MFNWtKUb<ob<R=MDLCdNNZ:K:\\M>D:;b`gBnCaZRLCTJcDXoXuuT=SEOiX_wUWOIGUiHgRZeipUDQKVbWWEEUGeW`kSMOC[=g?USQcH`GcN_fN=GTsbHKW>KS[WgNCgcCSb_h\\ED<CfZggRcbr_GdSwGQGG]e@cddchVSf`Ub\\aSGeGOMreaBQstO;YG;Gr[bn?Y]qwqquqswrAyxwHIYVu]w<GiAmtMIxSWxHiyywvXWYHiiiYDQugXeXf;Uj?V;Qb?WwGKCfeDR=hZBe?rf;Uj?>:bfAxCUU?kDf;Uj?V[sGjEx\\ir;OtLItHqs<ObtGtKqs<Oit?uBAco?rf;]?roogwOUAkBZgitOCDkb@ocOatNasboxloDRgBEkDf;UR_ePOux;HJsSuccGgvmyvIyWhqw@CHKScosHqufVsFXoSaOvHYUC_rfaV@QUZmDf;UjPr^<N?<XLiwgiwJmVcaXtlMIxqnQtE=n>iMvYvvaoxpYPQX\\eOjT;Frcq_f^_Y^]>HdOFfug[Npl`XeA>fbGr<VuWvoViqwh`Gx]pNbYHaHOuOAfiwgbhev@ikhdZo\\fZ[oxZ@oaxnoQx_QyqyuiAqvG]TAsCXaEvj@P]TN]rY_f_oWYr=Apq`uY^mnO[Iahd_[APxDorIp`pnbbyae@rf>ejTj?H];hRyHW_ua]ssCYlavmiF_cyGiuoix?Ux`isuqTuAh]aVnwduaXYkwPGfuaXLIsrMXAwvLSe_aIIAcI[CPofvsV;[F[ce\\?vjWcaUh<uVg_tKCw@CxygThoWF=hogW[mkTPmThyk@U\\DL=lq`hpo]mpQrMtKqIKuXssuv:IjwIkRmOTYWixJQAo?lrkUuNatn<UXmUt<PjQseto:PPMQQSiuqhXVhpeut<tuk]xbXvgIvlLmmhpcHK>]V]mN`=o;PtLQx;IRvDMaLKDuTQutmuswtxSxsvmvBIl]TpyTV\\tNGUWPAxYmtyDxdQwdQodPo@qr@MxFaXNUwodJVYtaUS_ypYlJ_TlLavoEQcyjpMLn@Pq\\OmxpPtOxHMTXkpqliUY?YlCan]AqTlOqHU_UTmUxeXYqQRTln]`YsIslIo:aUyqUddjS\\MmxNXXpZFqPpwBAcqGgc>\\^@cvoyBA[LieRit?NsOX_Qof\\FdBywtnhf?kyAiu_rbVyuywyxamG_H?mY^sR`cWOwY@ycXdaPyuYyUWeUge?oar>wVAcmP_nN[Tfnqhqw_hCQlFg__I[VYlhv\\h^_R?P;geWfhOCgybpmwm?Ha=vNqttqcEIx\\quXIv^eFXwChyuyCgQYe[giXoYtEulgV_cF_Sx`isuoDg;HFWuWwgyocsUDDkhOYb>agVoBbsxaqW\\QHAUX@se[gXwIr=wB[sEpiCW_eZ_feSGSOvfAWhUr<SRD_SwoghoWEqdg[saKU^avCYdIaImaCbAHLUEw;ik]G=[gdKuHcEdawfarRGuqwWYmEeUEPqHtwHTuyCibEsBveUOggYKUj?XwKbFYV:=hJ[wbOgDwfWOfTQFJMIdSGs]yeEvJATyESCCyW]YoQwTqsHibEsDTSdFqhgUuxKybYFf_f=WXcaHqabxCeAQt\\sYVWt[uUpohtoI:IUmwI>uTCerOqtPqs>qv[cxnWeFuUYidLQTnOc^IF;wiSGSksdT;vxiGdIBeWEpaIpaCnmfwSYpaWWQiOcx=SyRsyD=UKicZEbkoSxcycySaIXOEBs[hpGttAr?Qt<KsWYvSyU;AURWujQtEuGGsTtoXIkeGgv_AUmcxp_hBcFKSc_YSl_xUExWWFgGfhoGbEGbmfnER;EXQugjYvSyw]at^ehXiEVsDtOijcy\\]cVmgdOvf[X\\arCwRYcYNufU=URiuLQtu_VCQDaof`MIoSdq?FH[dmgiwgEBEWcEgMsS<CImmvFmGXaWjaVEWhJaXZIfLabASxNovAmY\\WTIiFWaWvkwkiIqkx`sTxAVSMX^IskKB?gyF?IticZaeOuDikUvGYmqVNyClaydYgioRQwHAUEnMTnUbasxRsbBOyFiFWEgA;WTcUaCiBIchuGfWBbwD]Wu\\Myw=r@YbR=BvUgw=Y]qrNwxFuWieEBEyrCwMmcMUbGGFlMtc_hhyhhqi<wBJAR_SgQURqgDNit[IUsAs]kwmOxKOuKITvCfqoTh_UCcCDkEvSWdMWBWR<[hQsTJ?byqyEyUjsGcafywywYFVAGToF^is<SRXUIuSI;?iSQESCUyoYwIv=yb=uXKCfmIV_;R?Ufpmxhiuu;Ir;csiVYqiwMYWQskUCGyYtOf<[sjgwZgE;ICgASGmGVOV;OtqAfmYbdSrUuCx;yZGVKsHqsH]gvjSyNEV;?dB=EcQBYcwfaWhQw\\ssdgTe_e@oBGYtPqt`QwcusIaryuDf;U:gX>sXP_UHGx`kCYiboYdpQYFecqOvBcSq=wRIHBogb?g:OsKayGgcGGtkyDnWBrwY?OwDKX\\ohcYEUKEyWrcCwpsxqYHg[hQGXXGdgiBugCnahq=VK[eqyuyuIU=dmaGbiIwcIbuRB=giwgIKYbEGMcs;of^UI<uBZWwUcvLiY=iUUkCISHQ=fNabg_tjYWEUSEKt_USCGXccW;iup]yDOHX_RmCYkgTviDWYBS;R<=sUUuhgUZaVYeVW?bl]reIeZyRPefI[excYT[IbcBg;HvYu_igqCT^=s;QiRScDQekqvQSshqXOWRAocbOhqubOgu_yUwOEvAEu[w;wcvKxBesnqTcoDhwYBaReWHtMFWsX^kYWsSwwUGoI=SEBKY?mba_w;KtXKvPEEqEyVqcgaIIEIReulwH\\WSmaERKT[gSPGumMxXgyWgeXqIZuiQyF;yuZSxj[F]=b]wyQsDQSVOaSxmyTmxawIGKyDac:GTjmwHWI:IYg?Gj[C]ewmMrJ]BHGRRsS`ccKMKHyTXEY?LuELQIxWghUupLttSNhkeupXTQp=xZIrLPlLxrxiwYqmcMW\\awFPoiAwbAP\\YM;XkiIRbAv^xwAUsnmJAtXgQuoMwNIOKpjVLwRHPjaQuAy]YR=URFiLs=Vo@NITxwAW=MYy@ye<QSDYrxYyqYnEL\\QYIpkeIxPaTdiXIYQBQQ[eO[pV^uxCpYNLS@hp>dXhDYbeX=<UW<wjpgo?mpIjSV_WFliFiX>dkiZ@GiKp\\nA\\`_s;PqtF_iVuEfj<AaSIwQPZ_fohpoHo^M@]O>tqQqn?r=HhR`tX?eMAwp_ZWAmAImPNg:?hKH_xha;x^X@i\\^p_@]Y>sp@tK`_:ItAA`jAdQ_yOXg]npBniG?iePkZgmZIkcAjoqjxo__t=iiS=GT=Gjshx]yxgguyVbEuKMInIeheD?QG_iYlEr<gUgcGLYSKYV=gVq?X=GrR]wYMIS_cuwrCgts]GYkHgMWHETUIbxalexW_]kAPpgYuWIy@IwjMlkat@mqtayOyppYYHEO_tN[qx=QwwdMxdNWPkatuD]kgprj`xy\\XbALgLoyeYbINRQLLIsqhWUqX\\arCDv>AL`arELQtMPneLxeUqpWWtvm]sgAmsInOQTc`LnDMQqkj`J`@vWAsUPJ=ak:iYvlyE`ojmkjlvV]vO]PPXvAAqquPqtWieW?AKaAo`psAhOltqixwFileTuQaoKdu@IQeUtwuxsPUAEuYDRY<mfYODlkodPoiN^HXciKGLkc`JlyUrEv?Dp?=NYdpFTK=hpC`YXHQmmJv\\sd<YjPtAQr;Dq?yqZaUx`YsevjAPbaUOXMK=kLTUIlSnLL`uSfDlJyKR\\v\\lUtltLdyd=vAPq\\hM=lSFiujxWjqT>=M<Hv=TKl<Rx\\xGivHpkeTvo@N^Tpv`sULwYdv<PvAmJ\\mmsLNoUkTxRw\\wSYpAptOLo?dylIoapm\\EYkQKg`JJIR;uSOQQFhweAYkdynxNbEQoxwXik^aUoevq]U`hxlYTIERU=wn@RAUuceo:DmByk]pkC=jGdV;UO]moYxwgIJZhUHlmvDKp\\nupodPoq@qRuTH\\V<EumeY:dK;htbpQEDxRYVIhPrmTQ\\XILxSatNarvMV;HKCyrIxMaeq[hx[hqZpunQM<ux==tSTUiQW`xNY`llXnpaR`poa]QKEmbeyWXQqMTEXnD`klXUrHlg<PbhlnAqmAUkmXm\\KkLWi@YVPKhlYmqnGuvp]wjQqGmvx`kyAwgiS?iRTXTJuTaaQyaqBIjJPpgImmqUCurCEYQTxXyv^mwQAr:DUp`WSYMi]q[TWGmXB]s\\xQrEx?ypyMxMlxRMrDhLe<PFtmwmxSMLkqwgYsPhV`aJFyQGdKWiuWyquiuiAp<lrRtvViPotuvmYv=VnHj`\\xEEJP<YUAQVToV=WuHqqukM\\VriVYlLVPvaMjpQnVxnYpYTTp@<wV=KRLqDun?TQTAfWgrnG^GX`xWrMxchNen@[yFfPVft@mKArMWvGoevW[MGqKo_FixJ@^_ihoVoDyqKw[nIa;pkowZa@inh^jAdYxjNx\\CPrlHfT?tQ@nAWaNf[Y>ebqmrybYfiN^xUInZQaiOdhnceQqhG`YWxnqZwQdfp\\Px\\rvx_a`WWyOqknaxJvbiohAakqxkA^msI`VQhnit?pn_xxbIfENvCfe\\ff[xmIXyRQth^_QX]eWlFOj<qfO`q;w`:OZN?^@V]xgjvoZDGbHvxkItI_nwWqw@pe^vY^\\bi_tAr:XdqYcXQjx@e\\>wqPqfAtcHn_wfwO^y^_uvtg__uhxAhuRq\\R@uUo^HitmOkaapV@y;vduafrO`Hg[Yx]Evx^xgIInqg\\sXZmV^anmpoZ@Nc?onu_rEYrWwuRAjUyj[gbO@y;vjVYa[NZgiZUOsRiv@w]@va`XgxYihauWnrr@rn>y;yZY>h;GaWhlSGbX_pQ`oYFoI^pYOov_yPy^qfacGln>yewsRxpXotx__HhdtImFFmRXa:wlsOc:AppQgAwb=H`nacQ_oP>x=Qt<nuMYt=yhkPp@vc\\n\\MPq`xxBAxqH]tPcQxdU?fZXfXy`lqrnGyhIwMynSwat`qRgiE_[[>wbQ[^X`@o^UhZ`N`=giZ?l=QiMIi`X`Ii`p`\\yNwa^pXAf]`^vWmF^y>Y^bYnxWZ;psWApZg]lvan^nnOeIf_SOlGo^]Q]Jwglp\\KFmbqcfxivQstaZWhyGNm@wnefh_N\\>xqAYtopbYOuZix;xavIvINlP^ohNy:axuHnwA\\m_ZMQjCVocAmZ>epOaOahyIcrqyKwxvFmQy[;w]tnhFQwjVg;HakAdVYp_PrrO`knw]_hQYp?_]mvkaxal^yZxkHphDYq`nwsIprVmCAqPvyPQt_XcG`w\\odc`]XnqoqtUW`vffyO`nGZnn_n^unYw_v[D^k]Qk]w[]`dbAe]Oj<hpvTCAIliVNOCmkvKKx=QHOSY\\;GM[XTEC^aCoIb?OwRcIu;wFYVPYRtOHMcwUyhIoXhmHWeVBUtk=WRWG=_elmHEAYW[DtyTbUCdQxdUeIGh:]fIsTPmRq[R>wCFKR^ov??c_yIQYyTWdpgRjouj?vlkVIAWtWf\\=C=gvLoUXgGg?tTqfyCSQidKShAoh=GI?GVckXIotnCrtOteGEkGD[CFbYrEcWeMT^cgJqYBUiy;YoABxitxkSsutrsRb]egShu?cqkg?kHhqFasbu_Y^?xnctOiyuyu;QhbkevWY]qR[aUrkfl]UDKyd]IwkbumfOmbM[XiutdSFyqeaYi^cRhqV;USwqvqsiSEh\\eB`[VSWYImevCEdwbjkByGxGwW:qr\\QyF[IbYrtwx<GBTwGQ?uZIEQ=VkssmgIywgIWrL=wOeTOmwVavS=i;eYJ[IfqTTMUDKInShjYT`oxU=Wt_EEWsKyB@meZ]uEkvfSF@KXVeBsmHr=xxeC`IRIacVWGCuHlAv?UyFWB@IVmKHAAGi[v>odQex?AXu;I<yB@IIQAsUkdyqRiOBNMYcwBeiGnoWa[feayAoCDaWcebdYEKGchmRSwBYsYVAgecehGfcMblCfgWXT[FXuEBkhoCOtHNoMlHuRiHQ^xKjiUNXSQynJYSDMygpxtHng<pkMQFIXqEKMHMD`wPauWPS_hyULvsalMmrFLJkuwU=ka]n;tOitMFXyVHwrtVR`kvAqaYS=MKrmtQdLkgylhfEAnJpiwYwYwgSQp_HqsHef>`hyjUV_`_kH>wgns`?v[QsF?]jps@hi`_wIic;NxM^rkaumwsGNq_hkEQeRfhDYyrNdRfhGis@hu]_r<IsVpk_QoHpp>g\\\\GlJ^]q`nmWcAAx;HsfOnw?fZXhAW[vnoiG_fnlIwn\\xwoXhii\\N_hKxsv^ZaYnx@pmIfE_qe`kn>llOwNIoCff@?t]^iZGrW_bna^gvjAFwWv_pIbYimRNeX_lEfxSW\\pvy`QbaOqohpoHrlGaxXpgOpcg^YIiaWyTh`JIrFyywv`WarZv\\WohZa]y>]EoZWOw^VuW?ZFhoa?o^poM`enQnPHkcanNYuBp[Dy^\\Xy:Hty_pFa``_]UvrkWjuAr\\qyP?f`n\\K_y[Gkhqw<YQUgQCVYsy_yHQkYwivF]ewmVOYwKIx`cGjsS<WcQcxnafwAy@=tuYvegiJuCQmv;WBAkwXAe\\]ixuRvCiRIEc=x:oem;Uf;t[=tPwIUQiOKUDcYr;e]]cG=wgsXiUX^]bP;eXWxkss=qIUoYyms=OBeUWeUvSdQ?eRjpOkLosmOAtS[xJw=V[XPfHvQ=txTmkmpT=Mw<w=TYWyuuEJwaXpaXGdSJquXqsRTPkyrnmUv@OlMNlAX[Qr?llQAxo`jXqkb]t=dwyLs>`MWxM[xjhiKDpxGpQAlrNdXXhnx]kuXL?tPvQQZyKDamJpmg`U[hWnDon`V?`NaYeiIr\\PlMW\\^IwGqw=Vbt@oM@ttPffy]eQgwQyoApoYaAygMP]wgdIXb]xk_sj_v;UxUcyAUs>aSn;DIWuuiDi]UE[SWWBmuw:wsf]B`]YZMuiiTlOIcaivGtqIbkytFyxt_UGgvygvTuISsERmvV;u@_Gx;hTutAkTmkhIIvxmX_ExpiwUItQEiM;w>kX<CHbKYuUY[UVkwWuoDReR?oIioRuaWtIhoWtO=VlUbNsSUAsR_tIegYWytQy<wVp?cUoY??hb_XhgTGsyM=ws;H]uIDufUCBBuTHYwjKuXKyFSsjICw=VBuh\\Of\\SeI[GR?dKuCVUeJOBsKIyKUmcRU=VKkIj[tU_RsYwoqI]UicgYKogfqUdicPaWoqDTuv@cIUEi<gx]WUpKU`?t[mtAYdCYbxIeg[IKqpfMXjLu\\<mhpughlT`lMPjvTp<DQJyVbYJ;uMRpW]`WwATndVjhR^]kRlQjTNyLuAltwUjwuS:Qjl@rVtSierqLkxTSuLXH\\sUIVDawVlVjukYyxu<qyArFTqV`StlwktOQUKH\\l`dqW\\qk>rDXwEhvv_hQGrRfqeFeQWvRhqHnxMVyd^xJygyWyUWvQ_ofFpp_hoWmA^p:a`=nbMfiZWxyqpqG^\\_llg\\VwdQwjsXje`g@QsMnmsovSvruV_:YlR@yc^tqO^aAoPIi[Yvhi_npys@_]gudAynQ_IX`Qv]Rnf;PuTQuCvwpiluwgyYgUqpKvytGpZxcBWcPyh[njXXZEfiQOw;XiAQhYp]Ei\\dYZuP\\ciw^ignqc^IajhmWhmNX^q_^LxmK>xLfZU^eMfv\\ylDqbIqdh@fwAgWPmuxqGvgCxyLxf`QmTPmdYweivRygnqeYHnQ_xShokX`eniY>[tAr<pqq@fxFn?vsVHaM^_u`uMA_w?beYpZx[FwtahjUfu<VhVwqNwbvQpR?fRnsCAiQvpvqwsouKGdBHmBx[dWnk@yHvwXikEPuup\\IXrhPaj?^sHtwhgansa>wZ`^\\GmSVfWQqoWaqownwaPgtTQ[VVgY_hCYhAwkhOunHggQuoXmqvg`?fsn^LiuuxhyYd_gvEnycgeT?ft`aepDoYkYcY]hgQuOSeXICwIW?CXBiUgmuGeChIFGmwMoDNitOAy:IhQKwiOH^GbA[xXeHa_fwYHtEFxWdkqT<WDsmCFGhvERXKvI=YJiuW=smSDvsV`[ItsItiubIIWAvA_V]cuhsIh;FnaER=WsuF_YBEsx]?GskrwwVTsb`;VNGdxssQ[IQuducecqiqquumBUofFmHgcU@mHh[ClkrpMSaUHv=V;yXcYBEkLnIJGuwG<uIULtArnQSMhNoLqkpqaHTQhQbEjKpuYMT\\UVbyWnuMRXrrMKquyjQXOeT`lv\\dw@EN\\=KrpvSMmQiXFHxEx[gGo^AaXv`HnsuWs[_\\\\XaBPfk@iv_wVH]jGqc>]di[ePqOV_P_ajAr=FoNOmTYk]xkqA\\wAyAyaXWw;yxxxxwh[Un`v@x<Y[lGmwi]q_ui_m]a_SgqT^ay>y;YkZyqiWt``iT?fl?`?osBWaxieInrm@oaxf:Y`=`uRvylIrhQrFQwx>]m?_agibOnkqfnAmD`w:OnZIvs`]@Ad:XlxYZwGn\\v_oQ^WOjgh^PWeiIpZhv<gisnwgoixiwP_uMhft@syIl]nwS>n_YsQXwwPegwdcwiO?dQGfLP`P?tLVZGVs:i[\\vdkHtvvjuwywyxmf_gY_<`]kxjJg_bgjZx[f>ii_n^hZE^_ri_I^oC@wYGeypiwvu?v[PqyapcWPgeWgmWbF^ixIn]wrrw_NN]<wtU?gF^fnfehad^oyIv]x^oW^dHxhH@anhk\\XtLXyl^uiHmrYnB`xR__PItKI`n^`^ffNv]RpoM`gYiaKNatAjhWgpIjCfiZGnwA\\efcvp]fVq^hjx>NqU@qC^_eS;RSyvAQV;IHbssfUvNQtaABwauHoRAIW^cyVqcsmGw=XjktJAXhox\\WSqQyjUVsAV\\igwQSHmbYieIiYeiWeeYKqc@iE\\sEmasKibQsvaKVluxv;whiiUkyC]rEoEJwyayIyIiKwSWWRsiTayYFmvKuHKuRGiXbyyUDu\\xobxKgxJ@MmZmSEqvdTKSMwHYx:PuEXvAmsdDknMvZWuLQ^q^w?Vf\\fkMi_RfnlitAqmXp]sv^>x^wac[xfw?scfv[v^PAs\\xs_o_ZgqRNqMOwK_hCXay@icQx@AeDWhaIeOvrIaq[Iar>abhvrVvIiqkYaW^s=Aptq]RHpyYrvHkFGnmqjwVb`paiijool`Xwi_]>fgifw=Vln@x>w[hGgcGvi`qyvlsWvsvjhqwhGsUf\\p?gwH]`@hh^xlwyeAutx[baq?Fws`n?`yQ_dN?tqPgRqkNa`Ly`^axUvquO\\fxiNilPyi\\yb^I[GFwaaw=NuYI_P>mhxyZwanwguGgb?nFo]Uvlv>sWInF>dfvdYY^GflCitF?hFNwm_jPpgcIbtnvJXZj`q\\IsYoimG^iyxspklykxykvvb>yoO^fmIoBaqfNeknvwGwRwieOcIyhj>a=PcLFyRgviiukv`lw\\\\ivHpkpqxpGp>@iaHdbIy[^sMOlYnuUV[torHOdu__TWhZokbF_cylngxvqcS`iQV`<Ix]yrsPv@YqvVm_vxHIpfylw@bdGt:_pO^uwFcya]R`tCfgxfiHxZnQytQi`XeR?ugfxCW]Qwsn_nAvsuG]nGgQWvFYlPIflWaEvuGxnMYeqpg__d<qpgWoFXax>eX?uaOlMnuqv_oum=S[oiSGCcAfjaEqoiICccEhV=yOMsOKGMIEusXVysqevV;EmUGlSbwUReMEWyeWcxgaHCyfrif^wCZogJYHSyBrwtAMiPYcnwe<Uy[_dQ[x=wBIKwlwxy_ylSsXKuySbu=EbSIgUwZAYnKs_EXEWErMFPSuKAUHgfLiXTwYbAh<uyRyc;sHwAwAuIl;SYCY:GhQ]y[ybHwygcf<YxaibXMw?UHYgIumxvIy?uIwIwy?BygypYuEqtvufNyclmrrmFHwvGevGgwgyRByyqyUuQuOwXYiYuiwqwR@mRPmwSYx\\qiweGX[fLIw\\ssQEx=yExAxyuIqIIvgv@AWwWyhyshcU<cYmqvGAe<GrmQUdcx@YEi=xWitY]y\\SXLYccKw=yCkSs:oEImF;WVpORhUuAWct]iuwdxOyOyg<wvHySwMCBkF`wXDQsTaiiaiZcuyuywywyywIwY^iRn?dpirLODFevaysYwVT]vsyvHocMyWxGYcQt_;vjgBQoGWMSsYiuMYJwHLIryArwwxXWqVYmimqSYUWeKVXoo]nu`ymymAmJMIwnqu]iqtdUvXqxeytQplPPcdjBayLylx`vTpp?irZXpHijXeXeio^xV<dTw]MiQqYYVmQvH<lCemyiqwhqfXu=tJH\\KQmtEUMkpuLDk>Iv=yjyYxWyX?xMsyqeqpUqxhiuE`XSXOPiWkQsiuTQtLm`UbemCEtyqWsQX^hjFHNA@sUTw;ikMMohpypQsOxKy\\yjLUaLbcweLIumX_qpg?``BafwY`yIrxHpb@pjydyPyOVuNFw\\gk<Yit>eQ?oy_xCgea^ge?s[`cBvymymyayWptP?x=pkYwayAx=AsanevgwoxtxyvPYeCYxIX[P^iVWdM^wAPtyFlIOsNPpYv_hnfSIeIQxhVw^^pfif]Xnq`ulWkXVjVWo<xbPiot_sT_u\\Yh=vrmAS[UUkDBqdOQBMWYeMbrsRHkF<_XtcRUGCD;RUYR]qiEiTx;SXGr;IRSyu]iY=OXyIVcAr:eXoeHNcD^acsufvYHnKxgSGDIBaAuQMWvUYB]YIKhXgfw[fgil=uxgYuqAtLhNluKXTt_imZ\\lmarMtSRDW;mNiQpLhLqdV`TvkeOAmqLYKV]w[loKxTh`UCdMcLPCYpausXlJvUPieRwdQtaqQpsmHRQPYPAqfiPC`YGXnXHlVHth\\Oi\\W;EqeEtkyswHxDlSSUvExK@]wCHsLiuo<V\\DjpiPlqxMhKU<Raqy:HxhppqeKt`x=muGeLfYMlikkLudMmAHWCeU_]y?PoHDTpiREpLCtQj=T>xmx`p;tw=<LbhTwmrEuwuUwx=MmaxSLQ:Lx`XsxusGOZWOuTOsBqyM^qYG\\_wtphphYZlxxLF``g_hGxeQ]IfaUN[rarOYoQWjkhj\\GrKYpOh[aa[Ey_iO[Wohb@kMOmvXoyxZP^lcxwD?vqAr:Iq]AwKGlJNsiXx=iiUVZqNxV^vKIw`QxhHdh@[SHq^X]fxsf_xVVu<Ar=HgGg_iGy_xsJ`cCxnB@oUGyxxeHhiCwtmpgwaetIcXYsXfuAn^fYxPpgcY\\LIpXplR>kS@vYPZWx][a[xwtq^b;hj@YrRFpkFnk@h[@q[FexxpnxiyqiD^`xxyeHasiebnj]ovlofuN^WAnvA[SP\\rW\\dxtaplhpxIov\\v^Y?ihIyFqdEWdEyf>vibweTPeunbnxtNXw[Ne`axmI]NVa?PiHq`pf_hST?INer?WCQ[vlYhVmugeWgevp=C]kraMFiMxgSyVohJSR]aun]gB]Rf[e@icmQItkt:uwmgblYVG_smYRtker;GMuYqMGNwFNog]eU<EyAqIV?wMiVI?WdEwdAiTcUQEfvItratdyuZICLiXf;Yh[yKAY>eIFyxZMssaskgi`GGY]VdmecsxqceIIGQYhKqw@eET]u\\ITrAR`YYA?t>[f=ETkkgoKVxOBy?tGkYKWRp?Fy_feCXKAfe]YFwYZMvvybXgCGIsAkV>QFT;YJyE]=y`]xqiinwCFurS?WXCxwasUwHIoI=_WvKU`=gI=WDaxSodycSvMToyIcCysisaasOysFutyaSqWyNQwsSuR[H@IWpGxeEwQyGWwu<[SaOT:yCWKI>wVUqgW]i?CckGu>If]=r]suSsDs;Yw_iv=DIMiKESY=q<Iv[Iv[XVAAOMtW;mn>\\o=xjvqwupQyQv?YkCXRxlyOlYCxkoLxIAMBDy>EKrIjmtyBYP^txoixktWv@u[LnX\\vhetChLRmkkMysLv@QmKms;HPT`QUpqw`wYxkPqpT=wT`XtDYU@PIYP^tu\\qLIYQhusyds[MQndK`mSbaKRiQrMxFySUUUUlm^Xx>qs?XKQ<j=EMjQnrDl]tJ;hpvqt;HJdiR@]vsxl:xr\\=SBxPVDZkYvAIoNFsInfXxpKf_<oiTid^XcVH\\MilqGmhvcg@wmX_Of\\O^sUfgr@kt?[TAkMwx;hvrojvNfiqevqiKnyRFp=qn`YiUXZrafIAxyi[maig^v[oo\\YltXq;QvmO\\AP^;pe[iu\\akXordy]DPcb^vAfnpppP?i>o[cailN]<ockq\\N^hOxlqNrdNvqh]Xfib^m[Nf:r:ESLCTJcDw]Tv=VZKvYExiEEsIS^KDWisGivxWwE_IreVEqdgKdh_SZaYquwXiiJ;SX;w<]fl?V;QB`grvgsfwCfsDy=UHoYlkeGUeJyE=mEDUsWyI>=iHoxcgCUwSayYLexVWci]SEysKydPKsi=sKIyS_gvICe`V_aSpxmFajvaY_yVdUvEQvm]KaPv[hS]TYN]rCuL?`PZyuThKXhwxyrpamjXqIUqv<vwIYHxxcAxqatw<vh]qY]O@dV]tp?DRiLR=DSxMmUvnvofkIjComSx`hAb<?c@qse?GwrvWEKacuOWa_w=SCy;sFGXL=YxaIyOgSKcmmsl;wvswV[v[iIBCRxeSIqbeyVpkEXwhGKsQaxf]ys_v[sdnuf=uHdiGUeYHMcXWSwav<ysiQiGaUxmFUuCsQgyIwByeCoyjcepuhraeAyWxhsYEyDyT:Xvc\\X\\anMxyEyUxQPj]QCYXiQn?]nYiXQiUZXvC\\q\\Pn<`yweYwuKv<x:Qt_XS;xmrDX\\YqYiyhDyKpljYnuPrhINyIumxViiqqquy=yrYVlalOhlweq`pjliNIdTOeTFYMLEVXlWshvP\\q?ipQQn<QRBUyfyVyTSVLxoMUKixjIkmxVKPN@qkkUjp=k<TlkeRW<KVuuAIL;ut]]SrmV;tsPpw^EwyUonuN^xVexvVTwymsayVPHwmptYLXfxvCqkMUSq`XJyQpiPM<wqmusuvYuytytZtyAtUrQj`ynpQnV<vs`nuUygypElOpqtNLxyEwLiNbUQsXj`tWx<pCQyCQNviNs@tmQlqHmxXRTQvmUJqhwhEXExU]<YXhYZUTpYl?=yEMX[xv_aMFmPyIRx@wPtmU<qguXEXUvQM;yN[YJuTYpaOfqQSyYn]qu`L=XyLXLLYRwqUMynaUlydvvMUthnUMq_XUtiPEuJXmPtLYUiWqhJw=V[yu=TmCXxGlxvXk_XUtBWxSPpsj]yVus;HLDTquXQm`U:\\W;Qj?TKP\\OFQJNtLQTx`@vJYy<uQShwo`XvlUMhvxQXEXUVQvPtprhraULYxQxqMtAoZlKAeuq<QMtX\\IUVQvPjglm]lMETtXETD\\SYHlHduQ`prhrglWkUkU]YP<xf=lOyJ`Er<h:\\UEATZUo:Qj?T;[g<ShZE::::SXFkyYGy^?t<sva;eu<J:<j^PNano<?f<3<\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlZ2RGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiRnJkYqLSUpQ0hJTERSRU5HNiI= CHRISTOFFEL Symbols Univ.-Prof. Dr.-Ing. habil. Josef BETTEN RWTH University Aachen Mathematical Models in Materials Science and Continuum Mechanics Augustinerbach 4-20 D-52056 A a c h e n , Germany betten@mmw.rwth-aachen.de Abstract A lot of tensor operations are included in the MAPLE tensor package. Examples are illustrated in the following MAPLE program, where the metric tensors and the CHRISTOFFEL symbols have been calculated for cylindrical and spherical coordinates. Keynotes: Metric tensors; CHRISTOFFEL symbols; Cylindrical and Spherical coordinates Definitions The CHRISTOFFEL symbols of the first kind are defined by restart: with(tensor): Gamma[kij]:=(1/2)*(g[ik,j]+g[jk,i]-g[ij,k]);

LCgmSSJnRzYiNiRJI2lrR0YlSSJqR0YlIyIiIiIiIyZGJDYkSSNqa0dGJUkiaUdGJUYpJkYkNiRJI2lqR0YlSSJrR0YlIyEiIkYr

where g[ij,k]:=Diff(g[ij],xi[k]);

LUklRGlmZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkJkkiZ0dGJzYjSSNpakdGJyZJI3hpR0YnNiNJImtHRic=

is the partial derivative of the covariant metric tensor with respect to curvlinear coordinates. Note that the CHRISTOFFEL symbols, in general, are not tensors unless the coordinate transformation is affine. Cylindrical Coordinates cylindrical_coord:=[r,phi,z];

NyVJInJHNiJJJHBoaUdGJEkiekdGJA==

covariant metric tensor: g_compts:=array(symmetric,sparse,1..3,1..3): g_compts[1,1]:=1: g_compts[2,2]:=r^2: g_compts[3,3]:=1: The components must be stored as an array of size equal to the rank of the object, with all index ranges beginning at one and ending at the dimension of the space. Only "square" ranges are allowed. The index character is stored as a list of positive and negative ones (1 or -1). A positive one (1) in the ith position in the index character list specifies that the ith index of the object is contravariant, while a negative one (-1) in the ith position in the index character list specifies that the ith index of the object is covariant. For example, [1,-1,1] specifies that the indices 1 and 3 are contravariant (written as superscripts), while the second index is covariant (written as subscript). g:=create([-1,-1], eval(g_compts));

PTYiSSZmYWxzZUclKnByb3RlY3RlZEdFXFtsI0kraW5kZXhfY2hhckdGIzckISIiRilJJ2NvbXB0c0dGIz02JEkqc3ltbWV0cmljR0YjSSdzcGFyc2VHRiM2JDsiIiIiIiRGMEVcW2wkNiRGMUYxRjE2JEYyRjJGMTYkIiIjRjcqJEkickdGI0Y3

D1g:=d1metric(g,cylindrical_coord): Gamma[kij]:=[ijk]; CHRISTOFFEL[first_kind][cylindrical]:= Gamma[kij]=Christoffel1(D1g);

NyNJJGlqa0c2Ig==

LzcjSSRpamtHNiI9RiVJJmZhbHNlRyUqcHJvdGVjdGVkR0VcW2wjSStpbmRleF9jaGFyR0YlNyUhIiJGLEYsSSdjb21wdHNHRiU9SSRjZjFHRiU2JTsiIiIiIiRGMUYxRVxbbDM2JSIiI0Y2RjMiIiE2JUYyRjNGNkY3NiVGNkYzRjZGNzYlRjJGNkYyRjc2JUYzRjNGNkY3NiVGNkYzRjNGNzYlRjJGMkY2Rjc2JUYyRjZGNkkickdGJTYlRjJGNkYzRjc2JUYzRjNGM0Y3NiVGNkY2RjZGNzYlRjJGMkYzRjc2JUY2RjNGMkY3NiVGNkY2RjIsJEY/Riw2JUYzRjNGMkY3NiVGMkYzRjNGNzYlRjJGMkYyRjc2JUYyRjNGMkY3

In matrix notation we arrive at: Gamma[1][ij]:=matrix(3,3,[[0,0,0],[0,r,0],[0,0,0]]); Gamma[2][ij]:=matrix(3,3,[[0,r,0],[-r,0,0],[0,0,0]]); Gamma[3][ij]:=matrix(3,3,[[0,0,0],[0,0,0],[0,0,0]]);

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRiciIiMiIiE2JEYqRidGKzYkRidGJkYrNiRGJkYmRis2JEYmRidGKzYkRiZGKkYrNiRGKkYmRis2JEYnRidGKzYkRipGKkkickdGIw==

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRiciIiMiIiE2JEYqRidGKzYkRidGJkYrNiRGJkYmRis2JEYmRidGKzYkRiZGKkkickdGIzYkRipGJiwkRjEhIiI2JEYnRidGKzYkRipGKkYr

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRiciIiMiIiE2JEYqRidGKzYkRidGJkYrNiRGJkYmRis2JEYmRidGKzYkRiZGKkYrNiRGKkYmRis2JEYnRidGKzYkRipGKkYr

Spherical Coordinates restart: with(tensor): spherical_coord:=[r,phi,theta];

NyVJInJHNiJJJHBoaUdGJEkmdGhldGFHRiQ=

g_compts:=array(symmetric,sparse,1..3,1..3): g_compts[1,1]:=1: g_compts[2,2]:=r^2: g_compts[3,3]:=(r*sin(phi))^2: g:=create([-1,-1],eval(g_compts));

PTYiSSZmYWxzZUclKnByb3RlY3RlZEdFXFtsI0kraW5kZXhfY2hhckdGIzckISIiRilJJ2NvbXB0c0dGIz02JEkqc3ltbWV0cmljR0YjSSdzcGFyc2VHRiM2JDsiIiIiIiRGMEVcW2wkNiQiIiNGNSokSSJyR0YjRjU2JEYxRjFGMTYkRjJGMiomRjdGNS1JJHNpbkc2JEYlSShfc3lzbGliR0YjNiNJJHBoaUdGI0Y1

D1g:=d1metric(g,spherical_coord): Gamma[kij]:=[ijk];CHRISTOFFEL[first_kind][spherical]:= Gamma[kij]=Christoffel1(D1g);

NyNJJGlqa0c2Ig==

LzcjSSRpamtHNiI9RiVJJmZhbHNlRyUqcHJvdGVjdGVkR0VcW2wjSStpbmRleF9jaGFyR0YlNyUhIiJGLEYsSSdjb21wdHNHRiU9SSRjZjFHRiU2JTsiIiIiIiRGMUYxRVxbbDM2JUYzRjNGMiwkKiZJInJHRiVGMi1JJHNpbkc2JEYoSShfc3lzbGliR0YlNiNJJHBoaUdGJSIiI0YsNiVGMkY/RjIiIiE2JUYzRjNGPywkKihGOEY/RjlGMi1JJGNvc0dGO0Y9RjJGLDYlRjJGM0Y/RkE2JUY/RjNGMkZBNiVGMkY/Rj9GODYlRj9GM0Y/RkE2JUY/Rj9GMiwkRjhGLDYlRj9GP0Y/RkE2JUYyRj9GM0ZBNiVGP0Y/RjNGQTYlRjJGMkYyRkE2JUY/RjNGM0ZENiVGM0YzRjNGQTYlRjJGMkY/RkE2JUYyRjJGM0ZBNiVGMkYzRjJGQTYlRjJGM0YzRjc=

In matrix notation we arrive at with(LinearAlgebra): Gamma[1][ij]:=matrix(3,3,[[0,0,0],[0,r,0],[0,0,r*(sin(phi))^2]]); Gamma[2][ij]:=matrix(3,3,[[0,r,0],[-r,0,0], [0,0,r^2*sin(phi)*cos(phi)]]); Gamma[3][ij]:=matrix(3,3,[[0,0,r*(sin(phi))^2],[0,0,r^2*sin(phi)*cos(phi)], [-r*(sin(phi))^2,-r^2*sin(phi)*cos(phi),0]]);

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRidGJiIiITYkRiYiIiNGKjYkRixGJ0YqNiRGLEYsSSJyR0YjNiRGJkYmRio2JEYnRicqJkYvRiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiM2I0kkcGhpR0YjRiw2JEYmRidGKjYkRixGJkYqNiRGJ0YsRio=

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRidGJiIiITYkRiYiIiNJInJHRiM2JEYsRidGKjYkRixGLEYqNiRGJkYmRio2JEYnRicqKEYtRiwtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiM2I0kkcGhpR0YjRiYtSSRjb3NHRjVGOEYmNiRGJkYnRio2JEYsRiYsJEYtISIiNiRGJ0YsRio=

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRidGJiwkKiZJInJHRiNGJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGIzYjSSRwaGlHRiMiIiMhIiI2JEYmRjQiIiE2JEY0RicqKEYsRjRGLUYmLUkkY29zR0YvRjJGJjYkRjRGNEY3NiRGJkYmRjc2JEYnRidGNzYkRiZGJ0YrNiRGNEYmRjc2JEYnRjQsJEY5RjU=

In a similar way one can find the CHRISTOFFEL symbols of the second kind. Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.