3.2 Vector and Tensor Mathematics - PolymerProcessing.com

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. 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.

Vector and Tensor Mathematics 23 3.2 Vector and Tensor Mathematics Thevariablesusedtodescribephysicalquantitiesareofanumberoftypes, includingscalars,vectors,andtensors.

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