Definition of vector and tensor. 1. Vector Calculus and Index Notation. Vector. Special symbols: Kronecker delta. Symbolic and Index notation

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1 Dfnton of vctor nd tnsor. Vctor Clculus nd Indx Notton Crtsn nsor only Pnton s Chp. Vctor nd ordr tnsor v = c v, v = c v = c c, = c c k l kl kl k l m physcl vrbl, sm symbolc form! Cn b gnrlzd to hghr ordr tnsor Vctor s st ordr tnsor clr s zroth ordr tnsor un Dun, UC Lctur, Vctor Clculus nd Indx Notton un Dun, UC Lctur, Vctor Clculus nd Indx Notton ymbolc nd Indx notton ymbolc or Gbbs notton Indpndnt of coordnt systm Indx notton, =,, Enstn s summton convnton = = = = Coordnt trnsformton (rotton) c cos( x, x ) = = c un Dun, UC Lctur, Vctor Clculus nd Indx Notton Old systm Notton unformty dummy ndx Nw systm x = c x, x = c x, = c, = c pcl symbols: Kronckr dlt 0 0 = =, =,,; { } = =, c= b= b = b = b = A = A k = k = k = sotropc ck c = Lws of k cosn un Dun, UC Lctur, Vctor Clculus nd Indx Notton 4 tl gos hr

2 pcl symbols: ltrntng unt tnsor() sotropc tnsor k 0 f ny two ndxs r qul = f k =,, or f k =,, or = = rst r s t rst r s t c= b= =kbk b b b ( b c) = b b b =kb ck c c c Isotropc for r.h.sys c= b c =kbk k c= c = k k AA k = 0 un Dun, UC Lctur, Vctor Clculus nd Indx Notton 5 clr Zro vctor A A k = p = k A = λ + μ ( + ) + ν ( ) kl kl k l l k k l l k un Dun, UC Lctur, Vctor Clculus nd Indx Notton 7 pcl symbols: ltrntng unt tnsor() k = k = k l m n klmn = l m n kl km kn k kmn = mkn nkm = = 0 = = = 6 k n kn n k kn k k k k Algbr wth tnsors c... =... ± b... c = α n n n n c b n n n n c kk n = Out product, dyd for vctors = b contrcton = b nnr product nsor dntfcton thorm un Dun, UC Lctur, Vctor Clculus nd Indx Notton 6 un Dun, UC Lctur, Vctor Clculus nd Indx Notton 8 tl gos hr

3 nd ordr tnsor: mtrx form() n n= ( n n n ) = ( ) n n = ( ) rnspos s not th sm s blow t = n ( t t t ) = ( n n n ) t n t = n ( ) t = ( ) n t n un Dun, UC Lctur, Vctor Clculus nd Indx Notton 9 ymmtrc nd ntsymmtrc tnsors () ymmtrc tnsor so Antsymmtrc tnsor =, = 0 = 0 ( ) =, = = 0, : = 0 k k t = n = n 0 un Dun, UC Lctur, Vctor Clculus nd Indx Notton Dul vctor kdk = d = k k, d = : u= u= u nd ordr tnsor: mtrx form() trnspos C= AB =, C Ak Bk C C C A A A B B B C C C = A A A B B B C C C A A A B B B un Dun, UC Lctur, Vctor Clculus nd Indx Notton 0 t = n = n, t = n = n xay = A: xy sclr ymmtrc nd ntsymmtrc tnsors () = + ( ) [ ] ( ) = ( + ),symmtrc [ ] = ( ), ntsymmtrc d = = k k k [ k ] = + d ( ) k k, dul vctor un Dun, UC Lctur, Vctor Clculus nd Indx Notton tl gos hr

4 Prncpl xs nd vlus Drvtv oprton n An = b=λn :prncpl drcton; λ :prncpl vlu A n = λn = λ n ( A λ ) n = 0 A λ = 0 rl prncpl vlu (t lst on)=> prncpl drcton xsts dstnct prncpl vlus=>orthogonl prncpl drctons prncpl-xs systm=>dgonl form tnsor un Dun, UC Lctur, Vctor Clculus nd Indx Notton = c = c c k l k k k l l l φ ( φ) = = φ x u = (on ordr hghr) ( u) = = ( ) = = u = = u dφ = φ dx k ( u) k k k du = u dx du= dx u un Dun, UC Lctur, Vctor Clculus nd Indx Notton 5 hr ndpndnt nvrnts I nvrnts = = tr = + + = + + I = ( l m m l ) l m = + + = + + I = dt = = λ I λ + I λ I = 0 un Dun, UC Lctur, Vctor Clculus nd Indx Notton 4 Intgrl formuls of Guss nd toks Guss s thorm V dv = n d k... k... toks s thorm n d = n d = t ds st s tk... st s t k... C k un Dun, UC Lctur, Vctor Clculus nd Indx Notton 6 tl gos hr 4

5 Lbntz s formul On dmnsonl vrson d bt () f( x, t) dx dt t () b f db d = dx + f (,) b t f (,) t t dt dt Gnrl form d dt ( x, t) dv = dv + n w d t k k... V() t V Du to boundry moton t un Dun, UC Lctur, Vctor Clculus nd Indx Notton 7 End of Lctur Homwork: Rdngs: Pnton s Chp. Excs: Pnton s.4,.5,.7,.8 (optonl) un Dun, UC Lctur, Vctor Clculus nd Indx Notton 8 tl gos hr 5

1.9 Cartesian Tensors

1.9 Cartesian Tensors Scton.9.9 Crtsn nsors s th th ctor, hghr ordr) tnsor s mthmtc obct hch rprsnts mny physc phnomn nd hch xsts ndpndnty of ny coordnt systm. In ht foos, Crtsn coordnt systm s sd to dscrb tnsors..9. Crtsn