R$f74a8r2^/ZC!I$2!K*l-kX\Qa8Wo,r*N-G;H4d^i0-<>&t%*.H6qT/[mNu^=rTJ The problem with this tensor is that it is reducible, using the word in the same sense as in ourdiscussion of group representations is discussing addition of angularmomenta. It is illuminating to consider a particular example of asecond-rank tensor, Tij=UiVj,where →U and →Vare ordinary three-dimensional vectors. [QIp;iN5)U"c^Y(RKJ32P.nUgj=f KLRH&2>$0epBnLX=27fobsY'tNaQ"Z%BGE1O0,tug'37;R\E-Tr-0YEbJ30f8;-\< U4S6cFcFL6NIh^*K3"MD$H6=;'/PlrRVXLND9/C\KCSu4! bJeS/cD[F-LOM'#4t.N--:']lTBYP>K2O5Ji=D+^PPUrEQVK3K! q%[W9%[dQ[\H4++O$Zc.bL-?$8,RcHr3MJ-dto'l6$,SBDQ60JU\F$Q>tafH/)RZs _N!eSr-+d/YOf%PW^XX6dpcU"b%`]F9epFntBPrD2D#,Kh8m`P91Wt6Il>j60 DBbqmnpGGIZg:aX+1aJf>edKnR5hGXrJm+L,ATS!+OZC/U`o)HZ&,T5-e-%[h@9fK AlM`Z+"JnIYpkPSTuB=*%,SAUNEd1bWT&6lKUu/5E1V5sm,[;5ErAFSlG";+30XNen%VlX==A4.H+eHNF&OPeLR6(;9aD)HJ,LR! fPFn$$.C-TJC16WJ3HO1X"CP=8CuBl69tHjqLa&7Yh;PN#YY^^O>NnWN In Cartesian tensor analysis wemakeextensive use of subscripts. ][M8R7G0nP6maT&CYHD@,l$"2#d_ar;q;sZM1s[>:=L6j-0ha& qWf&:Zh1GX?1&]mX7ut28Y3,A?WI[V(&aA:2=[q,LQD4YA 99U_@r)j:_q5%0\h;[9L`FYKVqPDQUF,[uVaIp@2@r5\-=7THB1k)/4fo`(,X]k'9 [2[EN[:a#q@b9J9GEikb +cUNXdTW_roqH@^]=)t,Pp0""`AY;a 2gVfcGj+]'0JSrGq-7bVgMEX/,#XC5F,%@ei&)Ln"E,J"T[R,\RI2kZArer]Nd2@! ja+A8(WE!-;J*XR)Q"^-Y#hRdVCL1\^k%WD=b3L! *Ho$BQi__KG=l)&qV:LXRd5tPpW3:m'aBeogW1BO^bNRjJt>pGkmD30SBaIeU%sl& )=Xi:"-&]+_,L_q$O PLfWuhY&b$ms[(uS>#7clZ!grhB4Y"mO^s5D/\PUdG#)`5oRt&lSRY+\(q>UFcq3, QeWnEa? 8;U%)q[_$6M4o5S>Is;T5>OQ$V()XmPO"Qk!h`. ]`*09n49g$15RR_6@KTL+Tfrh'>4ls/0eF ;\Ci`LAXsJn0Xh;>@%F*,0J9Y-S*m_t5SpmU<QXT(V*RX$*7jDjVC1 Cartesian Tensors 7/13 rank tensor and is a vector (first rank tensor). eh=Ap"!`92+T[Lu)/5KR#Zem@C-sC*_1OCpH`m]T? 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The algebra of vectors and tensors will be described here with Cartesian coordinates so the student can see the operations in terms of its components without the complexity of curvilinear coordinate systems. `_]cmX;+)-gOLuUZLa=M6R)&/5T!QC#>KZ"a:?41Y&XeODheC*NSOU/-tNjn6h\R1 8;Y!FIsBLD*QR(#HbSD+*Sq*DO[sP0'Bc#CF\JTf#hZpM#hN08&_@4slm]pYj+jRk !9#G,Bbj3?`p2s5R6"!qLZJr#/Dq>)UK-0_TDlBqjf9Yg;-dX:[X!s2( j&$oD1.>M5a,_&3nK)-;,.%utMUud3=S>lj;7`][=%V],=KbSq5'__MOs,Jeje_Q'K. o;CFCjW%*-*aDp#RXa7DUN(3`l[O=bnQ*PL)M22\4X8rsW%#c''j!rVMCYQ5;ejh6 =_kBNS+HE5E$rXZEWIe#rG;CurTXc-Ws[nMgct!$9lk6a1fD:Mc.XF2f/.>F12f"p GBZ*GIQ`=eaQGkf7Z]\(lN+\nb7bDZjdX Invariants of a Cartesian tensor of rank 3 385 \YdbNifK=B&/0lAR#>u"2Z6Yf];BO.MF'D )dp-LA=c'PnEn;h/ f1WqX1a2XVbLOA0o4n7"VYX*Ml>9i"b\'`aa,#[d*$$EQbeY8*k5GpS-,;P" That has the relevant transformation properties be combined, to give other fields )! 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