Vector Calculus, Linear Algebra and Differential …

1 VectorCalculus, ,2019 WethankChetBalestra,DanielBettendorf,Sco ttBrodie,CalvinChong,WayneFincher,RadiGr osu,KennethHahn,ShengyuanHu,JakeJanuzell i,DanKemp,CalebKoch,JonathanKoppelman,Jo shuaLevin,AdamLutoborski,BradyMcCary,Rav iRamakrishna,VictorReiner,OliverSangster ,VictorReiner,YiRen,JohnSmith,TanLei,Geo rgeWoodrowIII,andthestudentsinMath2240,s pring2018(inparticularAmoghAnakru,Nathan ielBannister,andSamWeiss) :errata;minortypos; [ ,2017] : ~aavectorinRm shouldbe ~bavectorinRm . |~a (~b ~c)|,not~a (~b ~c)|; [ ,2017]4thparagraph,line7: sayingthatthexmhavealimitin[0,1/2) shouldbe ..havealimitin[0,1) . :Thesecond|xij a|< shouldbe|yij a|< .Page113 Fifthparagraph,firstline: Thedisc{z2C||z| R} ,not Thedisc{z2C|z R} Page132 Inthemarginnotethatstarts weshouldhavereferredto lengthsofmatrices ratherthan absolutevalues.]]

2 Page134 Inthelastequationoftheproof(justbeforeth eexercises)[0]shouldbe0(the0isscalar,not amatrix). ,themeanvaluetheoreminonevariable, ;theproofofthatpropositionshowsthatifthe lineartransformation multiplicationbyA 1isbijective,itsinversemappingislinearan dsohasamatrix; , , correspondingrow (althoughonecoulddefinepivotalrowusingth etranspose).Page171 Toavoidtheuseof pivotalrow (seetheerratumforpage170), ()everycolumnofAcontainsapivotal1()every rowofAcontainsapivotal1() ..whichhasapivotal1ineachcolumnbutnotine achrow . 0sinnxsinmxdx=12Z 0 (cos(n m)x cos(n+m)x) dx=12 sin(n m)xn m 0 sin(n+m)xn+m 0 = ,inthedisplayedequation,+..aksinnkxshoul dbe+ + : for~ 1 shouldbe for~ 2 ..thecorrespondingentriesfor~v2are 3, 2,and0 shouldbe.

3 Thecorrespondingentriesfor~v2areare 1,1,and0. Page197 Toavoidtheuseof pivotalrow (seetheerratumforpage170),wewillreplaced thepivotalrowsofAarelinearlyindependent by therowsofeAcontainingpivotal1 sarelinearlyindependent .Page212 Ifandwhenthebookisreprinted,wewillmoveth ediscussionofdimension, ,immediatelybe-forethesubsection Matrixwithrespecttoabasisandchangeofbasi s . ,Vissaidtoben-dimensional,butwiththecurr entorderofdiscussion,dimensionisn tyetdefined,andwedon tyetknowthat{v}and{v0} , 1{w}and 1{v}arelinear, [addedMarch25,2017] ,wespeakofg(A~v),butgisdefinedon ,thesetofunitvectorsinQ,andthereisnoreas ontothinkthatA~ , :LetQ Rnbethe quadrant ~w ~0,setQ def=Q {~0},andlet ~w2 ,then~w ~0and~w6=~0,so( )A~w>~ :Q !

4 Rgivenbyg:~w7!inf (A~w)1w1,(A~w)2w2,..,(A~w)nwn ; (~w)~w A~wforall~w2Q ,andg(~w) (~w)=g(~w/|~w|)forall~w2Q .Sincegisaninfimumoffinitelymanycontinuo usfunctionsQ !R, iscompact,gachievesitsmaximumon atsome~v2 ,whichalsoachievesthemaximumofgonQ .Letusseethat~visaneigenvectorofAwitheig envalue def=g(~v).Bycontradiction,supposethatg(~ v)~v6=A~ ,g(~v)v A~vandg(~v)~v6=A~vimplyg(~v)A~v=Ag(~v)~v<AA~ (~v)A~v<AA~visstrict,thiscontradictsthehypothesisthat~visanelementofQ atwhichgachievesitsmaximum:g(A~v)isthelargestnumbersuchthatg(A~v)A~v AA~v,sog(A~v)>g(~v). ~w0= 3p27p2 , ,g:(~w)~wshouldbeg(~w)~ ,Aw/wishouldbe(Aw) [addedMay13,2017] :A g(~w)~w ,notA g(~w) ~ [addedMarch25,2017] , [addedMay22,2017] ,theterm4(u2 u2)2shouldbe4(u2 u02) , ~hn=|an+1 an| shouldbe ~hn=an+1 an.

5 :Inthelastline, thenonzerocolumnsofeBformabasisfortheker nelofA shouldbe thecolumnsofeBcorrespondingtothezerocolu mnsofeAformabasisforthekernelofA .Wearealsochangingthenotationtoreflectth enotationusedinthesolution,soAisn m,notk n, : asmoothk-dimensionalmanifold ,not asmoothn-dimensionalmanifold .Page292[addedMay22,2017]Lastmarginnote: Thedisplayedequa-tionshouldhaveDnF(z),no tDn(z). : Choosex2U ,not Choosex2g(U) . :Thecompositions and bothmakesense,butitis ,not , [D ((y,0))]isRm TxM?,notTxM TxM?.Inanyeventualreprinting,wewillrewri tethisparagraphasfollows:Thederivative[D ((y,0))]isthemapRm TxM?!Rnthatmaps[~a,~b]to[D (y)]~a+~ [D (y)]:Rm! :U!V TxM?suchthat =id:U! [addedAugust1,2017] :intwoplacesinthethirdline,~vishouldbe~v 1.

6 Basis~v1,..,~vnofRnwithA~vi= ivi,where 1,.., k>0,.. ,part3: IfA\B=/ forA,B S ,not IfP(A\B)=/ forA,B S . ,partc:TheformulaforH(x)ismissinga1/2;it shouldbeH(x)=12(1+(f0(x))2)3/2 f00(x) 1+(f0(x))2f(x) . ,parta: thenA0isstillsymmetric shouldbe thenBissymmetric .Page438[addedMay22,2018] ;theratio-nalsin[0,1] :ShowthatifA Rnhaswell-definedn-dimensionalvolume,the n@ shouldbe .Page513[added,May22,2017] [addedMay22,2018] ;volkissuchafunctionVbutitisnotunique(fo rinstance,theproductofthelengthsofthevec torsisalsosuchafunction). : ( ). , ,thecolumnsofM>. ( )toshowthatvolkistheuniquereal-valuedfun ctionVofn krealmatricessuchthatforallorthogonaln nmatricesPandallorthogonalk kmatricesQ,wehaveV(M)=V(PMQ)andV( 1~e1.)

7 , k~ek)=| 1 k|foranyn krealmatrixMandanynumbers 1,.., [addedMay13,2017] :Bydefinition,apiece-with-boundarymustbe compact, +y2 [addedMay13,2017] : Forwhatvectorfield~Fcan'bewrittenW~F? ,not Forwhatvectorfield~Fcan'bewrittendW~F? ('isa1-formanddWFisa2- form ,sotheoriginal wordingdidnotmakesense).Page653[addedMay 13,2017] :Wisdefinedasa boundedopensubsetofRk ,butsinceWisopen,it snotclearthatitmakessenseto give@(W\Z)theboundaryorientation .Insteadweshouldwrite letWbeapiece-with-boundaryofRk ,andinthe4thline,change withcompactsupportinWto withcompactsupportintheinte-riorofW .Further, , Since'=0outsideW shouldbe Since'=0on@W . : When|y0+~k|2V shouldbe Whenthepointy0+~kisinV . : if ,not ifandonlyif :Thereshouldbeno s between consisting and al-ternately.

8 Page38[addedSeptember4,2017]Inthesecondl ineofthesecondparagraph, becausehaven tdefined shouldbe becausewehaven tde-fined .Page51[new, ,2019] :Intheprintversionofthebook, ,(1 A)shouldbe(I A).Page109[addedMay27,2017] :Delete is : Nextwewanttoshowthatfhasaminimum. ,thereshouldbeno s after given (theextraneouss snodoubtcomefromanattempttosavethefile). Page214 Secondmarginnote,line5: subspaceE Rn ,not subsetE Rn .Page227[ ]Page227:Thelastmarginnoteshouldstart IfAisarealmatrix ,not IfAsarealmatrix .Page296[addedJuly4,2017] : anarbitraryC1mapping ,not anarbitraryC1mappings . , :The2xinthemonomialrowshouldbe2x(xinmath mode)Page318[addedJuly4,2017] , [addedMay22,2017] : iden-tity shouldbe identify.

9 Page359[addedAugust1,2017] [Df( (v0))][D (v0)]=6 Page376[addedMay22,2017] : face shouldbe eigenface .Inthelastline, egenvalue shouldbe eigenvalue .Page420[addedAugust1,2017] : probabilitydensity ,not probabilityofdensity . Iffis ,not iffbe . f E(f) g E(g) . :|f(x1) f(x2)|< ,not|f(x1) f(x2|< Page451[addedMay13,2017] ,theword Function shouldbeinthefirstrow, [ ,2017]Inthefirstmarginnote, , therinrdrd playstherole shouldbe therinr|drd |playstherole .Page510[ ,2017] ,ontherightside,thereisanextraendparenth esis:(f )(u) shouldbe(f )(u).Page534[addedMay13,2017] :weshouldhavewrittenM Rn, [addedMay22,2017] :Bx(M), [addedMay22,2017] , [addedMay22,2017]Firstline: overthepiece ,not throughthepiece.)

10 ,thetwocolumnvectorsshouldbeinparenthese s:P0@xy1 x y1A !D1 z}|{2410 135, !D2 z}|{2401 135!Page599[addedMay13,2017]Thefirstmarg innoteismissinganendparenthesis: !(Px(~v1,..,~vk) shouldbe !(Px(~v1,..,~vk)) .Page612[addedMay22,2017]Line4: require ,not rquire .Page614[addedMay13,2017]Lastline: obviously ,not obvioiusly .7 Page705 , : , |g(y1) g(y2)|,not|g(y1)| g(y2)|.Page733[addedMay22,2017] , , :eishouldbe~eiandejshouldbe~ [addedMay22,2017] , whereais couldbereplacedby thefunctionagives .Onpage42,partbmightinclude Thepipeformsatorus,withr 1 .Page79[addedSeptember13,2017] [~a ~b,~a,~b],butbythemarginnoteonpage78,thi sisthesameasprovingitfordet[~a,~b,~a ~b].Usingthelanguageofantisymmetry( ),goingfromonetotheotherrequirestwoexcha ngesofarguments, , isnotwelldefined;therearetwoanglesbetwee nalinelandavectoranchoredatapointofl:one in[0, /2],theotherin[ /2, ].)