3.2 Vector and Tensor Mathematics

1 TensorMathematicsThevariablesusedto describe physicalquantitiesare of a number of types,includingscalars,vectors,and orthas been madein the notestoindicatethesetypes consistently as follows:s= scalar;lightfaceitalicv= Vector ;boldfacewithsingleunderscoreT= Tensor ;boldfacewithdoubleunderscore:( )Scalarsare usedto represent physicalquantitieswithno directionalqualities,such as temperature,volume,and usedfor quantitieswhichhave a singledirectionalquality such as velocity and (we willconsideronlysecond-ordertensors)are associatedwithquantitieswhich havetwo directionalcharacteristics,such as a momentum VectorOperationsGiven a coordinatesystemin threedimensions,a vectormay thus be repre-sented by an orderedset of threecomponents which represent its projectionsv1; v2; v3on the coordinateaxes1;2;3:v= [v1; v2; v3]:( )Thethreemostcommonlyusedcoordinatesyste msare rectangular,cylindri-cal, and spherical,as described in theCoordinateSystemsNotebook.

2 Alter-natively, a vectormay be represented by the sumof the magnitudesof itsprojectionson threemutuallyperpendicularaxes:v=v1 1+v2 2+v3 z3=3Xi=1vi i:( )Theunitvectors 1; 2; 3are x; y; zin the rectangularcoordinatesystem, r; ; zin cylindricalcoordinatesand r; ; in of theseunitvectorspoints in the directionof the indicatedspatialcoordinate,andhas a magnitudeof the vectorandtensoroperationsdescribed below are generallyapplicableto thesethreecoordinatesystems,withthe exceptionof di erentialoperators. Di erential operatorsinVectorandTensorMathematics24c ylindricaland sphericalcoordinatesmust be handledmoreexplicitlybecausein thosecases iare not constant in direction(withthe sole exceptionof zin cylindricalcoordinates).Themagnitudeof a vectoris given by:jvj=qv21+v22+v23=vuut3Xi=1v2i:( )Additionand subtractionof vectorsis easilyexecuted:v+w= (v1+w1) 1+ (v2+w2) 2+ (v3+w3) 3=3Xi=1(vi+wi) i;( )as well as multiplicationby a scalarsv= (sv1) 1+ (sv2) 2+ (sv3) 3=s3Xi=1vi i:( )Thedot productof two vectorsresultsin a scalar:v w=v1w1+v2w2+v3w3=3Xi=1viwi:( ) TensorOperationsA tensoris similarlyrepresented by an orderedarray of ninecomponents:T=264T11T12T13T21T22T23T3 1t23T33375:( )Thediagonalelements of a tensorare thosewhich have two identicalsub-scripts,whilethe otherelements are ofa tensoris obtainedby interchangingthe subscriptson each element:TT=264T11T21T31T12T22T32T13T32T3 3375:( )VectorandTensorMathematics25A tensoris described as symmetricwhenT=TT.

3 Onespecialtensoris theunittensor: =2641 0 00 1 00 0 1375:( )Thedyadicproductof two vectorsresultsin a Tensor ,as follows:vw=264v1w1v1w2v1w3v2w1v2w2v2w3v3 w1v3w2v3w3375:( )Thisleadsto the de nitionof the unitdyads,of which thereare nine: 1 1=2641 0 00 0 00 0 0375;( ) 1 2=2640 1 00 0 00 0 0375; etc:( )In a similarmannerto vectors,tensorsare easilyaddedT+U=264T11+U11T12+U12T13+U13T 21+U21T22+U22T23+U23T31+U31T32+U32T33+U3 3375=3Xi=13Xj=1(Tij+Uij) i j;( )or multipliedby scalars:sT=264sT11sT12sT13sT21sT22sT23sT 31sT23sT33375=s3Xi=13Xj=1 Tij i j:( )Thedoubledot productof two tensorsresultsin a scalar:T:U=T11U11+T12U21+T13U31+T21U12+T 22U22+T23U32+T31U13+T32U23+T33U33=3Xi=13 Xj=1 TijUji:( )Thedot productof a tensorwitha vectoris:T v= 1(T11v1+T12v2+T13v3) + 2(T21v1+T22v2+T23v3) + 3(T31v1+T32v2+T33v3)=3Xi=1 contrast,the dot productof a vectorwitha tensoris:v T= 1(v1T11+v2T21+v3T31) + 2(v1T12+v2T22+v3T32) + 3(v1T13+v2T23+v3T33)=3Xi=1 general,(T v)6= (v T), however, theyare equalifTis a tensoris de nedas:jTj=s12(T:TT) =vuut12 XiXjT2ij:( ) simpleintroductionto vectorsand tensorsis providedby: H.

4 Anton,ElementaryLinearAlgebra,4th Ed., JohnWileyandSons,NewYork (1984).Numerousproblems,somewithsolution smay be foundin: , Schaum'sOutlineSeries,McGraw-HillBook Company (1959). F. Ayres,Matrices, Schaum'sOutlineSeries,McGraw-HillBook Com-pany (1962).An excellent discussionof vectorandtensornotationwhich is particularlyrelevant to polymerprocessingis in AppendixA of: , ,O. Hassager,Dynamicsof PolymerLiquids,Vol. 1, JohnWileyand Sons,NewYork (1987).Copyright2001.