Div， grad curl and all that (已读).pdf

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1、6Div,gradcurlandallthat6.1Fundamentaltheoremsforgradient,divergence,andcurlFigure1:Fundamentaltheoremofcalculusrelatesdf=dxover[a;b]andf(a);f(b).YouwillrecallthefundamentaltheoremofcalculussaysZbdf(x)dx=f(b)¡f(a);(1)adxinotherwordsit'saconnectionbetweentherateofchangeofthefunctiono

2、vertheinterval[a;b]andthevaluesofthefunctionattheendpoints(boundaries)ofthatinterval.Thereareequivalentfundamentaltheorems"forlineintegrals,areaintegrals,andvolumeintegrals.Invectorcalculuswedealwithdi®erenttypesofchangesofscalarandvector¯elds,e.g.r~Á,r¢~~v,andr£~~v,&eachhasitsown

3、theorem.We'vediscussedlineintegralsbefore,mostlyinthecontextoftheworkdonealongapath,butlet'sremindourselvesofthede¯nition:ZBXnF~(~r)¢d~r=nlim!1F~(~ri)¢d~ri;(2)Ai=1inotherwordsweadduptheareaofallthelittlerectangles"F~(~ri)¢d~riconsistingofthevectorF~atthepoint~ridottedintothepathel

4、ementd~ri,seeFigure.Rememberthepointisthatalthoughwearedoinganintegralina2Dspace,weareconstrainedtomovealongapath,sothereisonlyonerealindependentvariable.1Figure2:Lineintegral.Examplefromtest:Considerthetriangleinthe(x;y)planewithverticesat(-1,0),(1,0),and(0,1).Evaluatetheclosedlin

5、eintegralII=(¡yx^+xy^)¢d~r(3)aroundtheboundaryofthetriangleintheanticlockwisedirection.Sincethevector~rhascomponents(x;y),themeasureisd~r=^xdx+^ydy,so(¡yx^+xy^)¢d~r=¡ydx+xdy.R1Onleg(-1,0)!(1,0)wehavey=0,sointegralis(¡y)dx=0.OntheR¡1R01leg(1,0)!(0,1)wehavey=¡x+1,sointegralis-(¡x+1)d

6、x+(1¡1011y)dy=+=1.Onthepath(0,1)!(-1,0)wehavey=x+1,sointegralisR22R¡¡1(x+1)dx+0(y¡1)dy=1+1=1.Sototallineintegralis2.0122Let'sgobackandlookatthelegfrom(1,0)!(0,1)again.Wecouldhaveparameterized"thislegasx=1¡t,y=t,0

7、gralasZZZ(0;1)11F~¢d~r=(¡tx^+(1¡t)^y)¢(¡x^+^y)dt=(t+1¡t)=1;(4)(1;0)00sameanswerasabove.6.1.1Conservative¯eld.Recallwesaidthattheintegralofanexactdi®erentialwasindependentofthepath.Thisresultcanbeexpressedinourcurrentparticularcontext,lineintegralsofvector¯elds.Supposeyouhaveavector

8、¯eldF~whichcanbeexpresseda