Thedifferencebetweentwopoints X and Y ofspace ℰ isdefinedby v = X Y , (1.1) where v isthevectorwhoseoriginisin Y andendsat X .Allthevectorswhichcan bedeterminedthroughthedifferencesbetweenpointsbelongingto ℰ formtheset V associatedwith ℰ .Theset V isa(real)vectorspace,wherethetwobasicoperations inherenttothenotionofvectorspacearedefined,whichare(i)theadditionofvectors and(ii)theproductofavectorbyarealnumber.
Theadditionoperationbetweenapoint Y ∈ ℰ andthevector v ∈ V definesthepoint X ∈ ℰ suchthat(1.1)isverified.Thisoperationallowsustoestablishabiunivocalcorrespondencebetweenpointsof ℰ andvectorsof V .Infact,wecanarbitrarilypicka point O from ℰ ,andthenforeachpoint X ∈ ℰ thereexistsauniquevector v ∈ V such that v = X O
Space V iscalledthree-dimensionalprovidedthereexistin V setsofthreevectors {ei }={e1 , e2 , e3 } whicharelinearlyindependent1 andcanspantheentirevectorspace V ,thatis,theycangenerateanyvector v ∈ V throughthelinearcombination v = ��i ei . 2
Anyoftheselinearlyindependentsetsofvectors {ei } iscalledabasisforthe(real)vector space V
Beyondbasicoperations,multiplicationbyarealnumber,andadditionofvectors, thevectorspace V associatedwith ℰ isendowedwithaninnerproduct,alsocalledthe scalarproduct,operationbetweenvectorsof V .For u, v ∈ V ,thisoperationisdenoted by u ⋅ v ,whichisgeometricallydefinedbytheproductofthelengthsofthevectorsmultipliedbythecosineoftheanglebetweenthem.Thisoperationsatisfiestheproperties oftheinnerproductinthesenseofthealgebra.
1Vectors {ei } arecalledlinearlyindependentif ��i ei = �� implies ��i = 0for i = 1, 2, 3. 2Throughoutthisbook,unlessstatedotherwise,Einsteinnotationisadoptedtoshortensummation notations,resulting,forexample,inthefollowinglumpednotationwhenindexesarerepeated: v = ∑3 i 1 ��i ei = ��i ei
Also,andthroughtheintroductionofthisoperation,thespace V hasatopological structureinducedbytheinnerproductthroughthedefinitionofthenormoperation foravector v ∈ V
||v || = √v v . (1.2)
Makinguseoftheseoperations,vectors u, v ∈ V aresaidtobeorthogonalif u v = 0. Similarly,abasis {ei } of V iscalledorthogonalif ei ej = ||ei ||||ej ||��ij isverified, i, j = 1, 2, 3,where ��ij istheKroneckersymbol.3 Finally,abasisof V iscalledorthonormal ifitisorthogonalandthenormofthevectorsinthebasisisunitary,thatis, ||ei || = 1, i = 1, 2 , 3.
Thematrix [gij ], i, j ∈{1, 2, 3},definedby
gij = ei ⋅ ej , i, j = 1, 2, 3, (1.3)
isnotsingularwhentheset {ei } isabasisfor V .Inturn,fromthedefinitionitfollows that gij = gji ,thatis, [gij ] isasymmetricmatrix.Letuscall
Itcanbeprovedthatthistransformationproducesabasisfor V .Reciprocally,theapplicationof [gij ] over {ei } yieldstheoriginalbasis {ei }.Infact
gij ej = gij g jk ek = ��ik ek = ei . (1.7)
Anotherusefulresultisthefollowing
ei ⋅ ej = g ik ek ⋅ ej = g ik gkj = ��ij (1.8)
Acoordinatesystemconsistsofabasis {ei } for V ,notnecessarilyorthogonal,and anarbitrarypoint O of ℰ calledtheoriginofthecoordinatesystem.Whenthebasisis orthonormal,thecoordinatesystemiscalledCartesian.
Observethatthenotionofvectorwasintroducedindependentlyfromtheadopted basis,or,equivalently,fromthecoordinatesystem.Whenabasis {ei },andthenitsdual basis {ei },ischosen,theneachvector u ∈ V canbeassociatedwithatripleofrealnumbers {u1 , u2 , u3 } calledcomponentsof u withrespecttothebasis {ei },whicharedefined asfollows
ui = u ⋅ ei i = 1, 2, 3 (1.9)
Thesearethecomponentswithrespectto {ei } because,as {ei } isabasis,itresultsin u = ��i ei andthen
Likewise,givenacoordinatesystemin ℰ ,characterizedbythebasis {ei } andthepoint O ∈ ℰ ,wecandefinethecoordinatesofanarbitrarypoint X ∈ ℰ asthecovariantcomponentsofthevector X O from V ,thatis
=(X O) ⋅ ei (1.13)
Thus,thesamevector u ∈ V ,orthesamepoint X ∈ ℰ ,canbeassociatedwithdifferenttriplesofcomponentsandrepresentationsdependinguponthechosencoordinate system.4
Tounderlinethedifferencebetweenintrinsicandindicialnotation,notethat theinnerproductofvectors u, v ∈ V asafunctionoftheirdifferentcomponents resultsin
= I(Tv )= Tv , (1.34) andthenwehaveproved(1.30). Thefollowingnotationwillalsobeused
n = n ⏞⏞⏞ TT T n ∈ N, (1.35) with T0 = I. Thetransposeofatensor T istheuniquetensor TT satisfying Tu v = u TT v ∀u, v ∈ V (1.36)
Uniquenessisprovedassumingthattherearetwotensortransposesfor T,denotedby TT 1 and TT 2 ,andwhichwillbeassumedtobedifferent.Fromdefinition(1.36),eachtensor satisfies Tu ⋅ v = u ⋅ TT 1 v ∀u, v ∈ V , (1.37) Tu ⋅ v = u ⋅ TT 2 v ∀u, v ∈ V (1.38)
Subtractingbothexpressionsyields
Recallingthat a b = 0, ∀a ∈ V implies b = ��,wehave
Usingthedefinitionofthetransposeofatensoritisstraightforwardtoconcludethat, forarbitrary A, S ∈ Lin and �� ∈ ℝ,weobtain
(S + A)T = ST + AT , (1.42)
(�� S)T = �� ST (1.43)
Infact,takingarbitrary u, v ∈ V ,then
(S + A)T u v = u (S + A)v = u Sv + u Av = ST u v + AT u v =(ST u + AT u) ⋅ v =(ST + AT )u ⋅ v , (1.44) andso(1.42)isproved.Analogously,
(�� S)T u v = u (�� S)v = �� u Sv = �� ST u v , (1.45) andso(1.43)isdemonstrated.
Asitiseasytosee,thetransposeoperationisalineartransformationbetween Lin and Lin.Inaddition,givenarbitrary A, S ∈ Lin,itis
(SA)T = AT ST , (1.46)
(ST )T = S (1.47)
Infact,forarbitrary u, v ∈ V ,wehave
(SA)T u v = u (SA)v = u S(Av )= ST u Av = AT ST u v , (1.48) andwearriveat(1.46).Analogously
(ST )T u v = u (ST )v = Su v , (1.49) andthen(1.197)isverified.
Atensor S iscalledsymmetricif
Su v = u Sv ∀u, v ∈ V , (1.50) andinsuchacaseweconcludethat S = ST .Atensoriscalledskew-symmetricif
Su ⋅ v =−(u ⋅ Sv )∀u, v ∈ V , (1.51)
whichimplies S =−ST .
Thesetofallsymmetrictensorswillbedenotedby Sym andthesetofall skew-symmetrictensorswillbedenotedby Skw.Inparticular,thenulltensor O issymmetricandskew-symmetric.
Anytensor S ∈ Lin canbeunivocallyrepresentedbytheadditionofasymmetrictensor(called Ss )andaskew-symmetrictensor(called Sa ),thatis,
S = Ss + Sa , (1.52) where
Ss = 1 2 (S + ST ), (1.53)
Sa = 1 2 (S ST ), (1.54)
1.2Second-OrderTensors 9 are,respectively,calledthesymmetriccomponentandtheskew-symmetriccomponentof S.
Sincethetransposeoperationyieldsauniquetensortranspose,itfollowsthatthe linearcombinationofasymmetric(skew-symmetric)tensorresultsinasymmetric (skew-symmetric)tensor.Then, Sym and Skw arevectorsubspacesof Lin.Moreover, fromtheuniquenessofthedecompositionintosymmetricandskew-symmetric components,itisconcludedthat Lin canbewrittenasthedirectsumofthesetwo subspaces
Lin = Sym ⊕ Skw (1.55)
Consideranarbitrary W ∈ Skw andanarbitrary T ∈ Lin.Then,itispossibletoshow thatforany u ∈ V thefollowingholds
u Wu = 0, (1.56)
u Tu = u Ts u. (1.57)
Infact,forarbitrary u ∈ V wehave u ⋅ Wu = WT u ⋅ u =−Wu ⋅ u, (1.58) fromwhere(1.56)isproved.Similarly,andusing(1.56),wehave u Tu = u (Ts + Ta )u = u Ts u + u Ta u = u Ts u, (1.59) and(1.57)follows.
Thetensorproductbetweentwovectors a, b ∈ V isthesecond-ordertensor a ⊗ b thattransformsanyvector v ∈ V intovector (b ⋅ v )a,thatis,
(a ⊗ b)v =(b ⋅ v )a ∀v ∈ V (1.60)
Fromthepreviousdefinition,andgivenarbitrary a, b, c, d ∈ V ,weobtainthefollowingresults
(a ⊗ b)T = b ⊗ a, (1.61) (a ⊗ b)(c ⊗ d)=(b c)(a ⊗ d). (1.62)
Inaddition,given T ∈ Lin,itcanbeshownthat T(a ⊗ b)=(Ta) ⊗ b, (1.63)
(a ⊗ b)T = a ⊗ (TT b) (1.64)
Indeed,considerarbitrary u, v ∈ V ,usingdefinition(1.60)forwardandbackward, wehave
(a ⊗ b)T u v = u (a ⊗ b)v = u (b v )a =(u a)(b v )=(b ⊗ a)u v , (1.65) andthen(1.61)holds.Now,observethat
Figure1.1 Geometricconceptoforthogonal projectionovertheplane �� whosenormalvectoris n.
(I n ⊗ n)v v
whichyields(1.63).Lastly,notethat
(a ⊗ b)Tu =(b Tu)a =(TT b u)a =(a ⊗ (TT b))u, (1.68) andthen(1.64)isproved.
Analogouslytothedefinitionoftheinnerproductin V ,itisalsopossibletodefine theinnerproductin Lin.Considertwoelements T, S ∈ Lin whichcanbewritteninthe forms T = t1 ⊗ t2 and S = s1 ⊗ s2 ,respectively,with t1 , t2 , s1 , s2 ∈ V .Then,wedefine theinnerproduct T S in Lin as
T ⋅ S =(t1 ⊗ t2 ) ⋅ (s1 ⊗ s2 )=(t1 ⋅ s1 )(t2 ⋅ s2 ) (1.69)
Withdefinition(1.69),itisstraightforwardtoprovethat,forarbitrary T ∈ Lin and u, v ∈ V ,thefollowingresultholds
T ⋅ (u ⊗ v )= u ⋅ Tv (1.70)
Infact,putting T = t1 ⊗ t2 ,andmakinguseofdefinitions(1.69)and(1.60)weobtain
T ⋅ (u ⊗ v )=(t1 ⊗ t2 ) ⋅ (u ⊗ v )=(t1 ⋅ u)(t2 ⋅ v ) = u ⋅ [(t2 ⋅ v )t1 ]= u ⋅ [(t1 ⊗ t2 )v ]= u ⋅ Tv (1.71)
Letusdenote n theunitnormalvectortotheplane �� (seeFigure1.1).Thetensor n ⊗ n appliedoveranyvector v ∈ V gives
(n ⊗ n)v =(n ⋅ v )n, (1.72)
whichistheorthogonalprojectionof v ∈ V overthedirection n.Inturn,the second-ordertensor P = I n ⊗ n appliedoveranyvector v ∈ V yields
Pv =(I n ⊗ n)v = v −(n ⋅ v )n, (1.73)
whichistheorthogonalprojectionof v overtheplane �� . Itcanbeappreciatedthattensor P issymmetricandalsoverifies P2 = P.Infact
PT =(I n ⊗ n)T = IT −(n ⊗ n)T = I n ⊗ n = P, (1.74) and,forarbitrary v ∈ V ,itis
P2 v = P(Pv )= P[v −(n ⋅ v )n]= Pv −(n ⋅ v )Pn = Pv , (1.75) andthepreviousstatementshold.
1.2Second-OrderTensors 11
Tensorssatisfyingthesetwoproperties,thatis, P ∈ Sym and P2 = P,arecalled orthogonalprojectiontensors.Examplesofthiskindoftensorare
I, I n ⊗ n, n ⊗ n (1.76) Itispossibletoshowthat
dim(Lin)= dim V × dim V = 9, (1.77) andifwetakeabasis {ei } for V ,thesets
{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.78)
{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.79)
{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.80)
{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.81) aredifferentpossiblebasesfor Lin.Inthisway,anytensor T ∈ Lin canbeexpressedbya uniquelinearcombinationoftheelementofthechosenbasis.Thecomponentsoftensor T inthechosenbasisaredefinedinananalogousmannertothatforvectors.Hence,we have
Tij = T (ei ⊗ ej )= ei Tej , (1.82)
T ij = T ⋅ (ei ⊗ ej )= ei ⋅ Tej , (1.83)
T i j = T ⋅ (ei ⊗ ej )= ei ⋅ Tej , (1.84)
T j i = T (ei ⊗ ej )= ei Tej , (1.85)
wheretheinnerproductbetweenelementsof Lin isdefinedin(1.69).Also,thisinner productcanbeintroducedintermsofthetraceoperation,asshownbelow.Thisway, therepresentationof T ∈ Lin intermsofthesecomponentsisgivenby
T = Tij (ei ⊗ ej )= T ij (ei ⊗ ej )= T i j (ei ⊗ ej )= T j i (ei ⊗ ej ), (1.86) wheretheimplicitsummationofrepeatedindicesisconsidered.Coefficients Tij , T ij , T i j and T j i are,respectively,covariant,contravariant,andmixedcomponentsoftensor T. Inparticular,theidentitytensor I is
I = ei ⊗ ei , (1.87) againwithimplicitsummationoverindex i.Infact,givenarbitrary u = uj ej = uj ej ∈ V , wehave
Iu =(ei ⊗ ei )uj ej = uj ei ��ij = ui ei = u, (1.88) orequivalently,
Iu =(ei ⊗ ei )uj ej = uj g ij ei = uj ej = u (1.89)
ForaCartesianbasis {ei } for V ,thereisnodifferencebetweencomponentsof T.In thiscase,wesimplyrefertotheCartesiancomponentsofthetensor. Theadvantageofemployingcompactnotationisagainevidentwhencomparingto indicialnotation.Atensorisaconcept(lineartransformationin V )whichdoesnot
12 1VectorandTensorAlgebra
dependonthebasischosenfor V .Thesameisvalidforthetraceoperationandinner productbetweentensors.
• Tensor.Compactnotation: T ∈ Lin.Indicialnotation
Tij = T kl gki glj = T k j gki = T k i gkj , (1.90)
T ij = Tkl g ki g lj = T i k g kj = T j k g ki , (1.91)
T i j = Tkj g ki = T ik gkj = T l k g ki glj , (1.92)
T j i = Tik g kj = T kj gki = T k l gki glj (1.93)
• Applicationofatensoroveravector.Compactnotation: u = Tv , u, v ∈ V and T ∈ Lin.Indicialnotation(justsomeofthepossibleexpressions)
ui = Tij vj = Tij g jk vk = T j i vj = T j i gjk vk , (1.94) ui = T ijvj = T ij gjk vk = T i j vj = T j i g jk vk . (1.95)
• Compositionoftensors.Compactnotation: T = AB, T, A, B ∈ Lin.Indicialnotation
Tij = Aik Bkj = A k i Bkj , (1.96)
T ij = Aik B j k = Aik Bkj , (1.97)
T i j = Aik Bkj = Aik Bkj , (1.98)
T j i = A k i B j k = Aik Bkj , (1.99) andvariantsincludingthetensor g ij or gij .
• Tensorproductbetweenvectors.Compactnotation: u ⊗ v , u, v ∈ V .Indicialnotation
(u ⊗ v )ij = ui vj , (1.100)
(u ⊗ v )ij = ui vj , (1.101)
(u ⊗ v )i j = ui vj , (1.102)
(u ⊗ v ) j i = ui vj (1.103)
• Symmetriccomponentofatensor.Compactnotation: S ∈ Sym ⇔ S = ST ,andequivalently Sa = O.Indicialnotation
Sij = Sji =⇒ [Sij ]=[Sij ]T , (1.104)
S ij = S ji =⇒ [S ij ]=[S ij ]T , (1.105)
S i j = S i j (1.106)
Notethatforasymmetrictensorthematrixofcovariantcomponentsisalsosymmetric,andthesameholdsforcontravariantcomponents.Incontrast,thematrix representationofasymmetrictensorinmixedcomponentsisnotsymmetricingeneral.Indeed,since S i j = S i j ,weconcludethatthesamecomponentsaresymmetrically placedinthetwo(different)matrixrepresentationsofthetensor.
