identities18jjj.com

18jjj.com  时间:2021-04-06  阅读:()
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.
WhatwheelswouldnotspinMITOpenCourseWarehttp://ocw.
mit.
eduResource:CalculusOnlineTextbookGilbertStrangThefollowingmaynotcorrespondtoaparticularcourseonMITOpenCourseWare,buthasbeenprovidedbytheauthorasanindividuallearningresource.
ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.
mit.
edu/terms.

Megalayer 香港CN2优化线路VPS主机速度和性能综合评测

对于Megalayer云服务器提供商在之前也有对于他们家的美国服务器和香港服务器进行过评测和介绍,但是对于大部分网友来说需要独立服务器和站群服务器并不是特别的普及,我们很多网友使用较多的还是云服务器或者VPS主机比较多。在前面也有在"Megalayer新增香港VPS主机 1GB内存 50GB SSD 2M带宽 月59元"文章中有介绍到Megalayer商家有新增香港CN2优化VPS主机。那时候看这...

Sharktech:无限流量服务器丹佛,洛杉矶,荷兰$49/月起,1Gbps带宽哦!

鲨鱼机房(Sharktech)我们也叫它SK机房,是一家成立于2003年的老牌国外主机商,提供的产品包括独立服务器租用、VPS主机等,自营机房在美国洛杉矶、丹佛、芝加哥和荷兰阿姆斯特丹等,主打高防产品,独立服务器免费提供60Gbps/48Mpps攻击防御。机房提供1-10Gbps带宽不限流量服务器,最低丹佛/荷兰机房每月49美元起,洛杉矶机房最低59美元/月起。下面列出部分促销机型的配置信息。机房...

€4.99/月Contabo云服务器,美国高性价比VPS/4核8G内存200G SSD存储

Contabo是一家运营了20多年的欧洲老牌主机商,之前主要是运营德国数据中心,Contabo在今年4月份增设新加坡数据中心,近期同时新增了美国纽约和西雅图数据中心。全球布局基本完成,目前可选的数据中心包括:德国本土、美国东部(纽约)、美国西部(西雅图)、美国中部(圣路易斯)和亚洲的新加坡数据中心。Contabo的之前国外主机测评网站有多次介绍,他们家的特点就是性价比高,而且这个高不是一般的高,是...

18jjj.com为你推荐
曹谷兰曹谷兰事件 有吧友知道吗seo优化工具想找一个效果好的SEO优化软件使用,在网上找了几款不知道哪款好,想请大家帮忙出主意,用浙江哪款软件效果好porndao单词prondao的汉语是什么ip在线查询通过对方的IP地址怎么样找到他的详细地址?mole.61.com摩尔庄园的米米号和密码我都忘了 只记得注册的邮箱 怎么办-_-www.bbb551.com100bbb网站怎样上不去了sesehu.com68lolita com是真的吗www.dm8.cc有没有最新的日本动漫网站?www.1diaocha.com手机网赚是真的吗汴京清谈汴京繁华 简介50字?
双线虚拟主机 云南服务器租用 成都主机租用 合租服务器 韩国vps俄罗斯美女 秒解服务器 256m内存 128m内存 京东云擎 卡巴斯基官方免费版 域名转接 刀片式服务器 qq对话框 免费网页空间 流媒体加速 华为云服务登录 美国独立日 主机返佣 网站加速 博客域名 更多