e rim M v ® # a re
over a scurf*## 8 w ife •
ooatlawoa# and s in g le w is e d
8y
Th# lin e ia t # g r * l ©f th is function.
eofOM&t C Is tb s# g iv e n % »
(8*40 )
th # su rfa c e 8 Whioh i s beanded % # ©an T» a *4 # a ja rfc o f a tw o-iiiE ensi<«-*
*1
m
surf&o© %• th® jarapsr tfittie # o f th # eoordinat© system .
is th e mmt ©©op®!*#*# aysfcMt maA x
If
m A l i 8 a re th e v a ria b le co ordin ates
©a th # s u rfa c e i * th#® th e r « # ia .in t © © © riiaate# a t'* • •• /a t
a re O0a«h«sat*»
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ftm snrffcse- i mm a&m he
th e eoordi& eie lin e # *
into m m l I
tSCSft, formed %
A t A (|r !)* th e v e c to r lias th e o©®j>o»ents T i ***. 1
in th e a w eear<Sl»fcte- system*
fh© Vkxm in te g r a l o f f ar©w»! ABOM.
(tra v e rs in g ABCHfc mail 6 in the sea* 4 ir e # tis » } is glvs® fcy the sura of
th e four eeefpseen&e
%
s*
c%
■
a ;1
S * 1) SS8
a?,
•> U j ■» " r j
S'*3,
as
?g
S #
Along
A if
e le & f;
SG-j.
e i.m §
e©j
ftltrag M f
fh.@ swra is givem h j
dY<
i s~2
Nx ox *» «—»**■
S^Sx2
Along
a ”'1 A |
a#
dw r
th e re fo re i t fo llo w s th a t
g I .**
-
a is*
r lg
«*-* * * ^
ax■
S x1 8 x^:
5 *1 85s
■•here
d7*
*1 2 * x— l
M/it.
.jM r s e e n e e e s e w *
a?*
e©
te ee© © © © © ©
( 3 .4 E )
a*
M
■
’sr
*U
m *f3 £ *
a * 3, * r ,
ij
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(s * * s )
i§ * r « f« p e
851
(S «44)
^
th in equation is e q tjlm lw a t to
SI
M .
$ r15 4s**'
Xh* ff c s to r ^ fc i
S *3 6 * *
<S«46)
(S *4 § ) is aeeessary' h#*$ « # *
jjSq,
4 * * 3 *>
Sx1
S*3 •
i tb a t is * th e SJFSCt is rejMPSSeitfeMi feti 6 0 '^y th©
jareeenee ©f t l » *sfci**^rosssfsri© -fernsor € s *^ »
ft©Je»f « tfe.#©**#m I s ' »©*r o fe
fefctB** by In te g ra tin g e fe a tie a f l* 4 § ) mmr th e e n tir e m&rfkm % tM s
y ie M s
-
i
*1
(3 .4 6 )
«P
(8 *4 ? )
f t « *k - |
where
b t*
m '’""'“j’’ '** **^g" * th® ©si*! o f jf*
I t m t i m (3.*47} is lateen m
M*
(5*48):
S%©lw% theereau.
I n t e g & M t i t y sscaitioBS .for gars!.!®! disfflfcssiBsat of «. smlxt
By sowts o f Steles**- %he©r*»*. e®t*«tl<ra (3 *3 § ) can he expressed in th e
fetm
i
^ «■- *g*
r i j d* 13
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3 .4 S )
■ *4S *
|» w iw r ifeftt <P^ ~ <Pq » te n i-e it*
$ **.
*
8
d fj
la order th a t
■a %
- #
*-«
fee
m ast m o lsto *
m*
-w^ 'WM, ;i ,|,Ltj , ,
<3at*
gg
dxJ
(s«so )
f | and f j mast he tfee #«sp K »»t«' o f tfee- g ra d ie n t o f s e w
s e a le r
<p § ifcaa
I f JPjy .1# e«M»h « e « r l « f th e grmdt«a% o f a s e a le r fu m tim ,,0. th en th e
IIa ® i» te § r ftl
/
% 4 *^ i * * w e # end th e ««fti«?- f e a e t im f has a
®
k
unique valu e a t «&«& p e la t i ho t i f the: lin e In tftg g fflr / c
dx is mot
*«*■©-*. the- « ! « « « « f <? m m he ©ftMgarwi o n l y i n tire in f in it e ly m m lt region
8 m r r e t» ii» i ft f e r t ie n lf t r f o t a t *
I f th e v a lu e e f ft w a t e r n f t t ft p a in t P i« tfekea and d isplaced
p a r a lle l t « i t s e l f t e a aawrfey p o in t P*-* the- e o n a itie w o f p a ra lle lis m
-Iff* hy eq uation (3 *1 2 )» tfe&%
(5*68)
S i® © o n tih ia tt y ie ld s th e m rS a tlftft
4sftl •>
< 8 *5 6)
intow e
i
*■ * *
k
1
r*
s i c
»
« ♦
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( S *54)
f ! » lin e In te p m l o f to 5' ercw s i a f i n i t e closed petit. f * b o o a lin g m
su rface i can now be fo ra td 'and trta ^ o rw e d to a su rface la t« |p *a l'b y
wmmm o f I t & » * * theerws. j i t *
6 a1 *
C s.4 7)j .
j f|
these o f® *a tt« im y ie ld tfe® ohamgt
» *| I J
<3fi
r^
mm *
(3 *5 5 )
d *|
Cs* s6)
f o r te ta g ra b le
r ^ .w » t be s e ra * I f r j ^ is n o t s e re , th e
|» t « p * a t i« t l a SapeaaU&a#
4 h i»
l a tense o f
as d e fin e d by equation ( 3 .5 4 ) ,
by
\
r* “ ~ T i f”
*“
<5*k \
■*
yd®
V “)
)
* ...
( - r 1 1?]
d * \
"*
I
cJary
*
-2 2 £ .r *
•
c )ir
a& <3ar
( s - 87>
Bjr equation s (3 .6 3 ) and. ($ *6 4 )
<3#
..wifci iijnpy
- m -
d j*
m
m
~
I
m
J-
'
and
\ ®
— '| T »
£3*
.
^ r’®
m
«
*
Si®' s « b s tlt« ti'e a o f eq uation (3 .5 8 ) In to , ( 3 . 8 7 ) .g ive*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<8#68)
r '1. «“ *
lit
rite
»i
m s
whsrs
i
Sfil
" T T
lh
<?*
I * a coaapmmt # f & ts»®«r<*
The iis p l& e e w is t o f th e i» # t e r * w i l l he tm t»g*m hle I f tit®
1 mm a l l « **© •
te a « o f#
H i# fe a r th © r is r te n s o r 1. is fte fp fc o tly ee l le g the
1% cart S» as©& as a c r ite r io n for- th e it e s t t f ie a t t a a
o f th e d iffw rs m t ty y e s o f sjaaseej fe o lia ts a
f o r eita’ffiple m hag sero
© u rm tu re *
Ju st as a c o s tra m ria n t v a s te r was subjected t© a p a r a lle l tra n s p o rt
to a f i& t t # -dletttaee* so a ls o e«a a e trm ria a t weefcer*
The eex&ditlon ©f
p a m l S ells ® fo r a # © ra rt# » t w e t e r i s g lw *» t y ■.
m
4 • d r ^ *> r * . v.
1
1
jk
m
* *
3
“
*•
w here
I '&
a r **
^
_
, k
f **_ 4*. *
ni
m T”
v „
1k a *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Uw&K)
By fe e s u b s titu tio n o f eq uation (3 *6 2 ) in to eq u atio n (3 *6 5 ) , and
equation. (3 *6 6 ) in to eq uation (3 .6 4 ), fe e fo llo w in g r e s u lt is o b tain ed t
i
* *
/r **
Jj
>
*•
where
*
a
i
•*7 1
3,kh
,)*
i
r
3
• --*x
Jh
d*
i
r*
i
* •r
•
lfc
i
r*
Jh
nr
Jit
( s .« f )
As b e fo re * th e cu rvatu re te s te r- aast- be sero fo r In te g ra h i 1 i t y ; e th e r is e .*
th e m l « o f th e lin e in te g r a l o f a c o m ria n t te e te r is aero *
This
statem ent i s e q u iv a le n t to saying th a t th e order o f fo n ain g th e success lire
eo -m rian t d e riv a tiv e s o f a v e c to r is im p o rta n tj fe e result obtained from
two successive d lffe r e n tta tlo & s depends upon th e o rd e r o f d if f e r e n t ia t i on *
Si® roetfiod o f a tta c k used here can. h e a p p lie d to easy te n s o r.
The
g en eral r a le f o r a te a s e r « f a g in ® , e id e r is o b tained fey •aggressing; th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
teaser In * teeter frwSast term |~8«i* (2*25)] sad. W
wiȤ
produefc
rml® Iter
The Eicci-Slnstein e«rt?raete®i stsrvsttar® tensor ean tee easily forated
f m m e|
tey eeatrftst&ag %im i #»,4 k izuile®s to give
' '* ' '
1
i
_
„ si
„
jE S .iE ii*
r - 1 x -1 - j - 1 j - 1
s3 ,n
#j , i h
3xi
d3k
it
3h
u.
ji
This, tensor is used esetesslvely 1& relativity theory*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is ^
! ’ 5
it*
w
a rn
#s«stif
Th# fundamental -profarfciss of t©j»<sr# as they ere related to an
■iMTfbew# spate have bm m sufficiently elicited*
It now becom es
desirable to e & n m M m then la relit,tic® tt tab# IttiBdowental quadratic form
A t1 »
dx4 4 *3
(# * 1)
which converts the *®«rpfe«ws m p *m t»t® » eetrSeel or t-wassaii®. manifold*
Prom the a«B«awtri®*l point of view teaser# or w o t or s ere defined
at point® ia the manifold# bat# -as an
a ret tor at a specified
point 1b tee manifold eaaaot be ©owparei. In majpltwi® and direction with
sora© different asstor defined at the same point*
Sefor®- -wash quantities
as magnitudes (length) -of w®«b®r» and dlreetldB of vectors, and. the
I
angle between two vector© m m be «p®*tfi«&* it 1® m m m m m y- to- introducesome relationship between, the various n m l M # ® along the different axes
at each point of the spaea*
Uht® r»-'lati««liip is protiiea by -the fund­
amental quadrat!e form |Sq* (4 *l)~j.
This quadratic form, define# at
each paint of the a manifold want a • 1 dimes# isnal hyperewrf*®® m r r m m Atag the point*
All vectors originating at the central point and ending
®a the hy^reurfttee- are m m said to be equal .in length* or in absolute
nte*
In the quadratic form ds is the length or magnitude of the
vector nfeoae eon^oneete are <t***
%
applying the patient law to the fundamental quadratie form* it
follow# that g|j 1 ® & component of a tensor of the second order j this
tensor has, in general, a 2 Independent components*
It #o happen® that
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H j “ « jt*
therefore only d ” h
{4-2)
of n® component* ora Independent! that is,
■it m y m m trie te n so r o f tfe# *©©«mI order.
is
1 * will 'fetter t » c le a r that
t e is s p M s te y Ju st means t e s t f o r th© ease o f re e M S iiie a r ax e s , f o r
©3aw.pl© # t e a t tli© wumtom o f t t e «Rgl« feetetes two eompanmts das^ and
* e * does n e t depend np«a «ftt«te«r th® r a t i o o f th e pr©J©«%i«» o f d *1 on
dat^' to dbr^, o r ffe© r a t io o f th e
te te s a # th e
1*
mt
©at dar** to d ** 1®.
o f th e essltws o f %tm m g l« bwtmwm them .
Fmdteteot&l properties o f te e metric tensor
tfa® em poaoat® o f tfe® fte»da»®»tei a o tr io t m *© r ® m be arranged
is the tabular far®
X.
in
2
®m
* * *■•
Tl
s i»
/
i
i>21 H z * * * * «i» t
©©* #* * ##«'«©
♦
ir a
Hi®
®»t * * * * *a»
(**S >
m
of thee© ©te^p&seste is gifwa by
%1
« it
«tm
%1
Hz
« fa
Cd*d J
l« u l'
SLw%
* * »'* -fe—
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This dfttars& oftnt sweat be matpm4»4 by jsinerft w ith :r « s |* e t t© * raw or ft
®©:lwaa t© giss®
with f*®ps@t to r w i) *
« * X
wfasr#
(4.5)
^ 4 C *!® **!© © sritSi r©«p»©t t® ©@lwm i),(4 * 6 )
ift tfe
® #©
f«©
t©
r of the
life row
sail. Jilt
este#
fyott dftteriairt&nt th«#yj-# it is fawwtt that
ii| «Jk » <*#
tfeftt ^>~
» O, j / k
.(4*8)
e
than. th e ft tar® *%l»rt;i©m® can ba ©©8&«n««& la the- fonts®
f¥** _ «*Sk .fflft; ....k m
3
111
0
J « l
if j / k
(*# » )
' kt _
fc
*4 1 *
8 °3 *
1
0
if 5 • k
/ k
i f 4 j#
t « 8ft#y g^ 3 is © a ile d ifa® ftts ia s w a tftl ©©afcrftsftrfftset tensor? i t w i l l
p r© »
©f g*
u s e fu l Is
lad leftft la la t e r ftfteti© B «*
4 e ta rs l» a ® t
eq ual t© the i 3ff«y#« o f th e i«t«n^aii34% ' o f
th a t 1#,.
s#
**1
1*
Th# ssftgsitwi# # f ft
.the .ftftftiftr grodttet
The fuagamestel quftdtafttl© fomm 4m* » gg« 4»* 4 m $
the mgsitaii#
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i # o f th e v e c to r wfeo** «m pm m 3^9 m * 4m * *
Asso e la te d w ith •*# & p o in t
o f » « $ *« • is *
u lm g . th e axe© of whieh th e
r© fer«ae# © o e riis ftt#
ft may lift fftpfftft^B t® #*
J u s t ft ft fo r th e §.&*►
ftftitftftits fti w e to r 4 k * th e magnitude o f say e o a trfts ftrifts i v e c to r « & m
0P l^|Mr###^TP8sI
i i$
|ft|
i
4
■•» g j^ ft
{4*11}
If the veetor Mtfwmft to be « & m m tim m t v m t m iri# its mfpitede is, by
4*fSfti%l«a#
|f t |* • f 1^ % f t j*
(4 *1 2 )
I t »a»f becomes d fts iri,M « to «s#© elftte with. each e o w i i s t v e c to r ft
e o a trs w ftrie a t veeter* ft»d w ith «eeh c o u tr& m ria n t ve cto r ft c o v e rIa n t
wftbar#
the ft»«-©eiftt« © « a trftw ri« a t #eetor of a o o fa ria a t rector fcftft
the ecmpoaftfttft
v l * gk i iFk
(k * 1 « * * a ) ,
fr a a isfeiefe 'the o r ig ta ftl e o tw ia s t fotm 1#
^
*3 1
ea n rtn *
(4 *1 3 )
e a s ily obtained by m u ltip ly in g
w ith * " * M * * ° * * « “ r * , u l t 18th * t «'•*
f o n " " ln r-
re la tio n s *ad«&«
i
ai
wa
Ti *
8J
V‘
(4 *1 4 }
TJ w lrJ
I» th i# same m y * e©variant ®©ag»»#«ts «r e « t » w r t e l ©o»j>o»«t» «
be ft««oftiftte4 with may teaser*
%
ae lag the -ftfteeetat® e©»p®o«»tft of
Mi# v e c to r* ftqt»1sl«ft.« (4 *1 1 } a&4 (4.12) m n 1# simplified to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
wǤ,fW
jtt P * m* a *
(4 .1 8 )
C 4.16)
fhes# la s t two «4ta& t£en* saggss.% tfes s « * l« f p ro & is t © f w ro ts r a n a ly s is .
I f two -rectors w and f sr© s p s s if is d * tin ® •fe « ir s m l& r progM-et is
-i®ff sa®4 toy
'
( w ) • g j j n%
* ,g%$ ^
^
i
1
u 4 tr ■«* t*r
(4 .1 ? )
B gttstisa ( 4 . 1?) , la ta g s * soggest® tfe* -mmIn® o f th e s a g le between the
v t f t c r t a and v *
fro ® iflsstsr *» a ly s l® # th e s c a la r p e ia s t is d e fin ed fey
(« .t )
(4 .18 )
©os (<TV)
a
l a fcsras o f m lB M .m .9 giw m i a t h is t r # * ’fa&©»t.'s fts m ti« i (4 .1 ® )fee©o»s
(«•▼) » \/«
a1
• cos { vAr), (4.19)
from which.
(a . v)
eo# (a v ) *
_
_
«
y
^
L
£
_
v3)
(4 .2 0 )
is o b ta in e d .
Sia.iM -r r e la tio n s s s s ilf f o l i « fo r
two sets o f assosl& te ©©mfKaiants
mg -mmI wg»
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.
o f the... Aertoirataant .gjuthe a w ig g t M i o ftl
Sl3 i» a « , » » t of » ...o* order tenoor, therefor, it. formula
o f trm -m ttm m rtS L m . i« given b y
J £
is i
^
fho d e te ra is ft& t o f
d *>
(4 *2 1 )
% a"
m e t % • £® m »4- moi «K'pp#aai^l in
terms o f th e ■
&®b®mAsmat o f t t | «*& th e #t^ ra il3 a « % s o f th e tra n s fo rm a tio n ja a trie e s .
% a r ttlf t from d eterm inan t th e o ry *
*U
is !
% *
TsF
is!
<4*2t>
3 *3
I ** A 1 g*
wb&w®
dx1
A
d
m
d ?
(4 *2 3 )
<3B»
fli© resewMftao® o f th is - tr o a « fo *a fttlm f w » l i to t h a t o f a s e a le r
density i« a t mmm apparent*
© f both «i# m
% ta k in g tins. square ro o t of th e «&gattm<Ie
of equation (4 .225 ft typical t » l * r density
/ l
*
(4 *1 4 )
\/lT
1ft. fOT&MKl*
1% has boon
p ro v io w ly [® q . (2 * 4 $ f] tJm t th e jiratm et
o f a te n s o r c a p a c ity end ft t « a » r d e n s ity i« ft tw o . ts n s o r.
b»«CNM»A u s e fu l i a doftsA ag m e le n a a * o f m m &m &bl® e e l w
s a g a c ity .
% ehoosiag th e a o n lftr d s s s ity / g f
This p rin c ip le
o r volume
o f th e m e tric determ inant
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ftftftlttr -density w M eh i#
mm
m m tm
im lt ip lie d h y th e v e in s * elem ent d ^ # m
e & ffte ity * m %mm se&Jter I l ® » r o f *#r© o rd e r)
4v •
Is f m l i
/I"
A 't •
/¥
^
d»P * * . *
dsP
.
C c *i5 )
fiss #M*«lar dV represent* m
nm
lm
m
&nit ®f w lw *i tfe® scalar
density i/g~
*®ftlar
m n tw tfconfltt of a* a d*»it.y ©f volwsetri© €;«rte»t*
A
*&ie& iwolttded tkm density of Mtm wrnttmr or efenrge ia the
yoXu&m element might a ls o Jiftv* hern used ia ©quettaa (4*23)*
la till*
esse, the m m m cr oh&fg® 1» the ©lemestal vmltttm is gl-wa. ty
4at ♦ f d 't 0
(4 .2 6 )
where jO i * a s e a le r dene S ty e f m t t e r o r e h a rg **
tio g i o f g ^ j to th e d ia g o n a l fo rm
l a g en eral*, th e eem fOfgnt* o f th e ftasdsa»afcftl te a s e r ,g jj a re fisae*
t i oas o f p o *£ tl*B f I f ifa e tr valu es a t a g iven f e in t a re tafag®*
ta b u la r f « ia ©# re p re s e n tin g th e ©©sp©i»ntss © f g j j *
1
♦ *# *.
.ft
. « l f •■ *** S la
1
21 t t t * * * * S is t
ftft ft* * * * * * * ft ft ft** ft ft
i
t il
«1J
1
%1
{4*27}
-Sot ■*.*♦* e
5 - — *■
o m b » reduced to A iftgpaftl fo r a *
fh # M m gm m i fo ra o f the- g ^ j components
iw irs s e n ts p r « » t * 4 # a liy ®» J&elideext -«$*«« le e m lly m ssoetated w ith thegiw m p a is t*
1
pri&elple is ©ft#® «s#d la. r e l a t i v i t y th e o ry *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mQQm-
To o b ta in th e d iag o n al f e r n o f eq uation (4 .2 ? ) * mm a x is at* fo r
Phtefe th # *agai-t«<le 4 * | o f i t * d if f e r e n t ia l m y be e ith e r p o s itiv e or
n e g a tiv e is chosen ia o rd er to d e riv e so?re g u id in g p rin c ip le s *
The
is defime#. alo ag th e a x is a? .* • * 1 depending upon *rh*tho r d « |
la p o s itiv e o r n e g a tiv e . .T h* p r o j e o t t * o f da?- alo ng any a x is
JsL fit1*
is
(4 .2 8 )
£ )**
3a term s o f th e dae®1. t h * asgttitraA# o f
dSl ~ % J
i s , ty eq u atio n (4 .1 ),
' . «*. ** *..4 ^ C4E1) *
^acA
<3P-
But eq u atio n (4 *2 9 ) is e q u iv a le n t to
dSj « g n
th e re fo re , « * * * « i »
(4 .2 9 )
to th * d e fin itio n ©f * 1#
in
* * I.- '
(4 .3 0 )
Th* u n itie s along th e o M sad# x * are re la te d to th e defined e1 by
«•»
e j * •■—*i^ ,w # * *
S x1
(4 *51}
Squatimis (4 .8 0 ) m & (4 .S 1 ) provide th e desired p rin c ip le s .
A v e c to r S ir is p ro je c te d ©a th e a x is x 3* to g iv e
in g p o rtio n
S *X is d e fin e d Is;.as. (» * 1} d isem aienal spso©# form ing a
hyporsurfao© la an xM ttsM m &snal epaee. orthogonal to
6
S x 1* th e r e m ia -
S x ^ i th a t I s .
is d iv id e d to g iv e
g* -
8 # ♦ 6r k
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4 *8 2 )
« # i« *
where
8*x lie s in th e h y p e rs a rf**# d e fin e d by
€^j
S* * 1
S i? * &
(4 ..5 5 J)
l a term s o f 8 * * as expreased ia eqw ation (4 *8 1 ) the Magnitude o f 8 *
is g lw a by
<5 s2 « g . » -»^3L . 5 3E1 + 8 * *
■%3
3%*>
ill *
* 1^2 * % j
8 a^ ♦
8 *^
dx1
s ‘x i
(4 . 54 )
» ( OSj) 2 * ( o * • ) *
where ( 8 » j ) *
8 5 **)* • * ( Ox1) 2 1® glw*& % eq uation ( 4 .2 9 ) .
Th© new faadai3© a ta l q u ad ratic form
( S'* * ) 2 • gf j
8% 48 *r$
(4 *3 8 )
i® tre a te d i a t h * saw wagr as « * « th e o r ig in a l q m d ra tl© form
8 s2
• g jj
S x*
8
i a
i f f «* -
1
(4 .3 6 )
I * o b tain ed * th is process i s rep eated
Ci t • t
to y ie ld t h * g e n e ra l r e s a lt th a t
1,
(4 .3 ? )
The ta b le &m now he w r itte n .la th e d e sired d iag o n al form
1
t
£11 :'©
.
»
1
♦
1
1 t la
Tnii
©
.
* *.
Wt
,* *
0
I
*•*
0
i
/
a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
«u
*
1
38
* * **
i J 1
O'
*«**■»
'1
|
0
*. *. *. .*
a
O
(4 .5 8 )
2
0
3
' ---------------- ►
I t should be rw a s tw rw i th a t th is d iag o n al t& m 1*. m Rlntaiasd o n ly ia
tfa# ism ed ist# ©eighborli©®* o f th e p o in t
5*
th e divergence sad th e te p ls e lta
th # diverge^©# is «n © p m te f A k l i s a p p lie d t© a ten so r d e n s ity
sad no t to # te a s e r*
th e d e n s ity diverges©® © f s. eo ntravm rian t T e ste r
Is fe ra e * from the © w # rl« » t derlmtie# © f a esatretsmrlseat w e to r d e n s ity
% ©®atr#r©ti©s.§ tfe# r e s u lt © hteined Is # s s s ls r density*
I f A is #
e o n tra v & rta a t v e c to r density* its © w h ite s t d e riv a tiv e is given fcy
(2*5®)
A c o n tra c tio n o f in d ic e s and ehsstg# o f dummy in d ic e s produces
(4 *4 0)
la *
fh # s c a la r d e n s ity
th e v e c to r d e n s ity A j
i t is s##a to fe© independent o f th e a f fin e connection ©r th® m e tric
te a s e r*
I t sms astsd fa ssoMsa- ( lf * S } th a t
/g "
i® # s c a la r d e n s ity .
If
s c o n tra v a rle n t r#© t© r is m u ltip lie d l y # s c a la r d e n s ity * & e o n tra v a ria n t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-©fig**
v e c to r d e n s ity re s u lts *
The diverges#® © f th e v e c to r de®#!1^
way b©
fotsasd .%« .giv® th e s c a la r # e n a itir jjy e rg e n ii#
1 t h * produ e t o f *%»« t i «
Sine® l / \/"gf I# m. s e a le r
l/
(4.41)
%
, f
dm
x/"g~ give® an l&stfrl&xKt
a® th *
S lv v * 1/
(4 *4 1 ) and
ily a r ta a a a
CV T
In O artesiact e o o riim t© # .th® absolwfe# divergence is ia d ts tis fp is h s b l®
from th e ie n B ily 4 iv e r g a » # f both redme® to
(4 .4 3 )
Jm
An in te r® s tin g ap fsT ieaiiea m i th# dlvergenc# to th e mssoelat&d
eontrav& r 1an t r e e io r d e n s ity # f th e ©©sarisstt g ra d ie n t o f a s c a la r
fm setlo a r e s u lt* in th e la p la e ia a *
I f - f t ® a s e a le r fu a e tie a * i t *
g ra d ie n t 1*
d<p
(4*^
dm
From t h is e © v a ria n t vw eter # i® r# «s® h * formed t h * e e s tm m ria n t vsotor
»
/^ r
V I
Jm
g
a
(««*& }
'" T tS T
The a b so lu te divergence o f th is v s s to r d e n s ity is lam
as th # ta p la o ia n
o f (p j i t i s given- by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V> " 7 ^ "
"T k
(/*"
V
/T
2'
w
h
e
re V
gkm “4 % )
(4*48)
<**
re
p
re
s
e
n
ts ife
e#p«itw
\/s
* ta
<>*
knees* as Hi® la p te e la a .o p e ra to r*
-
The
<4 - 473
ia vuadefin&hle w i thout
a m e tric * ■
fbm div©rg«n#e o f any te n s o r d e n s ity m & a le # fee ferns© d fey a
c o n tra c tio n o f th® e m m tim t d e r im tiir # o f th # te n so r d e n s ity *
r e s u lt is o f l i t t l e
■g.-#
i»psrtmn®«f: tt# r # fo r # i t
This
is o m itted *
Displeesm ent o f «$$& *. o f le a g fti in ja e trie
Th© fundam ental q « a ftr« tie form ds2 •
standard o f orient& bl® le n g th a t each p o in t*
i * 1 d *$ e s ta b lis h e s a
I t now feeeomss necessary
to disouss th e d i» p la © « w s t o f u n its o f l@ng.th.
I t is assujaed th a t the
tra its o f lengths (u n itie s ) «& a re ©#»ti»«#»e- fu n c tio n s o f p o s itio n *
displacem ent
P am& P *.
6x
rep resen ts th e e e s to r t»tw»«n two neighboring p o in ts
Th® len g th o f ife is w o to r a t P is s ire n by
*•*’* % |
If
6x
The
6x± 6
(4 .4 8 )
is d isp le eed to f % th e v a ria tio n in I t s m g a ita d # Is assumed
p ro p o rtio n a l to th® e exponenta -of th e di splacement and to i t s le n g th *
th a t i s #.th© v a ria tio n
8 1* 1 %
8*^ #
(4*4®)
iter® fjj- is a component o f a e o m ris o it ^ e o to r p r o p o r t io a lit y co n stan t*
I f th© displacem ent 8 x
i-s c a rrie d around a closed p a th , the necessary
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
• v Q
eondi tio n th a t th e
have th # »*»® valu® A s a re tu rn e d to i t #
s ta r tin g p o ta t I s th a t
tio n *.
"1
be a eompmerrfc o f th # g ra d ie n t o f a s c a la r func­
fh is conclusion fo llo w s tram th # e e e e id e m tio n # o f se ctic n s
(1 11-11} to ( i l l -1 4 ) *
B a t, i f th # f*a g « syst®® o f u n it re c to rs is to-
be th© sass# a t a l l p o in ts ia .epaee* I t i# aeeessary th a t
% * 0.
C 4.80)
th is c o n d itio n e iia m e te rla e ® 8& #asm la» geNwaotry and th# associated
li#3«
«
.a
i#
a ©pace*
7.
C o m rla n t d iff® f# e -tl» l# An ll® «8«m l#» space
l a th # previous #®©ti®» Si®s«aal®» space w&e d e fin e d as a spec© fo r
w hich th # same tra n s p o rta b le standard o f
#ad#t© 4 a t a l l p o in ts *
Em® •=- existen ce o f th # sane tra n s p o rta b le standard o f l«& gth p e n a ita
•p a o tf ie « tt© » o f Hi® c o e ffic ie n t# o f a ffla ® ® oi»#© ti© a i s a s p e c ia lis e d
fo rm ,
ffe# tru e in creas# M a -m otor «. d u rin g a p a r a lle l tra n s p o rt is
g lrw i by eq u atio n {S *iS ) m
Bn1 * do1 ♦ r * ^ j
*c X
(4 *5 1 )
Hi® p a rtic ra ls r s e t o f eeord&aatee along w hich t ill# Ba* is aero is known
a # geodesi© c o o rd in a te # *
Along geodesic oeerdittftte# th e v a ria tio n which
i t is n m m m ry to gtv§ to th® itH8» r i© a l values o f th e compwaest# to
r e a lis e p a r a lle l tr*e » j> © rt Is giv«a by
du* ® - 37^,
u01 dr.^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4 *5 2 )
<m63*»
S iaee th # s a m standard o f len g th #adL#t« a t a l l jw lats*.. th e jm p ritu d # o f
a w o to r mm% rem ain
tatdm r traaaefeert.*
T h is re tja irs a e a i oak#**
i t m ceseary th a t
D
|«|® * ®Cif| w 1 ui ) » Q*
(4*6$)
A# Du3- mad Du$ a re zero * i t fo lio s ® tb a t
(4*S4)
th is r e s u lt ia a dir@ ©t #«»*<p.«a#e o f th # gauge lu m ria n c e * i t m y he
used as a erth eriO B fo p gaag# i» m r la a o « *
* * i. l * * H } -
f a r t h e r , since
* .3 a *k - r " k gta a *k - 0 ,(4 .6 6 5
th e re fo re
B« i i -
* r 31 ® i» « * *
* ^ j.ik
* r 3 . 3fc>
(4 .S 6 )
where
T 't t i k m
*
.
r i #j k *
»
e* * *
(4.8?)
(4.6 8)
From these r e l attracts th # f#H©*§i*tg r e s u lt is ebtad&a&ft
(4 .S f )
8*
The C h r is to ff#1 symtols
Setae in te r e s tin g @ w *® p « e ts r e s u lt fro ® the re la tio n s o f the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s e c tio n .
I f tfee- in d lo # # o f ♦ tw a tio n (4 .5 9 ) are permuted and
'
th® re s u ltin g re la tio n s eamM&ftd by add! M on sad s u b tra c t!o n , th® expression
* , r
J x?
1© obtained#
ow r
car
Ik # tBam ft&ty r
€ h .rig to ffe l» g
,,
*■**#
(4 -6 0 )
o f eq u atio n (4 .8 0 ) is k a o ® as
-mmbat o f t§ ». f i r s t k in d s i t is fre q u e n tly
r# p r« seated I f
^ .la
”
{ 4 - 615
I f th # fc ta d * * o f
° £ ,,» « .
(4»S I.) is ra is e d ,, th e C h ris to ff e l symbol .
1*
« * t is . V
n m a tip ly te g
r ^ i } by g1*
th # q « « w tity obtained is
“ r M
vsherej
f r « -
( i l *
is th e C h r is to ff# ! symbol o f th e seooad k in d *
( 4 -62)
fh # symbols' T
a re mere co n testants -& ere£oro th ey w i l l bm used on tit# fo llo w in g pages*
The syii& *l» F rep resen t a f fin e ®mm»e%im» ia s o n -m e tric a l geeawtary b u t
C h r is to ff# ! symbol# in m e tric a l g o « e try #
0*
Geodesies i a
A geodesic lin o s a tis fie s tfe# fa llo w in g # f n a tio n [®q* (S*S§)j$
^ 4
4 tz
*
r 1 ,. —
**
It
—
At
* 0.
® ti« eq u atio n a c tu a lly d e fte r# a am m .il #©.§*#»t o f a s tra ig h t lin e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( 4 .6 3 )
$h»
«SS**
grad ual ju n c tio n -of- these mm 11 m-gnmmts reppeeeats s geodesic lin e *
th e re fo re th e tfeg s eit to a geedesie lie s s a tis fie s the c o n d itio n o f
p a r a lle l d isp l& e a& en t*
In Mimmmm&m spa©e the t e g s a t to a geodesic a t
a p o in t F * a f t e r a p a r a lle l 4 i* p M o * @ n t to a p o in t IA* on th e geodesic,
co in c id es w ith th e tang en t a t P * $ m y w o tu r u making an angle 9 w ith
th e tangent a t P .Mates th e w m an g le a f t e r p a r a lle l displacem ent w ith
th e ta n g e n t a t P **
the geodesies possess seme w r y in te r e s tin g p ro p e rtie s •
They a re
th e lin e s o f n s ta tio n a ry ” len g th % th a t is * the- s h o rte s t o r lo n g e st path
between tw o p o in t# w i l l he geodesic.*
a re s ir a i ght lin e # *
In Euclidean. space th # .geodesic#
To aheer t h a t e ^ a a tia a (4 *6 3 ) d e fin e s the c o n d itio n
f o r s ta tio n a ry le n g th * th # fo llo w in g v a r ia tio n o f th e in te g r a l o f th e
mm le n g th is form edt
SS. “ Sj"'
dt *
2
S1 * d
s j
\/® iJ f f “~ •§£
**
sd t * D *{4 .6 4 )
s tie re
i
-
S
J dt
s d
At
.
(4 #€5)
ffe® v a ria tio n o f m om be w ritte n - a©
5 e **
5 *^ ♦
6 *^ *
(4 *6 6 )
Wheat eq uation ( 4* 86 ) is s u b s titu te d In eq uation (4 *6 4 ) and th # second
term is in te g ra te d by .parts*, th e fo llo w in g r e s u lt is o b tain ed t
5S; « •
j
|j||r
*•
6 IE3, d t •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4 *6 7 )
*■€§«*
For t h is in te g r a l to w m ifO i i t
J
§
L
—
(
I
is mmrnmmse# th a t
ji \ „ , .A
ft-,-,. „
■ mm
\ dm I
•
• . Jr *
f a aoV
*®
^ 'S fe # 0 ©
/
dwr
Wien s from ag nation (4 *6 5 ) is g s t« tit« t# i. 1m eq uation (4 * 6 8 ) and
th a d iffe r e n tia tio n s are :f « *"*» r» d .# th e fo llo w isig
_
is obtained*
ajc® a * j
i ^ * 5 * to 1 aj®
”7 3 * “
—
- f - s r i r d t~ ' ° -
( 4 - 6S)
i f th e duamy index J i s the seeosdi term e f eq uation ( 4 * 6 f) la
eh&agei to 1# t& « la s t two term s ea» te ptm pad to gin®
eij “ * (*7 ? ~* "iS) £ £ ■ °-
(8*7o)
But
^ % 1 iatiF1 dx1 _ 1 /
^ % » \ dx® ix ^ ,
+* ^ J ;» r* r *
C4*71>
th e re fo r© #q *m tl« a (4 ,7 0 ) eaa fee w i t t e a as
*. .
d t2
. 4 ( i M
* i a » . J !s » )
M
<»»■>
s i
(472,
( 4 - 72)
o r * by a s ta g a q u a tic s (4 *6 ® ),
g«
l | r + ^ .m
iT
r l r * 0-
' 4 -, s >
to o th e r f e m o f e%uati>m (4*.?S) ©a» t e obtained t y s ra ltip ly in g
equation. (4 .7 5 ) by g
and th en by a&ing eqwati©a (4 *6 2 ) j th is procedure
is represented by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The las h fo rm reduces to
A !* ♦ r
«3t
dt
(4 .7 4 )
a*
#fe
to e ^ u a tle n (4 *6 3 ) *
( 4 *63 )
Since
rep res en ts « geod esie, aa4 sine# e fa a tio a (4 * 74} re p
a lin e o f
'.s ta tio n a ry le a g tti., th e lis a o f a ta tio a a jry le n g th must
a geodesic lin e *
10*
Transfers a t io n of the., j» e d * s i® .to w
co ordin ates
I t is ia te r e s tte g a r t u s e fu l to aet© i « r eqam tifltt (4 *7 4 ) tran sfers®
to seme n&w ooer& ixftte eyst
mmdt
**»
.*
•
traasf#r«& M © B o f T
_ is y«»r®»
ml
seated by
•fe
r -i
1 _ifc / ^ ® il *. ^%j®.
^glm \
*
r £ F '
“ 35* — 3 ? )
2.
* W « l^
o%
. MMW 0 '
dx
r***0 I
o |6
.
'—if iiQ
<>x
bJL
d
\mS
dm
* 8
.^ s *
d r*\
a r
a *-1 d * " /
dx
sc
1 F
<)x^
1 ?
dx
c3x©
as1
dxS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4 * 78 )
m lfU
%• o o a s tts ria g &m o f th© %«*»« Is th e brftekets l a «Nftt«tia& (4 *7 § # an
osqpaasion, forsaala w M c l is ap p lisftb lis to th© romals ta g ten as ©an fee
o b tain ed *
I f ifa® f i r s t t mrm is els,©is©»# tb®» i t s expansion is represented
br
&
/-
iv
<sg»\
< )r>
1 7
-
j
/
* "
<**“
. js *
1 7 1 7
1 ?
■»<*,
a y
^
^
3x°
.
J *1
+ a% »
1 7 1 ?
^
a rt
,.
1 7
(4 - TC)
The tr e n s f orm ati on fo m n ia now ©an be r®4%te®t to
r ‘
al
r
‘
i £
®4
i 2
^
cfx^-
J £ ,
ds*
< « .„ >
where
fh e pr®»«wj® o f - th e wwssisd to r a ia th # %«®sfoi*»fcti-@K f o m n ia o f F*
ia d io a to e i t s »m «ts»«»jr t l a m e t e *
In order to tr m m fo r n
|4*:74)». ©aofe -to m is expressed ia
term s o f th # new e « » j^ e « M i. sad t h e ir tr« & e fe *» fttl< m **
The re s a lt ©an
is® mxmmriwmd ia the- oepaotisn
i f * H ■. i *
it 2
mi at
at
- I* 2 :♦
^ «*
r~ * i £
ed 4 t
iS i'i.ii.o
at j d3 *
(4 .T » )
Bkauation (4*79) demonstrates th at- the equation o f ft g®o#esi© is iamriftaat
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ia f o r a .
I t can a ls o fce in te rp re te d as a dem onstration o f th e f a c t th a t
th e in t r in s ic d e riv a tiv e o p e ra to r traz^ ferm e lik e a co-variant v e c to r,
11*
fee Metaasm -Chrl s te ffe 1 eu rm hu re /tensor
.
In s e c tio n ( X I I - M ) th e m sowm strleal cu rv atu re te n so r was d e fin e d ,
fo r in te g r & M lity o f th e dlapXaoflgjast o f a s e c to r f o r i n f is lte a im a l d is ­
placements th is te n s o r must he »®ro*
l a m e tric space th e ® m t u r e
te es or s o f eq uation s (S ,6 0 ) and (^ *6 ? ) assume s p e c ia l fa n s #
fe e p o s tu la te
o f gauge iamurlea-ee which o .h araeterises Memarsnian space perm its expres­
sion o f these tenso rs as fu n c tio n s o f th e d e riv a tiv e s o f g ^ *
The
c u rv atu re tensors are then eosspletely determ ined by th e g ^ and t h e ir
f i r s t and second d e r iv a tiv e s ,
ffm ti*
{3 .6 0 } sa» re p re s e n t th® cu rvatu re
te n s o r f o r displace® *® t o f a eem h ravarlaat v e c to r wh os® components a re
W* o r fo r th e a s s o c ia te e o v a rls n t v e c to r whose components a re Uj_,
Per
convenience eq uation (3 ,6 0 ) is rep e ate d as fo llo w s %
(4*80)
I f th e c o e ffic ie n ts r* a re considered d e fin e d by equations (4 *8 0 )
sad (4 *6 2 ) * eq u atio n (4 *8 0 ) d e fin e s th # li« m s e m « C h rls to ffe l cu rvatu re
te n s o r*
fe e i index o f eq u atio n (d«S0 ) m y he low ered* g iv in g
* > , * » , » * i 3 * £ , » • *1 3 ( ~ E
‘ -jjr
r -*
*
( 4 , 81 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Mom s
o
b
©special ireJatioasJiifS ©
u
sb
e mhmu. h
» exist b
etw
een the
©
«w
j©
n
®
sts of the foertfe order lieama-^SItfietoff'el tensor.
T
h
©first
%mm in eqnaMs® (4 ,8 1 ) s n b© w r itte n in another fo r® *
a?
r 1
A
-
(g |, r l ) « r 4 - i f i i
3
a*
axk
<*xk
«-iL»
7 ?
J"’
.*
5 ^
A
I
\,( 4 * 8 f )
A k /
5*
w here, % eq u atio n (4 * 8 9 )*
r *
*
(* -s s )
d*
B im M m iy .0 th e second term i » eq u atio n (4 *8 1 ) ©an he w r itte n in th e fo r®
s1^ « i L
dxh
r 1 • — i-.
**
r
,^*k
-
r 1
5**
* it5 ^ . (4 .8 4 )
/ r 1
'■* I
^
«V
By s u b s titu tin g the aio is f r e s u lt * in to . eq u atio n ( 4 .8 1 ) , the fo llo w in g
form o f R ie ia s n n -C liris to ff©1 e a m to r ® te n s o r is ob tained *
S3 » , » ’ - 5 7
fi.-*
*
T' L ( T' ^
*
’ ’’ i
♦ «»»»*> X
I-»1. , *► T-*^1
If , /I T“*^
I-*" ♦
at
# T-»
1 $ *U t
T-»^ «*
x ah
\
$%
-r-«5
I”
*,
ifc
T-»
J-* '
x j #*fe x A
(4»8S )
% osiaag equations (4.60) and (4.62, eq uation (4 *S i ©an be w r itte n
m
s
/
Ag*r
A
A
J2
\
i |L iJ n . _ 4 % — i * » — ife )
\<^xm^ x
n ^
fs.®1^^ £n.jafc
cJx^
d x m c>x
cJxk/
r*
~“Inn,K!h
^
,ii*V J k j
'^T^Jb
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4*s«)
fro m
tbi# eq u atio n it fo lio s -a tbftt
possess©# tk« following 4jpnotri«*i
* £ M & m sw ij.*»»
sjm,Jth * %i*Jss*
f t # f i r s t t m o f bfa«#«
(4*87)
expr### th® f a s t th a t B jb
is a n t i-
ayasw trl© in j»m and s#t j tli# la s t # % » « * !* expresses th e sym m etry
between th # two gretap® o f In d ie#® J» and k h .
Sins® ifc«
T ar© © « # t& n t# is B u d ideas #pws® w ith
C a rte s ia n re fe re n c e frasses, i t fo llo w s tfemt tfe® 1 iegmsnl&n-Chr is t o f f ©1
te n s o r is m m f o r t h is cas©*
1&»
fh # c o n tra c te d te n s e r o f l l # # l and E in s te in
fee- « J *« 4 li« » a n ia -c a iris to ff® 1 fo u rth o rd er te n so r
h# ecmtmuriMi w ith re s p e c t to tb # i «w§ b iattieo**
M4rm « , ©«®
fe e re s u ltin g second
order te n s o r
1
»4i
-
J*Jr*
rl
HHt
(*.<»>
1# cal le d tli® co n trac ted c u rv a tu re te n so r o f lie o t tm& lin « t « i» f when
th e components o f g a re constant • * th e f i r s t and la s t term s v a n is h .
ten so r m m ««#€ by Slnstoim to rep res en t th e g r a t it a t ie m l f i# M
This
® f* i»
tla * ® o u tsid e # f i » t t e r *
Th® te n se r
in g i t lay g * ® *
«ns easi ly b# w r itte n as a s&xod te a s e r by s m ltlp ly ­
I f th© mixed, t m m & r to co n trac ted to g iro * tii.se s c a la r,
th en fe e co n stan t o f
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•IPS#*
s - t ^ Snfc
I * ©hta.ia®&*
This soHMftmt o f s a m t o r e h#« «a i«p®rt*B»t r a le l a th©
tlMKtifgr o f r e l a t i v i t y *
13*
The ld # g t*itt,# » o f 8ia p a M
T h » » e re sera® id ® » titt« s
th e o e m rie a t p a r t ia l d e riv a tiv e s
o f th e © m rm tare te n s o r, fegtcma e s .Wm. lie a e k i id s a tlti© # *
ffeea®
M e a t! t ie s e re glvwa %
% uslaag a system o f
f» m g S W tgr
a re e a s ily «*Mrlf3*&#
co o rd in ates th # ten so r 1 # expressed i a th e
(4 *8 6 ) a t t&» f»i»% '® oa»ii«re€# then# id e n titie s
This preosH&rrw © lIM a s t# # th e
d e riv a tiv e s
o f tbo f e r ls t o f f a l s y a lo ls % SfflQweti-iag th e f i r s t d # r lm tiv » # ©£ th®' gf
I t redweas lo e a lly th#-' ao w aviaat d e riv a tiv e s to o rd in a ry d e riv a tiv e s *
CSae# th© id e n t ity i s a ta rtfio A f o r on© system o f axes i t is tru # f o r say
o th o r svstem* '
M
» lorml oo^di^at## of.Ilie
a
a
a
a
a
a
A ©suar&i&ftt* s y ste a
In
is
o a llo d sa a ffia ® normalco o rd in ate system
case th e oo&atlo&s o f th© d i f f e t m t i a l
rojwtsesifc path# th ro a rh th # o r ig in * ta k e th.®
la *
fo r geod esies, w hich
lin e a r fo r a *
a lo c a l S tteli& H m space
can bo a s s o c ia te d ,
to a f i r s t d#jpr®« © f #p |r«*iisfcti« »» , w ith m given p o in t in th© iMsediRt®
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
seigh?wtioe# o f th a t p o in t *
mm w e ll t» used •
An erthogiam l system o f re fe re n c e way $ m t
1m th is ease th e
q u a d ra tic form reduce*
to the form
i « * » (dT1) 2 ♦ (dT2 ) 2
where th e
(di?53-)2 *
iNSp*#*im%»- ffe# new e o e x d in a ts **
( 4 .1 1 }
I f s is a le n g th measured
alo ng a geodesic is s u in g trm m th e glwss p o in t and. l.^ th e d ire c tio n cosine
o f th e geodesic* th e mmrt&m&mm o f a nsarhy point# to a f i r s t ' order o f
ap p ro xim atio n * mr© g lean ty
#
• l ks
(4 .9 2 )
th e «<jsati«B o f a geodeeie fo r such. co o rd in ate s is g iro a by
4,s
line®
*
* r
a r ’ 0-
<4- 9 ! )
is proportions. 1 to c f th e la s t form o f eq o stio n can ju s t as
w e ll he .w ritten , la th e form
•# m
(4 .1 4 )
$*
As these Id e n t itie s Imm. t h e ir lo c i m il along H i# geodesies*, th e p a r t ia l
d e riv a tiv e s w ith re s p e c t to # o f o % « a ti« t (4 *9 4 ) a re soro# th a t is *
^
^
^
m o.
(4 *1 S )
Sense* f o r a r b itr a r y Trainee o f y # th e c o e ffic ie n t o f y * tt®
is z a ro |
■itiotefero
—^ S j 1 2 ♦ .j i O Ej l g ♦
d -^ r
d r*
» o.
Ay®
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4 .9 6 )
l « * r th* g i m
pai»%. th »
r
s r# In f in it e ly s m a ll, b u t n o t t h e ir
p a r t ia l deriwfciresj th e re fo re tfe® earvK tare %®»s»r o f equation (4 ,8 5 )
can assume -the £ * ll* ir i& g s ta p le f a r * i j t th e «m f co ordin ates t
,
. - i £ S * i a . _ iE & « 3 s .
jy in
d ^n
u .8 7 )
t he last two sjstea* of o^aatloas can b e sol-red far the
i» teras of the 1 and It* eeerdlmat*** yie-lding
- ilM
2 ,
* K ^ n i)
(4*#8>
For th* assumptions **d*# t h is © tm bisw e&n be replaced by
■^is* * '¥ ^ % 1 *b » * si®»al5
(4*9»)
in the Iwaedimb# i»i.ipjl»rl*©#d ©f t h * point*
% mk l a g u«* «f equations (4*59) and (#*§#), there ©an fee obtained
for the s p e c ia l © eerd iafct**
- %
* i&
*
<4 - 100>
h*e<we«» upon is te g ra tiffln ,:
**
+¥
* ^la,nk^ ^
^ *
(4*101)
to a f irst degree of ®pprox iambion,
The following important conclusion is obtained from the abor©
restilt**
if th* ©urmtere teaser is sat sera the spaa© cannot be
Suelidean* bat te a first degree of appr*at**ti«a it is g»«lid®aa*
Hi*
generalised tto*ory of relativity utilises »«©©«#stir# appr axiss&tions o f
th* kind just de«#rib«d*
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r*
s i f t i i s i c tm s m .
I* t th® u su al te n s e r a n a ly s is o f th® © o c rd iee tee * te a s e r « w p s « l«
a re r e fe rre d to th # d iffe r e n t Sale. « f th® e c e rd le a te e * dx^.* p e rta in in g
to tli# ©nwtfl© o f fevaneh*1# e«n««f tfeey may Is# r e fe r r e d , h o w w r , to
the d if f e r e n t ia l* of a**<%. d e ** of th© eurvee of ©a. a r b itr a r y e w p l t 1 (S) #
la, the latter ca se, th© tensor a n a ly s is I# © ailed la^rimsi© tensor
a n a ly s is .
In tills te n so r analysts th© ©es^pcaent* of tensors a re in v a r ia n ts .
l a -fee iataetn-al® theory there is no need o f a c tu a l co ordin ates p e rta la ia g to th© « w m $ 3 © 1| the ilf f e r w s t la ls of arc s u ffic e *
the ennuple
S# therefor©* -dees'aet have t© be ©set ©gmuftle w ith w hich coordinate® can
be associated.*
I t w i l l be assumed la ©feat follows th a t the eimupl# does
n o t have to be &tt enasifl© w ith which coordinates can be associated.
From a second p o in t o f view# the in tr in s ic tenser a n a ly s is m y be
ecneMered as the r e s u lt of © splcying the arcs o f the curves o f th e
arbitrary ©nnaple 1. a© seewheleaenie pawusaetNMre*
Tranoeauem ClS) used
para»et»r» in the study o f general ceaMeettm* end rasaheleeandc i » i f ® l d s *
1.
Systes® of- ©«B#ro#a«©e.
If
0 * a re th e a © eatlaeeus -to t d lffe te s tla b l© eo^p«w«ats o f m
o e a tim v a rim b v o lt v e c to r |3 *. a diep&eeem m t to th e d ire c tio n o f th e
©©©ter a t a p o in t F (x ^ ) s a t is f ie s th e -e fae M ee *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1: » Qx^
-m—a-
Thes®
(5.1)
» fu a M « a # a d m it » ** 1 i» a » p « I« a t s o lrtlo n g
. . . * P ) ** ©m
■C i • 1. *•■*, m - 1) *
( 6 *2 )
Is o f
uhere the «
■
*#are arbitrary o
e
m
s
ts
a
a
ts a
s
s
t the ssttrlar
d* 3
rank n • X*
o f curves *
(5*1) «*# said to dofiii® a
fh
rw
a
g
fc each point i* sfem the «Mnti«*ee 31 *r« not all zero, th
ey®
p
asses o
n
©« re of the e*st|p
rtj#ae# m.4, only one, Ifee tangent to th® curve
at Pf thl* taagoat has tfe
®a
s
a
a
s iirootloo a
s th# vector ^ at this point*
If n eoutr&mriant mib w m lm m ' {3
<
*
(c
x« i, *,*, a) are to h
e
independent, they m
ast aahtefy the deterataaaat
6f | / o
(s.s)
a t le a s t in a esatfeftltt ro g i« » o f th e spae© •
ffe##® n vecto rs d e te rs in # a
system o f n independent ®oiigrt*me©«, taaowa as « a earm pie, in such a m y
th a t through eaeh p o in t P th e re pas#*# a o e m t o f th e system h a rin g a#
tm -gent# a t ? th e diareetloa# o f th # a ia d # p « d ® a t u n it m e t or# passing
th ro u 1 t h is p o in t*
Sins® th® determ inant
eaepwaled by a tn o r# *
i
cs
9.
in d iffe r e n t from zero, it can he
th a t i s .
J
X
^
1
x eofactor of
in
&
(5 .4 )
/
I f both sid es o f eq uation (5 .4 ) are- d iv id e d hy
i -
I I V
| , th e r e s u lt i#«
$■
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( § * 5)
where 0 *
W |$<*4 | *
i# th # e o ffto to r o f
• * » « • *o jr
t » tli#
i# « rtfc *# to *t t@
J
$ 4
fo r x /
| d ir t ie d
J#. I t fo llo w s
ttwit
§*
<4
-
4
Hmmni r#latt® B «-
*
4
tb ftt
o«
y
(1 a
v g * »**
j , ,
as th # eoiap©R©ttbs o f a « # # ® ria » t w a it -rector®
•
<s-6)
O 01
V
m
$** *
©s® * • e o a s iie ro i
lb##® a vector s a re
feaowa as th # vw eter# © «^apt% * (o r *#© ij»ro © «l) to th # a u n it vecto rs
t« a g « it to th© e a r# ** o f th # « m « p l« I *
She special ©&«® of a jMtaalty erthogoafti aea-aall vector field# la
«a »*#$### is «all#& «a #r%#fsxB»i eagagle,
f*
T r*m s f© r» ti« e # t # ia t r ln s lo ©«#i
Th# trftasforsirttosa. f r e a th # o r iia a r y to tke ia b rtm s i# components o f
# te a s e r ©boys tli# standard, f e r a l laws o f te a s e r s a a ly s ia ,
I f a sm all
displacem ent v e c to r dx a t J*(x4 ) 1# trm s fo rm e d o r pro jo e te d oa th® co n p*aeac®s# i t s
compoaeats ftloag th # #«&gru«ae*s ar# g lv « t f
dx1 f
f g .t )
I t # e o !ffe a « ts 4 # ©long H i# s ® a p » « * ® p r® J# #t« I oa -fee co ordin ate
curve# » r# given tjr
■
-
$ * *s \
(5 ,8 )
f r o * th e la s t two sets o f «<|asti© as# i t fo llo w s t h a t
A 4 www*,
v i
ax4
V»
<)#
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
{§»§)
*481**
I f # in to # tra n s fo rm a tio n of* to # e«npcmmt« o f a te n so r which is th®
r e s u lt of a ©JMUftge frora ooertinatte® x * to 3"*v 3? is replaced by #*>* th®
tr« n « fo iw s ti® o becossss # 1®% tram to e o rd in a ry to th® to b rtits i® eesqprtasento*Th© law o f tr s a s fo w n tio tt f r o * th e o rd in ary, to toe in t r in s ic componearits
o f a pen©ml te n so r i«-# w in g
Jh
V ... ^
where f * 1* * * ***
», * * ♦ %
and ( f •§)».
<*.**'. ■ ..<)•*." <toJ'
I T T - i n
I X
^ x * * * * ! , . . . 1* __, ____
J, . . . A , * < 5 a o )
^
” *
is- m m m gtm m t ©f f along, to e congruences *
—~ r denotes d ir e s t! on&l d iffe r e n tle t io n to th e p o s itiv e d ir e o tio a o f an
■<3«r
■ a rb itra ry « « r w o f th e aengrannen** Th® re la tio n s IwrfcsHws. th e d ire c tio n a l
d e riv & ti-re s
and th e f w r t ln l d e riv a tiv e s ■■^ ■r a re obviously
7
7
‘ - f r
(5 a i)
From forasulae ( b * l l ) i t fo llo w s th a t
i f **
d»^.*
(5*12)
to to # tra n s fo rm a tio n s o f ao n -to n eo r o b je c ts such as th© c o e ffic ie n ts
o f eesneetioBS and to e C h ris% o ff»l symbols to® o rd e r o f p a r t ia l d i f f e r e n tia tio n must to c lo s e ly observed-* -to# n^s»^ex«nxttt.M li^ o f p a r t ia l
d iffe r e n t! a t ion o fte n totrodswse* a d d itio n a l te ra s to th® trs ri s f © rm tio n
o f n o n -ten so r o b je c ts •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
&»
In teg ro M . 1.1t y eeadl tie & e
Th® ist@ p*af3l l i t y eemdltleaBMti f a r th e p a r a lle l dlepSfceemeni o f »
s e a le r q u a n tity re q u ire th a t [s e e t* < m - « g
«ar4
<3x4
i!
<3x'^
th a t i« * th e order « f 'parti® .!
should s o t e ffe c t
to te r m
(8 .is )
Of
dxl
dsr® <)as4
o f th # ©©alar q u a n tity
r e s u lt o tta iB # # *
o f th # tire a ti.e w ftl
Sqwatios {S*15.) east b© expressed
by mmmm o f eq eetiea a (5 *1 1 ) *
In o rd er to o b ta in tt© r e e a ltia g ia t e p ’f t M Ilt y coaditlwase la th# in d ic e s
1 «sd | , © fm tio e s (S ,1 S ) nr#'
()x ^ zP
j£ i.
*)x P
{5»M >
low Tsdies © q » t l« t « ($.#11) mm s a b s titttt*# i s ©qswtioa ( 5 .1 4 ) , th e ex­
pression
f
d-sF
i# o b tain ed *
d z F d'sF
J ii ,
\ d a4 c h t /
dm%
5
*? \ J
J * L w (5.
c) * t / c)xP
I f © qm ttoii® {© •!§ ) »r© expasai©# and m u ltip lie d by
d wF
7 7
7
^
#
th e r e s u lt is th a t
£
d 0-liTwgo
3 ? i s j bp
<p
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<w*6§**
A ll th e terms aew t vanish ssesp t f « r P • f • r# tte s rs fs rs th e co n d itio n s
of
(5 .1 6 ) a re « p re s s s € # f*a « lly a * w e ll by
m w iiw w w
m
^
w m M m ia m ' v ^ m f c r n r
^
^
«*
i w
ij
i t »
c)s
*
(
0 . 17)
where
J* _
i|
•
f
\
^
^
^
-
lf: M . / < * . , d ^ \
^ *r /
^ s 4 \ <J«3 d%T j
/ -£ _ _ d ^
\ 7j
„
h
d ^ \
Ceas)
*
These r e la tio n s o b vio u sly g iv# th e necessary and s m ffle ie a t cond i t i one
th a t tp^ d.k. where <
pk * (p^{s*) » fee as ejeaet d if f e r e n t ia l (S* P* 54*6) *
4*
M e tric es in ie etie its
fp to t h is tim e no eaaslcNorstistt tins been g iven to c o e ffio ie n ts o f
connections o th e r ttiftn th e v&m &tzri* type*.
Witte ttee ia tro d o o tis n o f a
w t r io a r t « a « l« ibsxmrimmm til# b a sis o f tlem am ian. g ® « w tiy was determ ined.
Witte th e te tro d a e tis s o f a ® H f e « i« * io tr*B »fo*SB R ti«« c o e ffic ie n ts * th a t
is c o e ffic ie n ts fo r A M
~j£iL ^ IL *. / ~..^-«r
« *« *« )■ * r
eo nn setisa a re n o t synsastricj tto r # f« r e
th e c o e ffic ie n ts o f
/
1^ .
When a m e tric is
th e s e iw tttis a th a t th e emmwimmt d s rim b iv s o f th e a s tr i©
h b M i s t i l l , a n t be aa% i« fiw 4 | b a t sin e# th e ayw astry ecm ditions o f th e
c o e ffic ie n ts o f eo n aaetiea do n o t s a la t# th e r e d o o tim o f them to C hris t o f f e l
symbols i s n o t possib l» »
H ie ©ow riim fe d e riv a tiv e o f th e a s tr le ten so r
w ith respaste t o an ssgagg&s is glv®» by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
—
i j r ■ - 3#
-
^ C ii
LJk - « ^ Li k
•
r
* *7 l *
0 «
where % ,#Jk '®3*®
< « •» >
~ 3*ik*
e e e ffle ie s te o f eorm eeti on .
Fro®, the ©os—
s id e ra tio a ® o f equations (4 *8 3 ) i t fo llo w # th a t
0 |o p «
C e^o)
- ' - 5 , i k) « * «5^sk • 0 .
/ ^ &» 4
\
% th© q u o tie n t law th© fa a a t it y f —j± |A . % #j k *■ Lj ,ik J n u e t be a te n se r
o f th e t h ir d o rd e rj th is te n s o r I s denoted by I
H »»
* % , i j - — 1 - % . j k - l i.U c •
a8
2 BJ » k l *
(5 .8 1 )
~
and
2 Bi , j k
‘ T
i
O8
" LJ . i k
" h . ji
•
By a d d in g th © l a s t two- « q » a t i« s a a d s e h tr a o tia g e q u a tio n
lo w in g te n s o r i s
o b ta in e d #
b3 . »
I q n a t im
{ 5 ,2 1 } th e f o l ­
* B* . J * * " m i ’ I
f e
4 ■ %
- - %
)
- W
( 8 ,2 2 ) M e a ls o be w r it t e n a s
3k , i j *
^ r k #y
»
C «*M )
where
* Bj , k i - ® i#J k
is a te n s o r * m ad
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5 .2 4 )
* 6-2
Is f t * C h ris to ff© ! symbol o f the f i r s t MutJU
%e#t j is a c tu a lly a com-
p m e s t o f t h * to rs io n ts n s o ri i t can h® sheens t© 1® equal to t h * a s t i syasastri* p a rt o f ft© eongNsaents ©f th® e s e ffie ls n ts o f connection*
% m M m o f s q u s tis a (5 *2 3 ) .and ft© eyssastry o f th e S hrtsstoff® ! symbol®*
(5 .2 6 )
an
fsn ao r in th # iatli® *® i | *
I t should t® aofcwi th a t f t * te r s io a te n so r -sat f t * c o e ffic ie n ts o f
© M » # ftffla hers th e -same tr*(«sf<M P »ti«a. laws fo r e r n p e s i # tw rasforw fc*
f t m s a s f o r c o o rd in a te tr « is fo r * * tls » ® »
Jaaoiber in te re s tin g . p o in t is
t h a t f t * spec© oorrsspm SSag to f t * * ® * ffl® leasts o f connection ©ass be
Tilm m m im . o n ly i f the ter®!®® Iw s s r Is n a il#
S«
q*oi*s.lq-® in tepa® ■of; f t * oomgruea©**
X f * € ir * * -t* d o u rv * 0 is given ty * i » * * { * ) * whmrm 9 I# ft®
j» r * B i» ftis nr® le n g th * sad I f s^ * » i e> n r# f t * o rd in a ry and ia tr tn s l®
eomgMme&fcs* r# S f*® ftir® ly #. o f t it * u n it v s s ts r te n gent to C a t an a r b itr a r y
p o in t P o f C# ft * ®
<6 * 3 ?)
Bat
C5.«S)
th erefo r®
(S *2S)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
**» » •
—
, v„i
« *3 J&Jlw #
<);i r
(5 *5 0 )
eq u atio n (5 *2 0 ) %«©«»»«
^sr
IT
4
4 /^dSI
• ft**
d®.
or
(5 .5 1 )
from t& is li
1
da
th * #qw«ht«n o f «t geodesie ia
spae© w ith co ordin ate
■pariftblo# was giirwa in. m fo ra
't*»w4
© -s4
.* _
**■<*•»»» .♦.
nhfflns ft
1« g i v * %
to
£
i-*
*r<&
*T&
a
» p*
(® *S f) *
/c
U>*S3;
l » t e ln m th# f i r s t in tr in s ic
d s r lm t lr # of ft tenso r fm aistlt-y is a»«M j«r te n s o r o f one h ig h e r eo varian ce
o rd e r, equation. {$«$$) transfers*,s to an o th er co o rd in ate system as fo llo w s i
o ir l
W
....
* 7 ^ j
*
*
<5 *M >
But a tra n s fo rm tio a to @ «gra#ao®« '<§$ms$w th e &®mm la w ; th e re fo re
S- ***Jl
D
4
a
• ■««*■? «*-» a«3 ,
Oft
^ 0t» DS
afaer# a3 is . fiv e ® ty eq aetton
/
—
(5 *55)
Ia te r m
o f th # eo&grue&ees th#
§ ® o d « le eq»sht«a is d « fi» e € by th e ir m e fo r w ttio a to e » example o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* * ® f «
e<mgyu#R##s of th # geoieai® #q«*ht#B l» t w i of th® «©#r€iit«t®8f tt»t
I s , th® geodesic
m
is tw a « o f th® mm grmm mm is given by
(6*3i)
eh er® , by eq u atio n -($ *2 3 )#
ji * r 3
ml
ml
f5
»1
( 6 . 37)
sap# th© anti»sym m © trie e e e ffl# i® a te o f eeossottim # r e s a ltia g from th®
tra n s fo rm a tio n *
f t so. feeppes# th a t w ith, re s p e c t to th # coordinates th©
c o e ffic ie n ts o f e earnest! ea «ar# redaoed to the S h r ls te ffe l symbols* feat
tii® tf m s f o r m t to a o f th # ®mrnt%emt ie r lm t f - r # to mi a r b it r a r y enxmple o f
esemgrmmmm Is tro d a e s # the to rs le s te a s e r S .
#,
frm s a fe rm t lasts- from., mm oaaagl® o f eoogrusaeos. .to > -geooad
f a th® previou s so o tlo s i t m m mhmm ttost i f a ears# 6 t * g iven fey
at* ■•■ s t^ (*) th a t th # orAteoyy o a t Interims i# eomjpoment*
w ait, te e te r t i ^ s ^
sad * * o f th e
to C a re g iven fey
«* - . f i i
«
end
re s p e c tiv e ly *
f t fo llo w s th a t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.38)
s&th resp ect t® »mm m w tpl# S#
S im ila r f ormolas hold when 0 is r®»
fe rre d to «&e$i*«r «aa*pl® S* in s te a d o f 1.*
n
ds*^*
th,# r e la tio n mr • •5-— suggests, fo llo w in g
as
rep resen tin g t& e
%
( § , p* 5 7 2 ),
©oaisoneoti, re fe rre d to I*, o f th e w r it
s e c to r tangent to tli* g f » » i ewrwe 0^ * of the hth ee»grp@pee o f 1* by
V| 1 *
*
and th© o o n tra m ria n t
(6*40)
r e fe rre d to 1 *',.. of the wait ve c to r
timipfit to th# gmmmA eurse %
o f th e hth. o-ongrasaa## of M %
th e n by ap p ly in g th e prteofptos of ecguatloa (§«3®},# th© following
lo tio n s a re ob tain ed *
-J ^ L , »
dm1
nipgi-inii mijiTiieinvi*
^
%
«s
- j ^si
»m wi»<iy.i■>■■■* !«■
7e*
.«* .ibs?
. .^s* 1
»i»«r« * i iwa»mig i
**'$
w m o w m m ih
'
Jsh
means of th® principles of
,
^s*1
mmmnmmm»m.
» fr
77
J x5
*
($.#$} end (S*s|* it follow#
fro ® eq p etto a (s » 4 2 ) -teat
a *?
7 *7
d* 1
^ e *h *
or
k
T T T
d a **1 d *3
c
a
*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l a a s im ila r way i t 1# p js t ib l# to o b t& is th e r e la tio n
-J il
< )» •)
^
4
.
b l
.
(6 #44)
Equations ($.«$$) a«A (6 *4 § ) •hat® th a t th e sys'teas o f q e a a titie s ,
■-»-««««• » ,d - “■- t »
d w **
<?s*J
a re e « J « g itt© t® «a® a a s th e r*
.Aeeardiag to ©qaaMeaa (5 ,4 3 )*
fe ^
»>
c^8 *!<
d « *n
(5 .4 6 )
a »3 ’
th e re fo re
s* j . A
. i d
j . J ,
°h
SS #
(6 .4 8 )
^
. 4
4 ^
< )» i
.
let a a i a i l a r way th e w l a M «
da1 * ■■»— r d»*3
(5 .4 7 )
©am b© ©&*•&»*&«
fji# p a r t ia l I t r t m i i w a o f a tw a o tim , w ith r© *pe© t to two sets o f
m ria b l.© # s * 5- aad
a re re la te d t t o e t
( 8 .48)
- J S - . J X - J L C t
s*
<3«fi
<3#*
dm **
3®^-
th© *» ©tpatieras d e fia © th e r©X»%i«a« b atm en tfe© d ire e tie s s a l d e rim tli© © ,
S<pahi®»© (6 .4 8 ) * (5 ,4 7 )* and (5 .4 8 ) show th a t th e tra a s fe ra a tle n
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safes o f & te n s o r, r e fe rr e d to * * eatmuple I j to th© e©p»
JSr©» th e c<
tfeo oouol low * o f % «© «r at
of *
f
fo r
:g « « n l to o o o r f t r « a s f « * » . %y th e r « l *
£>®$m
»®i * » •
h| ♦■•■a
J g ta ,
< )***"
'■ * »
'." - " ^ I iiIil'.ij ju ":: m n :
^t, * * * in
Ji » ** Jw|
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(S *4 § )
m* im tm mm*mm re ia tb d
tmmu mmjt&m
to
Y ecto r a n a ly s is ®&b to c o ^ ld e r® # as a s p e c ia lis e d breach o f -Hi#
m e tric a l te n s o r a n a ly s is f o r **r® « f i r s t * and seecad order tensor© fa
spue#.
Being a s p e c ia liz e d branch o f te n s o r a n a ly s is ,
r e c to r a m ly s ls d iffe r s from te n s o r a n a ly s t* in s e v e ra l im p o rtan t a s p e c ts .
I t i # based upon ea i n t u it ir e d e fin itio n , o f a r e c to r , th e r epre sente,t i on
o f second o rd er a n ti-s ym m e tric ten so rs as p seu d o -recto rs. tfee us® o f
th e symbolism o f to # *4 » 1 * e y e n te r ( V *
i
♦ J
and t®
liia ite d to three-dim en®i«m &l $ ft*li£ # a a space w ith id e n tic a l end unehamgis g w i l t len g th s along th© v a rio u s re fe re n c e sonns*
Hies© s p e c ia liz e -
tio n s s im p lify to e m to e a a tic a l procedure® # f to e r e c to r c a lc u lu s , h u t
r e s u lt :1c le s s r ig s r #. warn- eo n fu sio a. *» # added co m p lic a tio n f o r th© study
o f th e more xsgaer&l te n so r a n a ly s is .
- lo u g h ly epM acteg. l a re c to r a n a ly s is a re c to r i® d e fin ed a# * * .
q u a n tity to ic h fee® d ire c tio n aw l m agn itu de.*
This d e fin itio n f a il® ,
however, to com pletely © toractorl® ® a re c to r * i t is a c t ® w ffio i® a tly
cle ar' and comprehensive.
A® fcrjW ghaa point© out (1 5 , p . 2 ) , i t f a i l *
to e x p la in t h a t is sawwRt by
d ir e c tio n .’*
As exam ple* © f la c k ©£
c la r it y # f such a d e fin itio n © f a -tw a to r. to® r e c to r product © f two v e c to rs
Is 'ttmNMHMKetotit»2'''toe e a r l o f a v e c to r depend® upon to e sense o f r o ta tio n
o f to # re fe re n c e axe® j to # r e p to » « to ti® » o f r o ta tio n by v e c to r*, se eatiisc *
leads t© un true re s u lt® .*
m # . « w t w * A d * ©toy to e M a i a lg e b ra ic law *
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o f te n s o r a n a ly s is a re
" p o la r” w o to r s f th ey a re tru e f i r s t order
te n s o rs «
V ectors smelt a s ifc# me®tear prodmot and th© em rl a re " a x ia l"
re c to rs .
'f t# a x ia l w e to r s are n o t tro o re c to rs b u t are pseudo-vectors
fa m e d fro © tw ® sweseMl order -aatd -a^ w sstrie te a s e rs *
S t# a x ia l re s to re
a re seaslMsns to th e sense o f retatM sn o f th # re fe re n c e axes i » a tr a n s fo e m tlo a o f C a rte s ia n axea* tfeey twrtw?© m . wroasl w a te r s as 1eng as th e
» *© . .«#»©# o f r o ta tio n # f .th# r« f# r« » e « .teas (e tttta r "rig lit-feaad o d " or
"lafb-laM atifci*. refareas© ayataaa) is sa&s&s&MHftg hm t, i f th e «#»#© o f
vetBf&N! ia ibttn& A * th # axtnJ. mtAmem # s a p #%»*■
l a t h is chapter «a atte m p t i s msA© t o c la r if y mmm o f th e math,©m t i c a l fo u n d atio n s o f re c to r a n a ly s is im tis e tr re la tio n s h ip s to ten so r
a n a ly s is by w in g th e
1*
l»#t«n*« pseudo-tensors and te n s o rs .
Second orA er a n ti-s j^ g a e trie ten so rs as a x ia l w o to rs
flm a x ia l w o to r# o f w e to r a n a ly s is a r# d e rtre d fro ® •©©end order
s u a tt-s y a a e irie tenso rs i a tbre©
space.
In th re e dim ensional
© actor space th e eosagKsieota t | j o f a soocand order antl-syrraae t r ic te n s o r
oaa be arranged M tii# fo llo w in g a rra y *
I
tn
%i
%
t
*
%
*»
%»
J
X
1
2
t
*■
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
{«*!)
B a t, sis©# %|_j » «* % i *
eaa fe# w r it t w a In the f« m
I
2
©
0
* %8
J-
.
1
♦ is
% 'U
1 • Hz
m %5
$
%3
s
0
s
(«* 2)
*►
—
w ith © sly tfcr«# i:M «p®»4«fe eorapenaat#*
I f a. choice o f th© o rd er o f
tfe© indie©© {th e « r6* r 1# $ 0 S fa r example) , w hich s w rn t® to ohooslng &
jritril® f« M l sens© o f r e ta tle a ® f th e mm® (rtg kt*9 m m i® & system) is usule*
-thea w ith each component b ^ j a oonpanoaaife 1* mm be asso ciated by ©fcaeeiag
th© 'Is d ia o * so th a t th e ® r i« r t * J * fe i s deduced from th # order 1 , 2 , 3
by an e « a p e m u ta tio s u
th e ta b le ( 6 . 2 ) eaa now b© w r itte n as
I
1
©
TS
1’• « *
- 2
©
1
f1
*
©
S
( 6 * 3)
J ----- ^
a pseudo-tensor*
The ©awt© o f th e eegoeghattwe a d d itio n sod a m l t i p l i e a t t * tro u b le s
is e a s ily w »£*rsteed a f t e r a trs B s f« « » M © » o f the ex p o n en ts o f (6 .2 )
( 6 * 3) to a n w re fe re n c e system is m &m *
sE^" th© ©®S8|MSaS®tll o f th e te a s e r t a re
r# fr® S te t# 4 by 1® w ith a * b * o#, a a « m
1 ., th e tra a s fo r ssation form ulas fo r th e \
la a new co o rd in ate system
H t*n * stace
can b«
je r m b a tio a o f th e order 1# 2 ,
h beemm
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* # § * * 9 5 -
J S L . & 't
*
.
S T
bx
/> j?
©p
S t
b x$
C«f 4 )
o r* % fx p ® 4 ta g a n t
^
jSF.
U * 1 a *>
a 1
J T
* I d .l8
<3** <>JlJ
■S1
t^JE1
<£k \ n a
b-9?}
,J E j S
U *1 ^
I ' ■"■■■■■■ -
-: -
Th# c o e ffic ie n ts ©T tti® f
* £
bx
-i
A
M
bm$
where th© o rd e r
- j £ ^ S .W
U
bj )
W P P * * * " !* * # . I
(6 .5 )
A,
warm e lm m tly th® m issy* ©f th© determ inant
S&
*3
bx
b^
bx 1
\— 2
ox
.«■$
bx
b*%
dwfi
bW
J*L
bx1 b ^
perraotaticaa..
j # \ . i
bW
-b ^
is deduced fro ® th e o rd e r 1, t # I b y mm mwrnm
The « t p i o f th e t e r w
ia . th e tw e a tie a 1© mmde * o r -
€ ® p m M n g wpem w hether I #. m# a i« mm evmm mr an odd perm utation o f th©
I* , $ m S*
S o lvin g ©qpAtion {-•«#) fo r i t s m inors, i t ia p o ssib le to o b ta in
m .a
♦ X
J 2 .
bx3
< )**
•1
Z T
1
A
-ii
bx
These minorm can 1© su b s titu te # , in to ©qu&ti«& (6 .5 ) be ©fetala
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(« •? )
«*$£«»
* v e c to r d e n s ity *
fttis 1©. th # trfM s fe rs a tlo e fferomla o f th # f f i t is
e le e r ljr no t th a t o f a te a s e r*
I f i t w ere »<st f a * th® fa c to r A ~ #
h m m m r0 IM « tre & e f& rn & tlo a f e r a u l* s e n M be th a t o f a tr u e f i r s t erd e r
te n s e r*
It so i»j»p#®s th a t fo r reet«»p i.liM r coordinate systems th #
of the diseettea m m im m Is** & mis® ef
spaa sfeetfeer th# tm» eets of i m
or -1 depending
have th# same or opposite seas© of
th e m am ##»m
of rotation is m ain tain ed ,
f transform s between snsataxtgalar e e e r& im te systsa as a tr e e first o rd e r
te n s o r*
th e q u a n tity f i s o f -ft* a x ia l w s to r ty $ » *
J e s t as a ©oatimmriastt pse\id«»m«%#r was efete.iaed frost a dotahly
e e m rl& a t a a tl-w s y ^ w trie te a s e r* #e sis© can a c o v e ria n t pseodo-m etor
s im ila r ly be ob tained f r o * a doehly e e a tra ifa ria a t a n ti-s y m m e tric t a is o r .
th e re s tilh * ©©atpamfele to «f«*rt£«& (6 * 3 )* th a t w u M be ebteijsed is
t lJ *
\
\d %
* - ™ ¥ j| *
dm
» f fc,
dm
(6 *9 )
j
a m e te r c a p a c ity *
Zm T e e te r « a l mm%m areflpet#. ia v e c to r a n a ly s is
i a im po rtant epeim tteo la m e te r a n a ly s is is th a t o f th e m o to r
pro d u ct*
l a te n s o r a n a ly s is tfj# prodoot o f two e o a h jra m rie a t ve c to rs is
gleee hy
«*■
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( 6 . 10)
*# ? < *
a seeend © rder te a s e r*
te n so r,, * 8 ^
B a t* t » o rd e r to form a com p letely an ti-s ym m e trie
must be sx&ri&netsd f r a * ♦ ftm tia s i ( 6 * 10) *
th e re c to r prednet
-of t « s s r -an alysis e o rres p m d la g to th a t o f ro o te r s n s ly s i* will th en ta x
*. „ i
v l.
wftere w 1^ is a esag&etely
(6 .1 1 )
tensor*
By. using the principles
■of eqpsstiutt {« -*# ), th e reotor p*«wfas©t bmmmrn
««*$
a ro o te r d e n s ity *
f a ♦ J^Sj~ Wsj « ¥ k ,
(6 .1 2 )
As b e fo re , i f th is faepreseiea Is lim ite d t© r ig h t -
handed C artesian. re fe re n c e fram es i a th rs « -d ia s » a s i^ a l « p w *#. th s n
>
A
* *1 *
l a th # em bolism o f m eeter a n a ly s is using th® u n it re c to rs
&*- S# "k# along Sams x # y # s# r s s p s e tir s ly * th® r e c to r product o f two
re c to rs u and r esu© be w r itte n in th e form
A
’w
* 1-
V
* i(« ® r® «
* k (« ^ U*— tt** t^ ) *
(-0 * 1$ )
I t csn b# yeaarleei th a t the p rise i-y le® iw o lr w l ia th is s e o tie n can
1m sxtfesnded to Snelud* th e reeioy p*©iia©ts o f © om risat- re c to rs by ©sing
t h e ir associated © o n trftm ris a t w e t « *
She s e a l*# prodwst- o f two s s e te rs fcae been d e fin e d j j l t * (4.1S )Jj
i t i « #. hm m nm r* a s » r ie e l% « $ u t« *3 e n t fo r Csurtssisn re fe re n c e tmmmm to
ft e o n tra e ti on o f th e te a s e r p ro is o t o f two ye© t© r»*
is f i r m
The s e s ls r p re & je t
by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<u . r ) «
- 1 »3
«
ttS
W I!5- 'W j.
C i* l4 )
iia e s
• 1 if i » j
th e n u m r ie e l welwee o f «
fo r
ts
i
e r e Im dlstiagBlsfe-ftfele*, and s im ila r ly
«nl
te d V |t t h e r e f o r * Ifee »©*tlay p *e to « t i s eacpr*»*it>le i a any o f
tb © fo U e e l& g
fe rw s t
« l vi * u l
{«as>
“ i Ti - ui
These re l& tlo ® * K » re iy demmstrisfce tlm t tts* eom rSiuat and co n traw a rian t
compoaeats o f a r e c to r w ith rs e p e e t to 6» rte « l« & roferesss# f t m m
ere
In d is tin g u is h a b le *ad lu t^ e fe a a ^ a b l® -*
1*
ffcelB®*® ttoereaii.to^^seetor^aas-^eie tm m
H i# p rin c ip le s displayed- by
lis p o rte a t
( 6 . 8 ) and ( 6*S ) tow® * eery
%& Ham repreeatsfeetiea o f Stoke*® tfeeore® .
P re v io u s ly *
it- was ehowa t h a t th e pr®«J«fit o f a te a e e r' d e n s ity and a te a s e r e a p e o itf
fie ld e d a tr e e te a s e r .
I f * s w ft« i- « ie n e » t € # -5 i # represented % *
w e t t r e a p ie it f # asd i f th e r o ta tio n Cor m a rl) r ^ j i « represented as a
weeto r d e n s ity * th*a- t h e ir prodtw t can » i» ts A m i t * previous terns o r
eto F fceter*
Ifa e ia g
( 6 * 16 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
"where
< « a *>
*
and
(8 .18)
ISiterS
< ea#>
ftfc » A
then by » n fe s tlt« ii© » in
(S *3 § ) th© ©«©-t©y«M&ngajr8i8 fo ra o f
Stoke* s. th«er«m l e ©bta llie d *
(6 .20)
fh e most ©•©&! fo r a of S toke*a tfaeor*© ln « l*ro ® the d e fin itio n © f
ih© f© ta ti© & ( « r em rl) i » t© ra » o f th e spsA elte o p e ra to r V
•
the
Gompori©nts o f eq u atl on (6 .1 8 ) a re «© »sM «r«d as d e flatin g the fMtead©**
w e t o r 1# % a s s o c ia tin g th © w i t re c to rs i # J # k w ith th # axes x , y #: * *
r e s p e e tlr e ly , i t S# c le a r th a t i » th e ■ v s o to r-^ a .ly s i* symbolism
i
J
k
,J S l
d%
dm
In tro d B sin g ©quatidn
( 6 *21 )
(6 .2 1 ) in to eqwrtiiao ( 6 .2 0 ) # th e re ob tains
(6 *2 8 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t- s ix «
dbc * fg %
*
4**
(6.23)
mnA -
CV * t )
^
dai
~
*
* (il££ _ j 2 jl\ d<r
I a x ■ dw )
90
th e d o t
between th e m e te rs attestin g f o r e m a la r 'p r& ta e t*
( 6 ,2 2 ) re p re s e n ts th e s y a b e lie
4*
(6,24)
% tssti® »
fo ra o f Stohe’ s theorem *
fh e greMtient.*.. earl#. ttrs fg e n g e , a n i L ap lacisn in re c to r ~analysis fe:
l^ e r io tts ly |® t* ( 2 ,1 3 ) j # th e g ra d ie n t o f * s c a la r f m a t t m was
<ft»ffeed a# - ^JL # m e re f m s a s e a le r f e in t fw e tto tt# i t m s shews to
dx%
be a iy jr is a l #o m ri«a% m e te r * th e ©«a|M»®»t.s o f th e gpadteKt i » th ro e d i« m s i« » l spas# ««a fee s » l t l p l i e i by t h e ir co rresfo ad iag n a il vecto rs
and lo p a m m te i by
9 * 1:
+ 4
l o t * a y ra h e llm lS y * t h is is s f i a l to
Srad 9 * V (p * i
*- 1C
(« •» )
V <p* th e re fo re
* $ ***^ “ * k
*
(6 ,2 6 )
The a b so lu te divergence is a tru e s c a la r an# is d e fin ed by
[®q. ( 4 . 4 2 ) ] .■■
B fe
v *
~
where r Is a ©OBtram riasat m e te r *
C /S
e *%
( 6 . 2?)
!» m e te r a n a ly s ts fo ra f o r th re e *
dissensions! s ja e e with. C a rte s ta a re fe re n c e £rm m m 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t$m& e l a t i o n ( 6 *2 ?) ©an be w r ttte a ms
H v r m J 2 ~ * J z L *
dx
dy
“ 2Ti *
w
hieh* 3a iwmm of the eperatear
S f-r
-* •
# i# tfalwltal t©
♦ ■k
* J
-$j»
*MM0NWafM®
(#*2f)
'^fe-
i r
* J
♦ it
( 6 «so)
^ -
dy
w
h
ere the <
t©
i mhma&s f#r e eealar f*re<
i&
e$*
o f », s c a la r fw m tiettt *w&# g lT *e as j^ lq , (4 .4 6 )J
V <P
I
JL
(\/i~
Ce.3i)
S h i* fin set &o& als o takes ft s p e c ia l fe r a in tferee^isw atsieostl space w ith
C artes ian re fe re n c e fra n e e *
% using e % m tis a * £§*2S) and
* 1 if k * m
• ©■ if8k
m
^ n?
c>;
<y
„ .
(t*8g)
a g#
Sometimes in w e to r a m ly s i® i t is d e s ira b le to use o th e r than
Cartesian. referen ce f n o M *
I» th is ease i t is necessary t e ms® form ulas
(6 *2 7 ) «ad (§mS%) #©r tiw d iw rg e n c # wad i«p3ft$i«n* r e s ^ e tife ty *
In tense ©f th e la it ie s mg along th e th re e a x e s , th e fundament®. 1
fa a d r a tls -fa ra bee owes
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ds2 •
43^ #o® # £ |*
(8 *5 4 )
where #lj 1b the eagle betweea the i axis and the J mAam It a#e««*»
•arily I*©Hew® fro® ©<p*t.ioa (8,54) that
4g |j. *
# j ©os % J»
(6 *3 5 )
and
« tt » * f *
<«*«e)
llfsmtS,*® (6 *3 4 ) * ©as, be e x p ress ed *th erefo re *a s
4 ®f ' * % J :i3si
*
v^l^T * v ' f j j *
©as % j . dat1 da^
(6 ,3 ? )
la th e sp ecial ©as© o f ©rltxegcm l coordinates ©<pmbi«a (6 *3 ? ) reduces to
the fo llo w in g simple form
ts 2 •
f l t f i 3t1‘} 2 *.
C«<*ss)
la c y lin d r ic a l ©©ordinates, fo r
dsf * d rf ♦ r 2 *» 2 ♦ is 2 *
« li * **
*$, * l |
% i * **»
©g * **f
{«
% t •
«g * 1«
(6 ,4 0 )
Fee th re e dim ensional space w ith ©rthogsn&l er<rvi lin e a r coordinate® ,
th e fp ra iie a t,
ead l* p ls o l« a e s s tw s o c ia l « M mere fa m ilia r
fees*® l a term s &£ th e e^*.
These sjseeial f& ras e a s ily fo llo w fro ® th e
p rlm c ip le s J « e t e lic it e d *
From wfwt to.® preceded i t
is c le a r th a t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
J
• t * / S li *
(*•*!)
«B*t that the g ^ (t ^ j) ere sere for orthegoml ©©ordinate*.
lay rector
h a * either eo'wriftafc o r e o a tm m rl« a t esap€»s»ttfc*. expreesihle la term* o f
ife# #£*
%
eqtufttieaa { f* 4 1 } fta i {4*14} i t f o llo w that
*1 * %3 ^
< « *« )
©r
tt3- * {« **'}* t t |*
(8 .4 3 )
Fro® timmm r e la tio n s th e im g aittti© ©F a v e c to r fo r orthogonal c © ordinate*
1*
U 2 *
* g j^ C'l*^)2 »
( it j) 2
* {ejm1}2 * C#!1 n±) 2
©1* last two fo rs » © f
(-6.44)
(# *4 4 ) waggeet d e fia ia g th e e©®B©R««ts of
a re c to r .to the form
% ** -®i
m #11 wt * V ^ l
(6 .4 6 )
fh «a th e le n g th o f th e vecto r i® m erely eqpaftl to th e earn o f th e equaree
©f 1he a 5-.
ffa« ©©av^tism ©f efaetiom (6 *4 5 ) lee&e to the f o i l w in g repreeeata**
tio n ©f th e gradient of a eeal& r faaet& «m
(g re 4 <p)» ** ~ i ..~JL.
.
® i dm r
( i d s w a a tie a )
Th is is iii« usual v e c to r**® ,ly s is form o f th e
(6 *4 6 )
f o r © rihogoaal
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
c u r'd .lin e a r 8 « r f i a i t * « in tfc rw ^ is a m a ia a a l apa® *,
Similarly th e c u rl
o f a w a t e r for arth&gcmaX auririlfaaar eow dS&ato* im t&rao^disHm oio&al
space ia aaepratsifel# ia ilia fallowing .isasmart
(C u rl £)%. m *—
• <pJLwi»
® i*J
® i#J \^ x
{© . © ^ f ») «*
* *
( « j F j) -
(» i F p j
,
(6 .4 ? )
afear®' (c u r l f )^. rep resen ts t t » fc aaapoaaat # f a w l f and i # j # k a ra
persm tations ® f f* . 2m S .
the itw rg e e a # of" a w a te r (a c tu a lly a p s e u d o -w e to r) can a ls o b®
dafinad i® the » p » ia i tans*#. S a fa rria g again ta e f i t t t ®
(8 .2 ? ) ©ad
using tba ral&ttoa® f o r ori&ogeaal eurrlliaaar eoordl&atea that
« ij
#§
©
©
0
»I
0
0
0
-
(* 1 « * • » ) * .
(6 *4 6 )
4
0
O
« *g
2
A
0
0
- <n •* •*)
C e*4t)
Tk )
<6*S0)
-3
«*
s
th a divergence baeoases
f® utilig® tha s p e c ia l eoapmiaats o f
awlttply waft divide
by
,&rid
(6 ,4 5 )* i t 1® meess&ry to
for
then
© (ju stice (S.*6©)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Sgn&ttoct (f.*§1)
tbo «s»»l s^t©r»«««lysi@ form for
enrri linear mmmpAismtmo i» 1 iir« @ ^ ii» M i© » a l spa©*-.#
Us® 'XAj&aoima im tfeo ossml mootorMwimlyolo form fo r cmlfeogoaml
e u rv i lin e a r <Hsord£mmto« is om slly obtmimmt % isiio om botltutltim o f th#
sp o o la i fmlmtimaa- o f
(#«£$} # (6 * 4 5 ), mod (6 *48} tu t© eq uation
(6 * 3 1 ) j th # r e s u lt is th a t
®f
V p <p
*% •% **
di
J*
la s t f« *r soot ions g iiw tfco t w in itffforsne#® totwoes tfe# usual
■so©to r
and te n s o r a n a ly s is .
F w rtli«r ro lm tiaash ifM i can bo
.© flag tb e M m m iirro lre d is th ese mootloom ^ b u t ‘tt.oso
s u ffic e be im ftiem t* the method o f approaeh*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
• 1 0 6 **
H w rly a»th©&« o f a p w o i %# dpacusl©©! p r ©.bloats depended ,,uf«®
p o s tu la te s ta w lv is g w o to r ^ a » « titi# s ia S®elid#«» spa#©*
mad© p o ssib le th e le g i# # ,!
Such s e t hods
© f iyuwsiciyl, th e o rie s w ith o u t
i»tr® <tii® iag th e ©o&eopt o f satargf.* barb as ita J ly r i^ a lr e d e la b o ra te p h ysical
maA ■
iiife # ri^ ® t» ti# » »
concept m m Tgy ( ’‘ v is
W ttb th e £ » % r© ie # M « o f th e s c a la r
th® qu estio n ©f w hether a dyxw aieal tiie o ry
in v o lv in g d e riv a tiv e s o f th© *s « rg y ©ould be dwri-swS a ro s e ,
la 1788,
i©i§f»ag# ta tr e ^ e e ft i a M « immrn-' w*®©asaff«« A a a ly tiq u e ” ( 11) a method
which re s u lte d i a a a a ffir m a tiv e answer t© th® q u e s tio n ,
e s s e n tia lly reduced d fM » l« s to & problem to
The aethod © f jr©e#d«r© to 1® used to th ie ch ap ter is based upon
'to® j®«tel<rt»i
o
jz k
Mte
')■ * H k
(k " 1,
> •0
*
*
# -0 ,
(f*l)
© tier* D/fot rep resen ts th® i a trto s t® d e r im tiir e w ith re s p e c t to th© p a ra *
m eter t # X§ : is th® k eenfKnateatt # f th® r e s u lta n t d* th e e x te rn a l for®#®
©» th e
iS 'tfe® i n e r t i a l c o e ffic ie n t
o f tfe® k th p a rti© 1© o f
©ad a i s * $ a * l t o th r© # tim es th # w a t e r o f p a rtic le s i a
th e
second
This fundam ental p o s tu la te is %fe® te n so r fo r a of Keeton*©
o f m otion; th e re fo r© i t #*s» be tr« « *fo ra # d to any o th er
i t i s as sussed t© re p re s e n t th e © qpatlena o f a c tio n o f e ith e r
M parti® !•« la tk re e ^ d i^ ru s lo n & l ISucltdeaa spa©© o r of one p a r tic le in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
• lm * >
S S -d iw B S l om ul { n * 31 )
spsee*
From t h is fundam ental p o s tu la te
va rio u s re la tio n s and p rim e ly !# # o f so aalytieal dyB*y»ic« w i l l be obtained
m -&ummm4iMg $a ges«
1.
l» g r« ig e *e
o f srofeiaitt.
I e $ a a t l » t
©f » t l « © e re q u it# e a s ily otefcaioed f t p * th e
futsdsasatftl p o s ts le ts { i f *
en erg y*
( f .1 } ] % i& tro d & elac th e eoneept ©f k in e tlo
S i# k in e tic energy T o f & system o f m p a rtic le s is d e fin e d by
* - i -V ( * £ ) * * I
(^ e ) “•
( 7 -2)
The d e r i m t i e ® e x p r e s s e d i a te rw i o f T * 1® g ivea by th e ©qua tie s
is
H
Te®
die
AM
*1 **
A*
d#.
« * - -§ -> •
<»•*>
Woe i f eq u atio n (7 *3 ) is s u b s titu te d i& (7 ,1 5 # ttsm the eq uation
^ (" ^ j
is o b tain ed *
(7 “ t )
Sthes th is eq u atio n is «xps.«ded by msisg equation ( 3 ,3 2 ) ,
th e express! on
*1§~:
r e s u lts *
'**'
"$ § **** * ^
C7 *5)
th e le s t term on the l e f t ia equation (7 *5 ) mm. be changed to
p i
<*f . r£a» w
-1 feat - J f C
m
-p i
x fat
.c5f T h l
-r
«js
^
5»*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7 *6 )
Bsefe. sine©
x
aq u atio n (? *6 ) ©an be
3f
-*«®«
' mJL ®
ds“
(1 *7 )
S» %h© f « r»
df
* p ^ T’
( ^
h .» * i f
IS * H r.
■
* Pa
/ .dT
)n ~
*
6f
- t e i\/ a t
a ? g
(7 .8 )
s
j
Wmvmf-m® © qaatiott (7 *8 ) can » # # » » th© for:®
=3.
T“^
—
yAr-i **«
3t
i rr -nfimJiftr"//
a f jsoc
*22*V *r
t-■•-f- «*
ra•-^
{ ? •!© )
S u b s titu tin g mqgmtt&tm (7 *1 0 ) la t # (?»$)#. tfe# equations
«
a re ©bt&teed.
/ at \
I " .A S ) '
\ a* /
(7 *1 1 )
,
«
Eb.es© a^pMSttsxt ar© faKHsa « s f*
m equation© o f
©'Qter eeordiartee
ttaaafco m as fo llo w s *
at
d j af
dm
y^-k
dx
(7 *1 $ )
® r# by ©qua t i obs (2 *1 4 ) ©»& (4 *? # )|
dm
at
“I d :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7*1S)
f
espy* *■■<§ % | * * * * *
(f * M >
Lagrange*» equation s r#f»es«ti% 4*» » ® t i « o f 1 p a rtic le © la threeA a » « l « » l »pt©«:* b at S% t r i l l is® afeow® la t e r th a t th ey a ls o » y re p re s e a t
th e
o f «* *$ s o trlo *2 . elre»it -with
a Imdtepsmlsmt meshes or
of ©bo ©l0 ctro-®e©haaioal system* such as a rotating electrical aaaehiae*
with a degrees o f .freotom*
&©«©#trieal iy# I w w ,
ropressm t the m otion of a. free feint la a
l * p « ge*« epsttites
m e tric space j the
in e r t ia l coefficients fowteg th e scm^omemt* o f th e metric tm s o r * la
sow®. cases th# particle say met be free* t u t ssay he co nstrained to a
certain surface*- these ecmstraimeel m otions will be glsemssod la th e n ext
soetlsm *
Zm Comsrteraimed actio®
So f a r la %M« trea tm e n t o f te a e o ra a l l timasfmmaatimas have been
flroii one »«€lao»alo«l oooriliato ^vfcssi to sm sthor* th© tx a a s f ©rssati on
«f Imgramg®** equations [ lq * C ? .ll) awi ®|* ( T .lS ) ] is an example*
Oe^oraetrioallsr,, Lapmapt*« equations as girea. roprssoafe the free »oti«m
Of & p a r t ic le la an »«41»s b s 1 ^ * 1 spa©#*
S«m »tta»s# however, a p a r tic le
is mot co m p letely free hot is esa».^paim©.t to ® w
on c e rta in surffeee* or
in sub—©paces o f the m -dlw aasiaaal space* such iaotimm e re called con­
s tra in e d motion®*
th© ©qtatietig of oomstraint are m sthsssatleal equations
stating re la tio n s h ip s , between the co o rd in ates or t h e ir differential# to
such a way as to limit the motions to motions c o n s is te n t with the e«®»
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-410*
— fo r each r« d « # fi© » b y one o f th© aaaher o f c o o rd in a te s ,
tfe#r# is m #%tt«fci«a « f eom«%f«to%f
tim e * s ft
Tw© eases «h#ul& be co n sid ered ,
o ftts M m » o f
■or in t * w
o f o o a s tra ta t smy ©osst&in
F ir s t , when th e
ar# #xpr«® ««i tn. term® o f H t# c o o rd lm te v a ria b le s
o f d if f e r e n t ia ls of th© o o o rd im t# v a ria b le s by equations
A ic it a re e x a c t d iffe r e n t;iftlg , the r e s a ltia g %n«aal®#l system is said to
b® h m im m m m * th e re fo r# tfe®
O f to o « ® y v a ria b le s *
o f c o n s tra in t rep resen t th e us©
The soooad g w s s iM lity is th a t n o n -in te g ra b le
o f o o iw lam lat ta tsvma o f th # d if f e r e n t ia ls o f th® coordinates
e x is t so th a t © © iw a p o altiig r*la % 4 « ® between th # coordinates cannot he
o b tain ed *
Wtom th is is th e ©as#,. th # « * # t r « ia ® i dynam ical system Is
said to b® n<m^ol©no®Qftft*
These taro eases can fe« represented in te n s o r
I®aipmg®*
The ro d tte tio a ©f th # masbsr o f ooordioatss l a a holosom i# system
can b® expressed by a .ro#tom gol*r fer#»sf® «w itio» re p re s e n tin g a tra c ts fo r m tto a f r o r m old o o o riisia t##
to » xmr coordinates <5* w ith th # la s t
t
r o f th® co o rd in ates
n m ll# fft« r h®l#ws»i®tts ©qa&ticais o f c o n s tra in t
( r < » ) can b® expressed by th e r equations (o f rank r )
Fj
fb®#®
* # .
(m e )
can b© solved fo r to y C» -* r ) independent v a ria b le s ., which
way re p re s e n t new eeerdiw at® #*
I f th # la s t y # f tfe# new e e o riia a t® #
q3- « r# asswswl n u l l , them
• 0f
a *r< # . « Fg tx1,!; ) • %
#
** Pr (st*»t) ■•■■#*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7 .1 6 )
<4*1*
These e q u a tio n * j t f *
(?.1 8 )j can be
s e lf# # fo r the at3, in term s o f th #
%* and t * th® so lu tio n . y ie ld s t l » b e q n a tle a *
(fan
fr a a ffe ic h th # re e ia s g s l& r tra a s fe x tja tlo a s a trliE
Si® determ inan t o f th©
mferiac
a *11
a * 1 1® o b ta in e d .
is s in g u la r j th e r e -
<3f3
for© th e i& e e re * tra a e fo rm ti© ® m a trix p u t*. ’©saaot he unique. Since
a 5
LaacJ
th e elem ents -£-§L ©.re n o t - defined.;. tr m e fo s m ti ©n« o f e o n tra v a ria n t
a#
teneo re are n o t p o s s ib le . I f a saetri© is d e fin e d f o r th e n-spaee# th®
e ont r a r e r i artt In d ie e s o f a tm m &r say be changed, however, to cover le n t
in d ic e s and the tre a s fe rm tls n s perform ed*
Jfon-holoammvM aquatic®,® o f e o n s trs in t a re £wxmt±m.& in g e n eral o f
tlie » ■eeordiaa&te v a ria b le s end e it h e r th e m d if f e r e n t ia l o f th e co ordin ate
v a ria b le s o r th® .in tr ia s ie d e riv a tiv e s « f th e m co o rd in ate v a ria b le s .
r noa-holonoffious equations o f c o n s tra in t e x is t* th e y » y
If
be expressed
i a tli® fo ra
f^Cae1* dx®* t ) • o
Cj * 1 , ..*** r ) .
(7 4 8 )
fte e e e q « ® ti« M say be solved f o r any |» - r ) ind®p«mi®nt d if f e r e n t ia l*
•which m y re p re s e n t th e d if f e r e n t ia l* o f a r e * ds®* o f an ©naapl© o f
em g rn e aee **
I f t h * la s t r o f th e se d if f e r e n t ia ls ds1 a re assumed n u ll,
then
W 6We* ^ ' * %Cx®* f e 3-* d t ) » ©I
deS* r#g •
• e e e e e e e e e
A*®
dx:3-, d t) - Of
e e e e e e e e e e e e e e e e e e e
• fy C *3-# d *^ # d t ) ». 0 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mx
equations (7.19) *r* ssl**4 f«r the m old differential s, the
»
4 ^ m $1 (ds3-#
a re ob tain ed •
As *
dx
«4* 4
(?.2C
eqpStiOBi ( f *20) may as sterns th e form
*
whore th e c o e ffic ie n ts
♦ f 4 t (k * * # * .
•»
mm,I f:, ■assay %@ jhsiettoia® ©f th e T& ri& bles se
e i tti® param eter t* .
c o n s tra in ts w i l l b® gtvea. fa r th e r
eo B S id em tio e ia a s»©®#®Ntiiig.
r m m 9-
for Holonomio
A tra a s fo m a tie a o f tagr*®®®*® e * ^ ia t i* s J it.* (7 ,1 3 ) ] from an n«
epeee to a subspae® seas e a s ily be accom plished«
Is ls g th e bran s f om a t I on
, th e t r a » # f « a a t i- « is gi-em b y th© (a ** r ) © w p & tlo a #
** X±
(i. » 1,.
a)
{7*21}
or
« ...
/.d f f'
(7*23)
~—~ <*
tfeere
A
Q-f •m "**
-M. "•“**?
I t should b« c a r e fu lly noted th a t Is equations (7 .2 3 ) th a t
f o r th ® t ts ®
(t«
stands
d © rtw tiT ® fs lls s lo g t h # sootiest § th a t is .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
gqtt&tisas ( f ,* t i.) « r« S au sag e*# aHgs&tlsas fo r at iy » a i-© ® l system w ith
hQ lm m m x® c o n s tra in ts } they i^ frs s w fe
motion, @f a p a r tic le in an
(n «* r ) d ia a a s te iia l
lag ran g s’ s equations f o r &«$&&*»£« ifs te m m & he 9xjmm4®& t y sv.h~
.«M tw ting. fo r tb s k in e tic ©aargy f * and :p t-rfo tw l»g th e ii^ ie s t e t opera­
tio n s *
Using
«“
|e
< ? •*« )
in sq uation (?»E3) * % © r# ebtal& s
*
r
415
where th© eo astrm lats depend e x p lic it ly upon ti® © *
c o n s tra in t do s o t e « ttt« ia tins©- © s p lie ity
’ V
I f th * equations o f
“ eyeUe** m rls h l©
p a r t ia l d e riv a tiv e s w ith re s p e c t to tim e [ * , . (7 .2 5 ) ]
war ** — »*»»'
m
' d f1*
I a t h is efts*
♦ e e il.. Q^#
a #
<7-27)
th en ill©
e re s e re , th a t i s .
{7#20}
.
(7*t7)h®o®aes
* ij
* ( ^
" 1
— J ^ ) 4“
4 P * <»3 .
<7 - * »
which i s © q n im le n t t©
S ij
where
37^^^^
♦
IJ ,„ P
4-
i”
is th e C h ris to ffs ! gymhol of th e f i r s t
ffas-ss ©apaiAed. form s hse-enas wmvy u s e fu l ia
< 7 .3 0 )
-
k in d [E q. (4*80)~J *
p im e tle a l a p p lie s ti one
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4*.
T r to s fa rto tlto . e f l A R r m m * s ©quat l ems to m
of eoagriaefieea
tograttge’ s msgmtimetM o f m otion ©R» b© expressed w ith re fe re n c e t©
an ©sample ©f « s a p n t » t s ia s to a d of tru e ©©©rdinatos*
fo accomplish
suefe © tr a a # fo » » M « a th a p rin c ip le s © o tlto ad I®- ch ap ter 7 w i l l be used.
togrftag©*# equations f o r -fame ©o o rd im to e a re give® by eq u ati on©
(7*13)*
% a tr*s*f«rsKtl«jsi h© «a «a»apl* «f ©©»greume«« they becam
e
(7»&1)
Shi® form to a c t as oosweiiieBt as a f o r * a t t lis ia g th e k in e tic energy
expressed to to m s © f th e m m d if f e r e n t ia ls <ts^ and. th e ©M v a ria b le s
**■* sod th e d e riv a tiv e s esrpweeei. as d ir e c tio n a l d e r iv a tiv e s .
I t should
be .noted th a t a trs a « f© x a a tl« a o f th e ty p e m««d in eq u atio n (7 .3 1 ) is a
trs a s fo rffl& tio s between d if f e r e n t ia ls o f arc as th® u n d erlyin g v a ria b le s
m y fee «a d efia« 4»
l a ta w s ©f th e a sew d if f e r e n t ia ls sued th e a ©M
v a ria b le * th # k in e tic © » rg y east be expressed by
(7.3*)
% a s ia g th e p rin c ip le s o f d tr e e tio a a l d iff© r e n tia tio ® o f ch apter V0
1
wMI
-*-■
^-^1
tE&tjf
-|. I. . Hjlfc M—‘irtu JS
h
(7*33)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m td
^ It1
^e*9
dx1
<7.«$9)
When «f»ftt!«»® ^7*84} a»d C?*JS) ftr® subatltwt#! la »i«ati©a (7*31),
s b I abba. now® te w y ia if© # # ar© atta»g®»# tfa« equation®
«r* attala©#.* ifsasa ©
taafctao# ©
fm
otim »r® w^pm-mm^mA to term
s of s
m
«3n*3tpl« o f asogrvMMNM*
flx is la s t f a r * © f th # a q u a tio n * o f m otion 1#
«fe im l« *rfe i© « .'tm m gtv«a by « M t t » 3b*t (au# p.* 4 8 ) »
©*
fh». ©fuatim® of w&bt<aa far »<»^bleaos&© wymtmsm
1m tli« e&e© © f fealafsm ie :%®iei©s3, ayabama tti® ma&hmr o f independent
e©«r&laftb*« re q u ire d t© a p w iify th® eo ® t% »r»ti« is ©f a system a t say tla ®
is equal t© tb s » « *« « * o f is g rs e s o f f r s s ia i © f th® system* iS r a«a»
feblm-bsd© systsm s, however., th # »«®t»#r o f i»d® p#»d«tt coordinate® needed
to a ja a a iiy th® e m n ttg m m ttm a t s a y timer may h# g yaatar th a n th e number
o f degrees o f freedom , ©w&ag to th© f a e t th a t the system is subjected
t® o o n g tra irits wfeielt a re supposed to do no warfe, and abiefe ar® s tre s s e d
by & m b s © f s©a&*^t®p?mta® ® x jr® *« i® a * © f th® fo ra
$*3 + f 1 d t * O
( t * r * 1 * *.**# m# 3. ^ r < n),
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(? ,3 7 )
• i i § *
*" *”
“ d « * f1 ’ * * * • fOT0MM5“ o f « *
tba
H i# fc*
s1 mi
& m h a sym&m o f (m * r ) e q m fc im s , i f -ft® system ha# ho Im te g ra l*
w h atso ever, * » i i f th e fcera# la dfc mm »©gl®©fc#i,- is sa id to rep res en t
a s in g le r*d £ a m is £ « m l a « N fc o l© » ^ ® «$*«# i » ®» » *# p a e * (2 3 , p . I t ) *
A a o a -*iio i-» « ie system m y fc© .j»gar4«4 as a m tja o t t© th® ©oraSifcioas
o f the f ora. «3^pr##a«& fcy ♦q o a ti« .s -
or i t m y to regarded as a
systtts acted « p w 4y -e e rta ia a d d itio n a l e x te rn a l fo rc e s * nam ely, th®
fo r# ® * which sswfc fc© e x e rte d by th # e©m »tr®t»t® i a o rd er to compel fcfc®
system to f u l f i l l , th® e o a d itie & a *
W*& by Is t t la g
% assaying th e lo tfc # r p o in t o f v ie w ,
§»*• be th® work dsn® <n tb # f s l « a fcy th e a d d itio n a l
fo rc e s l a a displa#eas«afc 8 * #«fcj««fc to th # © o s iltia a s o f c o n s tra in t,
and fcy lo ttin g - %
6 x 4 fc® th® oorvom&tm&ing, work f o r tfe« o r ig in a l e x *
t * m ® l fo r#® #* laLgraxg®*'® # q p atl«M i mm fc® w r itte n in th # fo ra
I S ( g j f r ) ’ “p r * h
* *V
<7 ' S8>
S titt though fcfe# fo re o s X:* | a re ■tuHaxmm^ th e y a re sack th a t ia a d is p la c e meat eons la te n t w ith tti# r# # % m la ts th e y d® no work*.
F o r a l l ta la e s o f
th # r a tio # o f d iffe r e n tia l® tfelefc s a tis fy #q»#fcl«sw (7 * 3 7 ), i t fo llo w #
th a t th® work don# by th e re s tra in in g forces must fc® zeros th a t i s , th e
r e la tio n
X *I
wh#r® th®
8 x4 *
p * T&% 5 * * #
^’ sar® maA®b®rmA.M®& m u ltip lie r ® , ® ast
@ f»«,tiaa« o f » o tl® a o f fcfe® system * r #
(n *» >
fc® s a tis f ie d *
the-
Hoar mpwrnmmboA by th # (a ♦ r)
equations
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
• I lf *
I
- -J p L
r
3*.
Q m 1#
r)
♦ *3 * o
th© n * r
***♦ p •
* 7%i ♦
( 7 *40)
a re ae** m2 # *■**» asm an * p1 »
p2 *
fh© s o lu tio a s o f ih * * » equation® rep res en t th® perforiam © * o f
th© *y*t*B U ,
l a in te r e s tin g p o in t o f view ©tt&Stt* whoa th© s rn iltip lie rs p1 a re
e o n *id « r*4 * # s u ita b le fm otora to mtem tfee f i# M
tT ^ eonju g a te to th e s o w rta a t © B eter*
o f © o m tram rtaat ireetor®
• p1 X 4.- c o n s is t o f u a it ingotors.
th e re b y * aa ©susapl© o f o o afra*© © © is ob tain ® * w ith re fe re n c e t© which
equation® {T*.3T): take th # f o r *
i s 5* *
when th©
te ^ * O'
Cj
• 1 #. *.***, n ) ,
(? *4 1 )
T^*s a re sero $ th© rsesdboteg «. » r «fi ff« r# a % ia ls o f a re do n e t
van ish *
A fu n c tio n $ is an in te g r a l o f eq uation (? * 4 l) i f —£&» «* Qt
dm&
f o r i • 1#; 2 , ♦*.** r * th a t i # # i f and o n ly i f -ti» fw a ily o f hyj»r»urf& © ««
i# ** constant ©©stain® th.® f i r s t r ©e®gr».«a©«s o f th© eimupl©.*
th en th e
systw s* o f equations be h© s a tis fie d is g iven b y fo llo w in g th® procedure
use* fo r equations (T * 3 6 ),
fc(-Sr)-a-S:
* %
(T*42)
and
to
wtier® the la s t r ©f th * d#
1
are ©ero due te o rth o g o n a lity b e t* * * * th e la s t
r ©eepeaeat® o f “a and t e •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*120*
If
(7 *4 9 ) is o x p m io i ty p ab stitttfeisg
f * §
loc^s * “
■* *
m i Igr f « r f « s i» g th e iM ie a ta s #
t !m fo llo w in g ex­
p ressio n fo r th e o % t» tie » * o f m etisa is e b te ln e it
• • e<
s
* -J
♦
m
L
5
'-m m
®
*
vm
S*
s*5 »
(7*49)
-th is expression I s «w © tl»r form o f th e equations o f ao tio ® of a n o nboioacwiio A yw aio iiJ system *
% r© t».ratog ■to tfc# fe«4«j»»% al fo s to la te [lq « ( 7 * l f ] e a t by tr a s s fcensing i t to «s staemple o f © o » g r«e »«s# itt® fo llo w in g eq uation is
ob tain ed *
le
«X
_^
i t )/ '4d ri rr *
-ilf
*
*t^ t *
<7*4t)
H im ■tti.s e x jre o s ie a is e i^ e a ie i, i t tefeee th e form
■I. |
s4 j
jjjh
* H . «*?
^ 8 r - * j r * Ei *
C7#48)
where
*»■•■»? “
r t» « ?
* * T
la t h . M t r l c . < * » . . * « ,
O
“ Si , « p
»
* “ 3 7
-
Jsi )
$ i >“ P
la t h . t a r a i . t « a a r .
<’ • « >
If a tr« a -
fona&tion o f ® q m ti© » (7 *4 9 ) to anotiior eonfgrnese# were * i ®
fo llo w in g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the siethod ® f chapter V0 to e s th # resulting
scmld h-avo th e ®aa®
form ae ©H$giatie& (7 *4 5 ) j to # r« fo r# th e tr e » # fe rM .tic « need s o t he g iv e s *
th e form given fey eq^stism (7 *4 $ ) is toe preferable form from th© view­
p o in t o f eppiieatleo as le e * Safeor 1# In vo lved in ealentoting I t s cess-*
poaents tiiaxt ia
Hi© ©oafoaesta o f either th e form given fey.
eq uation (7 *4 2 ) o r to o t given fey » f a * t ie » ( 7 .4 5 ) ,
Soiaa o f tfe# p * ia e if l» * g l m
la. to ie © lif t e r w i l l fee a p p lie d to
th e a n a ly s is o f a ty p ic a l #l#«troN W »e3M »ie*l p*©fel«»».' th e ro ta tin g
e le c t r ic a l tofefeto**. to tim ass*
-ffe
e e fn a tle n e of pertorsMmee
o f to®r w ia tla g e le c tr ic a l. soehtoe- will 'fee tswoeferaaed from* s e t of
lie tm rm ia n co ordin ates t o «n eonaplo © f congruences to o rd e r to s im p lify
ife e ir e o la tio n *.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<•4*0#
» « m :» s m s m m m m m w r n t
w m s*
Th© geaer& l
« p p » r « » # o f on© ty p e o f r o ta tin g # le e t r le * l
wmab&m 4# so- s im ila r t© th a t o f m o th e r ty p e th a t i t 1® o fte n w r y
dHUTfist&t * e l i s t b i p l A
Ir e s
h s tm s s tfe* m rim m - ty p e * m m appearance a lo n e .
measurements mad# a t th e © e ll te rw im i® o f th e aaefein#
f a i l t© fu rn is h « « .ffi® l© » t a d d itio n a l d a ta fo r th e «®«spl#t# id e n tifie s ^
tio a o f th e machine*
le a d * w
A l i t t l e eatpeniomc® w ith e le o tr ie a l aswdtlaery soon
to th e © © »© l«sl«i th a t th e e s s e n tia l d iffe re a c ® t e l w n th©
v a rio u s types o f asefela#® i® th e way 4a whldh *h# © o il® w
nested*
ia te ro o s **
Xr®» (® * 10) has hssed h is Wrep»@ s«w%fttlw* ssehiass ©a. th is
.e s s e n tia l d iffe r*® ® # t M
has considered th e e s rie s s typ e# o f machine® as
r ® s » iti» :g f r o * tfe# method o f fe te to o n a s e tim o f th e c o lls o f h is repre**
-seastatlse wwsltlm m *
Iro n *® s»th#*i»tt*#3, to o l® mam those o f te n so r
a a a ly s is f h is metSied ® f reasoning was th a t © f d if f e r e n t ia l geom etry*
th e
w ta to r wishes to a p p ly th e method* o f th e S&fcrissl# te a s e r a n a ly s is o f mu
©mwple o f #©ttgroisa.ee® to th e r#tfttisg"*#a«h£ae proKLata*
The equations o f
m otion is . te n s o r twwm m m mmmm& to re p re s e n t th e p»rfarm sse# o f a
general #l*M*treeneehis®ie«bl ©yet®®# thw hsfer# th e y w ill, re p re s e n t a
ty p ic a l e le c tr o -^ e h a n io a l « y *ta » * th e r o ta tin g e le e t r ie e l Machine*
J&thettgjb a© s p e c ia l s tre e s is placed ©s eleetro-ew ohsaaieal equivalent® *,
th e r#ea@e&&t£ei» of- t h e ir e x i s t * * # l® necessary is. o rd e r to fo rm t h is
© R ifle d th e o ry o f e le c t r ic a l a»d saw hsaieai system s*
Its a p p ll« a ti* » © f th # general, method to tfe* -smiysis o f as ta ta e tio s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
—1 2 1 —
V
wrfcor is giron.*
this.. fc f f X lc a t i* , is ©a sttssspt to formulate i a the language
o f th® in t r in s ic tenser «i3» ly « l.e mmm of S tanley*® results (El), his
g en eral ss#tfe.©d o f «p pr-»eti being ©issilar to that used % P ark (1 6 ) and
other©*
1.*
A n a ly tic a l approach to r o ta tin g e le c tr ic a l pMdylae#
Hi© i i f f e i r m t i a l
ib is * . Mfarwaaat tfe# fftrftoran ao* o f
e le c t r ic a l c ir c u it© a re ©midges**. ia f o r * w ith those which w p r© « *» t tb s
performance o f »«® b*»t® «l c irc u it© # aSsc th e re is a a»«-tc-® « e cor—
re s pondcnc e between th e ccsccfb® .of © Ic e tr ie a l and o f igsebs&ie&l sy»te® s*
Soovistg th© analogous concepts iaafc®# i t p o ssib le f o r one to s e lf®
m echanical systems i a barm® o f © fu im le s t e le c tr ic a l s y s tc w ,. or 1A 0®
vm rm *
I t f® a ls o p o ssib le to ©c-lre ia te rlia & e d ©©eha&ieal and e le c t r ic a l
syste&s w ith o u t reducing tfe© aystem to e it h e r * » © fu ls m ls a t e le c tr ic a l
o r an ^ a i m l t u t , m «$laitleal -aysttta*
T b i* la t t e r method •*« & * p ? #f» ra b l#
to th© a u th o r| th e re fo re i t w i l l b© w a d to s-olse a ty p ic a l e le c tr o iwefeaaio&I sp st«a# th e ro ta tin g , e le c t r ic a l m achine*
a c tu a lly an a n a ly s is o f a a r t i c u l a r
i& thoagh i t is
te a t sys-fwa, th is
tr e a fe e s t should be o e astto red as a trsatsasHfe o f a aetltod o f a m ly s is
©blah a p p lie s to © l»© trl© aX« moolbaaiml.,. «ad ©lectro -ro so h en ical system®
a s s p e c ia l cases*
For r»f#r#a© « fu rp « © » # ® «© o f th© forw&X an alo gies
between electrical s»d «® eh»»i«al c ir c u it concept# a r * g ir» n te th e
fo llo w in g ta b le *
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*1 E 3 *
1» th is ta b le i t abeuM fee sotsd th a t a l l swalogsw® % b*iaatiti## ®*n# re p re ­
sented by tfe® s m
syafeijli tfesy -w ill he tr e a ts # M m t ie s lly la tfe®
mtfe»i&*i&QiS iis ftt fe llo w s .
For ft g en eral # l# ® % r© -» s lia a io ftl system i t is n@®»#®»ry to know
tb « fo llo w ia g tta s # se ts © f Assign o « o #teat» la order to p * r f« n t *&
a n a ly s is o f tfe# system*
tfe# sm slst«###8« tfe# in ductan ces, i*ssJwdi»g
th e t a e r t if t l # « # £ £ ie l« » t» j arid th e safrtfcsitaa*®##* ia e lw d iiig tfe# #«&»
p lia n e e e .
Wmt ft rs is ttia g am eliln# o n ly two s e ts o f d esig a eo sstaa'ts,
th e iMttetsaa### and tfe# re s is t*® ® # # , m v m ttj a re re q u ire d beeae## th #
Sftpfteltftfto##' &r« so ftzaall*
For e e siw aleije® , i# * ig a © eastsats sfeesld fee
arran g e# la «os» t&fesiliwf form# as a® exfte.pl®* th# la^aotaao## a re re p re ­
sented la tfe# fo r®
(sa)
ft ft-ft ft iftftft * ft-ft ft"ft ft ft-ft- ft ft ft
Bn1
% g * * * » &rm
^
J
* o « t ro ia tia g ; e le e tr ie ftl ssftohl**# i * f fee gresped la to m m o f th e
fo llo w in g ty pes *
those w ith s li p * r i» f f t i tfeoe# w ith eeo aatftter#*.
la
a lte r n a to r I s a ty p ic a l a M p -rin g ssu& iaftf am o r tla a r y sh u n t-e x c ite d
4 *o dynamo re p re s e n ts * ty p lo a l eosassitft-tor ssftchia#.*
lis » a a l« a i geom etrical
»i$li©Ae a re s u ite d f o r tfee re p re s e n ta tio n o f saohtee® o f th e s lip -r in g
typ e as tra n s fo rm t ie s * o f ft ■pftftleu.lftr--isa«feis»#- b u t n m -liejn arm ia®
g # .« # trte ftl methods a re r e t i r e d fo r th e r epr ©sent a t i oa o f a i l m achines,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ta ts lw iis g bath
and eom m tator ty p e s , * « tr im s f © rm tio n s o f a
jN k rtle a la r sa&ehlm*
Sias© th e commutator type o f m ash i** -differ* f t p * th * o l i f r l » g
typ® e s s e n tia lly o n ly tm. t h * mmmme o f © s te rn a l connection o f I t s c o ils ,
i t * •■ « * reasonable- to h a lt * * * th a t «a© ©an h * considered as a trs n s fo r m tio n o f th© o th e r .
This is th e © a **, t o t tt©n»®l«raarmian methods
o f a n a ly s is a re re q u ire d «
In f a c t , *© # t m m hlm n can to reduced, fo r
purpose© o f a n a ly s is , to th e eooBantatsr type by « s u ita b le tr a n s fo r m tio ft o f co o rd in ate s*
The «d««ntag» gained % aueh a tran© form at!on is
a e iin p lifio & tio n th a t ju ra t! t * aa e a s ie r s o lu tio n o f th * equations o f
perform a© #.*
Blonde 1 fu rn is h e d th® clu e f o r th e typ e o f tra n s fo rm t i e * needed
to reduce s lip - r ln g aaeh in a* t© e q u iv a le n t eesasatater n a s h im s •
o r ig in a l psper C l) Blond®! ett(gge*t*4 t h a t th e « » t e »
In an
re a c tio n o f «&
a lte r n a to r is e s s e n tia lly no d iffe r e n t from th a t o f a &«* m achine.
%
co n sid erin g tw o-pole E&cfeines, he 4 *m m *tir*t*4 th a t s h iftin g th # trash#®
o f & d -e machine m s a a le g o n a . to chaagiag the power fa s te r o f th e load
©a an a lte r n a to r *
As th©
re a c tio n in a d -e B aw feias.bii been
eonaidered in term® of #§®»ga®ti:®iiig -and «r®@@*mgB®tissi»g eompenent®,
along and p e rp en d icu lar t o , r * « i# « M m ly :Jt th * lin e jo in in g th® f ie ld
p o le s , B londel m s 1 *4 to e m a ld ir th© s m t t t i© re a c tio n o f an a lte r n a to r
drawing balanced c u rre n ts to b© d iT id e d -Into cosponents alo ng and per-*
p e n d ien la r t o t h * lin e jo in in g th # f ie ld poles.,
1® a ls o assumed th a t
each component was produced by i t a own e w rre n t*
This assumption make*
I t p o ssib le to a s s o c ia te -a s e t o f c o ils w ith each eopponent o f c u rre n t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
These e e l l * •h m ld be etetleraw ry w ith reep eet to th e f ie ld c u r c u itj a ls o
th e y should be sjfreaged so th a t t h e ir wajpoott© axes correspond w ith the
d ire ® tie s o f th # eegypme&ts © f arasatwr® re a o ti© a wfcieh th ey a r e to
rep resen t#
Im p lic it ly oontssi'iiea to t h is typ e o f re s o lu tio n is th e
assumption th a t Hie reso lve d eesspmeat* o f arm ature r s a o ti on re pis'©sent
th e sarae s o rt o f spa o © «d lstr i buti© as o f magnetomotive fo rc e #
tre a tm e n t th®
fo r th is
^S te a s iitd e i 8pae@ ~distribut!on is assumed*
If
Blonde 1 * s resc&vd&SK is «#$*«**#& m tte ® s *tl© iilly in te n w o f c u rre n ts *
i t . can be considered as a lin e a r irsaa® f© ira«bi« f r o * on® s e t o f cu rren t#
to an o th er s e t *
Walt i t
is knows th a t th e d if f e r e n t ia ls o f co ordinate
v a ria b le s tran sfo rm to to e same m y a# do e u r r m ts f th e re fo re B londe!*s
re s o lu tio n d e fin e s a ls o & tr a a a fo r m tie n from mm s e t o f d iffe r e n tia ls
to another s e t *
Jk c lo s e r eaestoH atim o f th e tra n sfo rm atio n s in volved
bring® out th e f a c t th a t i f th e a n g u la r p o s itio n o f th e ro to r o f a
nateh-toe is considered as a co ordin ate v a r ia b le , th a t th e re la tio n s
between th e d if f e r e n t ia ls eamset b® In te g ra te d to © btato th e u n d erlyin g
v a ria b le s ;| l a o th er words* an to irto s i® trsaisfo rm atio n 1® in v o lv e d *
to a t is * a traasfarxaatlssi betswwwi a s e t o f coordinates and mn ermuple
© f ©oagptmraeae*
ih e statem ent# o f th e p re v ie w paragraph ©a» b® c la r if ie d somewhat
by co n sid erin g a s p e c ia l ease#
VA
s in g le ' phase a lte r n a to r w ith a
s in u s o id a l s p M M M Ito to ib e tlm o f arm ature and w ith a s ta tio n a ry s a lie n t
f i e l d w ith p ro je c tio n axes alo n g and psrp@ndieu.lar to it®
th e re fo re a moving a x is mm b© associated w ith th e
d ire c tio n o f to® r e s u lta n t arm ature re a c tio n *
This moving a x is w i l l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
b# denoted by t 1 , and th # s ta tio n a ry axes #411 b© denoted by # * cad s2 #
cleag -cad p © rj# s d l© a la r t© # re s p e e tly e ly * th© m g a e tie a x is o f th#- fie ld #
I f th # » g m 3 *r f © s itis ® « o f th # sisviag & xm .tar# 1© mccsurcd between
% « m g p e tte w d a .o f th e ©m atter# © o il cad tb® *a g a # ti« » i s
of th # fie ld #
th cc th e tr a m fo .r s » ti» « between th # tins© * # £ • • ©f change o f th© m rl& b les.
and bet*we«s. the d iffe r e n tia l# o f th© © a rlc b l# # cr® g tre » by
|3- »- oos # # * ♦ c ia # s2 ,
and
dq1 * «os 9 d«^ ♦ * i » # i® 2 #
r« s p © « tir© ly »
(8 *2 )
fo r those e l a t i o n * to b# © m e t d if f e r e n t ia ls i t I©
E ® fle s « y th a t
cad
'
.
.^..^...
a#
# *,
'
i t is c le a r th a t «i© equation® (M*S) do n o t represent- © m o t d iffe r e n t t ills *
A- isor® gen® m i Bl©adel«dy|>« o f trc n s fe n a c tio n Is cnssspllfied by
th© © fp lio c tie ® o f B len d #!*# p ri» « Ip l© s to- th e tbmo-pbSMi® a lte r n a to r .
As b e fo re , © iw asoidel « $ # # e ~ ^ lc trib # 4 4 « j# o f th#' xmgtm tom ieti'e# fore©
© f K i m t r e r#a© tl#fc t« a s i m i *
ffe« f i e l d © frem it is a g a ia ©sew ed to
1# o f th© © eld est twewftol# ■ty p e w ith s ta tio n a ry axes ©long cad perpcadle^
a la r to it© am g m tic cad#*
t t r t b c r * I t I® csransd th a t th # an g u lar
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
p o s itio n of* 69ÿ r o to r i s Meaawred between th e n a g a e tie a x is o f any one
o f th© phase s o ils o f th e ariiR ter® as & re fo rm s ® and th e tm p io tie a x is
o f fe e f ie ld *
The o th e r two phase s e lls o f th e *ra » ta r® « r# spaced a t
120 degree Is b e r m ls fr® » th e f i r s t pirns# s o il.* I f fe e s ta tio n a ry
4
axes a re donated by fee v a ria b le # • and fe e moving axes by th e v a ria b le s
I 1* fe es th e trsasferasftticaa from th e statie aaa ry to fe e moving axe® is
g iv e s fey fee eq u atio n *
dq^ * cos 0 ds^ ♦ e ia 0 d»®
dq® * eos{® ♦ -120) ds^ ♦ s in (0 + 120) d«®
(8 .3 )
dq® * coa (# *■ 130) da®1 * s 'in (0 ■* 120) da®*
l a g « e r » l# however» in case fe e mm o f fe e fe re e ph®e«-*®urr«o.ts is mot
s e re , ft th ir d Is&epsoadent s fe tio a a ry e o s rd is a t® is re q u ire d *
This ad di~
tio n a l e e e rS ia a te oaa he s m e l to have eq u al fr e je e tie n s on a l l th®
moving axes ; i a th is case eq afttifltts (8 *3 ) w ill, he rep laced by
dq* ** eoa 0
* s in 0 ds® ♦ dsS
4q® *» ©os- C® ♦ 120) d®l + sin. (0 + It© )
d«® + is ^ ( 8 ,4 )
dq® «* eos {© — It© ) i » l * s is {0 — 120)
is® * da®*
I f eqwatieaw (8 * 4 ) a re solved fo r th e d # l i» term s o f th e d q ** ft
tra n s fo rm a tio n o f fee eaapsaeate dq^" to th e ©exponents. ds^ is ob tain ed *
Hush » s o lu tio n y ie ld s fee fo llo s d ja ^ equations
■* j| cos 0 dql * || e®« (0 * ISO) dq® + ^
is® • | | s is . ® clq^ * |
d»® «
©os ($- 120)dq®
s ia (0 ♦ 120) dq2 * *|> s ia (® » 120) dq8
dq1 + *| dq* * *| d # .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8*5)
S i#
im equation# <& .$) must fee conjugate to those
l a # q » t i® s (0 « 4 ) feeaause th e y r e s u lt frost th# a e lu tlo a o f equation#
(8 *4 ) .
l a o th e r w ords, i f . fe e m m tti.0 i.0mtm in equation # (-8*4) -are
denoted
th en fe e 'ftps m tm isust obey fe e re la tio n s h ip s
<8 .6 )
H it ii these f i n a l re la ti& a a h lp s cm hand,. th e aethed o f an a ly s in g general
r o ta tin g saefala®# earn b# resuEwd-#
The equation#- of aotion for dym &im JL system# hmm feeea assumed
to a p p ly to e le e tre -w o fe im lc ^ systems*
I f * # lf p -r ln g asohia® (on®
w ith sieving. xwfereme exes) is w m ly s M a# a elip-ring type o f mach in© ,
fe a t i s , i f i t is n o t referred to s ta tio n a ry mam® b u t to axe® sseriag
•with fe e o o aln eto ra ,, fe rn th e «q«§»tie»# C f*2i ) re p re s e a t i t s perform ance.
It is necessary, hw®TOr, f e a t th e q u a n tltie ® Q-j fee d e fin ed fey
(8# 7)
M i e re fe e q u a n titie s @j mrm e it h e r a p p lie d torque® o r a p p lie d vo ltag es
and fe e q u a n titie s 3t|j. q * « r e e it h e r r e s is tiv e torques o r r e s is tiv e
v o lt age-drops *
By safest!fe& lttg aq u atio n C®*T) in ( 7 * 2 i ) , th e fo llo w *
la g equations rep reeen tim g th # perfoHawse# a re ob tain ed *
(6 * 8 )
la order to determ ine th e pmrttmmm®® o f th e g iven s lip - r in g nsushine
expressed a lo n g jaevi&g re fe re n c e axes i t feeea&es n ecessary to so lve
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<029*
a set. © f equations o f its® fo m gives by eq u atio n (f3 .8 ),
For r o ta tin g * la e til< M l-m c h in e e * i t so k a p p a s tb a t th e q u a n tlti® #
g ** are*, ta
a o t ©oastant b u t th a t th ey depend upon th© p o si­
tio n of t ! » rotori therefore the a p p lic a tio n . & f expression (8 * 6 ) results
in a s e t o f d if f e r e n t ia l equations w ith c o e ffic ie n ts whieh are fu n ctio n s
o f th® rotor p o s itio n *
Sash equations a re usually d i f f i c u l t t© »®iv®f
therefor® i f a transf©rm ati on a M e h would reduoe th# ooefflelsnte to
constants o r to simple forms could he found* It ie p o ssib le th a t much
of th© difficulty sdght b© remored*
th # Blonde! type of tra n s fe r m at! on
[jSq. (8,4) «®d (8.5)] w i l l do j« # t t h i s .
I t is necessary, h e w e m * to
transform- th e q u a n titie s in equation® ( 8 * 8 ) to ©qttiv a le n t q u a n titie s
along in tr in s ic referea©®- frame* •
&
previous ehapters*. th e method o f
trmsforaatiea; and th© resulting m-expanded form of th# performance
equations were givea [% • (7 ,3 6 )]] 1» lag rsag iaa for® ..
The new performance
equations are given by
(8*9)
c)qm J q 3-
<^q^ ^q®
where th© primed q u a n titie s a re tfee transform ed q u a n titie s ,
flies© ®q«a-
tiOKS d e fin e , In g e n e ra l, th® m otion o f a p o in t la a n o B -S iem isiiea
m e tric spa.ee .
When th e y a r e s o lv e d , equations ( 8 * t ) re p re s e n t th e p e r*
formane© a machine which i s e q u iv a le n t through a tr a n s fo r m tl on to a
given. «lijM**i»g a«chine8% asi fo r «x*uspl*» th is equivalent maehlae a lg h t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*180*
to® a
ty p e ms happens to to® th® ease I f a Blend®! type o f
trm s fo ra s tis s a . i s used*
E*
o f th® eowgCBsent*. o f th® ipdwe.’teftae® and tt*o re s is ta n c e tensors
flu® inductances, tootlte s e lf wad * I b » I # o f th® c o ils o f & ro ta tin g
© le e tr ie a t
£®p«s&». la g e n e ra l* t?pon th® p o s itio n o f th© r o to r *
l a o rd er to s -i*tp lify their m th « a s *tl« *l re p re s e n ta tio n a® w a ll as th®
re p re s e n ta tio n o f o th er ^ a a titio s s * i t i s neeessary t® in tro d u ce sojbs
assum ptions*
1*
fh® fo llo w in g a s s m p tio its a re tsadet
toystwiwi®#. and eddy cu rren ts a re n e g lig ib le *
f*
th® variation » f -self-* « r mutw l*iM tic ts n o ® s w ith th e p o s itio n
of th® r o to r fo llo w s sin© m wm m *
S*
The srw rtar® is smooth* and has balanced s in u s o id a lly d is ­
tr ib u te d w inding® *
4*
fe lly two f ie ld p®l®s e x is t f o r synchronous or fo r d -c machines*
to*
la d tto tio s -ty p » s&obin©® ham a uniform a i r gap and balanced
s in u s o id a lly d is trib u te d r o to r w in d in g s*
■
6*
!« # is t« n e # eheng®» da® to h e a tin g mrm n e g lig ib le *
7.
Three-phase a rm tn r® «1 re n it® *r® used unless otherw ise s ta te d *
.
Th# # ® !f—indtwta®*® « f an artsiitoer® phase is a maxima® fo r a s a lie n t
two-pol® m ofals© tfe#a ft® d iro o t « x i* is lin e d up w ith th e jaag&oti®
a x is o f t h * phase and & x t o i n w l»r©
w ith i t *
th ©
fuadratwp® a x is i s In lin e
I f th e ro to r .p o sitio n is measured from th© m agnetic a x is o f
phase 1* then th® ao lf-tn d u etan e© ©f phase 1 is given toy
% “ I * # * * « * • ^ ®#
where
i s gr®»to«r th an
(8 .1 0 )
th is fu n c tio n a l re p re s e n ta tio n ©an to®
determ ined e ith e r toy c a lc u la tio n o r t e s t *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i t a d ia f t y , fo r phases t and S th e s # lf*l» d « e ta a © « s are gives. by
Lg ■•*
e hjg ©o# 2 (9 ■** X 2 0 ).
<8.11)
%
#
(8 ,1 8 )
aad
** * *
%
cos
*< •
♦
12° ) *
the wfcttftl tRdiwehaaces between. Us# arrsatwre phase c o ils va ry is a
s im ila r tray*
But before the » t » l litftaetiittses can tea d e fin e d it is
necessary to d e fin e the relative d&reetlesui of th# currents*
P o s it lire
mttrmx&m m m d e fin ed In the same r e la t iv e d ir e c tio n j that la, p o s itiv e
c u rre n t ia phase 1 tend# b# produce a flia»liaM:
ge in phase 1 in a
d ire c tio n ojpisitt to that which la prodstead by a p o s itiv e c u rre n t im
the m utual taductspsees between the aanaatara phase colls
a re given by the fo llo w in g equations*
(8. 18)
cos l { 9 «*■ 8 0 ) ,
*» -• %
* Sm cos (29 » 120)"i
/
(8.14)
%f. * * *a * % ®«® * •*
M 1S *
%
+ %
{8 ,1 8 )
®e ® (2© * 120).
fh e masinwiR value# o f the vartatioa of one o f these m utual inductances
occar at doable the tm qm m cf which ©orrespeads to the r o to r speed•
A
p a r tic u la r v a r ia tio n , th a t between phase 1 and phase 2 for exam ple, has
i t s «3Ki»w valw.es whsa the m ag n etic-axis of the field bisects, th e angle
between th® adjacent c o il sides of phases 1 and 2 .
l a practical oases,
th e v a ria tio n s of th e mntual-irdwcfc&nees are al«®«t eq ual to these o f
the s e lf-in d u c ta n c e s •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The nrn&mm. au tua l~to d u etaa c© between any jiw ® # o f th e arm ature and
th® main f ie ld c ir c u it w i l l e x is t wh«& to# d ir e s t aaci* o f to® f ie ld 1®
to lin e w ith th # rofi.gn.etio a x is o f th e p a r tic u la r phas# s o i l ,
th e w a ria -
tlo n s o f th ### WBfewal—todwetea### fo r th e tor«e-jh»® ® s a re given b y r.
®lf **
*% f *
«##• #|
%
% f * ®f
(8*1®)
®#<^ # *
**°).»
( 8 , 1?)
o#®C^ *
120}*
(8,18)
■She # # lf» la d u e to t» * o f th e » t o f i e l d c ir c u it is co nstant te d i®
to case a d d itio n a l w in din g# e x is t @m e ith e r th e s ta to r o r ro to r o f
# waehtoe th e in.duetan.ee w a rie tle n s sen Tsm represented by equations
s im ila r to those Jwst g iv e n *
If an in d u c tio n ty p e o f waeltto# 1® b # t o g co nsidered , th en both to®
r o to r and th # s ta to r a re assumed to h# « o o to and to co n tain s in u s o id a lly
d is trib u te d balanced w inding®* th e re fo re to # s e If-to d u c ta n ces and th e
im inm l-in du ctan ces between, s ta to r c o il® o r between ro to r- c o il® a re
c o n s ta n t.
Th© m utual todw ctB w ## h#tw #*n to # s ta to r and to e r o to r c o ils
a re th e o n ly inductances which v a ry * th e y a re given % th # fo llo w in g
cos # *
m
m. „
da -
(8 . I f )
oo® (0 ♦
1 2 0 )J
(8 .2 0 )
cos (# ***
12 0 }*
(8 .2 1 )
IS O ) |
(8 .2 2 )
CO®
(0
-
#©»- %
#o« (« *
(8 #C9)
120) f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8 *2 4 )
®2Nfe ** *
008
+ 180) f
(8 *2 6 )
Mgg - 8 m m (0 ~ lt© > .f
lg
(8 *8 7 )
• M eo® 6#
The numbers %m 2# * d
tmmSmr® 4.* 5 , and 6
S re p re s e n t th© s ta to r {mmmtw?®) c o il® j th®
th© r o to r © o ils *
o f th e m r ia t io a o f those w rta a l Satokctan®## *
* 1# th e m odana m in e
I t should t » noted th a t
(8 ,8 8 )
la fo ra to g th##® m r ia tio s s o f s ta to r -to -r o to r iad u#t*ae© «# * p o sitin ®
c u rre n t la phase ■!, fo r ©asapl©* wmm assumed to pro too© & flux*»llnkag®
ta .tfa # sasse d ir® « ti® & mm m p o s itiv e e u rrs n t i a f&a®» 4 *
'The m lu ® o f th # aswteat o f in e r t ia o f th e r o to r Is m is# a c canponeat
o f tfa# in d xctanc© te n s o r$ i t Is c o n s ta n t.
I t w i l l b© represented by Jr »
S u ffic ie n t ia fo rjs a tiffls i s now on h a s t so th a t. & ty p ic a l inductance
te n s o r mem. he form ed*
She e©«pom«a.ts o f such an inductance te n so r
should be chosen ttpom' asKKig th e m ricm s
such # ©ay as t o rep res en t th # gtvw i laaohla©*
o f th is s e c tio n ia
Equation (8 .1 ) rep resen ts
a method o f © r it ia g th© eoBponents.
Th© eenpeeenta ©f th e re a is ta a e © i#»#© r a re e a s ily for&wd*
Th®
m u tu a l-re s is ta n s e s a re m aoally sero la. a r o ta tin g e le e tr io a l sm ehinej
th e re fo re th e *# rem ain o n ly th e so I f -r e s i stan c# s o f th e vario us c e ils *
load the f r ! « t l« a and ©issiag# re s is ts a e e o f th e r o to r *
Th# eeapeaeata
o f th e ro a is ts a e e te n s o r should be arranged i a a form s im ila r to th a t
used fo r th * inductance te n s o r*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3*
tti# ts a n o tto o motor
mm w w M -ro to r in d u c tio n m eter Is f » lt # e a s ily represented by
H ie methods o f th is s & a p te r*
l a o rd e r to rner# c le a r ly dem onstrate th#
f r la c lf t s s ma& methods
th is p a r tic u la r ro ta tin g asteliia# # i l l
be analysed i a seme d e t a il*
ffa# Inductance te n s e r fo r an in d u c tio n motor o f t& # w®aad«rotor
type is formed from th e oemps&tHfcs glir#ti ia th e previous s e c tio n *
It
1# represented by
t
**%
-%
t
f
-*w
-%
%
.
M eo*©- *'(©##
K ««a
9*120} (9+120)
» cos
K cos
(9+120) S cos© (©-120)
q4
* COS#'' M cos ,
(9+120)
M co#
1 ©089
.(9-120)
r. cos
M CO#
(9+120) ( M S 0 )
%
0
0
*8
0
/
M eo#
©
q1
{#*120)
M cos
C©*ito)
IS COS#
0
**
0
-Mfe
O'
qS
JL
%
H cos M cos U: eos#
(9-120) (9+120)
■©
#
*®fe
h
0
Q
A
< 8 .» )
SSL
0
t6
0
<ie
*r
q?
3—
Where th # v a ria b le # * * * f1 * and #
a re stator variables corresponding
to phases 1*. 2, and 3 # r # o » # « tir « ly t an& # # * * # and q6 are the rotor
variable# eorrespoadihf to phases 4, 5, ®a<t 6 m respectively; also
is equal to 9, the angular position o f th# r a t io *
Use m
utual-tn&
uetone#a
b
etw
een stator p§*##«# o r between r o to r p
h
ases do sot v a ry j a ls o th e
s # lf» ia d o e ta » o # « a re eoaofcaat*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•is®**
'the re a ls ta a e e te n s o r is .glum by
Qi
■t
i
IT
m
.
»s
#
»*
«S
7
W
/
0
0
q1
0
0
0
qlt
0
0
0
0
*s
0
%
0
0
0
<d
0
0
0
0
0
0
0
0
0
0
®fe
0
«•
0
0
0
0
0
0
%
*7
9JB L49
*
0
V
' 0
0
0
0
0
0
%
0
0
0
.
.
C i*iO )
nfeor® 8^ 1# th e re s is ta n c e o f a phase c e ll o f th© arm ature s ta to r * 1^.
i# th e re s t stance ©f * r e t o r © o il* amd Eg* Is Hie re s is ta n c e which
rep resen ts te e f r ic t io n and windage le s s *
Th© n l w i o f g j j ® ai
as g lu m by equations (8*& 9
(8 *30 )
could he s u b s titu te d la equations ( 8* 8 ) ©ad th e equations o f performance
o b tain ed *
Tim re s u ltin g equations a re q u ite d i f f i c u l t to so lve because
souse c o e ffic ie n ts a re tr ig m o ia e tr f# fw o etlo afi o f th e p o s itio n o f te e
rotor* If the two equations o f c o n s tra in t
q*
ssd
.
q4
* qZ
„
+ q5
+ qS » 0
(8 *8 1 )
_
* 4* * o
<8*S2)
a r# in tro d u c e d , th e e q u a tio n * earn be s im p lifie d somewhat, b u t th ey a r e
s t ill
q u ite in vo lv ed *
Levin© ( I t ) has g lu m a tf« s tw w t o f th is
s o r t*
By Introducing th e linear t rm is fo re s .ti on represented by equations ( 8 .4 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130—
and (S -S )# te e l e l t l p
{to eto 4i«.g torque), re s is ta n c e s , to to etaaees
(to e to d to g t o e r t t a l
and c u rre n t# ean fee re fe rre d to &
s e t o f © rteo goiiat #@ ©i€to&t« axes « h l« h a re s t a t i oaaary w ith , re s p e c t to
th e s ta to r .
f l m re q u ire d tferteferttattes® ts a trto is formed by ap plying
equations (8.4k) to tooth th e s ta to r and th e ro to r c ir c u its *
For th®
•fa s te r, th e c o e ffic ie n ts a re constants j th e y a re o b tained toy p la c in g #
eq u al to se re degrees to eq u a tio n s *
f o r th # r o to r , th e c o e ffic ie n ts
ar© fu n c tio n s o f 9$ th e y a re t@mm& toy ©haaglag t ie
v a ria b le s to
equations (8 *4 ) to th e r o to r ra ria b te # o f th # in d u c tio n m otor*
A fte r
th e two tt© a -e in g » le r conjugate t r a a s f o n m t i* m atrice s h a w been, ©toteto ed *. th # equation o f e © » # tra in t g iw a toy equation (8 .3 2 ) is in tr o ­
duced to form th # foU enrtog te e ® inj»l«ur e©»J«gai® saatrie«#t
s1
s2
**
«
sS
s5
#
/
1
*§
0
0
s
1
0
0
0
1*
i
0
0
0 •
1*
s is 0
0
**
0
#
0
0
0
0
«ia(*+2S0)
0
#
0
0
|
1
etoCfa-lSO)
0
fl6
t
0
#
0
0
0
1
#
§
CO®
&
dm1
r
1
0
H
*
0
0 ;
#
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ct.ss)
q. 1
«4
I
©
1
© .
Q
©
©
1
• f
s
f
1
©
©
©
©
I
0
0
©
0
■#**
1
-1
a®3
©
©
0
t
©
©
® § ©0*9
|i»«rift*U0) «|sin(®«l20)
©
s ta #
§ » i n < 6 « 0) |*lti(#*120)
©
°l
©
"
©
&
1
84
(8 *8 4 )
»
There &*•• Sanaa f® 8 ® i¥ l« £ a & » j^ a & n t a f f l l e f n e ltitg e s ( in e lu d in g
tor<jue) f o r a wooasd-rotor ls d t » t i« i m © hi»® , tro t o n ly fo u r a re e«s ternary,
A tr a n s f or a» tio n o f th© vo ltag es 1® g ives by
s * i •* e
1
(8 .3 5 )
w
do1
<
«fe«r® «<*,
©a© o f th e aj»j»-tl0i neltft&BS* and « * i# or® © f th e
transfers® #; neltag©#.*
Using th is tmwa.%*# th * fo lliw d n g new voltage®
mtm ©btfidaedt
(8 . 36 )
1 * | 9s * 7 * s '
€* • © ♦
I
•<§ «.» »
f
S
#i *
i
*3 * *1 * * 2 * • »
5
is*® ? )
®» *
(8 .3 8 )
( 8 .3 9 )
4 ' * *? *
Th© re s is ta n c e te n so r rnxst a ls o t *
It® n&m e « p « j© a t#
a re given by
• It
( 8 .4 0 )
P
J .i
<M
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
liM&m th® va lu e s <&{ th® r# « i« tim es® a s given %y eq uation (8 .5 0 ) a rs
s u b s titu te d in « q s a » ti*i (0*40)* th # f# ll© i* i» g s e t « f
is
*1
1
1
Ry "
*
* tt
1 %
■0
0
0
0
0
Ij fs na
0
0
0
0
0
0
a5
0
0
»4
§ ft
0
s 5'
Rjj*
*
0
0
0
0
®
0
0
0
0
0
0
0
■0
3 Ra
s1
0
#
C e *4 i)
w
'
I s
f
%
0
th e indw ettee© te n s e r aa®% b e
in t is® saws way as was
th e re s is ta n c e te n s e r* last th e re a re aeaiy # » r» »©»»ser© e o stp o n e n t# j
t h e r e f o r * « a e xp a n d e d fe r ® o f th e tr a n s fe r a fe tlo c i feriw sla l a
f u l . ■ S in c e
t.
,
q u it s h e lp *
,
(8 *4 2 )
Hhmst i t fo llo w s t lia t
t
k <3s1
(j £
\ Js*
, j a u ♦ - i a f f . <kf ♦
<)s^
t)*3, d»3r
j
£
.
j
£
<3*^
j
£ .
c)#3
<^s
apr
j
£
cjs5-
dfl£N
j
£
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
j- 4
Mi
w h*m
1
co
s@
<>** *
■"''"
<)®L d i r
m
,f
*
'"W "
^u«
d®
♦
d®$
d®^
_ .
„
M .„A£. .ist *
1
\
as
4
v
0®3
»
v
d *%
tli» eowffloi^iffes f w » C®»SS5
if
d®3
,
#
c)«^
M
-Tirrnrr.OTjji
dm1
3®*
#»'&afcitn.t»d 1» «% m tlesi ( 6 *43 }
fo r a l l th® pesstbl® oosRfeimtioaa o f 1 m A J * tfe® fo llo w in g a rra y o f
e®*p«isiri%® is oMata#®#
*6*
.»
s
a*
8®
©
/
§cv% '
t
"g* it
* 6 8|
>/s „
- 1 %
Mm
• 0 %
i|gk
s{Lm
*mm
}
-ft
it-
©
0
©
q
f *
©■•
8®
©
■©
0
a®
( a .44)
S
0
;
|r n
©
0
'
§
§ V
©
®
0
o
■
|c%«fe )
-V]| J%
0
a4
0
a6
\f%
. t ȴ
©
1< V
%>
©
Jr
©
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Saw th a t t l * isaw se ts o f
o f th e re s ls ts n e * and th e
iadw etsae# ten sore ar© amiJskbl® i t i s mmmmM&ry ia o rd e r to o b tain
t l » p « rf© i» e te « eq uation s to s u b a fittit® th® vo ltag es and th® tran.sforam t i aa s o s ffts ls n t# Sato sq p stio n ( S * t ) # sad to perfon® th® In d ic a te d
o p e ra tio n s *
Sin®# a w
o f Mi® nm t s « B f « « t 8 o f e ith e r th® induetanc©
o r the rs s ls ts n o * ta n s e rs a re f a r t i a j s
o f th o ossr& is& tes* th # lab o r
in v o lv e d in s o lv in g th@ p rfm im m m equations is co n sid erab ly redueod*
For k * 1 , Mi© eq u atio n o f p r fe r a m e o * askin g us® o f equation
(8*0), is
T * m 4* ’ i K
♦ V
» * - vf
♦ H ** # 4- * 1
»* -
S B . 4s
* | c ® g * ®t } #
Ce.45)
o r# l§r ro p la e ia g one of th e d e r iv a tiv e * w ith » ® p e t to tim e hy p,
§ [% *(!&
% p *8 » 8 %
♦ % )p ]® 1 * ^
p. #
♦ |
S 9 S4
» * 1 '* '|N®iS. +
{ 8 *4 6 )
For k » 2 , end k « S# re s p e c tiv e ly * Hi# fo llo w in g r e s u lt# o b ta in in a
s is ila r way*
-
§ % f « l * § [ % *
"
• « M
& p s1
f
%
¥
+
%)p]®2 ~
** # I ^
P
I3 * | m
s
p s5
(8 *4 7 )
I 2 * $ [ % ♦ ( I * - 2 ^ ) p ] Ss
* *1 * * t *'**•
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8*48)
I t should he noted th a t fo r fe • 1 , t m and S th a t a© te rn s i a 9 are
in v o lv e d *
Fnws a l l appearances* eq uation s C:a *4 S )# (8 * 4 ? ), and (0 .4 8 )
sdght he re p re s e n tin g & s ta tio n a ry astwerle#
there a re th re e rsw aiai& g eq’Ba.tioa# to d eterm in e*
Two of the fchree
rnqvrnti.ms a re e X e o trle e ir e u it e^atles# im re liria g stations! impedances*
and the third is a msehamleaX ©iremit equation. vh ieh represent# the
rotor aseehaaioftX e ffe e ts *
f o r t • 4 #»d g ^
» * * i§
t o r s i cm. term
fie ld #
t *gg
+
(J* s ia
©os # *
( j| #in(9-120)
|tH
S im ila r ly * f o r fc • 4 and g
sin
s in (#+120)
ee» (©-120)
9
cos (#+120)
*
(8 *4 9 )
* **> ® i 5
» g '*j^ * th e to rs io n term y ie ld #
cos # *
(*|- s in (9»120)
~ s in (9+120)
©os(9+120)
(# -I2 0 )J » sS •
( 8 . SO)
therefore the e*ptatleaa of perform m m ® t o r k *» 4 sen he w r itte n ia the
fo rm
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
| M
V »%* | | %
♦ |{ %
♦ i \ * \ > p ] »4 *
* % ) .4
«bF • *
i® + | i S i $ * 0
(8 ,5 1 )
HaHw^r for fc» ®
* tb s fe£le*in& equation is attained*
I M * &
-
«> » i * * | j \ ♦ (1^ *
* §C % ♦ % } I
* | l
§ Sl * Qm
(8 .5 2 )
I t juswris necessary b© e & le u la te th e terras s f th© torque e q u a tio n j
•that is te say*: th e eq uation e f ■p»r§&rm.nm fo r k- * 7 .
th is p a r tic u la r
s q u a ti on has th e fo llo w in g s iifftl# form *
( ly *
p) I
» «7
(8 *5 3 )
fh e qnlcrfei'ty e ^ * i t rnhmM. h
e
- reaaahered* is the applied torque
on fcb.® r o to r *
tb s ccesplet© s e t o f s ix d iffe r e n tia l. equations a re c o lle c te d below*
*§ [% + ( \ * % )p ] a1 •
9 #g • « % p fes ♦ *| f
®1 ~ j (®2 + «g.)
- ~:Vf Ma p ®l * § [ %
*
b2
(8 .4 6 )
- /S Sm p i S ♦ ! « , * • .
•g-CCg - ©g)
p • * ♦ » [ % ♦ ( i^ * £MR)pJ
' (8 *4 7 )
i
®*
#1 ♦ • $ * *» ♦
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(8 .4 8 )
♦ i \
• f x t :|l ♦ f
* % )p ]«
S j. 9
r
J| 11 j* S®
f
« » ,» • * * I
i, •£
_
+ | M S m* » 0
<e«si>
[®b * (H . * * b ) 0 • *
♦ K^) 4 » * ♦ ||> 1! 4 0- m o
** *
(8«83)
(e *§ s )
* "Jj. p) © * ©y
3ie«© © R a tio n s re p re s e n t th e complete p erf© raaa ee* both tr a n s ie n t
and stead y»stat® , o f th e tadwefioaa a a A ia # *
I t i s neoeesary, h w e v e r,
in order to o b ta in a complete s o lu tio n o f th ese e l a t i o n * to speolfy
e ith e r applied v o lta g e s , in c lu d in g th e ro to r to rq u e * sad s u ita b le i n i t i a l
c o n d itio n s , o r a p p lie d e w w a fc i, in o la d in g the r o to r v e lo c ity , and
s u ita b le i n i t i a l c o n d itio n s *
Tim equations m ight he given a s o rt © f p h y s ic a l in te r p r e ta tio n *
fh e s ta to r ® « rr« ite have been p ro je c te d on e rfe o g e a a l exes, taro o f
iib lo h a re alo n g said p e ti^ n d le u la r to * r e s p e c tiv e ly , th e H ».g w tie a x is
o f ptoe## 1, end th e H i ir e o f w M oli m ight be o o a s M e re t p erp en d icu lar to
H ie plane o f ‘fee o th er tiro .
JL separate and jtadependesib s o il w ith v a rio u s
s u tw a l—ljid tto ttv ® coupling® m ight be asso ciated w ife each o f these «x«®»
To each. o f th e # # © o ils a d iffe r e n t vo ltag e would be a p p lie d *
R o tatin g
in s id e o f th is s o t o f th re e © o ils , an oth er s e t o f th re e © o ils eouM he
p ic tu re d *
fheee ro ta tin g © © lie sho«M have w a ta a l-ia d u o tiv e couplings
between ferns® tve® ■and between. th e s ta tio n a ry s o ils # a ls o th ey should be
p ic tu re d as connected to- ocHssntater bars w ife two e x te rn a l brush-con*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■
—1 4 4 —
n e c tlo a s , arranged s ta tio n a ry ia ©pace, along. and p e rp en d icu lar to *
re s p e c tiv e ly , the m sgastie a x is of % ♦ s ta to r phage 1 *
th is p ic tu re
ia so®® ways resembles th a t o f a d~c m achine*
Ita ta llg r th e th re e vo ltag es a p p lie d to an in d u c tio n machine a re
balanced, 'th a t i a * t h e ir Instantaneous s ® is equal to s e ro *
A le e ,
u s u a lly o n ly th re e w ire s le a d to th® msmhltmt th e re fo re some s im p lifie a ~
tle n a ta th e equations ©an be » ! # ,
The sum o f th e a p p lie d v o ltag es
being zero makes i t p o ssib le t o e lim in a te from, th e equations one o f th e
.th ree ap p lie d , voltag es j o n ly th re e w ire s e x is tin g makes i t necessary
th a t the sum o f th e th re e lin e cu rren ts b# ze ro *
I f th e a w o f th e
lin e cu rren ts is equal to a e ro , m o th e r equation, is e lim in a te d * nam ely,
eq uation (8 *4 ® )*
There Terrain o n ly fiv e sim ultaneous equations to s o lv e .
These restaiain g fiv e e q u a timm a re o b tained by p la c in g s5 equal to zero
in equations (8 .4 6 ) to (8 .5 8 ) .and lay dropping eq uation (8 * 4 8 )*
These
statem ents mm based upon th e -m zM m m M m in v o lv e d *
fine group o f tra n s ie n t eexM&tiems which m ight be tre a te d by th is
method co n sists' o f th a t f o r which th e r o to r -speed is co n stan t.
In th is
-case © -mm be re p la c e d by a constant* th en th e s e t o f equations ©an be
solved by H eavisid e o r Lapl&eis® -tra n s fo rm methods* J u st as -any s ta ­
tio n a ry netw ork*
fe e same method which has bee® used here stay a ls o he a p p lie d to
two-phase sad single-pba® # in d u c tio n m otors.
Of course th e number o f
sim ultaneous equations in v o lv e d w i l l be le s s th a n those re q u ire d f o r th e
th ree-p hase ease*
Hi® r e s u lts obtained by th e method given a re s im ila r in many resp ects
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**141***
te .th o * #
by S ta n le y (2 1 ) .
S ta n le y s s # i *. n m -ia m r ia a t type o f
t * e a * fe r m tla & i th a t is * S tan ley trm s fo ria s d a l l q u a n titie s by means
o f th e s a w 4 * 1 o f
w ith o u t using Mm con­
ju g a te s e t, so t h a t i a tils a n a ly s is the- expression f a r pemmt M 4 n o t
m ain tain th© fs im r la a t farm
9g I* * **t i1*
'» * meMxo&a. wem- n o t those o f te n s o r a n a ly s is b a t « f an. # » i la s s
r e s tr ic te d ©lass- from m a trix tt»© ry*:
As m r e s u lt b is performance
equations w ere .t w w t e t s im p le r ia f orm
th o se g l r m
h e re «
I t is in te r e s tin g to mote th a t th e performance equations given
w ig h t be used w ith s s a s ts a t r o to r speed and. a ls o s it h th re e-p h as e sinu­
s o id a l v o lta g e s ia o rd er to- #i» tai» the s te a d y -s ta te psrfontw ase*
la
t ills ease* vario u s re a e ta a *# * -eeald t # d e fin e d ' and a s e q u iv a le n t e lr e a it
ooaM be- d e v is e d .
S p e c ia l c o n s id e ra tio n m ight be g ivaa to th e ease o f p u ls a tin g v o lta g e
am i to rq ue v a ria tio n s t o ebbala, so w very ta te r e a tln g eq u atio n s, b at
th is s p e c ia l c o n s id e ra tio n w i l l n o t be taken up h e re *
fe e method given ta r # la n o t
s te a d y -s ta te
of aaohlaea*
t© th e d e term in atio n
o f th #
fe e « i » l method* o f a n a ly s in g
machines a re baaed on acme s o r t o f e q u iv a le n t e le c t r ic a l e l r c u it which
happens to rep ^ easat th e
fo r &mm a p e e ifio s te a d y -s ta te condi­
tio n s , l e t th ey « r« a©% s u ite d fo r -the determ ination, o f tra n s ie n t eo n d itd e o .**
Method* fo r d # ta r« i» ijig tr a n s ie n t porforaHmo* o f machines seem
to be o f JmerewMtiag in te r e s t to th e en g in eerin g p ro fess io n as design
li» lte ti.o o .s e re .o ften determ&aedl by tra n s ie n t eead itiem *-*
I t is d e s ira b le
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th e re fo r© t h a t an
hmm aaomg h is Jteihsds o f a tta c k mj« » en g in eer*
ta g p ro h la w sow® method fo r th® d e term in atio n o f tr a n s ie n t co n d itio n s
of- © le e tr te a l m achinery*
fh©
giwwt h«r« ha© am add! tio n & l a d -
-earntag© ia th a t i t is based tm a m iif i c a tio n o f e le c t r ic a l and m echanical
system s*
Besides* th e wrttoot ©an be a p p lie d w ith l i t t l e
change to o th e r
p re b le a * such ae © l© c tro *a c o u s tic a l {sr©bl®«s*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ix,
mmmm m
:■fit® f i r s t ft * ® p a rts o f M
cgkcissxcbs
i paper ar® d e ra ted to tii® fundam entals
® f te n s e r a n a ly s is * heU i th e u su al and th® in t r in s ic , and th e a p p lic a ­
tio n o f thee® fu a d im m ta ls 'to d t f f # r « t l a l geom etry*
Many o f th® con­
cepts o f te n s e r W m & fy m m introduced In th e tre ® tw e « t o f d if f e r e n t ia l
■gmmmmtiry-* ■ ^kSm
®»e fs&Iawsd in. o rd e r t o ta k e adm ntag© o f th e
geos® t r le a l method © f wmmm&mgrm .A fter m h r io f In tro d u c tio n , th e
fundam ental d e fin itio n s and alg eb ra!® ss m ifm latien a o f ten so rs In them­
se lve s w ith o u t mSmm&em to an y g ® « s trl@ « i space a r« ©onsldared*
p »e*rt#*t#»«w r* a re in tro du ced and re la te d to %ra» te n s o rs *
A lso
ffeaa* under
t l m headlag © f d if f e r e n t ia l .g i»s® try * » s ti« rt tre a tm e n t o f »© n-*@ tnieal
geometry is la » -l« A « i*
fh # %ra® ehasg® in a tenser, component te e to a
m m ll Akmmg® in asard tn at*® or la tr ln a i® v a ria b le s is d e fin e d *
th is
d e fin itio n leads- t» th e d e fin itio n « f th® ©cwwri&at and the in t r in s ic
t e r i r s t i ’f®® w ith re s p e c t to & fy ta a e trl® *!' a ffin e e ©raneet i on w ith an
undefined w e trl« # t&® eh eiee o f th e tasferl© feeing l e f t open*
aetri,® "is- iatn o d u ©e& fey
g jj.
apaoe.
L a te r , a
q u ad ratic fo r a ds^ **
w hich eonwwrte th e 'M tfeerb® Mserfiien# spas® in to a m e tric a l
A fte r th® l a t f w t e e t i * o f th e m e tr ic , many o f th # p re v io u s ly
l®tr«xtn®a# <xme®pts « m m
s p e c ia l form es fo r ®i®g»fl«, th e spsw etri© '
a f fin e eseoMNertdeah. Imhmnm* a
ff e r is to ffe l symbol*
More
© m s it« ra M « n i s g lv m to th e s p e c ia l Riemannian ssetrl® geom etry*' t& a t
I s , th e & » trl® gesaaetiy w ith gang® ia « a r i« # « and m.. m$mm%g l# © oim eetlan.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
th an to th e m m & M m m m kiam wetrt© geometry, that i s , th e m e tric ge©«etry
with an im tran s p o rta b le metric, except la an infinite® i m l region, aa&
a general eessaeetieKi* 1?fe« lawrtast a»tii«i© of the intxlaeie tenser
a n a ly s is In general
a ga»#rml #«R«eti«s th e re fo re « non*
lie m a m i& a netrie g e o m e try m y he d e fin e d fey the ceasept® o f the
intrinsic teaser a n a ly s is , in eoasoetiem with the f-undamental q u a d ra tic
fOi'Aii*
lm p a r t T t o f th is p ap er, some o f th e oo »#eft« o f th e usual. w eetor
a n a ly s is a re r e la te d t e tw sser a n a ly s is fey making use o f th e pseudo*
re c to r re p re s e n ta tio n o f second o rd er e a fe i^ y a B w trie te n s o rs *
tu ck eat**
ai».pis as th e p«££*#fc# Hi© it r e r f p t c e * th e © « r l, th e la p la e la n , and
Stole©*® theorem w ere
eo m id er& txc n .
I s p a rt V I I , some o f th e Im p o rtan t © a a ly tle a l concepts o f te n s o r
dyxMuales a re .g itw a*
The metiss©i o f procedure Is based os th© I n i t i a l
p o s tu la te th a t H ie afeselttt# o r ia tr is s l® d e rlm iim s o f th e g e n eralised
wmimmximm is equal to th e
fo rce® *
% exp«PBiisg th e in tr in s ic
d e ri-ra tiT © and % in tro d u c in g th e ©e»«©-p% o f k in e tic energy,, la.grange’ s
equations a re transform ed f r o * mm- coordi®at© syste® to a n o th e r, and from
a co o rd in ate system to m « n » f l« © f ©®agr«#»«©s*
Both holososiie and.
non**© lonoaio cornstra in e d d ja w iic a l system# a re discussed*
P a rt f i l l o f t h is paper i« coacornod w ith ©leetro -m eo h an ical system s,
c h ie fly th© r o ta tin g e le c tr ic a l s a e h ia # *
th e more im p o rta n t analogous
connected w ith o le e t r lo a l end s e e h « jie a i aywtwas a re ta b u la te d *
Mm dynam ical ® q p a ti« i® o f m otion expressed 3m tarn ® o f co o rd in ate -variable®
s a t in t r in s ic m ri& b le e a re assumed t o re p re s e n t th® perfO M aae# o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e le e tro -ffis e h a n ie a l syabeiss*
th e method o f a n a ly s is re q u ire s th e teowledge
o f c e rta in e le e t r io a l -asd Bwotwuaical d esig s e o iis tw jts {ind uctances*
in e r t ia l e e e ffic te n ts # r w i s l t o w , . a t® *} o f the system ra d e r c o n s id e ra tio n *
A general method o f « a 4 y « ia g r o ta tin g # l® ® trf© a t ra e h ia *# * ia*»
© lading th o typ® o f v a ria tio n o f tts® e le c tr ic a l design co n stan ts* to
gim m *
ffeoa th # g en eral ratfood I® afg& lsd to th e In d u s tira machine*
A# tli® perf©rasas® equations in ts ra s of. lio a s n & la s co ordin ates c cmt&in
fu n c tio n a l # o s ffle l-e s & *# a jro rto w o ly in tro d u ced typ e o f tra n s fo rm tio n
which r e fe r s
is made*
system to * sp»®ia.J s e t © f n ® - lie m m iim coordinates
th is tr r a s fo r m tio n fs s a lts - S» an e q u iv a le n t s e t o f perform#®®®
equations w hich c o n ta in «*tigr e y e tlo v a ria b le s .
th ese e q u iv a le n t per­
formance equations east be solved q u it® e a s ily f o r most oases.
la th e
ease o f constant r o to r speed 'the equations assume «a © s p e c ia lly s in g le
form * in v o lv in g awwtowk ee sffleleztfes* in -term * o f th e e le c tr ic c u rre n ts
and t h e ir ra te s o f ebmagSf th e re fo r# e ith e r th e Weavis-Ad® o p e ra tio n a l
methods o r th e X a p la w ir a -tr s ia ^ r a method* o f s o lv ia g lin e a r d if f e r e n t ia l
eq u atio n s w ith sansbsat e e e ffio ie a t# can be a p p lie d *
Skperiens® has demonstrated th a t ia p ra e tte a l a p p lic a tio n s th e
ty p e © f t r a n a f e t m t l« ® is mere f le x ib le end le s s re ­
s tr ic te d th e n are th e in v a r ia n t tr& i aform ations i a w hleh c e rta in lin e a r
o r m u ltilin e a r a lg e b ra l© form s ar@ m aintained in v a ria n t as is th e case
o f te n so r m a ly s is *
l a th e s p p lis a tls a of te a s e r methods to th # in d u e -
t io a r a t e r th e fm dam ental q u a d ra tic f o r * ds2 » .g y fix * #x^ and th e
lin e a r fo r a mg. i * a re to o eioraples o f In v a r ia n t fe r a e *
% comparing
th e r e s u lts o f th e author*® in v a ria n t tra n s fo rm a tio n method o f s o lv in g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t&« im&miAm. «rfe«r w ith th e
©f
(2 1 ) s a s - t e w i t u t
actthod* i t i s c le a r ifa a t S ta n le y **
y ie ld e d
a nor© ©isfri® gnrap mi pm rim w m m m
Itafeer* work i a t h i * f i e l d a tg tit iwsXwl® m s ? w
a p p lic a tio n s o f
th® wotfooi© 0 m m her® o r a a « E f» ;a s i« « f tfe#©® method® to in c lu d e tfe®
#i© ©ry o f groop®# tngN&oggr advanced m t r t x and determ inan t tJuwwy* and
f a t h e r g e n e ra lis e d 'gM$K4£#£*s»
<&* ihavelvtag wavea mr» in v o lv e d I n t h *
p h y s ic a l fl&tar** o f -Hi® mWimmMmm of •r o ta * lag e le c t r ic a l naehiaea*
and a « S ekro d iag er’ s wave
aoofaa»ieii fo llow ® fro ® advanced dynam ical
setboitts,, I t aaems th a t jw fe a p i th® sjettiods o f S ehrodfager* « wav© mechanics
n ig h t be a p p lie d to r o ta tin g # l® « tr io « l' m*ektm#xy+
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
X .
IG fS M T H S C IT E D
A,
B
A a d r*. Method* o f e n l:« a 3 *ti« a of th e a rm tra © re a c tio n of
a lte r n a to r s , Papers of* th e In te r n a tio n a l E le c tr ic a l Congress*
S t* L o u is . 1*6SS• 1004*
2.
B r i l l onto.* I m b » Les tte *« i* rs ©n W eantqu© © t am e la s t ic it e *
Sasson e t C i© . B a ris * I f S i*
.3*
C & rtaa, 1 * Sur le v v a r ie tie s a connexion a f f in e ® t la th e o rie de
la r a l a t i v i W general is® ©, A n n ., Seal© lo r e * §# e e r ie , 4 0 *3 2 5 4 1 2 , m s * 4 1 *1 -2 6 , 1924# 4 2 *1 7 -8 6 , IM S *
4*
f ih r is t e f f e l, 1 * 1 * ttber d ie frm s fo re & tia a d er horaogenexi d if f e r e n t ia l
- auadraok© sw eiten Grades. G ora. fra* d ie rein © und angesr. M ath,
(C re ll® ) 7 0 *4 6 -7 0 . Iff®.
§.
a ld in g to n . A , S * th e m athem atical th e o ry o f r e l a t i v i t y *
U a tv e rs i ty B ra s s , C te trid g s * 1923.
6#
lin s t e in . A, the fo u n d atio n -of th® gen eral th e o ry o f r e l a t i v i t y .
I a L e rs n ts , H * A * ih e p rtn < A fl*- o f r e l a t i v i t y , p p . 109-164.
lo th a e a , London* 1923*
f,
E in s te in , A , The meaning o f r e l a t i v i t y .
P rin c e to n . 1923.
6,
Gran s t e in , W. C . The geoswtry o f llemeramisua spaces.
» M fe . I n « 3« *# 42-§® §, 1934.
9*.
Ir o n , G a b rie l.
dynamics o f r o ta tin g e le c t r ic a l
m achinery. Gora . o f M ath . te d P h y s ic *. 1 3 *1 0 3 -1 9 4 . 1934
Cambridge
P rin ce to n U n iv e rs ity B rass.
T ran s. Ajtseriean
10* Ir o n , G a b rie l* The a p p lic a tio n o f te n s o rs t o 't h e a n a ly s is o f r o ta tin g
m achinery. General E le c tric Seview , Schenectady. I f IS ,
1 1 . la p a & i© , G . t>. M eW iiqu© a n a ly tiq u ® .
. G « u th le r-7 i 1la r s , P a r is • IBS8 .
In h is Oeuvres *
I t . L e v in e , S . 3 * An a n a ly s is ©f the in d u c tio n m o to r,
6 4 *6 2 6 -5 2 9 . 193 S .
V o l. 1 1 - t t
fe te s . A . I . l . B .
1 3 . Mndff&y* S# » .*, M argm au, B . Foundations o f p h y s ic s , pp. 6 9 -7 2 .
Goto W ile y , M r T e rk * 1®S6.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M * t4 p s fe it« # t »
#©r Theori® de-r M M im & M t& A m i#- Smmt*.
f u r 41® rete ® w A a&gaw* M ath . (& r» 1 2 *)* f i f l - i i * . 18 74.
ISn* l£u3rn.&gha^a# F * 9 . 'Teeter a t» ly s is * 4 fcb® th e o ry o f r e l a t i v i t y ,
M m Bqptdtev f r M * * B altim o re* 1922*
1 6 . P a rk ,
H * % ®-rs»«% i.«a ..theory o f ®ym®kremm* ssaA iaes*
A.*1*B*B* 4 $ *rtft*4 *0 » U i » . .
fra a s *
I f * fre a M e ® , B* 1# F a B ia iw a ta l eoaeepfcs o f syaohroaous machine reac tan ces.
Trass * 4 *1 *1 * 1 * ¥®1* 8 7 , s a p p le a w t* If S f *
I t * M e e t , ® * , *»& I® v i- C iv it a , f * tlttli« 4 ii. d« e a ie a l d if f e r e n t ia l absoln
e t le w s a p p lic a tio n s . H ri& e m tlee lse A z a le a * ©4 s1 2 5 -2 0 1 , 6 0 8 ,
1901*
• •
I f * llessaraa, 1 * tUser d ie % f© «i® #«s* weleli® d e r @ *e a e trl* m* 9s*imde
I f eft*®,* S esiaw slt© H ath * Work®* 2nd e d itle a * $ $ * 272-288*
9 *a $ N tt* S elfsnig* IM S *
20*. JkdMMgtvgt*
A*, M * tftrsiefe® A aelysi® t e «ewre« le la t iv it lt s if n e o r i® ,
le a * A k *4 * As«t®r€®B. T e l* 12., !© * 8 * 1918*
2 1 * fts s a le y , 1 * f * J a s » a ly » i« « f tb e la d a e tle a te e te r*
V o l. 67* ireppleeMMst* 19S t*
3 m s * A *1 *1 *® *
1 2 * Tefelen* Oswald* In v a ria n ts o f q u a d ra tic d if f e r e n t ia l f « *
Ca ^bridge
T ra c t i a M ath* sad H a th * P hysios. Ho* 24 * Cambridge B h iv s rs ity
P ress, C m b rid p i* 1927*
£$« Traneeam t, II * - 9 * Les ©spaces non tolonomes e t le w a p p lic a tio n s
m » i | u s s * M em orial des
He* 76*
S a « tb i® r -? lllw « * .B ® r is , IS Ii* .
2 4 * H e y l* H*
18-11*
le i© * la fS s tlte s l al&eemflritacl**
.
H a th * I® i t a * 2 *894-411 •
§ S * If c y l, H * S ra y lta tie n sad e le c t r ic it y . l a Loren t t , H* A*
o f r e la t iv it y * p p . 199-2 16* Methtum,
1925*
th e p rin c ip le
2®*. W d lttafcer* 1 * f * A tr e a tis e m th e
AysKstiee o f p a rtic le s
*a € r ig id b o dies* p . 4 3 * Cas&rid g« 9 a iv e rs i ty P re s s , Caiabria g®.
19 27 *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S *
Ik e
a ijw e r®
•work was
ASSW i-liSDSaSfS
& A i w I « l p f h i*
to B r* F * ¥ .*
.&M express®3 h i *
wmAmr whose tfy » # tie » t h is
t® fr#f® se« y M* % -C®««y fo r h is h e lp fa l
oounaelj t*> Bar* * » $.» 'ill* * * -* f o r h i*, fiw a k c r itic is m ; t # tit©*® s t a f f
mmmhmtm o f Ut® ® l« *$ *le ftl e » ii» # r ia g # fhar®!®*, and mathematics
t®gmrfes*KEi.* who hare .g ir « a i i t * th e a u th o rf «a& to thos® wm.t I f rin g
*» & tftftt# * b * h **® € # w l® i* t h§a*© r © a& lysis ft& t r e la te d subjects a n t
u p ® o ® $ i © * * u s ii f t a i t h e
a itfe ® * ' h * *
s o f f 'W f c l j r d r a w n
*
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