e is identi64257ed with the vector that is obtained by translating to the point Thus every vector 64257eld on is uniquely determined by a function from Ra64257kul Alam IITG MA102 2013 brPage 3br Vector Fields Curl and Divergence Examples of vector ID: 50437
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VectorFields,CurlandDivergence Figure:Examplesofvectorelds RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence Figure:VectoreldrepresentingHurricaneKatrina RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence Figure:Examplesofvectorelds RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence ScalareldsAfunctionf:URn!RiscalledascalareldonUasitassignsanumbertoeachpointinU:ThetemperatureofametalrodthatisheatedatoneendandcooledonanotherisdescribedbyascalareldT(x;y;z):The owofheatisdescribedbyavectoreld.Theenergyorheat uxvectoreldisgivenbyJ:=�rT;where�0:ˆLetFbeavectoreldinRn:ThenF:=(f1;:::;fn)forsomescalareldsf1;:::;fnonRn:WesaythatFisaCkvectoreldiff1;:::;fnareCkfunctions.AllvectoreldsareassumedtobeC1unlessotherwisenoted. RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence IntegralcurvesforvectoreldsDenition:LetFbeavectoreldinRn:ThenaC1curvex:[a;b]!RnissaidtobeanintegralcurveforthevectoreldFifF(x(t))=x0(t)fort2[a;b]:Obviously,Fisatangent(velocity)vectoreldontheintegralcurve.Thusintegralcurvesprovideageometricpictureofavectoreld.Integralcurvesarealsocalled owlinesorstreamlines.AnintegralcurvemaybeviewedasasolutionofasystemofODE.Indeed,forn=3andF=(P;Q;R);wehavex0(t)=P(x(t);y(t);z(t));y0(t)=Q(x(t);y(t);z(t));z0(t)=R(x(t);y(t);z(t)): RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence DivergenceofvectoreldsDenition:LetF=(f1;:::;fn)beavectoreldinRn:ThenthedivergenceofFisascalareldonRngivenbydivF=@1f1++@nfn=@f1 @x1++@fn @xn:Denethedeloperatorr:=(@1;:::;@n)=(@ @x1;;@ @xn)=e1@ @x1++en@ @xn:Thenapplyingrtoascalareldf:Rn!Rweobtainthegradientvectoreldrf=(@1f;:::;@nf): RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence ExamplesˆThestreamlinesofthevectoreldF(x;y):=(x;y)arestraightlinesdirectedawayfromtheorigin.For uid ow,thismeansthe uidisexpandingasitmovesoutfromtheorigin,sodivFshouldbepositive.Indeed,wehavedivF=2�0:ˆNext,considerthevectoreldF(x;y):=(x;�y):ThendivF=0sothatneitherexpansionnorcompressiontakesplace. RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence CrossproductinR3ItiscustomarytodenotethestandardbasisinR3byi:=(1;0;0);j:=(0;1;0)andk:=(0;0;1):Ifu=u1i+u2j+u3kandv=v1i+v2j+v3kthenuv=(u2v3�v2u3)i+(u1v3�v1u3)j+(u1v2�v2u1)k=detijku1u2u3v1v2v3thelastequalityisonlysymbolic. RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence IrrotationalvectoreldAvectoreldFinR3iscalledirrotationalifcurlF=0:Thismeans,inthecaseofa uid ow,thatthe owisfreefromrotationalmotion,i.e,nowhirlpool.Fact:IffbeaC2scalareldinR3:Thenrfisanirrotationalvectoreld,i.e.,curl(rf)=0:Proof:Wehavecurl(rf)=rrf=ijk@ @x@ @y@ @zfxfyfz=0becauseoftheequalityofmixedpartialderivatives.Observation:IfcurlF6=0thenFisnotaconservative(gradient)vectoreld. RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence ScalarcurlLetF=(P;Q)beavectoreldinR2:ThenidentifyingR2withthex-yplaneinR3;FcanbeidentiedwiththevectoreldF=Pi+QjinR3:ThenwehavecurlF=(@Q @x�@P @y)k:Thescalareld@Q @x�@P @yiscalledthescalarcurlofthevectoreldF=(P;Q):ThescalarcurlofF(x;y):=(�y2;x)isgivenby@x(x)�@y(�y2)=1+2y: RakulAlam IITG:MA-102(2013) VectorFields,CurlandDivergence LaplaceoperatorExample:IfF(x;y;z):=(x;y;z)thendivF=36=0soFdoesnothaveavectorpotential.Laplaceoperatorr2:LetfbeaC2scalareldinRn:Thenr2f:=rrf=div(rf)=@21f++@2nfdenestheLaplaceoperatorr2onf:ForaC2functionu:R3!Rthepartialdi erentialequationr2u=@2u @x2+@2u @y2+@2u @z2=0iscalledLaplaceequation. RakulAlam IITG:MA-102(2013)