identities18jjj.com

15.
4SurfaceIntegrals57344ShowthatthespinfieldSdoesworkaroundeverysimpleinsideRcanbesqueezedtoapointwithoutleavingR.
Testclosedcurve.
theseregions:1.
xyplanewithout(0,O)2.
xyzspacewithout(0,0,O)45ForF=f(x)jandR=unitsquare0C*D*Aisnotneeded.
OnlynoticehowCaD:curlgradFisalwayszero.
ThenewestpartisD*A.
IfcurlF=0thenfFdR=0.
Butthatisnotnews.
ItisStokes'Theorem.
Theinterestingproblemistosolvethethreeequationsforf,whentestDispassed.
Theexampleabovehaddf/dx=2xyf=52xydx=x2yplusanyfunctionC(y,z)dfldy=x2+z=x2+dC/dyC=yzplusanyfunctionC(Z)df/dz=y=y+dcldzc(z)canbezero.
ThefirststepleavesanarbitraryC(y,z)tofixthesecondstep.
Thesecondstepleavesanarbitraryc(z)tofixthethirdstep(notneededhere).
Assemblingthethreesteps,f=x2y+C=x2y+yz+c=x~~+yz.
Pleaserecognizethatthe"fix-up"isonlypos-siblewhencurlF=0.
TestDmustbepassed.
EXAMPLE7IsF=(Z-y)i+(x-z)j+(y-x)kthegradientofanyfTestDsaysno.
ThisFisaspinfieldaxR.
Itscurlis2a=(2,2,2),whichisnotzero.
Asearchforfisboundtofail,butwecantry.
Tomatchdf/dx=z-y,wemusthavef=zx-yx+C(y,z).
Theyderivativeis-x+dC/dy.
ThatnevermatchesN=x-z,sofcan'texist.
EXAMPLE8WhatchoiceofPmakesF=yz2i+xz2j+PkconservativeFindf:SolutionWeneedcurlF=0,bytestD.
FirstcheckdM/dy=z2=dNjdx.
AlsodP/dx=aM/dz=2yzanddP/dy=dN/az=~XZ.
P=2xyzpassesalltests.
Tofindfwecansolvethethreeequations,ornoticethatf=xyz2isSUCC~SS~U~.
ItsgradientisF.
Athirdmethoddefinesf(x,y,z)astheworktoreach(x,y,z)from(0,0,O).
Thepathdoesn'tmatter.
ForpracticeweintegrateFdR=Mdx+Ndy+Pdzalongthestraightline(xt,yt,So1zt):f(~,y,Z)=(yt)(~t)~(xdt)+(xt)(~t)~(ydt)+2(xt)(yt)(zt)(zdl)=xyz2.
EXAMPLE9Whyisdivcurlgradfautomaticallyzero(intwoways)SolutionFirst,curlgradfiszero(always).
Second,divcurlFiszero(always).
Thosearethekeyidentitiesofvectorcalculus.
Weendwithareview.
Green'sTheorem:(2N/x-2Ml2y)dxdy$Fndr=jj(ZM/dr+dN/Fy)dxdy15.
6Stokes'TheoremandtheCurlofFDivergenceTheorem:Stokes'Theorem:F-dR=jcurlF*ndS.
ThefirstformofGreen'sTheoremleadstoStokes'Theorem.
ThesecondformbecomestheDivergenceTheorem.
Youmayask,whynotgotothreedimensionsinthefistplaceThelasttwotheoremscontainthefirsttwo(takeP=0andaflatsurface).
Wecouldhavereducedthischaptertotwotheorems,notfour.
Iadmitthat,butafundamentalprincipleisinvolved:"Itiseasiertogeneralizethantospecialize.
"Forthesamereasondfldxcamebeforepartialderivativesandthegradient.
15.
6EXERCISESRead-throughquestionsIn11-14,computecurlFandfind$,F0dRbyStokes'Theorem.
ThecurlofMi+Nj+Pk.
isthevectora.
Itequalsthe3by3determinantb.
Thecurlofx2i+z2kisc.
ForS=yi-(x+z)j+ykthecurlisd.
ThisSisae12F=ixR,C=circlex2+z2=1,y=0.
fieldaxR=+(curlF)xR,withaxisvectora=f.
Foranygradientfieldfxi+f,j+fzkthecurlis9.
Thatisthe13F=(i+j)xR,C=circley2+z2=1,x=0.
importantidentitycurlgradf=h.
Itisbasedonf,,=f,,14F.
=(yi-xj)x(xiandiandiThetwinidentityisk.
+yj),C=circlex2+y2=1,z=0.
15(important)SupposetwosurfacesSandThavethesameThecurlmeasurestheIofavectorfield.
Apad-boundaryC,andthedirectionaroundCisthesame.
dlewheelinthefieldwithitsaxisalongnhasturningspeedm.
(a)ProvecurlFndScurlFndS.
ThespinisgreatestwhennisinthedirectionofJJ,.
=flT.
n.
(b)Secondproof:ThedifferencebetweenthoseintegralsisThentheangularvelocityis0.
JJJdiv(cur1F)NBywhatTheoremWhatregionisI/Stokes'TheoremisP=q.
ThecurveCistheWhyisthisintegralzerorofthesS.
ThisistTheoremextendedtoudimensions.
BothsidesarezerowhenFisagradient16In15,supposeSisthetophalfoftheearth(ngoesout)fieldbecausev.
andTisthebottomhalf(ncomesin).
WhatareCandIr!
ShowbyexamplethatIS,FndS=11,FndSisnotgenerallyThefourpropertiesofaconservativefieldareA=w,true.
B=x,C=Y,D=.
Thefieldy2z2i+2xy2zk17ExplainwhycurlFndS0overtheclosedboundary(passes)(fails)testD.
Thisfieldisthegradientoff==A.
i[TheworkJFBofanysolid.
dRfrom(O,0,0)to(1,1,1)is(onwhichV.
path).
Foreveryfield17,JJcurlFondsisthesameout18SupposecurlF=0anddivF=0.
(a)WhyisFthegradi-throughapyramidandulpthroughitsbasebecausec.
entofapotential(b)WhydoesthepotentialsatisfyLaplace'sequationf,,+f,,+f,,=OProblems1-6findcurlF.
F=zi+xj+yk2F=grad(xeYsinz)In19-22,findapotentialfifitexists.
F=(x+y+z)(i+j+k)4F=(x+y)i-(x+y)kF=pn(xi+yj+zk)6F=(i+j)xR21F=ex-zi-ex-zk22F=yzi+xzj+(XY+z2)kFindapotentialfforthefieldinProblem3.
23FindafieldwithcurlF=(1,0,O).
FindapotentialfforthefieldinProblem5.
24FindallfieldswithcurlF=(1,0,O).
Whendothefieldsxmiiandxnjhavezerocurl25S=axRisaspinfield.
ComputeF=bxS(constantWhendoes(a,x+a2y+a,z)khavezerocurlvectorb)andfinditscurl.
59615VectorCalculus26HowfastisapaddlewheelturnedbythefieldF=yi-xkMaxwellallowsvaryingcurrentswhichbringsintheelectric(a)ifitsaxisdirectionisn=j(b)ifitsaxisislinedupwithfield.
curlF(c)ifitsaxisisperpendiculartocurlF41ForF=(x2+y2)i,computecurl(curlF)andgrad(divF)27HowiscurlFrelatedtotheangularvelocityointhespinandF,,+F,,+F,,.
fieldF=a(-yi+xj)Howfastdoesawheelspin,ifitisin++42ForF=v(x,y,z)i,provetheseusefulidentities:theplanexyz=l(a)curl(cur1F)=grad(divF)-(F,,+F,,+F,,).
28FindavectorfieldFwhosecurlisS=yi-xj.
(b)curl(fF)=fcurlF+(gradf)xF.
29FindavectorfieldFwhosecurlisS=axR.
43IfB=acost(constantdirectiona),findcurlEfromFara-30Trueorfalse:whentwovectorfieldshavethesamecurlday'sLaw.
ThenfindthealternatingspinfieldE.
atallpoints:(a)theirdifferenceisaconstantfield(b)their44WithG(x,y,z)=mi+nj+pk,writeoutFxGandtakedifferenceisagradientfield(c)theyhavethesamedivergence.
itsdivergence.
MatchtheanswerwithGcurlF-F.
curlG.
45WritedownGreen'sTheoreminthexzplanefromStokes'Theorem.
In31-34,compute11curlFndSoverthetophalfofthespherex2+y2+z2=1and(separately)$F.
dRaroundtheequator.
Trueorfalse:VxFisperpendiculartoF.
(a)ThesecondproofofStokes'TheoremtookM*(x,y))+=M(x,y,fP(x,y,f(x,y))af/axastheMinGreen'sTheorem.
ComputedM*/dyfromthechainruleandpro-35ThecircleCintheplanex+y+z=6hasradiusrandductrule(therearefiveterms).
centerat(1,2,3).
ThefieldFis3zj+2yk.
Compute$FdR(b)SimilarlyN*=N(x,y,f)+P(x,y,f)df/dyhasthexaroundC.
derivativeN,+N,f,+P,f,+Pzf,f,+Pf,,.
Checkthat++N,*-M,*matchestherightsideofequation(S),asneeded36SisthetophalfoftheunitsphereandF=zixjxyzk.
intheproof.
Find11curlF.
ndS.
"Theshadowoftheboundaryistheboundaryofthe37Findg(x,y)sothatcurlgk=yi+x2j.
Whatisthenameshadow.
"ThisfactwasusedinthesecondproofofStokes'forginSection15.
3Itexistsbecauseyi+x2jhaszeroTheorem,goingtoGreen'sTheoremontheshadow.
GivetwoexamplesofSandCandtheirshadows.
38ConstructFsothatcurlF=2xi+3yj-5zk(whichhas49WhichintegralsareequalwhenCofSorSzerodivergence).
=boundary=boundaryofV39SplitthefieldF=xyiintoV+WwithcurlV=0anddivW=$FdR$(curlF)dR$(curlF)ndsFnd~0.
.
.
1111divFdS11(curlF)ndS11(graddivF).
ndS111divFdV40Ampere'slawforasteadymagneticfieldBiscurlB=pJ(J=currentdensity,p=constant).
FindtheworkdonebyB50DrawthefieldV=-xkspinningawheelinthexzplane.
aroundaspacecurveCfromthecurrentpassingthroughit.
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