88Introduction to Tensor Calculus and Continuum Me

65x1.3SPECIALTENSORSKnowinghowtensorsaredenedandrecognizingatensorwhenitpopsupinfrontofyouaretwodierentthings.Somequantities,whicharetensors,frequentlyariseinappliedproblemsandyoushouldlearntorecognizethesespecialtensorswhentheyoccur.Inthissectionsomeimportanttensorquantitiesaredened.Wealsoconsiderhowthesespecialtensorscaninturnbeusedtodeneothertensors.MetricTensorDeneyi;i=1;:::;NasindependentcoordinatesinanNdimensionalorthogonalCartesiancoordinatesystem.Thedistancesquaredbetweentwopointsyiandyi+dyi;i=1;:::;Nisdenedbytheexpressionds2=dymdym=(dy1)2+(dy2)2+


+(dyN)2:(1:3:1)Assumethatthecoordinatesyiarerelatedtoasetofindependentgeneralizedcoordinatesxi;i=1;:::;Nbyasetoftransformationequationsyi=yi(x1;x2;:::;xN);i=1;:::;N:(1:3:2)Toemphasizethateachyidependsuponthexcoordinateswesometimesusethenotationyi=yi(x);fori=1;:::;N:Thedierentialofeachcoordinatecanbewrittenasdym=@ym@xjdxj;m=1;:::;N;(1:3:3)andconsequentlyinthex-generalizedcoordinatesthedistancesquared,foundfromtheequation(1.3.1),becomesaquadraticform.Substitutingequation(1.3.3)intoequation(1.3.1)wendds2=@ym@xi@ym@xjdxidxj=gijdxidxj(1:3:4)wheregij=@ym@xi@ym@xj;i;j=1;:::;N(1:3:5)arecalledthemetricesofthespacedenedbythecoordinatesxi;i=1;:::;N:Herethegijarefunctionsofthexcoordinatesandissometimeswrittenasgij=gij(x):Further,themetricesgijaresymmetricintheindicesiandjsothatgij=gjiforallvaluesofiandjovertherangeoftheindices.Ifwetransformtoanothercoordinatesystem,sayxi;i=1;:::;N,thentheelementofarclengthsquaredisexpressedintermsofthebarredcoordinatesandds2=gijdxidxj;wheregij=gij(x)isafunctionofthebarredcoordinates.Thefollowingexampledemonstratesthatthesemetricesaresecondordercovarianttensors.66EXAMPLE1.3-1.Showthemetriccomponentsgijarecovarianttensorsofthesecondorder.Solution:Inacoordinatesystemxi;i=1;:::;Ntheelementofarclengthsquaredisds2=gijdxidxj(1:3:6)whileinacoordinatesystemxi;i=1;:::;Ntheelementofarclengthsquaredisrepresentedintheformds2=gmndxmdxn:(1:3:7)Theelementofarclengthsquaredistobeaninvariantandsowerequirethatgmndxmdxn=gijdxidxj(1:3:8)Hereitisassumedthatthereexistsacoordinatetransformationoftheformdenedbyequation(1.2.30)togetherwithaninversetransformation,asinequation(1.2.32),whichrelatesthebarredandunbarredcoordinates.Ingeneral,ifxi=xi(x),thenfori=1;:::;Nwehavedxi=@xi@xmdxmanddxj=@xj@xndxn(1:3:9)Substitutingthesedierentialsinequation(1.3.8)givesustheresultgmndxmdxn=gij@xi@xm@xj@xndxmdxnorgmn−gij@xi@xm@xj@xndxmdxn=0Forarbitrarychangesindxmthisequationimpliesthatgmn=gij@xi@xm@xj@xnandconsequentlygijtransformsasasecondorderabsolutecovarianttensor.EXAMPLE1.3-2.(Curvilinearcoordinates)Considerasetofgeneraltransformationequationsfromrectangularcoordinates(x;y;z)tocurvilinearcoordinates(u;v;w).Thesetransformationequationsandthecorrespondinginversetransformationsarerepresentedx=x(u;v;w)y=y(u;v;w)z=z(u;v;w):u=u(x;y;z)v=v(x;y;z)w=w(x;y;z)(1:3:10)Herey1=x;y2=y;y3=zandx1=u;x2=v;x3=waretheCartesianandgeneralizedcoordinatesandN=3:Theintersectionofthecoordinatesurfacesu=c1,v=c2andw=c3denecoordinatecurvesofthecurvilinearcoordinatesystem.Thesubstitutionofthegiventransformationequations(1.3.10)intothepositionvector~r=xbe1+ybe2+zbe3producesthepositionvectorwhichisafunctionofthegeneralizedcoordinatesand~r=~r(u;v;w)=x(u;v;w)be1+y(u;v;w)be2+z(u;v;w)be367andconsequentlyd~r=@~r@udu+@~r@vdv+@~r@wdw;where~E1=@~r@u=@x@ube1+@y@ube2+@z@ube3~E2=@~r@v=@x@vbe1+@y@vbe2+@z@vbe3~E3=@~r@w=@x@wbe1+@y@wbe2+@z@wbe3:(1:3:11)aretangentvectorstothecoordinatecurves.Theelementofarclengthinthecurvilinearcoordinatesisds2=d~r
d~r=@~r@u
@~r@ududu+@~r@u
@~r@vdudv+@~r@u
@~r@wdudw+@~r@v
@~r@udvdu+@~r@v
@~r@vdvdv+@~r@v
@~r@wdvdw+@~r@w
@~r@udwdu+@~r@w
@~r@vdwdv+@~r@w
@~r@wdwdw:(1:3:12)Utilizingthesummationconvention,theabovecanbeexpressedintheindexnotation.Denethequantitiesg11=@~r@u
@~r@ug21=@~r@v
@~r@ug31=@~r@w
@~r@ug12=@~r@u
@~r@vg22=@~r@v
@~r@vg32=@~r@w
@~r@vg13=@~r@u
@~r@wg23=@~r@v
@~r@wg33=@~r@w
@~r@wandletx1=u;x2=v;x3=w:Thentheaboveelementofarclengthcanbeexpressedasds2=~Ei
~Ejdxidxj=gijdxidxj;i;j=1;2;3wheregij=~Ei
~Ej=@~r@xi
@~r@xj=@ym@xi@ym@xj;i;jfreeindices(1:3:13)arecalledthemetriccomponentsofthecurvilinearcoordinatesystem.Themetriccomponentsmaybethoughtofastheelementsofasymmetricmatrix,sincegij=gji:Intherectangularcoordinatesystemx;y;z;theelementofarclengthsquaredisds2=dx2+dy2+dz2:Inthisspacethemetriccomponentsaregij=0@1000100011A:68EXAMPLE1.3-3.(Cylindricalcoordinates(r;;z))Thetransformationequationsfromrectangularcoordinatestocylindricalcoordinatescanbeexpressedasx=rcos;y=rsin;z=z:Herey1=x;y2=y;y3=zandx1=r;x2=;x3=z;andthepositionvectorcanbeexpressed~r=~r(r;;z)=rcosbe1+rsinbe2+zbe3:Thederivativesofthispositionvectorarecalculatedandwend~E1=@~r@r=cosbe1+sinbe2;~E2=@~r@=−rsinbe1+rcosbe2;~E3=@~r@z=be3:Fromtheresultsinequation(1.3.13),themetriccomponentsofthisspacearegij=0@1000r200011A:Wenotethatsincegij=0wheni6=j,thecoordinatesystemisorthogonal.Givenasetoftransformationsoftheformfoundinequation(1.3.10),onecanreadilydeterminethemetriccomponentsassociatedwiththegenerali