Tensor Calculus. Tommaso Astarita
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1 Tnsor Cuus Toso strt Mn rfrns rs, Vtors, tnsors, nd th s qutons of fud hns, 989. orsno nd Trpov, Vtor nd Tnsor nyss wth pptons, 979. Fsh, Studnt Gud to Vtors nd Tnsors, 0. Tnsor uus T strt
2 Introduton - Pn Hstory. gr of Crtsn vtors nd tnsors. Cuus of Crtsn vtors nd tnsors. Gnr trtnt of tnsor gr nd uus. Conusons. Tnsor uus T strt Hstory ( Th onpts of tr tnsor nyss ros fro th wor of Cr Frdrh Guss n dffrnt gotry, nd th foruton ws uh nfund y th thory of gr fors nd nvrnts dvopd durng th dd of th nntnth ntury. Th word "tnsor" tsf ws ntrodud n 846 y W Rown Hton to dsr sothng dffrnt fro wht s now nt y tnsor. Th ontporry usg ws ntrodud y Wodr Vogt n 898. Tnsor uus ws dvopd round 890 y Grgoro R- Curstro undr th tt sout dffrnt uus, nd orgny prsntd y R n 89. It ws d ss to ny thtns y th puton of R nd Tuo Lv-Cvts 900 ss txt Méthods d u dfférnt sou t urs pptons (Mthods of sout dffrnt uus nd thr pptons). Tnsor uus T strt 4
3 Grgoro R-Curstro Grgoro R-Curstro ws n Itn thtn orn n Lugo d Rogn. H s ost fous s th nvntor of tnsor uus, ut so pushd portnt wors n othr fds. orn: Jnury, 85, Lugo, Ity Dd: ugust 6, 95, oogn, Ity Mthtsh nnn 900, Vou 54, Issu -, pp 5-0 Méthods d u dfférnt sou t urs pptons M. M. G. R, T. Lv-Cvt Tnsor uus T strt 5 Tuo Lv-Cvt Tuo Lv-Cvt, ws n Itn thtn, ost fous for hs wor on sout dffrnt uus nd ts pptons to th thory of rtvty, ut who so d sgnfnt ontrutons n othr rs. orn: Mrh 9, 87, Pdu, Ity Dd: Dr 9, 94, Ro, Ity Mthtsh nnn 900, Vou 54, Issu -, pp 5-0 Méthods d u dfférnt sou t urs pptons M. M. G. R, T. Lv-Cvt Tnsor uus T strt 6
4 Hstory ( In th 0th ntury, th sut to nown s tnsor nyss, nd hvd rodr ptn wth th ntroduton of Enstns thory of gnr rtvty, round 95. Gnr rtvty s forutd opty n th ngug of tnsors. Enstn hd rnd out th, wth grt dffuty, fro th gotr Mr Grossnn. Lv-Cvt thn nttd orrspondn wth Enstn to orrt sts Enstn hd d n hs us of tnsor nyss. Th orrspondn std 95 7, nd ws hrtrzd y utu rspt: I dr th gn of your thod of oputton; t ust n to rd through ths fds upon th hors of tru thts wh th of us hv to our wy orousy on foot. rt Enstn, Th Itn Mthtns of Rtvty [8] Tnsor uus T strt 7 Hstory ( Tnsors wr so found to usfu n othr fds suh s ontnuu hns. So w-nown xps of tnsors n dffrnt gotry r qudrt fors suh s tr tnsors, nd th Rnn urvtur tnsor. Th xtror gr of Hrnn Grssnn, fro th dd of th nntnth ntury, s tsf tnsor thory, nd hghy gotr, ut t ws so t for t ws sn, wth th thory of dffrnt fors, s ntury unfd wth tnsor uus. Tnsor uus T strt 8
5 Introduton - Srs Mny phys quntts r orrty dsrd y ony nur (postv, ngtv or zro) nd r d srs. y fxng syst of unt th gntud of sr s ndpndnt of th rfrn syst ut n hng y hngng th pont n sp. No sns of drton s ssotd to srs. Srs n oprd ony f thy hv th s phys dnsons. Two srs surd n th s syst of unts, r qu f thy hv th s gntud nd sgn. s n xp, th Tprtur T s sr nd, n fxd pont n sp, ts vu dos not hng. Cry y hngng th syst of unts ts vu hngs.g. 0 C= F. Tnsor uus T strt 9 Introduton - Vtors Othr phys quntts, tht r orrty dsrd, n h pont n sp, y gntud nd drton, r d vtors. y fxng syst of unt th gntud nd drton of vtor r ndpndnt of th rfrn syst ut n hng y hngng th pont n sp. Vtors n oprd ony f thy hv th s phys dnsons. Two vtors surd n th s syst of unts, r qu f thy hv th s gntud nd drton. Th oponnts of vtor (to ttr dfnd n th foowng) dpnds on oth th fr of rfrn nd syst of unts. s n xp, th for f s vtor nd, n fxd pont n sp, ts gntud nd drton dos not hng. Cry y hngng th syst of unts th gntud dos hng. In or strt wy vtors r th nts of vtor sp. Tnsor uus T strt 0
6 Introduton - Vtors vtor sp s st of nts d vtors stsfyng th foowng xos. To vry pr, x nd y of vtors n thr orrsponds vtor x + y, d th su of x nd y, suh tht: ) x + y =y + x (ddton s outtv); ) (x + y) + z = x + (y + z) (ddton s ssotv); ) thr xsts n unqu vtor zro 0, suh tht 0 + x = x, x ; 4) x n thr orrsponds unqu vtor x suh tht x + ( x) = 0. pr nd x, whr s sr r nur nd x s vtor n, thr orrsponds vtor, d th produt of nd x, suh tht: ) ( ) = ( ) x (utpton y srs s ssotv), ) x = x, ) (x + y) = + (utpton y srs s dstrutv wth rspt to vtor ddton), 4) ( + ) x = + (utpton y srs s dstrutv wth rspt to sr ddton),,, x, y. Tnsor uus T strt Introduton - Tnsors Th su nd produts (t st thr of th r xtry rvnt) of two vtors n dfnd ut (rs): "though th quotnt of two vtors nnot dfnd stsftory, tnsors rs physy n stutons tht th oo rthr ths. For xp, strss s for pr unt r. W hv sn tht for s vtor nd so s n nt of r f w rr tht w hv to spfy oth ts sz nd orntton, tht s th drton of s nor. If f dnots th vtor of for nd th vtor of gntud qu to th r n th drton of ts nor, th strss T ght thought s f/. Howvr us dvson y vtor s undfnd, t dos not rs qut n ths wy. Rthr w fnd tht th strss syst s suh tht gvn w n fnd f y utpyng y nw ntty T whh s f/ ony n th sns tht f= T." Th strss syst s tnsor nd t pprs tht (t st) two drton r ssotd to t. Tnsor uus T strt
7 Crtsn vtors nd tnsors Rné Dsrts: Rné Dsrts ws Frnh phosophr, thtn nd wrtr who spnt ost of hs f n th Duth Rpu. orn: Mrh, 596, Dsrts, Indr-t-Lor, Frn Dd: Frury, 650, Stoho, Swdn Tnsor uus T strt 4
8 Crtsn oordnts In n Eudn D th poston of pont n spfd y th thr Crtsn oordnts. fr of rfrn hs to fxd, s shown n th fgur, w t gnr pont O s th orgn nd drw thr utuy prpndur strght ns O, O nd O (wth postv sns shown n fgur). Tnsor uus T strt 5 Crtsn oordnts Th fr of rfrn s nory tn rght hndd. Tnsor uus T strt 6
9 Crtsn oordnts In n Eudn D th poston of pont n spfd y th thr Crtsn oordnts. fr of rfrn hs to fxd, s shown n th fgur, w t gnr pont O s th orgn nd drw thr utuy prpndur strght ns O, O nd O (wth postv sns shown n fgur). Th oordnts of P r th ngths of th protons of OP on th thr xs O, O nd O. Ths ngths r ndtd wth x, x nd x. Tnsor uus T strt 7 Crtsn oordnts If th rfrn syst s rgdy rottd th oordnts of P hng. Th rotton n spfd y gvng th drton osns twn O nd. Th nw oordnts r rtd to th od y: x x x x Convrsy: x x x x y ntrodung th Enstn notton ( rptd or duy suffx py su ovr th thr vus, nd ; th othr suffx, d fr, n t ny vu): x x x x Tnsor uus T strt 8
10 rt Enstn Thort Physst rt Enstn ws Grn-orn thort physst nd phosophr of sn. H dvopd th gnr thory of rtvty, on of th two prs of odrn physs. orn: Mrh 4, 879, U, Grny Dd: pr 8, 955, Prnton, Nw Jrsy, Untd Stts Tnsor uus T strt 9 Vtors Th poston vtor s n xp of vtor nd ts oponnts r th oordnts of P. W n thrfor th foowng dfnton: Crtsn vtor,, n thr dnsons s quntty wth thr oponnts, nd, n th fr of rfrn O, whh, undr rotton of th oordnt fr to o: W w dntty vtors y n undrnd syo, ut oftn od ttr s usd. Tnsor uus T strt 0
11 Vtors If th poston vtor s funton of t: x x t x x t Whr: x x Sn th drton osn r ndpndnt of t w hv: n n d x d x n n dt dt Thrfor th drvtvs, n prtur th frst nd sond on.. voty nd rton, of th poston vtor r vtors. Tnsor uus T strt Vtors Th poston vtor s n xp of vtor nd ts oponnts r th oordnts of P. W n thrfor ntrodu ts ngth or gntud: If = s unt vtor nd ts oponnts y thought s th drton osn. Thus Is unt vtor nd rprsnt th drton of. Cry ony two oponnts r ndpndnt. Tnsor uus T strt
12 Sr utpton If s sr th produt of ths sr nd th vtor s vtor wth oponnts. Cry th drton of s th s s tht of nd: Tnsor uus T strt ddton of vtors If nd r two vtors wth oponnts nd thr su s th vtor wth oponnts +. gn w hv: Thrfor th su of two vtors s vtor. W hv: Sutrton y dfnd y onng wth sr utpton: Tnsor uus T strt 4
13 Copnr vtors ny vtor whh s n th s pn s nd (wth dffrnt drton.. ) n rprsntd s: In oponnt for: y oong for souton of ths syst of quton ( nd r th unnown) w s tht th foowng ondton shoud hod: 0 Tnsor uus T strt 5 Kronr Dt Th Kronr Dt s dfnd s:, 0, Whn pprs n foru wth rptd suffx t rp th duy suffx wth th othr suffx of th Kronr Dt: Tnsor uus T strt 6
14 Lopod Kronr Lopod Kronr ws Grn thtn who word on nur thory nd gr. orn: Dr 7, 8, Lgn, Pond Dd: Dr 9, 89, rn, Grny Tnsor uus T strt 7 Unt vtors Th thr unt vtors tht hv ony on non-vnshng oponnt r th ntur ss of th fr of rfrn O nd r utuy orthogon:, 0, 0 Or: 0,, 0 0, 0, Cry w hv: Tnsor uus T strt 8
15 Unt vtors y onsdrng rotton gvn y n orthogon trx th s vtors r trnsford: Thrfor: y oprng th sond nd thrd nt w hv: Tht s th w of trnsforton of vtor oponnts. Tnsor uus T strt 9 ss of non-opnr vtors Th ntur ss s not th ony ss. y onsdrng thr nonopnr vtors, nd (gn thy shoud not hv th s drton) ny vtor d n xprssd s: d Whr th onstnts n dtrnd y sovng th foowng nr syst of qutons: d d d Sn th vtors r non-opnr th dtrnnt dos not vnsh. Tnsor uus T strt 0
16 ss of non-opnr vtors If M s non-sngur trx th vtors: M M M r non-opnr for: M Dos not vnsh f M 0 ry th "rrd" vtors for nw ss. ut suh gnr trnsforton ts outsd of th Crtsn vtors whh r onrnd wth ss of utuy orthogon vtors. Tnsor uus T strt Sr produt: Th sr (or dot) produt of two vtors s dfnd s (whr s th ng twn th two vtors nd ): os nd rd " dot ". y rng tht th ntur ss unt vtors r utuy orthogon w hv: Thus: Th sr produt s nvrnt undr rotton of xs s n drty vrfd: Whr sn s n orthogon trx Tnsor uus T strt
17 Sr produt: If nd r th unt vtors n th drton of nd : os os nd os Is th proton of th vtor on th drton of th vtor. If th vtors r orthogon thn os = 0 nd th sr produt: 0 Th sr produt s ry outtv ut so dstrutv wth rspt to ddton: Tnsor uus T strt Vtor produt Th vtor (or ross) produt (rd " ross ") of two vtors s dfnd s th vtor nor to th pn of nd of gntud sn drtd n wy tht, nd ( ) for rght-hndd syst. Cry th vtor produt s not outtv: Th gntud of ( ) s th r of th progr two of whos sds r th vtors nd. Tnsor uus T strt 4
18 Vtor produt Th vtor produt of th ss unt vtors r unt vtors nd: 0 Thrfor: Or Tnsor uus T strt 5 Lv-Cvt syo In thts, prtury n nr gr, tnsor nyss, nd dffrnt gotry, th Lv-Cvt syo rprsnts oton of nurs; dfnd fro th sgn of prutton of th. orn: Mrh 9, 87, Pdu, Ity Dd: Dr 9, 94, Ro, Ity Tnsor uus T strt 6
19 Vtor produt vry vu notton n ntrodud wth Lv-Cvt (or prutton) syo. 0,, -, f f f ny two of,, r th s s n vn prutton of, nd s n odd prutton of, nd Thrfor: It s sy to s tht th Lv Cvt syo n usd to ut th dtrnnt of trx: Tnsor uus T strt 7 Trp sr produt Th trp sr produt s dfnd s: Th vnshng of th trp produt s th ondton for o-pnrty of th thr vtors. Th trp sr produt n physy ntrprtd s th vou of th prppd wth sds, nd. Tnsor uus T strt 8
20 Trp vtor produt Th produt ( ) s d trp vtor produt nd, sn s vtor nor to th pn of oth nd nd ( ) s vtor nor to ( ), ust n th pn ford y nd. It n vutd strtng fro th dntty: For = th frst r s dffrnt fro zro ony f, nd,,,. Thrfor: Hn: prutton Tnsor uus T strt 9 Vtor dntts p p d d d d d d 0 d nus Jos dntty Pythgorn thor Tnsor uus T strt 40
21 Voty du to rgd ody rotton Suppos tht rgd ody rotts out n xs through th orgn wth drton gvn y. If s th ngur voty th rotton n dntfd y th vtor. Lt P pont n th ody t poston x. Thn drton of PR of gntud x sn. In short ntrv t th pont P ovs to R nd th vtor PR= x s: x x t In th t t 0, on fnds: t 0 x t v x x s vtor n th Whnvr th voty of pont n rprsntd s vtor produt of onstnt vtor wth th poston vtor thn th oton s du to pur rotton. Tnsor uus T strt 4 Sond ordr tnsors: vtor ws dfnd s: Crtsn vtor,, n thr dnsons s quntty wth thr oponnts, nd, n th fr of rfrn O, whh, undr rotton of th oordnt fr to o: Sry w dfn tnsor s n ntty hvng 9 oponnts n th fr of rfrn O, whh, undr rotton of th oordnt fr to o: pq p q W w dntfy tnsors wth dou undrnd syo, ut oftn od pt ttr s usd. Tnsor uus T strt 4
22 Sond ordr tnsors: Th oponnts of sond ordr tnsor n wrttn n trx for: Th prvous quton () n tnsor for os: Whr s th trnspos of (.. ). tnsor s d sytr f. Cry sytr tnsor hs ony 6 dstnt oponnts. tnsor s d ntsytr f. Cry n ntsytr tnsor hs ony dstnt oponnts. Tnsor uus T strt 4 Sond ordr tnsors: Cr shoud tn n dng wth non-sytr tnsors ry th prvous quton () y sdng nd shoud orrty ntrprtd. pq p q or roust syoogy (oftn d dyd notton) s otnd f w do not drop th unt vtors fro th qutons: p pq In ths s t s r tht p s th trnspos of. q p p q q Tnsor uus T strt 44
23 Exps of sond ordr tnsors Th Kronr dt s sytr tnsor: pq p q pq pq Whr th st quty s onsqun of th orthogonty of. Th oponnts of r th s n oordnt systs thrfor s n sotrop tnsor. If nd r two vtors thr tnsor produt s sond ordr tnsor: pq pq p q p q p q s nory d dyd. Othr portnt tnsors r th nrt, strss nd rt of strn tnsors. Tnsor uus T strt 45 Sr utpton nd ddton If s sr thn th utpton nd ddton of tnsors r dfnd s: y dfnng th sytr nd n ntsytr prt of : s It s vdnt tht ny tnsor n rprsntd s th su of sytr nd n ntsytr prt: s Tnsor uus T strt 46
24 Tnsor ontrton nd utpton Th oprton of sung th oponnts of tnsor ovr two of ts nds s d ontrton: For sond ordr tnsor th ontrton s sr nd s d th tr of th tnsor. Oftn th notton tr() s usd to ndt th tr of th tnsor. If nd r sond ordr tnsor thr tnsor produt s fourth ordr tnsor wth oponnts n th rrd oordnt syst: pq rs p q Tht s th nogu of pq p q nd, thrfor w usd n th dfnton of hghr ordr tnsors. Th ontrtons of fourth ordr tnsor r sond ordr tnsors,.g. Tnsor uus T strt r s p q r s 47 Tnsor sr produt Nory th sr produt s usd nstd of th ontrton: nd for th othr thr: Ps not tht.g.: us Tnsor uus T strt 48
25 Tnsor dou sr produt In gnr w hv: Th dou dot produt (or dou ontrton) s dfnd s th sr: : : ross produt owrs th su of th ordr of th tnsor oprnds y ftor on wh sng dot of two nd dou dot y ftor 4. In nothr wy: n rrow ( ) s on (undr)n nd h dot ( ) two ns. Tnsor uus T strt 49 Th vtor of n ntsytr tnsor: n ntsytr sond ordr tnsor oponnts so y ntrodung vtor 0 Sn: hs ony thr ndpndnt on n wrt: Whr us oth nd shoud dffrnt fro nd. sds oth vn nd odd pruttons ontrut to th su. Th vtor of n ntsytr sond ordr tnsor dfnd s: On hs: v 0 : p q qp 0 : p q qp s thrfor Tnsor uus T strt 50
26 Th vtor of n ntsytr tnsor: Ry: Whr th ondton of sytry hs n usd: Tnsor uus T strt 5 Cnon for of sytr tnsor For prtur vtor nd tnsor t y our tht thr sr produt hs th s drton of. In ths s w hv: Or, n oponnt notton: 0 Tht s hoognous syst of thr nr qutons for th thr unnown. souton of ths systs xsts ony f th dtrnnt of th offnt vnshs. Thrfor, th foowng hrtrst quton hods: I I I 0 Whr th quntts I r d th nvrnts of th tnsor. I tr ( I I ) Tnsor uus T strt Q 5
27 Cnon for of sytr tnsor W so hv: I tr tr tr Cry r th hrtrsts vus or gnvus nd th orrspondng th hrtrsts drtons or gnvtors. It n shown tht whn tr()=0: I Q s tr tr s s Tnsor uus T strt 5 Cnon for of sytr tnsor If th gnvus r dstnt thn t n provd tht th gnvtor r utuy orthogon. y norsng th nd hngng th syo to w hv: p p p p q pq Thus p s n orthogon trx. y hngng th rfrn syst ordngy w hv: p q p p q p pq Th tnsor n ths nw oordnt syst hs dgon for: Th hrtrst drtons r nown s th tnsor prnp xs. Tnsor uus T strt 54
28 Vtor nd tnsor dntts : : : Tnsor uus T strt 55 Hghr ordr tnsors In gnr w dfn tnsor of ordr n (.. n ts undrnd) s n ntty hvng n oponnts n provdd tht undr rotton to nw oordnt fr thy trnsfor ordngy to: pq...t p q... nt...n If th ntrhng of two nds dos not hng th oponnts th tnsor t s sd to sytr wth rspt to ths nds. sr dfnton for ntsytry hods. Th gr of hghr ordr tnsor rns prty unhngd: C C Tnsor uus T strt 56
29 Th quotnt ru Th quotnt ru s usd to prov tht nn quntts r th oponnts of tnsor. If nd r vtors nd thn r th oponnts of tnsor. Th usfunss of ths s tht = y rs n ny phys stuton n whh s nown tht oth nd r vtors thrfor y th quotnt ru w n sy tht th quton hods n ny rfrn syst. W shoud prov tht: pqq p pq p q y dfnng th frst rton s stsfd nd sn oth pq p q nd r vtors: q q qq p p pqq p p p p qq pq p q q 0 Tht sn nd r ndpndnt provs th thss. Th s proof y sy xtndd,.g C Tnsor uus T strt 57 Isotrop tnsors pq p q n sotrop tnsor s on whos oponnts rn unhngd y ny rotton of th fr of rfrn. Cry sr r sotrop. Thr r no sotrop vtors nd th ony sotrop sond ordr tnsor s th Kronr dt. 0 0 Frst w t prutton of th xs: W hv: ut for th sotropy w hv: nd sr rton for th othr oponnts. y onsdrng prutton nd rfton of xs: Th off xs oponnts shoud zro Thrfor: 0 0 Tnsor uus T strt 58
30 Isotrop tnsors n sotrop sond ordr tnsor s ry onntd to sphr. Ry sn x x s th quton of qudr surf tht s nvrnt of th xs rottons. Th ony sotrop thrd ordr tnsor s. S th rs oo for dts. Th produt of sotrop tnsors s n sotrop tnsor ut t s qut ntrt to fnd th gnr sotrop fourth ordr tnsor gn s th rs oo for dts. onvnnt rprsntton of th gnr for s: T pq pq p q q p Th sond tr s sytr wth rspt to th frst nd sond or thrd nd fourth nds nd th thrd tr s ntsytr wth rspt to th. p q q p Tnsor uus T strt 59 x vtors Th ony trnsforton tht w hv onsdrd s rotton of rght hndd Crtsn syst. sght xtnson woud ow so rftons.. rght hndd oordnt fr s trnsford n ft hndd on. Th trx s st orthogon ut wth ngtv dtrnnt. Cry, th vtor produt, tht strty dpnd on th ho of rght hndd rfrn syst, s not nvrnt wth rspt to sng rfton ut hngs th sgn. vtor whh hs ths hvour s nory d n x vtor or psudo vtor. It n shown tht th Lv Cvt syo s psudo tnsor. Tnsor uus T strt 60
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