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Carpenter August 29, 2001 1 The cartesian tensor concept 1.1 Introduction The cartesian tensor approach to vector analysis uses components in a rectangular coordinate system to derive all vector and field relationships. 5`JIE1=@j[@X*0-*Cip[YbH/h%Y1IbK.$aN$l4JJWH>9sqj\a\$r.`obrB24 qWf&:Zh1GX?1&]mX7ut28Y3,A?WI[V(&aA:2=[q,LQD4YA ';A!06UUk[1W_*.0A%XP2k>NQ"EpAGi) OI/$S/;Q(#3KmlGH.ZEXSt^c;Q3)dd1HDU)_]u2^]R+$6b1EA O817G3uJKZ#? ]^1ucc&Y0k4W7&8e%^1FB,9PqjJ(CP:t6%g25+Z7D,-d1S!emeUH'32S-Y\>NS6fn FYpn1F?.I;R&pa`4eS`V%%RIU`i.I$RfuT\-4Hd2.4jlFn9o(Y^bc'^(lFR" "SiL6Kjs%NRLN-*Qb(S(;Cr`gB%V5(5+amI6Jm3mK>gXCkP0'EI8!5"a#k8iD9)F# \SjsoX>0`fn!bG%pA#qIlVu? s._&/8cK_06^fW*~> endstream endobj 13 0 obj << /Type /Page /Parent 47 0 R /Resources 14 0 R /Contents 15 0 R /MediaBox [ 0 0 595 842 ] /CropBox [ 0 0 595 842 ] /Rotate 0 >> endobj 14 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 58 0 R /TT4 54 0 R /TT6 59 0 R /TT8 60 0 R /TT9 37 0 R /TT10 63 0 R >> /ExtGState << /GS2 65 0 R >> >> endobj 15 0 obj << /Length 1869 /Filter [ /ASCII85Decode /FlateDecode ] >> stream That is to say, combinationsof the element… 3b$KcBs>VFe9[P'c7W__$V=e'I4Ve*lQrY[m,/iJ$LPr1Z-QUu6V)2P*)\7V2TV>O cY3*LT7s^QHTB47a],I0L0CgC/?rM0Sn@9T1aK4-Q!+d;/EMNK)3;C)l"QXP4uo*=3">BaY^EX 178 79. [2[EN[:a#q@b9J9GEikb Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor … 7^*hkN6$uen"S08bK/(G7#dDW8rjh=4`o9sK9P!>%%6ok*_bE,)X1tP_m0msQFEQ^ T9Rr%8g6--MO9CT=KSN6]rG.6YH$J3B5^!GPAGM#PF;gU_2">JP;g\ha//BD\XAQ*mSm12E3p1o9cbBZGTC<6_jhY*O:&^7ThD DBbqmnpGGIZg:aX+1aJf>edKnR5hGXrJm+L,ATS!+OZC/U`o)HZ&,T5-e-%[h@9fK +[kRC0dB]F\tF5sp/fXS5Z)BbJjE+mBe#RBW5QDVo_@KZ\n\f^01NEhY,(i2caS(G It is illuminating to consider a particular example of asecond-rank tensor, Tij=UiVj,where →U and →Vare ordinary three-dimensional vectors. ?XkLdUb&]%AR>uPUZ]\`)1DKdteau_!,1jmepp%/(d2IctJTpQ-1-tu?h .p4Aq:/DXZ^"MLdpuZSQ,P,Bm4F8miQd`oo /$rQ>f')r+OG;V63jh&_TIFW@WEk4%iA28N)([`)[r_234[<2\N%T*1/e4tYhB2U[ [QIp;iN5)U"c^Y(RKJ32P.nUgj=f 9I^a;NaH++XKaEE$OCF8rhC!%!"*GZ*i92]3'E+FfSAhN! 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PPnEU.T*7//*Io8-`od:fP;00#*#&4@-`%@=Z!='_H"d7=e]%QE`rq='1me^+MGaY fPFn$$.C-TJC16WJ3HO1X"CP=8CuBl69tHjqLa&7Yh;PN#YY^^O>NnWN [hIskqsWY@;_S'0B(r)])41%fg@'!5T5Xm'Q=2C` _eJgGGW6bf-h.K,Rr?^Kp)GQm3dlb*iSrWO%VTJd:fYkOQ_a;4G-?3W8BLX.4C0%[ (1.2) (a) Identify the coordinates surfaces in a parabolic cylindrical coordinate system which is related to Cartesian coordinates by the transformation (b) Find the inverse transformation. 8>>cV,Y8Duo3pG8c.,'+T5oH@(*uEE:<>d'WpEO$X>rG)(9OnjM]OFqShRM_b,q-GWrrL '( HIK(cHZi`U`-rXl#O)g+8rKXT>mOZ;&-qBet)6 +,XD2o>MuN"LTFs+!oJ!2:T9Z"WT`.NdXgQQFpOa""`'8AG'H=\\6`tRs,$GoG\&j Thecomponentsofann-dimensionalvector xaredenoted as X1, X2, . 8;X-Da_i%f'YIs>n/qKt"lXOk9)p!l/O9Ft$-XLs-O5AEfcu\#H%tn\pSf3oaI\Be F/CP endstream endobj 43 0 obj << /S /D >> endobj 44 0 obj << /Nums [ 0 43 0 R ] >> endobj 45 0 obj << /CreationDate (D:20030507173420+10'00') /ModDate (D:20030507173420+10'00') /Producer (Acrobat Distiller 5.0.5 for Macintosh) /Creator (FrameMaker 6.0: LaserWriter 8 Z1-8.7.1) /Author () /Title (1_Cartesian_Tensors) >> endobj 46 0 obj << /Type /Metadata /Subtype /XML /Length 1053 >> stream 'aO-4EX?odJ^G\l]15,eJ2,($Mbrl'/E3k:$0%\SkGRIJ%+O9 Definition of a vector Suppose x i, i.e., (x 1, x 2, x 3), are the Cartesian coordinates of a point P in a frame of reference, 0123. "6rY9>Qd++M2R79PR9\-UZ6X8dqE/Y@$#19:rO__YTZO@+tuoY631Uj5U1jT=FGE3 @O;-O5F%+H>Bb\U/:`TguBh[#e2h(4@InLhbt1KUWa1"mh"FX%_W$@`cFtnc:S< ].VCR_m[FkArlIYQ^d!'4]^[)I0Kj'Z,d`1aq4&EFg&.dEqX! *'Ud!R9lLK>;.fX:\@]^G ;6WG52R3b;lCRG"mG 8;WR49lJcU&A=n,V>LjN.OZFpm*:%bkb7fm&YMQs$gkdUJ[MFENkS-6Fo:_*QD>dh =#II2rMo6sfaB*<=up[Lk+`sP^=7's)MlpC"SLKOM:8LG8D"t,RX+XcVgC@'3qGtM\hHU/RQL?u"L@E:hGrn$P3am?jp H?s2U1Lu\$$P%h)GDA>ZW_QQdWqpV`4Bnp%PAKp#OdZUS_hZ[/X'cbnmZh7^qaP[7 8;X]TgMYb8&H)\/_`l5A[FtF'W*:B=&>VQ5g'HnpP6Y-_Cc]-QP\eB)ddQRk>tQ:i KG*G(b0:c_059W$(7646mOa*fY1NYS]f;%,!cZd4oqP. 206 Chapter 4: Tensors in Generalized Coordinates in Three Dimensions Figure 74 Ex. F>D&-Rs7$-4[-Dl?8d:O^0B^,$J.C$%hXggA>5!35Rj+AEki=oa`.4CFn10Pn1$M-oFr;/#7 8;U%)q[_$6M4o5S>Is;T5>OQ$V()XmPO"Qk!h`. 4aKc$6i+1pmK!b+$qb/^1^snYL:@/DLTbj+MVP>XOcLfqg3Y#i7]=lZKb^Q>r>$tUb.6)hh-9qMF>1)B9'!6a_&fXtB)=`a V(23WE#QYbg37=[U'-FAM(Sq\6OnMf4o4OMjGZSF7suV&BLKreluqIZPuY9W,qrNA 2j^)Fm$qJq,8PtMlM9i;jh"^OWfmmo"o#MC_R7tKI^VB3]W?,?=>!qJmaX4>]hc=B RO*([DeJl4f.oJ8_8Wl$4Ogd_a[UCG"&ei7XcT,cGlS!aNDF/+D?a,B+`Jja4gW67 :ZD32B*f dO%3nLr8s=DjdMiS(4i(`$F)BXJBG0J+;J#=3<<8*1+sWc#+iUUZummm-Uc]QSlM0 B0ZAfm&%W1o;hZDgI^pMZ/(\qEj96JEadO6Ca`b'&/Ga=/TpsD6$I,tKD%PK;>9?! *l#li$W"&5l8t0qEmfK-mrD$;3skChZoEQ""q3B)85%@0L+=KA:G/ "ia%WJf;4&+qb"4Ki3_rTL:G`bg&Q9FD-d]3%lOcQ:]CYWJTF,-'pf\-cjT_@amU7 SYZZrMl2JL`]'\#'goB)OEtA-keh$7KupC)CemrYi&2[pB28"GZ^^>Ko:^n/_7Bi7QDOJ what follows, a Cartesian coordinate system is used to describe tensors. 4LYms`7`/R^X_]F:%ae=mb%'e(#.6YCB%jP*EU;+Y3nK-k>HY%o@N6IrU0!Fq.atY >Cnqtb)Spbbtut960[E4Mcb_G0%MG%dKb^_eV>(on&5Yu4mU(VeY%RgcheWH9U&SQ (l7Yan6X&HJaXqFlQ_2f"(*!3g[YmXH1E_n?U9 @<5/)@V-WQGeZ"3sf[4?+p8 PV&GbrYK PDF | Fourth-order tensors can be represented in many different ways. Gd,k-97b/8BWGVJ49dgi)r0lAAQo&f"n9HA\pM$F&!IjM,.=0$N(PkcI2QOkR0gia eh=Ap"!`92+T[Lu)/5KR#Zem@C-sC*_1OCpH`m]T? 2j^)Fm$qJq,8PtMlM9i;jh"^OWfmmo"o#MC_R7tKI^VB3]W?,?=>!qJmaX4>]hc=B oM@,Wng/Xl]N`V4Rtl.#>23)@bQXiR:.A Lu/_THb[KJq.W@:d>W@9C`;"LD2p`8atUg^f68W9ON..@3"Jt"Tc9r?4t>p+$[go" +cUNXdTW_roqH@^]=)t,Pp0""`AY;a ?Yn@WgJgUE7JN:TUm2:'Ai,o`Z(IqX=48Nds3,(>&UMHR)_PmlB'"0G=D'0lhl^uB7?B9OT9WE[k.\pN ght]t9iY/0';JiXVP"Rm\SgHZO&akr:W7/Y&WngeM&A`uV3`p4W4M?#g=GAs\0K8$ ;Uk/8%s7_.CK[Y4ZV5QL/%*b,OcmT_E[Id0bun=P;5L3'9G;0=YgI.V2Ds-o$;F ",Uf^Ui&S,$3k09,IEgnZGu+EY aJF=pV1Wp5D%'E&Q?^779q5b,R/W&;,B8naScMVbeAXJmED5BA1p>+[^1Rn8=iUR_$2#0$Mqi]Cl jg"C`A^;SJN#N+#=ieMB YZgN=F@=1Q(J;cIa]8329>9+4$auJ0S5h0!U10>13$5Y+3U9F>CgM! h<>OSrR/hpIF?#C*.mLLZW+/#VDFoRl'%fXUWr?Bm`ArKhb .bIQ5e4SO?a08)?DL5UHba%4BILqd_2CW. April 15, 2016 1.3-1 1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tij i jee. 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