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1、MathematicsforEngineersPartII(ISE)Version1.2/2007-07-11MathematicsforEngineersPartII(ISE)Prof.Dr.J.GottschlingM.Sc.M.SaleemVersion1.2(Date:11/07/2007)UniversitätDuisburg-Essen,CampusDuisburgInstituteforAppliedMaterialTechnology(IAM)Lecturer:Prof.Dr.JohannesGotts
2、chling1MathematicsforEngineersPartII(ISE)Version1.2/2007-07-11n1CurvesinR1.1ParametricdescriptionofcurvesWeextendthetheoryofderivativesandintegralstofunctionswhoserangearenvectorsinRinsteadofrealnumbers.Definition1.1.1:AcurveCinRnisthegraphofafunctionf:I®Rnfroma
3、nintervalI=[a;b]ÌRintoRnsuchthateachtinIhastheimage(1.1.1)f(t)=(f(t),f(t),...,f(t))ÎRn.12nThecomponentsf:I®IR,taf(t),i=1,2,...,n,iioff(t)=(f(t),f(t),...,f(t))arecontinuousfunctionsontheintervalI.Equation(1.1.1)12niscalledtheparametricequationofthecurveC.Thepoint
4、sf(a)andf(b)arecalledtheendpointsofthecurve.Remark1.1.1:Ifweidentifyeachpointf(t)=(f(t),f(t),...,f(t))ÎRnf(a)f(t)=(f1(t),f2(t),f3(t))12n···f(b)withitspositionvector[f(t),f(t),...,f(t)],12n[f1(t),f2(t),f3(t)]thenwecanunderstandfasavector-valuedfunction.Example1.1
5、.1:3Fig.1.1.1ParametriccurveinR(i)Supposer>0isarealnumber,thenthegraphofthecurveCgivenbytheparametricequation1-1MathematicsforEngineersPartII(ISE)Version1.2/2007-07-112f:[0;2π]®IR,ta(r×cost,r×sint)2isacircleinRofradiusr(Fig.1.1.2).3(ii)Letf:IR®IR,ta(r×cost,r×sin
6、t,ct),r>0,cÎR,be3theparametricequationofacurveinR.Thegraphofthecurvetracedbyf(t)astvariesiscalledacircularhelix2(Fig.1.1.3).Fig.1.1.2CircleinR(iii)Supposeg:I®RisarealvaluedcontinuousfunctionontheintervalIÌR.Thegraphofthisfunctiongisa22subsetofRandcanbeunderstood
7、asacurveinRwiththeparametricequation2(1.1.2)f:I®IR,ta(t,g(t)).Definition1.1.2:nLetIÌRbeanopenintervalandf=(f,f,...,f):I®IR.12nThecurveCgivenbyfiscalleddifferentiableifeachcomponentFig.1.1.3circularhelixf:I®IR,taf(t),i=1,2,...,n,iiisdifferentiableattÎI.Thevectorf
8、¢(t)=[f¢(t),f¢(t),...,f¢(t)]12niscalledtangentvectorofthecurvefatthepointf(t).Remark1.1.2:(i)Iff¢(t)¹[0,...,0],thenaunittangentvectortofisgivenbytheformulaf¢(t)[f¢(t)
