Sjamaar

Manifolds and Differential Forms Reyer Sjamaar DEPARTMENTOFMATHEMATICS,CORNELLUNIVERSITY,ITHACA,NEWYORK 14853-4201 E-mail address: 
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' Last updated: 2006-08-26T01:13 -05:00 Copyright ©Reyer Sjamaar ,2001. Paper orelectr onic copies forpersonal usemay bemade without explicit permission fromtheauthor .Allother rights reserved. Contents Preface v Chapter 1.Introduction 1 1.1. Manifolds 1 1.2. Equations 7 1.3. Parametrizations 9 1.4. Conguration spaces 9 Exer cises 13 Chapter 2.Differential forms onEuclidean space 17 2.1. Elementary properties 17 2.2. The exterior derivative 20 2.3. Closed and exact forms 22 2.4. The Hodge star operator 23 2.5. div,grad and curl 24 Exer cises 27 Chapter 3.Pulling back forms 31 3.1. Determinants 31 3.2. Pulling back forms 36 Exer cises 42 Chapter 4.Integration of1-forms 47 4.1. Denition and elementary properties oftheintegral 47 4.2. Integration ofexact 1-forms 49 4.3. The global angle function and thewinding number 51 Exer cises 53 Chapter 5.Integration and Stokes' theor em 57 5.1. Integration offorms over chains 57 5.2. The boundary ofachain 59 5.3. Cycles and boundaries 61 5.4. Stokes' theor em 63 Exer cises 64 Chapter 6.Manifolds 67 6.1. The denition 67 6.2. The regular value theor em 72 Exer cises 77 Chapter 7.Differential forms onmanifolds 81 iii iv CONTENTS 7.1. First denition 81 7.2. Second denition 82 Exer cises 89 Chapter 8.Volume forms 91 8.1. n-Dimensional volume inRN91 8.2. Orientations 94 8.3. Volume forms 96 Exer cises 100 Chapter 9.Integration and Stokes' theor emonmanifolds 103 9.1. Manifolds with boundary 103 9.2. Integration over orientable manifolds 106 9.3. Gauß and Stokes 108 Exer cises 109 Chapter 10. Applications totopology 113 10.1. Brouwer 'sxed point theor em 113 10.2. Homotopy 114 10.3. Closed and exact forms re-examined 118 Exer cises 122 Appendix A. Sets and functions 125 A.1. Glossary 125 A.2. General topology ofEuclidean space 127 Exer cises 127 Appendix B.Calculus review 129 B.1. The fundamental theor emofcalculus 129 B.2. Derivatives 129 B.3. The chain rule 131 B.4. The implicit function theor em 132 B.5. The substitution formula forintegrals 133 Exer cises 134 Bibliography 137 The Greekalphabet 139 Notation Index 141 Index 143 Preface These arethelectur enotes forMath 321, Manifolds and Differential Forms, astaught atCornell University since theFall of2001. The course covers mani- folds and differential forms foranaudience ofunder graduates who have taken atypical calculus sequence ataNorth American university ,including basic lin- earalgebra and multivariable calculus uptotheintegral theor ems ofGreen, Gauß and Stokes. Withaview tothefact that vector spaces arenowadays astandar d item ontheunder graduate menu, thetext isnotrestricted tocurves and surfaces inthree-dimensional space, buttreats manifolds ofarbitrary dimension. Some prerequisites arebriey reviewed within thetext and inappendices. The selec- tion ofmaterial issimilar tothat inSpivak's book [Spi65 ]and inFlanders' book [Fla89 ],butthetreatment isatamoreelementary and informal level appr opriate forsophomor esand juniors. Alargeportion ofthetext consists ofproblem sets placed attheend ofeach chapter .The exercises range fromeasy substitution drills tofairly involved but, Ihope, inter esting computations, aswell asmoretheor etical orconceptual prob- lems. Mor ethan once thetext makes useofresults obtained intheexercises. Because ofitstransitional natur ebetween calculus and analysis, atext ofthis kind hastowalk athin line between mathematical informality and rigour .Ihave tended toerrontheside ofcaution byproviding fairly detailed denitions and proofs. Inclass, depending ontheaptitudes and preferences oftheaudience and also ontheavailable time, onecanskip over many ofthedetails without toomuch loss ofcontinuity .Atany rate, most oftheexercises donotrequir eagreatdeal of formal logical skill and throughout Ihave tried tominimize theuseofpoint-set topology . This revised version ofthenotes isstill abitrough attheedges. Plans for impr ovement include: moreand better graphics, anappendix onlinear algebra, a chapter onuid mechanics and oneoncurvatur e,perhaps including thetheor ems ofPoincaré-Hopf and Gauß-Bonnet. These notes and eventual revisions can be downloaded fromthecourse website at())*+-,,/...0#132 )(0-4567899:08;<,=>/?213226,4 92>>8>,@A3B,CD7;8 E$0F()1G9. Corr ections, suggestions and comments will bereceived gratefully . Ithaca, NY,2006-08-26 v CHAPTER 1 Introduction Westart with aninformal, intuitive introduction tomanifolds and how they arise inmathematical natur e.Most ofthis material will beexamined morethor- oughly inlater chapters. 1.1. Manifolds Recall that Euclidean n-space Rnisthesetofallcolumn vectors with nreal entries x H IJJJKx1 x2 ... xn LNMMMO, which weshall callpoints orn-vectors and denote bylower case boldface letters. In R2orR3weoften write xHQPx y R, resp. xH IKx y z LO. Forreasons having todowith matrix multiplication, column vectors arenottobe confused with rowvectorsSx1x2...xn T.Forclarity ,weshall usually separate theentries ofarowvector bycommas, asinSx1,x2,...,xn T.Occasionally ,tosave space, weshall represent acolumn vector xasthetranspose ofarowvector , x HUSx1,x2,...,xnTT. Amanifold isacertain type ofsubset ofRn.Aprecise denition will follow inChapter 6,butone important consequence ofthedenition isthat amanifold hasawell-dened tangent space atevery point. This fact enables ustoapply the methods ofcalculus and linear algebra tothestudy ofmanifolds. The dimension of amanifold isbydenition thedimension ofitstangent spaces. The dimension of amanifold inRncanbenohigher than n. Dimension 1.Aone-dimensional manifold is,loosely speaking, acurve with- outkinks orself-intersections. Instead ofthetangent “space” atapoint one usu- ally speaks ofthetangent line.Acurve inR2iscalled aplane curve and acurve in R3isaspace curve, butyou canhave curves inany Rn.Curves canbeclosed (as intherst pictur ebelow), unbounded (asindicated bythearrows inthesecond pictur e),orhave one ortwo endpoints (the thirdpictur eshows acurve with an endpoint, indicated byablack dot; thewhite dotattheother end indicates that 1 2 1.INTRODUCTION that point does notbelong tothecurve; thecurve “peters out” without coming to anendpoint). Endpoints arealso called boundary points . Acirclewith one point deleted isalso anexample ofamanifold. Think ofatorn elastic band. Bystraightening outtheelastic band weseethat this manifold isreally thesame asanopen interval. The four plane curves below arenotmanifolds. The teardrophasakink, wher e two distinct tangent lines occur instead ofasingle well-dened tangent line; the ve-fold loop hasve points ofself-intersection, ateach ofwhich therearetwo distinct tangent lines. The bow tieand theve-pointed star have well-dened tangent lines everywher e.Still they arenotmanifolds: thebow tiehas aself- intersection and thecusps ofthestarhave ajagged appearance which isproscribed bythedenition ofamanifold (which wehave notyetgiven). The points wher e these curves failtobemanifolds arecalled singularities .The “good” points are called smooth . Singularities cansometimes be“resolved”. Forinstance, theself-intersections of theArchimedean spiral, which isgiven inpolar coor dinates byrisaconstant times 1.1.MANIFOLDS 3 ,wher erisallowed tobenegative, canbegotridofbyuncoiling thespiral and wrapping itaround acone. Youcan convince yourself that theresulting space curve hasnosingularities bypeeking at italong thedirection ofthex-axis orthey-axis. What you will seearethesmooth curves shown intheyz-plane and thexz-plane. e1e2e3 (The three-dimensional models inthese notes aredrawn incentral perspective. They arebest viewed facing theorigin, which isusually inthemiddle ofthepic- ture,fromadistance of30cmwith one eye shut.) Singularities areextremely inter esting, butinthiscourse weshall focus ongaining athorough understanding ofthesmooth points. 4 1.INTRODUCTION Dimension 2.Atwo-dimensional manifold isasmooth surface without self- intersections. Itmay have aboundary ,which isalways aone-dimensional mani- fold. Youcanhave two-dimensional manifolds intheplane R2,butthey arerel- atively boring. Examples are:anarbitrary open subset ofR2,such asanopen squar e,oraclosed subset with asmooth boundary . Aclosed squar eisnotamanifold, because thecorners arenotsmooth.1 Two-dimensional manifolds inthree-dimensional space include aspher e,aparab- oloid and atorus. e1e2e3 The famous Möbius band ismade bypasting together thetwo ends ofarectangular strip ofpaper giving one end ahalf twist. The boundary oftheband consists of 1Tobestrictly accurate, theclosed squar eisatopological manifold with boundary ,butnotasmooth manifold with boundary .Inthese notes wewill consider only smooth manifolds. 1.1.MANIFOLDS 5 two boundary edges oftherectangle tied together and isthereforeasingle closed curve. Out oftheMöbius band wecancreate intwo differentways amanifold without boundary byclosing itupalong theboundary edge. Accor ding tothedirection in which weglue theedge toitself, weobtain theKlein bottle ortheprojective plane . Asimple way torepresent these threesurfaces isbythefollowing diagrams. The labels tellyou which edges toglue together and thearrows tellyou inwhich di- rection. a a Möbius banda a bb Klein bottlea a bb projective plane Perhaps theeasiest way tomake aKlein bottle isrst topaste thetopand bottom edges ofthesquar etogether ,which gives atube, and then tojoin theresulting boundary circles, making surethearrows match up.Youwill notice thiscannot be done without passing oneend through thewall ofthetube. The resulting surface intersects itself along acircleand thereforeisnotamanifold. Adifferentmodel oftheKlein bottle isfound byfolding over theedge ofaMöbius band until ittouches thecentral circle. This creates aMöbius type band with a gur eeight cross-section. Equivalently ,take alength oftube with agur eeight cross-section and weld theends together giving one end ahalf twist. Again the 6 1.INTRODUCTION resulting surface hasaself-intersection, namely thecentral circleoftheoriginal Möbius band. The self-intersection locus aswell asafew ofthecross-sections are shown inblack inthefollowing wiremesh model. Torepresent theKlein bottle without self-intersections you need toembed itin four-dimensional space. The projective plane hasthesame peculiarity ,and ittoo hasself-intersecting models inthree-dimensional space. Perhaps theeasiest model isconstr ucted bymerging theedges aand bshown inthegluing diagram forthe projective plane, which gives thefollowing diagram. a a aa First fold thelower right corner over totheupper leftcorner and seal theedges. This creates apouch like acherry turnover with two seams labelled awhich meet atacorner .Now fuse thetwo seams tocreate asingle seam labelled a.Below isa wiremesh model oftheresulting surface. Itisobtained bywelding together two pieces along thedashed wires.The lower half shaped like abowl corresponds to thedashed circular disc inthemiddle ofthesquar e.The upper half corresponds tothecomplement ofthedisc and isknown asacross-cap .The wireshown in black corresponds totheedge a.The interior points oftheblack wireareordinary self-intersection points. Itstwo endpoints arequalitatively differentsingularities 1.2.EQUA TIONS 7 known aspinch points ,wher ethesurface iscrinkled up. e1e2e3 1.2. Equations Verycommonly manifolds aregiven “implicitly”, namely asthesolution set ofasystem 1 Vx1,...,xnW$Xc1, 2Vx1,...,xnW$Xc2, ... mVx1,...,xnW$Xcm, ofmequations innunknowns. Here1,2,...,marefunctions, c1,c2,...,cm areconstants and x1,x2,...,xnarevariables. Byintroducing theuseful shorthand xX YZZZ[x1 x2 ... xn \N]]]^,VxW"X YZZZ[1VxW 2VxW ... mVxW \N]]]^, cX YZZZ[c1 c2 ... cn \N]]]^, wecanrepresent thissystem asasingle equation VxWXc. Itisingeneral difcult tond explicit solutions ofsuch asystem. (On thepositive side, itisusually easy todecide whether any given point isasolution byplugging itinto theequations.) Manifolds dened bylinear equations (i.e. wher eisa matrix) arecalled afne subspaces ofRnand arestudied inlinear algebra. Mor e inter esting manifolds arise fromnonlinear equations. 1.1.EXAMPLE.Consider thesystem oftwo equations inthreeunknowns, x2 _y2X1, y _zX0. Here VxW$Xa`x2 _y2 y _z band cXc`1 0b. 8 1.INTRODUCTION The solution setofthis system istheintersection ofacylinder ofradius 1about thez-axis (given bytherst equation) and aplane cutting thex-axis ata45 dangle (given bythesecond equation). Hence thesolution setisanellipse. Itisamanifold ofdimension 1. 1.2.EXAMPLE.The spher eofradius rabout theorigin inRnisthesetofallx inRnsatisfying thesingle equationexegfr.Hereexehfji xkxfUl x2 1 mx2 2 m k/k/kmx2n isthenorm orlength ofxand xkyfx1y1mx2y2m k/k/kmxnyn istheinner product ordotproduct ofxand y.The spher eofradius risannn1- dimensional manifold inRn.The spher eofradius 1iscalled theunit spher eand isdenoted bySno1.What isaone-dimensional spher e?And azero-dimensional spher e? The solution setofasystem ofequations may have singularities and isthere- forenotnecessarily amanifold. Asimple example isxy f0,theunion ofthetwo coor dinate axes intheplane, which hasasingularity attheorigin. Other examples ofsingularities canbefound inExer cise1.5. Tangent spaces. Letususetheexample ofthespher etointroduce thenotion ofatangent space. Let MfqpxrRnsexegfrt bethespher eofradius rabout theorigin inRnand letxbeapoint inM.Ther eare two reasonable, butinequivalent, views ofhow todene thetangent space toM atx.The rst view isthat thetangent space atxconsists ofallvectors ysuch thatuynxv:kxf0,i.e.ykxfxkxfr2.Incoor dinates: y1x1m k/k/kmynxn fr2.This isaninhomogeneous linear equation iny.Inthisview ,thetangent space atxisan afne subspace ofRn,given bythesingle equation ykxfr2. However ,formost practical purposes itiseasier totranslate this afne sub- space totheorigin, which turns itinto alinear subspace. This leads tothesecond view ofthetangent space atx,namely asthesetofallysuch that ykxf0,and this isthedenition that weshall espouse. The standar dnotation forthetangent space toMatxisTxM.Thus TxM fjpy rRnsy kx f0 t, alinear subspace ofRn.(InExer cise 1.6you will beasked tond abasis ofTxM foraparticular xand you will seethat TxMisn n1-dimensional.) Inequalities. Manifolds with boundary areoften presented assolution sets ofasystem ofequations together with one ormoreinequalities. For instance, theclosed ballofradius rabout theorigin inRnisgiven bythesingle inequalityex exwr.Itsboundary isthespher eofradius r. 1.4.CONFIGURA TION SPACES 9 1.3. Parametrizations Adual method fordescribing manifolds isthe“explicit” way,namely bypar- ametrizations. Forinstance, xycos,yysin parametrizes theunit circleinR2and x ycoscos,y ysincos,z ysin parametrizes theunit spher einR3.(Her eistheangle between avector and the xy-plane andisthepolar angle inthexy-plane.) The explicit method hasvarious merits and demerits, which arecomplementary tothose oftheimplicit method. One obvious advantage isthat itiseasy tond points lying onaparametrized manifold simply byplugging invalues fortheparameters. Adisadvantage isthat itcanbehardtodecide ifany given point isonthemanifold ornot, because this involves solving fortheparameters. Parametrizations areoften hardertocome by than asystem ofequations, butareattimes moreuseful, forexample when one wants tointegrate over themanifold. Also, itisusually impossible toparame- trize amanifold insuch away that every point iscover edexactly once. Such is thecase forthetwo-spher e.One commonly restricts thepolar coor dinates z, { totherectangle |0,2 }~€|‚ ƒ2, ƒ2 }toavoid counting points twice. Only the meridian y0isthen hittwice, butthisdoes notmatter formany purposes, such ascomputing thesurface areaorintegrating acontinuous function. Wewill use parametrizations togive aformal denition ofthenotion ofa manifold inChapter 6.Note however that notevery parametrization describes a manifold. Examples ofparametrizations with singularities aregiven inExer cises 1.1and 1.2. 1.4. Conguration spaces Frequently manifolds arise inmoreabstract ways that may behardtocaptur e interms ofequations orparametrizations. Examples aresolution curves ofdiffer- ential equations (see e.g.Exer cise1.10) and conguration spaces. The conguration ofamechanical system (such asapendulum, aspinning top, thesolar system, a uid, oragasetc.) isitsstate orposition atany given time. (The conguration ignor esanymotions that thesystem may beunder going. Soaconguration islike asnapshot oramovie still. When thesystem moves, itsconguration changes.) Inpractice one usually describes aconguration byspecifying thecoor dinates of suitably chosen parts ofthesystem. The conguration space orstate space ofthesys- tem isanabstract space, thepoints ofwhich areinone-to-one correspondence to allphysically possible congurations ofthesystem. Veryoften theconguration space turns outtobeamanifold. Itsdimension iscalled thenumber ofdegreesof freedom ofthesystem. The conguration space ofeven afairly small system canbe quite complicated. 10 1.INTRODUCTION 1.3.EXAMPLE.Aspherical pendulum isaweight orbob attached toaxed centr ebyarigid rod,freetoswing inany direction inthree-space. The state ofthependulum isentir elydetermined bytheposition ofthebob. The bob canmove fromany point ataxed distance (equal tothelength oftherod) fromthecentr etoanyother .Theconguration space isthereforeatwo-dimensional spher e. Some believe that only spaces ofdimension „3(or4,forthose who have hear dofrelativity) canhave abasis inphysical reality .The following two exam- ples show that thisisnottrue. 1.4.EXAMPLE.Takeaspherical pendulum oflength rand attach asecond one oflength stothemoving end oftherst byauniversal joint. The resulting system isadouble spherical pendulum .The state ofthis system canbespecied byapair of vectors…x,y†,xbeing thevector pointing fromthecentr etotherst weight and y thevector pointing fromtherst tothesecond weight. x y Thevector xisconstrained toaspher eofradius rabout thecentr eand ytoaspher e ofradius sabout thehead ofx.Aside fromthis limitation, every pair ofvectors can occur (ifwesuppose thesecond rodisallowed toswing completely freely and move “through” therst rod)and describes adistinct conguration. Thus therearefour degr eesoffreedom. The conguration space isafour-dimensional manifold, known asthe(Cartesian) product oftwo two-dimensional spher es. 1.5.EXAMPLE.What isthenumber ofdegr eesoffreedom ofarigid body mov- inginR3?Select anytriple ofpoints A,B,Cinthesolid that donotlieononeline. The point Acanmove about freely and isdetermined bythreecoor dinates, and soithasthreedegr eesoffreedom. Buttheposition ofAalone does notdetermine theposition ofthewhole solid. IfAiskept xed, thepoint Bcanperform two 1.4.CONFIGURA TION SPACES 11 independent swivelling motions. Inother words,itmoves onaspher ecentr edat A,which gives two moredegr eesoffreedom. IfAand Bareboth kept xed, the point Ccanrotate about theaxis AB,which gives onefurther degr eeoffreedom. A AB AB C The positions ofA,Band Cdetermine theposition ofthesolid uniquely ,sothe total number ofdegr eesoffreedom is3‡2‡1ˆ6.Thus theconguration space ofarigid body isasix-dimensional manifold. 1.6.EXAMPLE(the space ofquadrilaterals) .Consider allquadrilaterals ABCD intheplane with xed sidelengths a,b,c,d. A BCD abc d (Think offour rigid rods attached byhinges.) What areallthepossibilities? For simplicity letusdisregar dtranslations bykeeping therst edge ABxed inone place. Edges areallowed tocrosseach other ,sotheshort edge BCcanspin full circleabout thepoint B.During this motion thepoint Dmoves back and forth on acircleofradius dcentr edatA.Afew possible positions areshown here. (As Cmoves alltheway around, wher edoes thepoint Dreach itsgreatest left- orrightwar ddisplacement?) Arrangements such asthis arecommonly used in engines forconverting acircular motion toapumping motion, orvice versa. The position ofthe“pump” Diswholly determined bythat ofthe“wheel” C.This means that thecongurations areinone-to-one correspondence with thepoints onthecircleofradius babout thepoint B,i.e.theconguration space isacircle. 12 1.INTRODUCTION Actually ,this isnotcompletely accurate: forevery choice ofC,therearetwo choices Dand D ‰forthefourth point! They areinter changed byreection inthe diagonal AC. A BCD D Š Sothereisinfactanother circle's worth ofpossible congurations. Itisnotpossible tomove continuously fromtherst setofcongurations tothesecond; infactthey areeach other 'smirr orimages. Thus theconguration space isadisjoint union of two circles. This isanexample ofadisconnected manifold consisting oftwo connected compo- nents . 1.7.EXAMPLE(quadrilaterals, continued) .Even this isnotthefullstory: itis possible tomove fromone circletotheother when b‹cŒa‹d(and also when a ‹b Œc ‹d). A BCD abc d Inthiscase, when BCpoints straight totheleft, thequadrilateral collapses toaline segment: EXERCISES 13 and when Cmoves further down, therearetwo possible directions forDtogo, back up: orfurther down: This means that when bcŽadthetwo components oftheconguration space aremerged atapoint. The junctur erepresents thecollapsed quadrilateral. This conguration space is notamanifold, but most conguration spaces occurring innatur eare(and an engineer designing anengine wouldn't want tousethis quadrilateral tomake a piston drive aywheel). Mor esingularities appear inthecase ofaparallelogram (aŽcand bŽd)and intheequilateral case (aŽbŽcŽd). Exercises 1.1.The formulas x t sint,y 1 cost(t ‘R)parametrize aplane curve. Graph this curve ascarefully asyou can. Youmay usesoftwar eand turn incomputer output. Also include afew tangent lines atjudiciously chosen points. (E.g. nd alltangent lines with slope 0,’1,and“.)Tocompute tangent lines, recall that thetangent vector atapoint”x,y •ofthecurve hascomponents dx –dtand dy –dt.Inyour plot, identify allpoints wher e thecurve isnotamanifold. 1.2.Same questions asinExer cise1.1forthecurve x3at– ”1—t3•,y3at2– ”1—t3•. 1.3.Parametrize thespace curve wrapped around thecone shown inSection 1.1. 14 1.INTRODUCTION 1.4.Sketch thesurfaces dened bythefollowing gluing diagrams. aa b b ababd c d a1b1a2b2 a1 b1 a2 b2a1b1a2b2 a1 b1 a2 b2 (Proceed instages, rst gluing thea's,then theb's,etc., and trytoidentify what you get atevery step. One ofthese surfaces cannot beembedded inR3,souseaself-intersection wher enecessary .) 1.5.Forthevalues ofnindicated below graph thesurface inR3dened byxn ˜y2z. Determine allthepoints wher ethesurface does nothave awell-dened tangent plane. (Computer output isOK, butbear inmind that few drawing programs doanadequate job ofplotting these surfaces, soyou may bebetter offdrawing them byhand. Asapreliminary step, determine theintersection ofeach surface with ageneral plane parallel toone ofthe coor dinate planes.) (i)n ˜0. (ii)n ˜1. (iii) n ˜2. (iv) n ˜3. 1.6.LetMbethespher eofradius™nabout theorigin inRnand letxbethepointš1,1,...,1›onM.Find abasis ofthetangent space toMatx.(Use that TxMisthesetofall ysuch that y œx ˜0.View this equation asahomogeneous linear equation intheentries y1,y2,...,ynofyand nd thegeneral solution bymeans oflinear algebra.) 1.7.What isthenumber ofdegr eesoffreedom ofabicycle? (Imagine that itmoves freely through empty space and isnotconstrained tothesurface oftheearth.) 1.8.Choose two distinct positive realnumbers aand b.What istheconguration space ofallparallelograms ABCDsuch that ABand CDhave length aand BCand AD have length b?What happens ifa ˜b?(AsinExamples 1.6and 1.7assume that theedge ABiskept xed inplace soastoruleouttranslations.) 1.9.What istheconguration space ofallpentagons ABCDEintheplane with xed sidelengths a,b,c,d,e?(Asinthecase ofquadrilaterals, forcertain choices ofsidelengths singularities may occur .Youmay ignor ethese cases. Toreduce thenumber ofdegr eesof EXERCISES 15 freedom you may also assume theedge ABtobexed inplace.) A BCDE abcd e 1.10.The Lotka-V olterra system isanearly (ca.1925) predator -preymodel. Itisthe pair ofdifferential equations dx dt Ÿžrx sxy, dy dtpyžqxy, wher ex ¡t ¢represents thenumber ofpreyand y ¡t ¢thenumber ofpredators attime t,while p,q,r,sarepositive constants. Inthis problem wewill consider thesolution curves (also called trajectories) ¡x ¡t ¢,y ¡t ¢-¢ofthis system that arecontained inthepositive quadrant (x £0,y £0)and derive animplicit equation satised bythese solution curves. (The Lotka-V olterra system isexceptional inthis regar d.Usually itisimpossible towrite down anequation forthesolution curves ofadifferential equation.) (i)Show that thesolutions ofthesystem satisfy asingle differential equation ofthe form dy¤dxf¡x¢g¡y¢,wher ef¡x¢isafunction that depends only onxand g ¡y ¢afunction that depends only ony. (ii)Solve thedifferential equation ofpart (i)byseparating thevariables, i.e.bywrit- ing1 g ¡y ¢dyf¡x¢dxand integrating both sides. (Don't forgettheintegration constant.) (iii) Setpqrs1and plot anumber ofsolution curves. Indicate the direction inwhich thesolutions move. Bewarned that solving thesystem may give better results than solving theimplicit equation! Youmay usecomputer softwar esuch asMaple, Mathematica orMATLAB. Auseful Java applet,¥¥ ¦'§N¨© , canbefound atª««'¥­¬F®®N¯¯¯±°‚²§D«'ª³°µ´¶·'©°F©D¸'¹ ®ºN¸»¶'©¦'¸®'¸»'¥D¥°¼ª«½²¦ . CHAPTER 2 Differential forms onEuclidean space The notion ofadifferential form encompasses such ideas aselements ofsur- face areaand volume elements, thework exerted byaforce,theow ofauid, and thecurvatur eofasurface, space orhyperspace. Animportant operation ondiffer- ential forms isexterior differentiation, which generalizes theoperators div,grad and curl ofvector calculus. The study ofdifferential forms, which was initiated by E.Cartan intheyears around 1900, isoften termed theexterior differ ential calculus . Amathematically rigor ous study ofdifferential forms requir esthemachinery of multilinear algebra, which isexamined inChapter 7.Fortunately ,itisentir elypos- sible toacquir easolid working knowledge ofdifferential forms without entering into this formalism. That istheobjective ofthischapter . 2.1. Elementary properties Adiffer ential form ofdegreekorak-form onRnisanexpr ession  ¾å IfIdxI. (Ifyou don't know thesymbol,look upand memorize theGreekalphabet in theback ofthenotes.) HereIstands foramulti-index¿i1,i2,...,ik Àofdegreek,that isa“vector ”consisting ofkinteger entries ranging between 1and n.The fIare smooth functions onRncalled thecoefcients of,and dxIisanabbr eviation for dxi1dxi2 Á/Á/Ádxik. (The notation dxi1Âdxi2ÂÁ/Á/Á Âdxikisalso often used todistinguish this kind of product fromanother kind, called thetensor product.) Forinstance theexpr essions ¾sin¿x1Ãex4Àdx1dx5Ãx2x2 5dx2dx3Ã6dx2dx4Ãcosx2dx5dx3,  ¾x1x3x5dx1dx6dx3dx2, represent a2-form onR5,resp. a4-form onR6.The formconsists offour terms, corresponding tothemulti-indices ¿1,5À, ¿2,3À, ¿2,4Àand ¿5,3À,wher eascon- sists ofoneterm, corresponding tothemulti-index ¿1,6,3,2À. Note, however ,thatcould equally well beregar ded asa2-form onR6that does notinvolve thevariable x6.Toavoid such ambiguities itisgood practice to state explicitly thedomain ofdenition when writing adifferential form. Another reason forbeing precise about thedomain ofaform isthat thecoef- cients fImay notbedened onallofRn,butonly onanopen subset UofRn.In such acase wesayisak-form onU.Thus theexpr ession ln ¿x2Ãy2Àzdzisnot a1-form onR3,butontheopen setU ¾R3 Āſx,y,zÀÇÆx2Ãy2 Ⱦ0 É,i.e.the complement ofthez-axis. 17 18 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE Youcanthink ofdxiasaninnitesimal increment inthevariable xiand ofdxI asthevolume ofaninnitesimal k-dimensional rectangular block with sides dxi1, dxi2,...,dxik.(Aprecise denition will follow inSection 7.2.) Byvolume wehere mean oriented volume, which takes into account theorderofthevariables. Thus, ifweinter change two variables, thesign changes: dxi1dxi2 Ê/Ê/ÊdxiqÊ/Ê/ÊdxipÊ/Ê/ÊdxikËÍÌdxi1dxi2 Ê/Ê/ÊdxipÊ/Ê/ÊdxiqÊ/Ê/Êdxik, (2.1) and soforth. This iscalled anticommutativity ,graded commutativity ,orthealternat- ingproperty .Inparticular ,thisruleimplies dxidxiËÍÌdxidxi,sodxidxiË0forall i. Letusconsider k-forms forsome special values ofk. A0-form onRnissimply asmooth function (nodx's). Ageneral 1-form looks like f1dx1Îf2dx2ÎÊ/Ê/Ê Îfndxn. Ageneral 2-form hastheshape å i,jfi,jdxidxjËf1,1dx1dx1Îf1,2dx1dx2ÎÊ/Ê/Ê Îf1,ndx1dxnÎf2,1dx2dx1 Îf2,2dx2dx2 ÎÊ/Ê/Ê Îf2,ndx2dxnÎÊ/Ê/ÊÎfn,1dxndx1 Îfn,2dxndx2 ÎÊ/Ê/Ê Îfn,ndxndxn. Because ofthealternating property (2.1) theterms fi,idxidxivanish, and apair of terms such asf1,2dx1dx2and f2,1dx2dx1canbegrouped together: f1,2dx1dx2 Î f2,1dx2dx1ËÍÏf1,2Ìf2,1 Ðdx1dx2.Sowecanwrite any 2-form as å 1 Ñi Òj Ñngi,jdxidxjËg1,2dx1dx2ÎÊ/Ê/Ê Îg1,ndx1dxnÎg2,3dx2dx3 ÎÊ/Ê/Ê Îg2,ndx2dxnÎÊ/Ê/Ê Îgn Ó1,ndxn Ó1dxn. Written likethis, a2-form hasatmost nÎnÌ1ÎnÌ2ÎÊ/Ê/Ê Î2Î1Ë1 2nÏnÌ1Ð components. Likewise, ageneral nÌ1-form canbewritten asasum ofncomponents, f1dx2dx3Ê/Ê/ÊdxnÎf2dx1dx3Ê/Ê/ÊdxnÎÊ/Ê/Ê Îfndx1dx2Ê/Ê/Êdxn Ó1Ën å i Ô1fidx1dx2Ê/Ê/ÊÕdxiÊ/Ê/Êdxn, wher eÕdximeans “omit thefactor dxi”. Every n-form onRncan bewritten asfdx1dx2Ê/Ê/Êdxn.The special n-form dx1dx2Ê/Ê/Êdxnisalso known asthevolume form . Forms ofdegr eek ÖnonRnarealways 0,because atleast onevariable hasto repeat inany expr ession dxi1 Ê/Ê/Êdxik.Byconvention forms ofnegative degr eeare 0. Ingeneral aform ofdegr eekcanbeexpr essed asasum Ëå IfIdxI, 2.1.ELEMENT ARYPROPER TIES 19 wher etheIareincreasing multi-indices, 1 ×i1 Øi2 ØQÙ/Ù/ÙÚØ ik ×n.Weshall almost always represent forms inthis manner .The maximum number ofterms occurring inisthen thenumber ofincreasing multi-indices ofdegr eek.An increasing multi-index ofdegr eekamounts toachoice ofknumbers fromamong thenumbers 1,2,...,n.The total number ofincreasing multi-indices ofdegr eek isthereforeequal tothebinomial coefcient “nchoose k”,Û n k ÜÞÝn! k!ßnàká!. (Compar ethis tothenumber ofallmulti-indices ofdegr eek,which isnk.)Two k-formsÝåIfIdxIandÝåIgIdxI(with Iranging over theincreasing multi- indices ofdegr eek)areconsider edequal ifand only iffIÝgIforallI.The collection ofallk-forms onanopen setUisdenoted by âkßU á.Since k-forms can beadded together and multiplied byscalars, thecollectionâkßUáconstitutes a vector space. Aform isconstant ifthecoefcients fIareconstant functions. The setofcon- stant k-forms isalinear subspace of âkßU áofdimension ãn kä.Abasis ofthis sub- space isgiven bytheforms dxI,wher eIranges over allincreasing multi-indices ofdegr eek.(The spaceâkßUáitself isinnite-dimensional.) The (exterior) product ofak-formÝåIfIdxIand anl-formÝåJgJdxJis dened tobethekål-form Ýå I,JfIgJdxIdxJ. Usually many terms inaproduct cancel outorcanbecombined. Forinstance,ßydxåxdyáßxdxdzåydydzáÝy2dxdydzåx2dydxdzÝ ßy2àx2ádxdydz. Asanextreme example ofsuch acancellation, consider anarbitrary formof degr eek.Itsp-thpowerpisofdegr eekp,which isgreater than nifkæ0and pæn.Ther efore nç1Ý0 forany formonRnofpositive degr ee. The alternating property combines with themultiplication ruletogive thefol- lowing result. 2.1.PROPOSITION(graded commutativity) . Ý ßèà1ákl forallk-formsandalll-forms. PROOF.LetIÝ ßi1,i2,...,ik áand JÝ ßj1,j2,...,jl á.Successively applying thealternating property weget dxIdxJÝdxi1dxi2 Ù/Ù/Ùdxikdxj1dxj2dxj3 Ù/Ù/ÙdxjlÝ ßèà1ákdxj1dxi1dxi2 Ù/Ù/Ùdxikdxj2dxj3 Ù/Ù/ÙdxjlÝ ßèà1 á2kdxj1dxj2dxi1dxi2 Ù/Ù/Ùdxikdxj3 Ù/Ù/Ùdxjl ...Ý ßèà1ákldxJdxI. 20 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE Forgeneral forms éåIfIdxIand éåJgJdxJwegetfromthis éå I,JgJfIdxJdxI éëê½ì 1íklå I,JfIgJdxIdxJ éëêèì 1íkl, which establishes theresult. QED Anoteworthy special case isé.Then weget2éîêèì 1ík22éïê½ì 1ík2. This equality isvacuous ifkiseven, buttells usthat2é0ifkisodd. 2.2.COROLLARY.2é0ifisaform ofodddegree. 2.2. The exterior derivative Iffisa0-form, that isasmooth function, wedene dftobethe1-form dfén å ið1¶f ¶xidxi. Then wehave theproduct orLeibniz rule: d êfg í$é fdg ñgdf. If éåIfIdxIisak-form, each ofthecoefcients fIisasmooth function and we dene dtobethek ñ1-form déå IdfIdxI. The operation discalled exterior differ entiation .Anoperator ofthis sort iscalled a rst-or derpartial differential operator ,because itinvolves therst partial deriva- tives ofthecoefcients ofaform. 2.3.EXAMPLE.If éfdx ñgdyisa1-form onR2,then défydydxñgxdxdyéÍêgx ìfy ídxdy. (Recall that fyisanalternative notation for¶fò¶y.)Mor egenerally ,fora1-form  éån i ð1fidxionRnwehave d én å ið1dfidxi én å i,jð1¶fi ¶xjdxjdxiéå 1 ói ôj ón¶fi ¶xjdxjdxi ñå 1 ój ôi ón¶fi ¶xjdxjdxiéqì å 1 ói ôj ón¶fi ¶xjdxidxj ñå 1 ói ôj ón¶fj ¶xidxidxj (2.2)éå 1 ói ôj ón õ¶fj ¶xi ì¶fi ¶xj ödxidxj, wher einline (2.2) intherst sum weused thealternating property and inthe second sum weinter changed theroles ofiand j. 2.2.THE EXTERIOR DERIV ATIVE 21 2.4.EXAMPLE.If ÷fdxdy øgdxdz øhdydzisa2-form onR3,then d ÷fzdzdxdy øgydydxdz øhxdxdydz ÷ëùfzúgy øhxûdxdydz. Forageneral 2-form ÷å1 üi ýj ünfi,jdxidxjonRnwehave d÷å 1üiýjündfi,jdxi ÷å 1üiýjünn å k þ1¶fi,j ¶xkdxkdxidxj÷å 1 ük ýi ýj ün¶fi,j ¶xkdxkdxidxj øå 1 üi ýk ýj ün¶fi,j ¶xkdxkdxidxjøå 1üiýjýkün¶fi,j ¶xkdxkdxidxj÷å 1 üi ýj ýk ün¶fj,k ¶xidxidxjdxk øå 1 üi ýj ýk ün¶fi,k ¶xjdxjdxidxkøå 1üiýjýkün¶fi,j ¶xkdxkdxidxj (2.3)÷å 1 üi ýj ýk ün ÿ¶fi,j ¶xk ú¶fi,k ¶xj ø¶fj,k ¶xidxidxjdxk. (2.4) Hereinline (2.3) werearranged thesubscripts (for instance, intherst term we relabelled kú
i,iú
jand jú
k)and inline (2.4) weapplied thealternating property . Anobvious butquite useful remark isthat ifisann-form onRn,then dis ofdegr eenø1and sod÷0. The operator dislinear and satises ageneralized Leibniz rule. 2.5.PROPOSITION. (i)d ùa øbû ÷ad øbdforallk-formsand andallscalars aandb. (ii)d ùû ÷Uùdû ø ùú1ûkdforallk-formsandl-forms. PROOF.The linearity property (i)follows fromthelinearity ofpartial differ- entiation: ¶ ùaf øbgû ¶xi ÷a¶f ¶xi øb¶g ¶xi foralsmooth functions f,gand constants a,b. Now let ÷åIfIdxIand ÷åJgJdxJ.The Leibniz ruleforfunctions and Proposition 2.1give d ùû ÷å I,Jd ùfIgJ ûdxIdxJ ÷å I,J ùfIdgJ øgJdfI ûdxIdxJ÷å I,J dfIdxI ùgJdxJ û ø ùú1ûkfIdxI ùdgJdxJ û÷ëùdû ø ùú1ûkd, which proves part (ii). QED Hereisoneofthemost curious properties oftheexterior derivative. 22 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE 2.6.PROPOSITION.d d  0foranyform.Inshort, d2 0. PROOF.Let åIfIdxI.Then d d  d å In å i1¶fI ¶xidxidxI å In å i1d ¶fI ¶xidxidxI. Applying theformula ofExample 2.3(replacing fiwith ¶fI¶xi)wend n å i 1d ¶fI ¶xi dxi å 1 i j n ¶2fI ¶xi¶xj¶2fI ¶xj¶xi dxidxj 0, because forany smooth (indeed, C2)function fthemixed partials ¶2f¶xi¶xjand ¶2f¶xj¶xiareequal. Hence dd 0. QED 2.3. Closed and exact forms Aformisclosed ifd 0.Itisexact if dforsome form(ofdegr ee oneless). 2.7.PROPOSITION.Every exact form isclosed. PROOF.If dthen d d d  0byProposition 2.6. QED 2.8.EXAMPLE.ydx xdyisnotclosed and thereforecannot beexact. On theother hand ydx xdyisclosed. Itisalso exact, because d xy  ydx xdy. Fora0-form (function) fonRntobeclosed allitspartial derivatives must vanish, which means itisconstant. Anonzer oconstant function isnotexact, because forms ofdegr ee1are0. Isevery closed form ofpositive degr eeexact? This question hasinter esting ramications, which weshall explor einChapters 4,5and 10. Amazingly ,the answer depends strongly onthetopology ,that isthequalitative “shape”, ofthe domain ofdenition oftheform. Letusconsider thesimplest case ofa1-form ån i 1fidxi.Determining whetherisexact means solving theequation dg forthefunction g.This amounts to ¶g ¶x1 f1,¶g ¶x2 f2, ...,¶g ¶xn fn, (2.5) asystem ofrst-order partial differ ential equations .Finding asolution issometimes called integrating thesystem. ByProposition 2.7this isnotpossible unlessis closed. Bytheformula inExample 2.3isclosed ifand only if ¶fi ¶xj ¶fj ¶xi forall1ijn.These identities must besatised forthesystem (2.5) tobe solvable and arethereforecalled theintegrability conditions forthesystem. 2.9.EXAMPLE.Let ydxzcosyzxdyycosyzdz.Then d dydxzysinyzcosyzdzdydxdyyzsinyzcosyzdydz 0, 2.4.THE HODGE STAROPERA TOR 23 soisclosed. Isexact? Letussolve theequations ¶g ¶x y,¶g ¶y zcosyzx,¶g ¶z ycosyz bysuccessive integration. The rst equation gives gyxc y,z!,wher ecis afunction ofyand zonly.Substituting into thesecond equation gives ¶c"¶yzcosyz,socsinyz k z !.Substituting into thethirdequation gives k #0,so kisaconstant. Sogyx sinyzisasolution and thereforeisexact. This method works always fora1-form dened onallofRn.(See Exer cise2.6.) Hence every closed 1-form onRnisexact. 2.10.EXAMPLE.The 1-form onR2$&%0 'dened by  $y x2y2dx x x2y2dy $ydxxdy x2y2. iscalled theangle form forreasons that will become clear inSection 4.3.From ¶ ¶xx x2y2y2$x2 x2y2!2,¶ ¶yy x2y2x2$y2 x2y2!2 itfollows that theangle form isclosed. This example iscontinued inExamples 4.1 and 4.6,wher eweshall seethat this form isnotexact. Fora2-formå1 (i )j (nfi,jdxidxjand a1-formån i*1gidxitheequa- tion damounts tothesystem ¶gj ¶xi $¶gi ¶xj fi,j. (2.6) Bytheformula inExample 2.4theintegrability condition d0comes down to ¶fi,j ¶xk $¶fi,k ¶xj ¶fj,k ¶xi 0 forall1+i,j,k+n.Weshall learn how tosolve thesystem (2.6) ,and its higher -degr eeanalogues, inExample 10.18. 2.4. The Hodge star operator The binomial coefcient -n k .isthenumber ofways ofselecting k(unor dered) objects fromacollection ofnobjects. Equivalently ,-n k .isthenumber ofways of partitioning apile ofnobjects into apile ofkobjects and apile ofn $kobjects. Thus weseethat/ n k 0 / n n $k 0. This means that inacertain sense thereareasmany k-forms asn $k-forms. In fact, thereisanatural way toturn k-forms into n $k-forms. This istheHodge star operator .Hodge star ofisdenoted by 1(orsometimes 2)and isdened as follows. IfåIfIdxI,then1å IfI 31dxI !, with1dxI"IdxIc. 24 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE Here,forany increasing multi-index I,Icdenotes thecomplementary increasing multi-index, which consists ofallnumbers between 1and nthat donotoccur inI. The factor"Iisasign, "I 4651ifdxIdxIc4dx1dx278787dxn,91ifdxIdxIc4 9dx1dx278787dxn. Inother words, :dxIistheproduct ofallthedxj'sthat donotoccur indxI,times a factor;1which ischosen insuch away that dxI< :dxI=isthevolume form: dxI < :dxI =>4dx1dx2 78787dxn. 2.11.EXAMPLE.Letn46and I4?<2,6=.Then Ic4@<1,3,4,5=,sodxI 4 dx2dx6and dxIc4dx1dx3dx4dx5.Ther efore dxIdxIc4dx2dx6dx1dx3dx4dx54dx1dx2dx6dx3dx4dx5 4 9dx1dx2dx3dx4dx5dx6, which shows that"I 4 91.Hence :<dx2dx6 =A4 9dx1dx3dx4dx5. 2.12.EXAMPLE.OnR2wehave :dx4dyand :dy4 9dx.OnR3wehave:dx4dydz, :<dxdy= 4dz,:dy4 9dxdz4dzdx, :<dxdz=>4 9dy,:dz4dxdy,:<dydz= 4dx. (This isthereason that 2-forms onR3aresometimes written asfdxdy Bgdzdx B hdydz,incontravention ofourusual ruletowrite thevariables inincreasing order. Inhigher dimensions itisbetter tostick totherule.) OnR4wehave:dx14dx2dx3dx4,:dx34dx1dx2dx4,:dx2 4 9dx1dx3dx4, :dx4 4 9dx1dx2dx3, and:<dx1dx2 =>4dx3dx4, :<dx2dx3 =A4dx1dx4,:<dx1dx3 =>4 9dx2dx4, :<dx2dx4 =A4 9dx1dx3,:<dx1dx4=>4dx2dx3,:<dx3dx4=A4dx1dx2. OnRnwehave :14dx1dx2 78787dxn, :<dx1dx2 78787dxn=A41,and:dxi 4C< 91=i D1dx1dx2 78787FEdxi 78787dxn for1 Gi Gn,:<dxidxj=A4C< 91=i Dj D1dx1dx278787HEdxi78787IEdxj78787dxn for1GiJjGn. 2.5. div,grad and curl Avector eld onanopen subset UofRnisasmooth map F:U KRn.Wecan write Fincomponents as F<x=>4@L MMMNF1 <x= F2<x= ... Fn<x= OQPPPR, 2.5.DIV,GRAD AND CURL 25 oralternatively asF Sån iT1Fiei,wher ee1,e2,...,enarethestandar dbasis vectors ofRn.Vector elds intheplane can beplotted byplacing thevector F Ux Vwith itstailatthepoint x.The diagrams below represent thevector elds Wye1 Xxe2 and UWxXxy Ve1 X Uy Wxy Ve2(which you may recognize fromExer cise1.10). The arrows have been shortened soasnottoclutter thepictur es.The black dots are thezeroesofthevector elds (i.e.points xwher eF Ux V>S0). xy xy Wecan turn Finto a1-formbyusing theFiascoefcients: Sån iT1Fidxi. Forinstance, the1-form SYW ydxXxdycorresponds tothevector eld F SWye1 Xxe2.Letusintroduce thesymbolic notation dx SYZ [[[\dx1 dx2 ... dxn ]Q^^^_, which wewill think ofasavector -valued 1-form. Then wecanwriteSF`dx. Clearly ,Fisdetermined byand vice versa. Thus vector elds and 1-forms are symbiotically associated tooneanother . vector eld Facb 1-form:SF`dx. Intuitively ,thevector -valued 1-form dxrepresents aninnitesimal displacement. IfFrepresents aforceeld, such asgravity oranelectric forceacting onaparticle, thenSF`dxrepresents thework done bytheforcewhen theparticle isdisplaced byanamount dx.(Iftheparticle travels along apath, thetotal work done bythe forceisfound byintegratingalong thepath. Weshall seehow todothisinSection 4.1.) The correspondence between vector elds and 1-forms behaves inaninter est- ingway with respect toexterior differentiation and theHodge star operator .For 26 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE each function fthe1-form df dån ie1f¶f g¶xi hdxiisassociated tothevector eld gradf dn å ie1¶f ¶xiei dji kkkkkl¶f ¶x1¶f ¶x2... ¶f ¶xn mQnnnnno. This vector eld iscalled thegradient off.(Equivalently ,wecanview gradfas thetranspose oftheJacobi matrix off.) gradfpcq df: dfdgradfrdx. Starting with avector eld Fand lettingdFrdx,wendsdn å i e1Fif sdxih dn å i e1Fifut1hiv1dx1dx2 r8r8rxwdxi r8r8rdxn, Using thevector -valued nt1-formsdx dikkkl sdx1sdx2 ...sdxn mQnnno dikkkldx2dx3 r8r8rdxntdx1dx3 r8r8rdxn ...fut1hnv1dx1dx2 r8r8rdxny1 mQnnno wecan also write s dF r sdx.Intuitively ,thevector -valued nt1-form sdx represents aninnitesimal nt1-dimensional hypersurface perpendicular todx. (This point ofview will bejustied inSection 8.3,after theproofofTheor em8.14.) Inuid mechanics, theow ofauid orgasinRnisrepresented byavector eld F. The nt1-form sthen represents theux,that istheamount ofmaterial passing through thehypersurface sdxperunit time. (The total amount ofuid passing through ahypersurface Sisfound byintegratingover S.Weshall seehow todo thisinSection 5.1.) Wehave d s ddfF r sdxh dn å i e1¶Fi ¶xi ft1hi v1dxidx1dx2 r8r8rxwdxi r8r8rdxndn å ie1¶Fi ¶xidx1dx2 r8r8rdxi r8r8rdxn d{zn å ie1¶Fi ¶xi|dx1dx2 r8r8rdxn. The function divF dån ie1¶Fi g¶xiisthediver gence ofF.Thus if dF rdx,then d s ddfF r sdxh ddivFdx1dx2 r8r8rdxn. Analternative way ofwriting thisidentity isobtained byapplying stoboth sides, which gives divFd sd s. Avery differentidentity isfound byrst applying dand then sto: d dn å i,je1¶Fi ¶xjdxjdxi då 1}i~j}n¶Fj ¶xi t¶Fi ¶xj€dxidxj, EXERCISES 27 and hence  d‚å 1ƒi„jƒn…u†1‡i ˆj ˆ1 ‰¶Fj ¶xi†¶Fi ¶xj Šdx1dx2‹8‹8‹HŒdxi ‹8‹8‹Œdxj ‹8‹8‹dxn. Inthreedimensions  disa1-form and soisassociated toavector eld, namely curlF ‚ ‰¶F3 ¶x2†¶F2 ¶x3Še1† ‰¶F3 ¶x1†¶F1 ¶x3Še2 Ž ‰¶F2 ¶x1†¶F1 ¶x2Še3, thecurl ofF.Thus, forn‚3,if‚F‹dx,then curlF‹dx‚  d. Youneed notmemorize every detail ofthis discussion. The point israther to remember that exterior differentiation incombination with theHodge star unies and extends toarbitrary dimensions theclassical differential operators ofvector calculus. Exercises 2.1.Compute theexterior derivative ofthefollowing forms. Recall that ahatindicates that aterm hastobeomitted. (i)exyzdx. (ii)ån i 1x2 idx1uu’‘dxi uQdxn. (iii)“x“pån i1”–•1—i ˜1xidx1uu ‘dxiuQdxn,wher episarealconstant. Forwhat val- uesofpisthisform closed? 2.2.Consider theforms™xdx•ydy,™zdxdyšxdydzand ™zdyonR3. Calculate (i), ; (ii)d,d,d . 2.3.Writethecoor dinates onR2nas”x1,y1,x2,y2,...,xn,yn —.Let ! ™dx1dy1 šdx2dy2 šuu šdxndyn ™n å i1dxidyi. Compute!n™!!uu!(n-fold product). First work outthecases n ™1,2,3. 2.4.Writethecoor dinates onR2n˜1as”x1,y1,x2,y2,...,xn,yn,z —.Let  ™dz šx1dy1 šx2dy2 šu šxndyn ™dz šn å i1xidyi. Compute”d —n™”ddud —.First work outthecases n ™1,2,3. 2.5.Check that each ofthefollowing forms›œ1”R3—isclosed and nd afunction gsuch that dg ™. (i)™”yexy•zsin”xz—3—dxš”xexyšz2—dyš”–•xsin”xz—Fš2yzš3z2—dz. (ii) ™2xy3z4dx š”3x2y2z4•zeysin”zey—3—dy š”4x2y3z3•eysin”zey—Hšez—dz. 2.6.Let™ån i1fidxibeaclosed Cž1-form onRn.Dene afunction gby g”x —™ Ÿx1 0f1”t,x2,x3,...,xn —dt š¡Ÿx2 0f2”0,t,x3,x4,...,xn —dtš¡Ÿx3 0f3”0,0,t,x4,x5,...,xn —dt šQ š¢Ÿxn 0fn”0,0,...,0,t —dt. Show that dg ™.(Apply thefundamental theor emofcalculus, formula (B.3), differentiate under theintegral sign and don't forgettoused ™0.) 28 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE 2.7.Let £ån i ¤1fidxibeaclosed 1-form whose coefcients fiaresmooth functions dened onRn ¥§¦0¨that areallhomogeneous ofthesame degr eep© £ ¥1.Let g ªx «¬£1 p ­1n å i ¤1xifi ªx «. Show that dg£.(Use d£0and apply theidentity proved inExer ciseB.5toeach fi.) 2.8.Letandbeclosed forms. Provethatisalso closed. 2.9.Letbeclosed andexact. Provethatisexact. 2.10.Calculate ®, ®, ® , ®¯ª «,wher e,and areasinExer cise2.2. 2.11.Consider theform£ ¥x2 2dx1 ­x2 1dx2onR2. (i)Find®and®d®d. (ii)Repeat thecalculation, regar dingasaform onR3. (iii) Again repeat thecalculation, now regar dingasaform onR4. 2.12.Provethat ®°® £±ª ¥1 «kn²kforevery k-formonRn. 2.13.Let £åIaIdxIand £åIbIdxIbeconstant k-forms, i.e.with constant coef- cients aIand bI.(Wealso assume, asusual, that themulti-indices Iareincreasing.) The inner product ofandisthenumber dened byª,«¬£å IaIbI. Provethefollowing assertions. (i)The dxIform anorthonormal basis ofthespace ofconstant k-forms. (ii) ª, «´³0foralland ª, «¬£0ifand only if £0. (iii)ªµ®«¶£·ª,«dx1dx2 ¸u¸Q¸dxn. (iv)ªµ®«¶£ªµ®«. (v)The Hodge star operator isorthogonal, i.e. ª, «¹£±ªµ®, ® «. 2.14.The Laplacian ofasmooth function onanopen subset ofRnisdened byºf £¶2f ¶x2 1 ­¶2f ¶x2 2 ­¸¸Q¸ ­¶2f ¶x2n. Provethefollowing formulas. (i) ºf £»®d ®df. (ii) ºªfg«£±ª ºf«g­f ºg­2®¼ªdfªµ®dg«½«.(Use Exer cise2.13(iv) .) 2.15.Letf:Rn ¾Rbeafunction and let£fdxi. (i)Calculate d®d®. (ii)Calculate®d®d. (iii) Show that d ®d ® ­¿ª ¥1 «n®d ®d £±ª ºf «dxi,wher e ºistheLaplacian dened in Exer cise2.14. 2.16. (i)LetUbeanopen subset ofRnand letf:U ¾Rbeafunction satis- fying gradfªx«À© £0forallxinU.OnUdene avector eld n,ann ¥1-form and a1-formby n ªx «£Á gradf ªx «QÁÃÂ1gradf ªx «, £n¸ ®dx,  £Ágradf ªx «QÁÂ1df. Provethat dx1dx2¸u¸u¸dxn £onU. (ii)Letr:Rn ¾Rbethefunction r ªx «Ä£ÅÁ x Á(distance totheorigin). Deduce from part (i)that dx1dx2 ¸¸u¸dxn £±ªdr «onRn¥¦0 ¨,wher e £Áx ÁÂ1x¸ ®dx. EXERCISES 29 2.17.The Minkowski orrelativistic inner product onRnÆ1isgiven byÇx,yȹÉn å i Ê1xiyiËxnÆ1ynÆ1. Avector x ÌRn Æ1isspacelike if Çx,x È´Í0,lightlike if Çx,x ȬÉ0,and timelike if Çx,x ÈÄÎ0. (i)Give examples of(nonzer o)vectors ofeach type. (ii)Show that forevery xÏ É0thereisaysuch that Çx,yÈÐÏ É0. AHodge star operator corresponding tothis inner product isdened asfollows: ifÉ åIfIdxI,then Ñ  Éå IfI Ç Ñ dxI È, withÑ dxI ÉÓÒ"IdxIcifIcontains n Ô1,Ë"IdxIcifIdoes notcontain n Ô1. (Her e"Iand Icareasinthedenition oftheordinary Hodge star.) (iii) Find Ñ 1, Ñ dxifor1ÕiÕnÔ1,and ÑÇdx1dx2 ÖÖuÖdxn È. (iv) Compute the“relativistic Laplacian” (usually called thed'Alembertian orwave operator) Ñ d Ñ dfforany smooth function fonRnÆ1. (v)FornÉ3(ordinary space-time) nd ÑÇdxidxj Èfor1ÕiÎjÕ4. 2.18.One ofthegreatest advances intheor etical physics ofthenineteenth century was Maxwell's formulation oftheequations ofelectr omagnetism: curlE ÉË1 c¶B ¶t(Faraday's Law) , curlH É4 cJ Ô1 c¶D ¶t(Ampèr e'sLaw) , divDÉ4 (Gauß' Law) , divB É0 (nomagnetic monopoles). Herecisthespeed oflight, Eistheelectric eld, Histhemagnetic eld, Jisthedensity ofelectric current,isthedensity ofelectric char ge,Bisthemagnetic induction and Dis thedielectric displacement. E,H,J,Band Darevector elds andisafunction onR3and alldepend ontime t.The Maxwell equations look particularly simple indifferential form notation, asweshall now see. Inspace-time R4with coor dinates Çx1,x2,x3,x4 È,wher e x4 Éct,introduce forms  É ÇE1dx1 ÔE2dx2 ÔE3dx3 Èdx4 ÔB1dx2dx3 ÔB2dx3dx1 ÔB3dx1dx2,  ÉË ÇH1dx1 ÔH2dx2 ÔH3dx3 Èdx4 ÔD1dx2dx3 ÔD2dx3dx1 ÔD3dx1dx2, É1 c ÇJ1dx2dx3 ÔJ2dx3dx1 ÔJ3dx1dx2 Èdx4Ëdx1dx2dx3. (i)Show that Maxwell's equations areequivalent to dÉ0, d Ô4 É0. (ii)Conclude that isclosed and that divJ Ô¶ ׶t É0. (iii) Invacuum one hasEÉDand HÉB.Show that invacuumÉ Ñ ,the relativistic Hodge star ofdened inExer cise2.17. (iv) Freespace isavacuum without char gesorcurrents. Show that theMaxwell equa- tions infreespace areequivalent tod Éd Ñ  É0. 30 2.DIFFERENTIAL FORMS ON EUCLIDEAN SPACE (v)Letf,g:R ØRbeany smooth functions and dene EÙxÚ¬ÛÝÜ Þ0 fÙx1 ßx4 Ú gÙx1ßx4 Ú½à á, BÙxÚÛÝÜ Þ0ßgÙx1 ßx4 Ú fÙx1ßx4 ÚÐà á. Show that thecorresponding 2-formsatises thefreeMaxwell equations d Û d â Û0.Such solutions arecalled electr omagnetic waves .Explain why.Inwhat direction dothese waves travel? CHAPTER 3 Pulling back forms 3.1. Determinants The determinant ofasquar ematrix istheoriented volume oftheblock (paral- lelepiped) spanned byitscolumn vectors. Itisthereforenotsurprising that differ- ential forms areclosely related todeterminants. This section isareview ofsome fundamental facts concerning determinants. Let A ãjä åæa1,1...a1,n ...... an,1...an,n çQèé beannên-matrix with column vectors a1,a2,...,an.Itsdeterminant isvariously denoted by detA ãdet ëa1,a2,...,anì ãdet ëai,j ì1íi,jín ã6îîîîîîîa1,1...a1,n ...... an,1...an,n îîîîîîî. Expansion ontherst column. Youhave probably seen thefollowing deni- tion ofthedeterminant: detA ãn å iï1 ëð1ìi ñ1ai1detAi,1. HereAi,jdenotes the ën ð1ì êòën ð1ì-matrix obtained fromAbystriking outthe i-throwand thej-thcolumn. This isarecursive denition, which reduces thecal- culation ofany determinant tothat ofdeterminants ofsmaller size. (The recursion starts atnã1;thedeterminant ofa1ê1-matrixëaìissimply dened tobethe number a.)Itisauseful rule,butithastwo serious aws: rst, itisextremely inef- cient computationally (except formatrices containing lotsofzeroes), and second, itobscur estherelationship with volumes ofparallelepipeds. Axioms. Afarbetter denition isavailable. The determinant can becom- pletely characterized bythreesimple laws, which make good sense inview ofits geometrical signicance and which comprise anefcient algorithm forcalculating any determinant. 3.1.DEFINITION.Adeterminant isafunction detwhich assigns toevery or- deredn-tuple ofvectors ëa1,a2,...,anìanumber det ëa1,a2,...,anìsubject tothe following axioms: 31 32 3.PULLING BACK FORMS (i)detismultilinear (i.e.linear ineach column): det óa1,a2,...,caiôc õa õi,...,anö÷cdet óa1,a2,...,ai,...,anöôc õdet óa1,a2,...,a õi,...,anö forallscalars c,cõand allvectors a1,a2,...,ai,aõi,...,an; (ii)detisalternating orantisymmetric : det óa1,...,ai,...,aj,...,anö ÷ùødet óa1,...,aj,...,ai,...,anö forany iú ÷j; (iii) normalization :detóe1,e2,...,enö ÷1,wher ee1,e2,...,enarethestan- dardbasis vectors ofRn. Wealso write detAinstead ofdet óa1,a2,...,anö,wher eAisthematrix whose columns area1,a2,...,an.Axiom (iii)lays down thevalue ofdetI.Axioms (i) and (ii)govern thebehaviour oforiented volumes under theelementary column operations onmatrices. Recall that these operations come inthreetypes: adding a multiple ofany column ofAtoany other column (type I);multiplying acolumn byanonzer oconstant (type II);and inter changing any two columns (type III). Type Idoes notaffectthedeterminant, type IImultiplies itbythecorresponding constant, and type IIIcauses asign change. This canberestated asfollows. 3.2.LEMMA.IfEisanelementary column operation, then det óE óAöuö ÷kdetA, wher e k ÷üû ý þýÿ1ifEisoftype I, cifEisoftype II(multiplication ofacolumn byc),ø1ifEisoftype III. 3.3.EXAMPLE.Identify thecolumn operations applied ateach step inthefol- lowing calculation.111 4109 154 ÷ 100 465 143 ÷ 100 415 113 ÷ 100 312 010 ÷2 100 311 010 ÷2 100 001 010 ÷Cø2 100 010 001 ÷ùø2. Asthis example suggests, theaxioms (i)–(iii) sufce tocalculate any n
n- determinant. Inother words,thereisatmost one function detwhich obeys these axioms. Mor eprecisely ,wehave thefollowing result. 3.4.THEOREM(uniqueness ofdeterminants) .Letdetanddet õbetwofunctions satisfying Axioms (i)–(iii).Then detA ÷det õAforalln
n-matrices A. PROOF.Leta1,a2,...,anbethecolumn vectors ofA.Suppose rst that A isnotinvertible. Then thecolumns ofAarelinearly dependent. Forsimplicity letusassume that therst column isalinear combination oftheothers: a1 ÷ c2a2ô ôcnan.Applying axioms (i)and (ii)weget detA ÷n å i 2cidet óai,a2,...,ai,...,anö ÷0, 3.1.DETERMINANTS 33 and forthesame reason det  A 0,sodetA det  A.Now assume that A isinvertible. Then Aiscolumn equivalent totheidentity matrix, i.e.itcan be transformed toIbysuccessive elementary column operations. LetE1,E2,...,Em bethese elementary operations, sothat EmEm 1   E2E1 A  I.Accor ding to Lemma 3.2,each operation Eihastheeffect ofmultiplying thedeterminant bya certain factor ki,soaxiom (iii)yields 1 detI det EmEm 1   E2E1 A   kmkm 1   k2k1detA. Applying thesame reasoning todet  Aweget1kmkm1   k2k1det  A.Hence detA1 k1k2   km det  A. QED 3.5.REMARK(change ofnormalization) .Suppose that det  isafunction that satises themultilinearity axiom (i)and theantisymmetry axiom (ii)butisnormal- ized differently: det  I c.Then theproofofTheor em3.4shows that det  A  cdetAforalln n-matrices A. This result leaves anopen question. Wecancalculate thedeterminant ofany matrix bycolumn reducing ittotheidentity matrix, buttherearemany different ways ofperforming this reduction. Dodifferentcolumn reductions lead tothe same answer forthedeterminant? Inother words,aretheaxioms (i)–(iii) consis- tent? Wewill answer thisquestion bydisplaying anexplicit formula forthedeter - minant ofany nn-matrix that does notinvolve any column reductions. Unlike Denition 3.1,thisformula isnotvery practical forthepurpose ofcalculating large determinants, butithasother uses, notably inthetheory ofdifferential forms. 3.6.THEOREM(existence ofdeterminants) .Every nn-matrix Ahasawell- dened determinant. Itisgiven bytheformula detA å  Snsign  a1, 1 a2, 2    an, n . This requir esalittle explanation. Snstands forthecollection ofallpermutations oftheset 1,2,...,n .Apermutation isaway ofordering thenumbers 1,2,...,n. Permutations areusually written asrowvectors containing each ofthese numbers exactly once. Thus forn 2thereareonly two permutations: 1,2 and 2,1 . Forn3allpossible permutations are 1,2,3 , 1,3,2 , 2,1,3 , 2,3,1 , 3,1,2 , 3,2,1 . Forgeneral nthereare n n 1  n 2    3 2 1 n! permutations. Analternative way ofthinking ofapermutation isasabijective (i.e.one-to-one and onto) map fromtheset 1,2,...,n toitself. Forexample, for n 5apossible permutation is 5,3,1,2,4, and wethink ofthis asashorthand notation forthemapgiven by 1 5,  23, 31, 42and 54.The permutation 1,2,3,...,n1,n then corresponds totheidentity map ontheset1,2,...,n. Ifistheidentity permutation, then clearly i  j whenever i j. However ,ifisnottheidentity permutation, itcannot preserve theorderinthis way.Aninversion inisany pair ofnumbers iand jsuch that 1 i j nand 34 3.PULLING BACK FORMS  i  j .The length of,denoted byl  ,isthenumber ofinversions in. Apermutation iscalled even oroddaccor ding towhether itslength iseven, resp. odd. Forinstance, thepermutation 5,3,1,2,4 haslength 6and soiseven. The sign ofis sign    ! 1 l"# $ 1ifiseven,!1ifisodd. Thus sign5,3,1,2,4% 1.The permutations of&1,2'are1,2,which hassign 1,and 2,1 ,which hassign !1,while forn 3wehave thetable below .  l  sign  1,2,3  0 11,3,2  1 !12,1,3  1 !12,3,1  2 13,1,2 2 13,2,1 3!1 Thinking ofpermutations inSnasbijective maps from &1,2,...,n 'toitself, wecanform thecomposition  (ofany two permutations andinSn.For permutations weusually writeinstead of (and callittheproduct ofand .This isthepermutation produced byrstperformingand then!Forinstance, if  5,3,1,2,4 and  5,4,3,2,1 ,then   1,3,5,4,2 , )4,2,1,3,5 . Abasic factconcerning signs, which weshall notprovehere,is sign  *sign  sign  . (3.1) Inparticular ,theproduct oftwo even permutations iseven and theproduct ofan even and anodd permutation isodd. The determinant formula inTheor em3.6contains n!terms, one foreach per- mutation.Each term isaproduct which contains exactly one entry fromeach rowand each column ofA.Forinstance, forn 5thepermutation 5,3,1,2,4  contributes theterm a1,5a2,3a3,1a4,2a5,4.For2 +2-and 3 +3-determinants Theo- rem3.6gives thewell-known formulæ,,,,a1,1a1,2 a2,1a2,2 ,,,, a1,1a2,2 !a1,2a2,1,,,,,,,a1,1a1,2a1,3 a2,1a2,2a2,3 a3,1a3,2a3,3 ,,,,,, a1,1a2,2a3,3 !a1,1a2,3a3,2 !a1,2a2,1a3,3-a1,2a2,3a3,1-a1,3a2,1a3,2 !a1,3a2,2a3,1. PROOFOFTHEOREM3.6.Weneed tocheck that theright-hand side ofthe determinant formula inTheor em3.6obeys axioms (i)–(iii) ofDenition 3.1. Let usforthemoment denote theright-hand side byf A . Axiom (i)ischecked asfollows: forevery permutation theproduct a1,"1#a2,"2#/... an,"n# 3.1.DETERMINANTS 35 contains exactly oneentry fromeach rowand each column inA.Soifwemultiply thei-throwofAbyc,each term inf 0A 1ismultiplied byc.Ther efore f 0a1,a2,...,cai,...,an 1*2cf 0a1,a2,...,ai,...,an 1. Similarly , f 0a1,a2,...,ai3a 4i,...,an 1*2 f 0a1,a2,...,ai,...,an 13f 0a1,a2,...,a 4i,...,an 1. Axiom (ii)holds because ifweinter change two columns inA,each term inf 0A 1 changes sign. Toseethis, letbethepermutation inSnthat inter changes thetwo numbers iand jand leaves allothers xed. Then f 0a1,...,aj,...,ai,...,an 12å 5Snsign 0 1a1, 61 7a2, 62 7/888 an, 6n 72å  5Snsign01a1,617a2,627888an,6n7 substitute22å  5Snsign 0 1sign 0 1a1,617a2,627/888 an,6n7byformula (3.1)2:9å  5Snsign 0 1a1,617a2,627/888 an,6n7 byExer cise3.42:9 f0a1,...,ai,...,aj,...,an 1. Finally ,rule(iii)iscorrectbecause ifA 2I, a1,617a2,627 888an,6n7 2<;1if 2identity , 0otherwise , and thereforef 0I 121.Sofsatises allthreeaxioms fordeterminants. QED Herearesome further rules followed bydeterminants. Each canbededuced fromDenition 3.1orfromTheor em3.6. (Recall that thetranspose ofann=n- matrix A2)0ai,j 1isthematrix ATwhose i,j-thentry isaj,i.) 3.7.THEOREM.LetAandBben =n-matrices. (i)det0AB1>2detAdetB. (ii)detAT2detA. (iii) (Expansion onthej-thcolumn) detA2ån i?1 0@911iAjai,jdetAi,jforallj2 1,2,...,n.HereAi,jdenotes the0n911*=0 n911-matrix obtained fromA bystriking outthei-throwandthej-thcolumn. (iv) detA2a1,1a2,2888an,nifAisupper triangular (i.e.ai,j 20foriBj). Volume change. Weconclude this discussion with aslightly differentgeo- metric view ofdeterminants. Asquar ematrix Acanberegar ded asalinear map A:Rn CRn.The unit cube inRn,D0,1 En2GFx HRn I0 Jxi J1fori 21,2,...,n K, has n-dimensional volume 1.(For n 21itisusually called theunit interval and forn 22theunit squar e.)Itsimage A L D0,1 EnMunder themap Aisapar- allelepiped with edges Ae1,Ae2,...,Aen,thecolumns ofA.Hence AL D0,1 EnM 36 3.PULLING BACK FORMS hasn-dimensional volume volA N O0,1 Pn QSRUTdetA TVR<TdetA Tvol O0,1 Pn.This rule generalizes asfollows: ifXisameasurable subset ofRn,then volA WX X RYTdetA TvolX. X e1e2A AX Ae1Ae2 So TdetA Tcanbeinterpr eted asavolume change factor .(Asetismeasurable ifithas awell-dened, nite orinnite, n-dimensional volume. Explaining exactly what this means israther hard,butitsufces forour purposes toknow that allopen and allclosed subsets ofRnaremeasurable.) 3.2. Pulling back forms Bysubstituting new variables into adifferential form weobtain anew form of thesame degr eebutpossibly inadifferentnumber ofvariables. 3.8.EXAMPLE.InExample 2.10 wedened theangle form onR2 Z\[0 ]tobe  R Zydx ^xdy x2^y2. Bysubstituting x Rcostand y Rsintinto theangle form weobtain thefollowing 1-form onR:Zsintdcost^costdsint cos2t ^sin2t RNW Zsint X_W Zsint X`^cos2t Qdt Rdt. Wecantake anyk-form and substitute anynumber ofvariables into ittoobtain anew k-form. This works asfollows. Supposeisak-form dened onanopen subset VofRm.Letusdenote thecoor dinates onRmbyy1,y2,...,ymand letus write, asusual,  Rå IfIdyI, wher ethefunctions fIaredened onV.Suppose wewant tosubstitute “new” variables x1,x2,...,xnand that theoldvariables aregiven interms ofthenew by functions y1 R1 Wx1,...,xn X, y2 R2 Wx1,...,xn X, ... ym Rm Wx1,...,xn X. 3.2.PULLING BACK FORMS 37 Asusual wewrite y a bx c,wher e bxcaed fffg1 bx c 2 bx c ... m bx c hjiiik. Weassume that thefunctionsiaresmooth and dened onacommon domain U, which isanopen subset ofRn.Weregar dasamap fromUtoV.(InExample 3.8 wehave UaR,VaR2 lnm0oandbtc*a)b cost,sintc.)The pullback ofalong isthen thek-formponUobtained bysubstituting yi ai bx1,...,xn cforalli intheformula for.That istosay,pisdened by  p aå I b pfI c_b pdyI c. Here pfIisdened by  pfI afI q, thecomposition ofand fI.This means pfI bx ca fI b bx cjc;inother words, pfI isthefunction resulting fromfIbysubstituting y a bx c.The pullback pdyIis dened byreplacing each yiwithi.That istosay,ifI a bi1,i2,...,ik cweput  pdyI a pbdyi1dyi2rrrdyik c*adi1di2rrrdik. The pictur ebelow isaschematic representation ofthesubstitution process. The formaåIfIdyIisak-form iny1,y2,...,ym;itspullbackpaåJgJdxJisa k-form inx1,x2,...,xn.InTheor em3.12 below wewill give anexplicit formula forthecoefcients gJinterms offIand. U V Rn Rm s xytuxv 3.9.EXAMPLE.The formula wx1 x2x aywx3 1x2 ln bx1 zx2 cx 38 3.PULLING BACK FORMS denes amap:U {R2,wher eU |Y}x ~R2 x1 €x2 0 ‚.The components ofaregiven by1 ƒx1,x2 „ |x3 1x2and2 ƒx1,x2 „ |lnƒx1 €x2 „.Accor dingly ,  …dy1 |d1 |dƒx3 1x2 „ |3x2 1x2dx1 €x3 1dx2,  …dy2 |d2 |dlnƒx1 €x2 „ |ƒx1 €x2 „ †1ƒdx1 €dx2 „,  …ƒdy1dy2 „ |d1d2 |ƒ3x2 1x2dx1 €x3 1dx2 „_ƒx1 €x2 „ †1ƒdx1 €dx2 „|3x2 2x2‡x3 1 x1 €x2dx1dx2. Observe that thepullback operation turns k-forms onthetargetspace Vinto k-forms onthesour cespace U.Thus, while:U{Visamap fromUtoV,…is amap  …:ˆkƒV„ {‰ˆkƒU„, theopposite way fromwhat you might naively expect. (Recall thatˆkƒU„stands forthecollection ofallk-forms onU.)The property that…“turns thearrow around” iscalled contravariance .Pulling back forms isnicely compatible with the other operations that welearned about (except theHodge star). 3.10.PROPOSITION.Let:U {Vbeasmooth map, wher eUisopen inRnand Visopen inRm.Thepullback operation is (i)linear:…ƒa€b„ |a…€b…; (ii)multiplicative: …ƒ„ |ƒ…„ƒ…„; (iii) natural:…ƒ …„ |ƒ Š„ …,wher e :V {Wisasecond smooth map with Wopen inRkandaform onW. The term “natural” inproperty (iii)isamathematical catchwor dmeaning that acertain operation (inthiscase thepullback) iswell-behaved with respect tocom- position ofmaps. PROOF.If |åIfIdyIand |åIgIdyIaretwo forms ofthesame degr ee, then a€b |åI ƒafI €bgI „dyI,so  …ƒa€b„ |å I …ƒafI €bgI „_ƒ …dyI „. Now  …ƒafI €bgI „ƒx„ |ƒafI €bgI „_ƒƒx„ „ |afI ƒƒx„ „`€ bgI ƒƒx„ „|a …fI ƒx„‹€b …gI ƒx„, so…ƒa€b„ |åI ƒa…fI €b…gI „_ƒ…dyI „ |a…€b….This proves part (i).Fortheproofofpart (ii)consider two forms |åIfIdyIand |åJgJdyJ (not necessarily ofthesame degr ee).Then |åI,JfIgJdyIdyJ,so …ƒ„ |å I,J…ƒfIgJ „…ƒdyIdyJ „. Now  …ƒfIgJ „_ƒx„ |fIgJ ƒƒx„ „ |fI ƒƒx„ „gJ ƒƒx„j„ |ƒ …fI „_ƒ …gJ „_ƒx„, 3.2.PULLING BACK FORMS 39 so ŒŽfIgJ   ŒfI   ŒgJ .Furthermor e,  ŒdyIdyJ* Œdyi1 Œdyi2‘‘‘dyikdyj1‘‘‘dyjl di1di2 ‘‘‘dikdj1 ‘‘‘djl  ŒdyI   ŒdyJ , so  Œå I,J  ŒfI   ŒgJ  ŒdyI   ŒdyJ “’å I  ŒfI ŒdyI@”•’å I  ŒgJ  ŒdyJ–”—  Œ  Œ, which establishes part (ii). Fortheproofofproperty (iii)rst consider afunction fonW.Then  Œ Œf x  Œf  x  f   x j f  ˜ x  f˜™ ˜  x*  ˜ Œfx, so Œš Œf›  ˜ Œf.Next consider a1-formdzionW,wher ez1,z2,..., zkarethevariables onRk.Then Œd iåm j œ1¶ i ¶yjdyj,so  Œ Œm å jœ1 Œ’¶ i ¶yj ” Œdyj m å jœ1 Œ’¶ i ¶yj ”djm å j œ1 Œ’¶ i ¶yj ”n å l œ1¶j ¶xldxl n å l œ1 m å j œ1 Œ’¶ i ¶yj ”¶j ¶xl ždxl. Bythechain rule, formula (B.6) ,thesum åm j œ1 Œš¶ i Ÿ¶yj ¶j Ÿ¶xlisequal to ¶  Œ i Ÿ¶xl.Ther efore  Œ Œn å l œ1¶Œ i ¶xldxl d  Œ i d   ˜i   ˜ Œdzi   ˜ Œ. Because every form onWisasum ofproducts offorms oftype fand dzi,property (iii)ingeneral follows fromthetwo special casesfanddzi. QED Another application ofthechain ruleyields thefollowing important result. 3.11.THEOREM.Let:U  Vbeasmooth map, wher eUisopen inRnandVis open inRm.ThenŒŽddŒfor¡£¢kV.Inshort  Œdd Œ. PROOF.First letfbeafunction. Then  Œdf Œm å i œ1¶f ¶yidyi m å i œ1 Œ’¶f ¶yi ”di m å i œ1 Œ’¶f ¶yi ”n å j œ1¶i ¶xjdxjn å j œ1m å i œ1 Œ’¶f ¶yi ”¶i ¶xjdxj. 40 3.PULLING BACK FORMS Bythechain rule, formula (B.6) ,thequantity åm i¤1 ¥Ž¦¶f §¶yi ¨¶i §¶xjisequal to ¶¦¥f¨ §¶xj.Hence  ¥df©n å j ¤1¶ ¦ ¥f¨ ¶xjdxj ©d ¥f, sothetheor emistrueforfunctions. Next let ©åIfIdyI.Then d ©åIdfIdyI, so  ¥d ©å I ¥¦dfIdyI¨ ©Y¦ ¥dfI¨ ¦ ¥dyI¨ ©å Id ¦ ¥fI¨di1di2ªªªdik, because ¥dfI ©d ¥fI.Ontheother hand, d ¥ ©å Id «@¦ ¥fI¨ ¦ ¥dyI¨@¬ ©å Id «@¦ ¥fI¨di1di2 ªªªdik ¬©å Id¦ ¥fI¨di1di2 ªªªdik­å I ¦ ¥fI¨d¦di1di2 ªªªdik ¨©å Id ¦ ¥fI¨di1di2ªªªdik. Herewehave used theLeibniz ruleforforms, Proposition 2.5(ii) ,plus thefactthat theform di1di2 ªªªdikisalways closed. (See Exer cise2.8.) Comparing thetwo equations above weseethat ¥d ©d ¥. QED Wenish thissection bygiving anexplicit formula forthepullback¥,which establishes aconnection between forms and determinants. Letusdothis rst in degr ees1and 2.The pullback ofa1-form©åm i ¤1fidyiis  ¥ ©m å i ¤1 ¦ ¥fi¨ ¦ ¥dyi¨ ©m å i ¤1 ¦ ¥fi¨di. Now di ©ån j¤1¶i ¶xjdxjand so  ¥ ©m å i ¤1 ® ¦ ¥fi ¨n å j ¤1¶i ¶xjdxj ¯ ©n å j ¤1m å i ¤1 ¦ ¥fi ¨¶i ¶xjdxj ©n å j ¤1gjdxj, with gj ©åm i¤1 ¦ ¥fi ¨¶i ¶xj. Fora2-form©å1 °i ±j °mfi,jdyidyjweget  ¥ ©å 1°i±j°m ¦ ¥fi,j ¨ ¥¦dyidyj ¨ ©å 1°i±j°m ¦ ¥fi,j ¨didj. Observe that didj ©n å k,l¤1¶i ¶xk¶j ¶xldxkdxl ©å 1°k±l°n²¶i ¶xk¶j ¶xl ³¶i ¶xl¶j ¶xk ´dxkdxl, wher e ¶i ¶xk¶j ¶xl ³¶i ¶xl¶j ¶xk ©¶µµµµµµ¶i ¶xk¶i ¶xl¶j ¶xk¶j ¶xl µµµµµµ 3.2.PULLING BACK FORMS 41 isthedeterminant ofthe2 ·2-submatrix obtained fromtheJacobi matrix Dby extracting rows iand jand columns kand l.Soweget ¸¹å 1 ºi »j ºm ¼>½¸fi,j¾å 1ºk»lºn¿¿¿¿¿¿¶i ¶xk¶i ¶xk¶j ¶xk¶j ¶xl ¿¿¿¿¿¿dxkdxl À¹å 1 ºk »l ºnå 1ºi»jºm½ ¸fi,j¾¿¿¿¿¿¿¶i ¶xk¶i ¶xk¶j ¶xk¶j ¶xl ¿¿¿¿¿¿dxkdxl ¹å 1 ºk »l ºngk,ldxkdxl with gk,l ¹å 1 ºi »j ºm ½ ¸fi,j¾¿¿¿¿¿¿¶i ¶xk¶i ¶xl¶j ¶xk¶j ¶xl ¿¿¿¿¿¿. Foranarbitrary k-form¹åIfIdyIweobtain  ¸ ¹å I½ ¸fI¾ ¸½dyi1dyi2 ÁÁÁdyik ¾ ¹å I½ ¸fI¾di1di2 ÁÁÁdik. Towrite theproduct di1di2 ÁÁÁdikinterms ofthex-variables weuse dil ¹n å mlÂ1¶il ¶xmldxml forl ¹1,2,...,k.This gives di1di2 ÁÁÁdik ¹n å m1,m2,...,mkÂ1¶i1 ¶xm1¶i2 ¶xm2 ÁÁÁ¶ik ¶xmkdxm1dxm2ÁÁÁdxmk¹å M¶i1 ¶xm1¶i2 ¶xm2 ÁÁÁ¶ik ¶xmkdxM, inwhich thesummation isover allnkmulti-indices M¹½m1,m2,...,mk ¾.Ifa multi-index Mhasrepeating entries, then dxM ¹0.Iftheentries ofMareall distinct, wecanrearrange them inincreasing orderbymeans ofapermutation . Inother words,wehave M¹½m1,m2,...,mk¾ ¹½jÃ1Ä,jÃ2Ä,...,jÃkÄ ¾,wher e J¹½j1,j2,...,jk¾isanincreasing multi-index andÅSkisapermutation. Thus wecanrewrite thesum over allmulti-indices Masadouble sum over allincreas- ingmulti-indices Jand allpermutations : di1di1ÁÁÁdik ¹å Jå ÆSk¶i1 ¶xj Ç1 ȶi2 ¶xj Ç2 È ÁÁÁ¶ik ¶xj Çk ÈdxjÇ1ÈdxjÇ2È ÁÁÁdxjÇkȹå Jå  ÆSksign½¾¶i1 ¶xj Ç1 ȶi2 ¶xj Ç2 È ÁÁÁ¶ik ¶xj Çk ÈdxJ (3.2)¹å JdetDI,JdxJ. (3.3) In(3.2) used theresult ofExer cise 3.7and in(3.3) weapplied Theor em3.6. The notation DI,Jstands fortheI,J-submatrix ofD,that isthek ·k-matrix obtained fromtheJacobi matrix byextracting rows i1,i2,...,ikand columns j1,j2,...,jk. 42 3.PULLING BACK FORMS Tosum up,wend  É Êå I ÉfIå JdetDI,JdxJ Êå JËå IÌ ÉfI ÍdetDI,J ÎdxJ. This proves thefollowing result. 3.12.THEOREM.Let:UÏVbeasmooth map, wher eUisopen inRnandVis open inRm.LetÊåIfIdyIbeak-form onV.ThenÉisthek-form onUgiven by ÉÊåJgJdxJwith gJ Êå I Ì ÉfI ÍdetDI,J. This formula isseldom used tocalculate pullbacks inpractice and you don't need tomemorize thedetails oftheproof. Itisalmost always easier toapply the denition ofpullback directly .However ,theformula hassome important theor et- icaluses, oneofwhich werecordhere. Assume that k Êm Ên,that istosay,thenumber ofnew variables isequal to thenumber ofoldvariables, and wearepulling back aform oftopdegr ee.Then Êfdy1dy2ÐÐÐdyn,ÉÊÌÉfÍÌdetDÍdx1dx2ÐÐÐdxn. Iff Ê1(constant function) thenÉf Ê1,soweseethat detDÌxÍcanbein- terpr eted astheratio between theoriented volumes oftwo innitesimal blocks positioned atx:one with edges dx1,dx2,...,dxnand another with edges d1, d2,...,dn.Thus theJacobi determinant isameasur ement ofhow much the mapchanges oriented volume frompoint topoint. 3.13.THEOREM.Let:U ÏVbeasmooth map, wher eUandVareopen inRn. Then thepullback ofthevolume form onVisequal totheJacobi determinant times the volume form onU, ÉÌdy1dy2ÐÐÐdynÍ ÊÌdetDÍdx1dx2ÐÐÐdxn. Exercises 3.1.Deduce Theor em3.7(iv) fromTheor em3.6. 3.2.Calculate thefollowing determinants using column and/or rowoperations and Theor em3.7(iv). ÑÑÑÑÑÑÑÑ1 3 1 1 2 1 5 2 1Ò1 2 3 4 1 Ò37 ÑÑÑÑÑÑÑÑ, ÑÑÑÑÑÑÑÑ1 1 Ò24 0 1 1 3 2Ò1 1 0 3 1 2 5 ÑÑÑÑÑÑÑÑ. 3.3.Tabulate allpermutations inS4with their lengths and signs. 3.4.Determine thelength and thesign ofthefollowing permutations. (i)Apermutation oftheformÓ1,2,...,iÒ1,j,...,jÒ1,i,...,nÔwher e1ÕiÖ jÕn.(Such apermutation iscalled atransposition .Itinter changes iand jand leaves allother numbers xed.) (ii) Ón,n Ò1,n Ò2,...,3,2,1 Ô. 3.5.Find allpermutations inSnoflength 1. EXERCISES 43 3.6.Calculate ×1, ×1,and,wher e (i) ØÚÙ3,6,1,2,5,4 Ûand ØÚÙ5,2,4,6,3,1 Û; (ii) ØÙ2,1,3,4,5,...,n Ü1,n Ûand ØÝÙn,2,3,...,n Ü2,n Ü1,1 Û(i.e.thetrans- positions inter changing 1and 2,resp. 1and n). 3.7.Show that dxi Þ1 ßdxi Þ2 ßáà à àdxi Þk ß ØsignÙÛdxi1dxi2à à àdxik foranymulti-index Ùi1,i2,...,ik Ûand anypermutation inSk.(First show that theidentity istrueifisatransposition. Then show itistrueforanarbitrary permutation bywriting asaproduct12à àjàloftranspositions and using formula (3.1) and Exer cise3.4(i).) 3.8.Show that forn â2thepermutation group Snhasn! ã2even permutations and n!ã2odd permutations. 3.9. (i)Show that every permutation hasthesame length and sign asitsin- verse. (ii)Deduce Theor em3.7(ii) fromTheor em3.6. 3.10.The i-thsimple permutation isdened byi ØÚÙ1,2,...,i Ü1,i ä1,i,i ä2,...,n Û. Soiinter changes iand iä1and leaves allother numbers xed. Snhas nÜ1simple permutations, namely1,2,...,n×1.ProvetheCoxeter relations (i)2 i Ø1for1åiæn, (ii) Ùii ç1 Û3Ø1for1 åi æn Ü1, (iii) Ùij Û2Ø1for1 åi,j ænand i ä1 æj. 3.11.Letbeapermutation of è1,2,...,n é.The permutation matrix corresponding to isthenên-matrix Awhose i-thcolumn isthevector eëiì.Inother words,Aei Øeëiì. (i)Writedown thepermutation matrices forallpermutations inS3. (ii)Show that A ØAA. (iii) Show that detA Øsign Ù Û. 3.12. (i)Suppose that Ahastheshape A ØUí îîîïa1,1a1,2...a1,n 0 a2,2...a2,n ......... 0 an,1...an,n ðòñññó, i.e.allentries below a11are0.Deduce fromTheor em3.6that detA Øa1,1 ôôôôôôôa2,2...a2,n ...... an,2...an,n ôôôôôôô. (ii)Deduce fromthistheexpansion rule,Theor em3.7(iii) . 3.13.Show thatôôôôôôôôôôô1 1 ... 1 x1 x2 ... xn x2 1x2 2... x2n ......... xn×1 1xn×1 2...xn×1n ôôôôôôôôôôô ØÕ iõj Ùxj Üxi Û foranynumbers x1,x2,...,xn.(Starting atthebottom, fromeach rowsubtract x1times the rowabove it.This creates anew determinant whose rst column isthestandar dbasis vector e1.Expand ontherst column and note that each column oftheremaining determinant has acommon factor .) 44 3.PULLING BACK FORMS 3.14.Letö÷x1 x2 x3øùûú ö÷x1x2 x1x3 x2x3øù.Find (i)üdy1,üdy2,üdy3; (ii)ü_ýy1y2y3þ,ü_ýdy1dy2þ; (iii) ü ýdy1dy2dy3þ. 3.15.Let ÿx1 x2 ú ö

÷x3 1 x2 1x2 x1x2 2 x3 2 øù.Find (i) ü_ýy1 3y2 3y3 y4 þ; (ii)üdy1,üdy2,üdy3,üdy4; (iii)ü_ýdy2dy3þ. 3.16.Compute ü_ýxdydzydzdxzdxdyþ,wher e isthemap R2R3dened inExer ciseB.7. 3.17.LetP3 ö÷r  øù ú ö÷rcoscos rcossin rsinøùbespherical coor dinates inR3. (i)Calculate P ü3forthefollowing forms: dx,dy,dz,dxdy,dxdz,dydz,dxdydz. (ii)Find theinverse ofthematrix DP3. 3.18 (spherical coor dinates inndimensions) .Inthisproblem letuswrite apoint inRn asö


÷r 1 ... n 1 øù. LetP1bethefunction P1 ýrþúr.Foreach n1dene amap Pn 1:Rn 1 Rn 1by Pn 1 ö


÷r 1 ... n øù ú ö




÷ ýcosnþPn ö


÷r 1 ... n 1 øù rsinn øù. (This isanexample ofarecursive denition. Ifyou know P1,you cancompute P2,and then P3,etc.) (i)Show that P2and P3aretheusual polar ,resp. spherical coor dinates onR2,resp. R3. (ii)Give anexplicit formula forP4. (iii) Letpbetherst column vector oftheJacobi matrix ofPn.Show that Pnúrp. (iv) Show that theJacobi matrix ofPn 1isa ýn1þ ýn1þ-matrix oftheform DPn 1ú ÿA u vw, wher eAisann n-matrix, uisacolumn vector ,visarowvector and wisa function given respectively by AúcosnDPn, uú ýsinnþPn, vú ýsinn,0,0,...,0þ, wúrcosn. EXERCISES 45 (v)Show that detDPn 1 rcosn 1ndetDPnforn 1.(Expand detDPn 1with respect tothelastrow,using theformula inpart (iv), and apply theresult ofpart (iii).) (vi) Using theformula inpart (v)calculate detDPnforn1,2,3,4. (vii) Find anexplicit formula fordetDPnforgeneral n. (viii) Show that detDPn 0forr0. CHAPTER 4 Integration of1-forms Like functions, forms canbeintegrated aswell asdifferentiated. Differenti- ation and integration arerelated viaamultivariable version ofthefundamental theor emofcalculus, known asStokes' theor em. Inthischapter weinvestigate the case of1-forms. 4.1. Denition and elementary properties oftheintegral LetUbeanopen subset ofRn.Aparametrized curve inUisasmooth mapping c:I Ufromaninterval Iinto U.Wewant tointegrate over I.Toavoid problems with impr oper integrals weassume Itobeclosed and bounded, I a,b .(Strictly speaking wehave notdened what wemean byasmooth map c:a,b U.The easiest denition isthat cshould betherestriction ofasmooth map ˜c:a",b " Udened onaslightly largeropen interval.) Letbea1-form onU.The pullback cisa1-form ona,b,and canthereforebewritten ascgdt(wher e tisthecoor dinate onR).The integral ofover cisnow dened by c    a,b !c   b ag t dt. Mor eexplicitly ,writingincomponents,  ån i "1fidxi,wehave c  n å i"1 c fi dci n å i"1 c fi dci dtdt, (4.1) so cn å i "1 b afi ct#dci dt tdt. 4.1.EXAMPLE.LetUbethepunctur edplane R2$0 %.Letc: 0,2  Ube theusual parametrization ofthecircle,ct&' cost,sint,and letbetheangle form,  ydxxdy x2y2. Then cdt(see Example 3.8), so(c)(2 0dt2. Acurve c: a,b * Ucan bereparametrized bysubstituting anew variable, t p s ,wher esranges over another interval ¯a,¯b .Weshall assume ptobea one-to-one mapping from¯a,¯bontoa,bsatisfying p+,s- 0for¯a.s.¯b.Such apiscalled areparametrization .The parametrized curve c/p:¯a,¯b U hasthesame image astheoriginal curve c,butitistraversed atadifferentrate. Since p +s 0- 0foralls 12¯a,¯b wehave either p +s &30foralls(inwhich case pis 47 48 4.INTEGRA TION OF1-FORMS increasing) orp 465s 7980foralls(inwhich case pisdecr easing). Ifpisincreasing, wesaythat itpreserves theorientation ofthecurve (orthat thecurves cand c :p have thesame orientation );ifpisdecr easing, wesaythat itreverses theorientation (orthat cand c :phave opposite orientations ).Intheorientation-r eversing case, c :p traverses thecurve intheopposite direction toc. 4.2.EXAMPLE.The curve c: ;0,2 <>= R2dened byc 5t 7@?A5 cost,sint 7rep- resents theunit circleintheplane, traversed ataconstant rate (angular velocity) of1radian persecond. Letp 5s 7B? 2s.Then pmaps ;0, <to ;0,2 <and c :p, regar ded asamap ;0, <C= R2,represents thesame circle,buttraversed at2ra- dians persecond. (Itisimportant torestrict thedomain ofptotheinterval ;0, <. Ifweallowed storange over ;0,2 <,then 5cos2s,sin2s 7would traverse thecircle twice. This isnotconsider edareparametrization oftheoriginal curve c.)Now letp 5s 7D?FE s.Then c :p: ;0,2 <G= R2traverses theunit circleintheclockwise direction. This reparametrization reverses theorientation; theangular velocity is now E1radian persecond. Finally letp 5s 7H? 2s2.Then pmaps ;0,1 <to ;0,2 < and c:p:;0,1<I= R2runsonce counter clockwise through theunit circle,butata variable rate. What istheangular velocity asafunction ofs? Itturns outthat theintegral ofaform along acurve isalmost completely in- dependent oftheparametrization. 4.3.THEOREM.Letbea1-form onUand c: ;a,b <J= Uacurve inU.Let p:;¯a,¯b<=K; a,b<beareparametrization. ThenL c Mp ?AN Ocifppreserves theorientation ,EOcifpreverses theorientation . PROOF.Itfollows fromthedenition oftheintegral and fromthenaturality ofpullbacks (Proposition 3.10(iii) )thatL c Mp ? LQP ¯a,¯b R 5c :p 7#S ? L P ¯a,¯b Rp ST5c S 7. Now letuswrite cS ?gdtand t ?p 5s 7.Then pS 5cS 7&? pS 5gdt 7U?A5 pSg 7dp ?5pSg 7V5dp Wds 7ds,soL c Mp ? L P ¯a,¯b R 5p Sg 7dp dsds ? L¯b ¯ag 5p 5s 7#7p 45s 7ds. Ontheother hand,Oc ?Ob ag 5t 7dt,sobythesubstitution formula, Theor emB.7, wehaveOcMp ?YXOc,wher ethe Zoccurs ifp 4\[0and the Eifp 4\80.QED Interpretation oftheintegral. Integrals of1-forms play animportant role inphysics and engineering. Acurve c:;a,b<*= Umodels aparticle travelling through theregion U.Recall fromSection 2.5that toa1-form?ån i ]1Fidxicor- responds avector eld F?ån i ]1Fiei,which canbethought ofasaforceeld acting ontheparticle. Symbolically wewrite?F^dx,wher ewethink ofdxasaninn- itesimal vector tangent tothecurve. Thusrepresents thework done bytheforce eld along aninnitesimal vector dx.From(4.1) weseethat cS?F5c5t7#7I^c 45t7dt. Accor dingly ,thetotal work done bytheforceFontheparticle during itstripalong cistheintegralL c? L cF^dx? Lb aF5c5t7_7`^c 45t7dt. 4.2.INTEGRA TION OFEXACT 1-FORMS 49 Inparticular ,thework and thetotal work areniliftheforceisperpendicular to thepath, asinthepictur eontheleft. The work done bytheforceinthepictur eon theright isnegative. c c Theor em4.3canbetranslated into this language asfollows: thework done bythe forcedoes notdepend ontherate atwhich theparticle speeds along itspath, but only onthepath itself and onthedirection oftravel. The eld Fisconservative ifitcan bewritten asthegradient ofafunction, Fagradg.The functionbgiscalled apotential fortheeld and isinterpr eted as thepotential ener gyoftheparticle. Interms offorms this means that adg,i.e. isexact. 4.2. Integration ofexact 1-forms Integrating anexact 1-form adgiseasy once thefunction gisknown. 4.4.THEOREM(fundamental theor emofcalculus inRn).Let adgbeanexact 1-form onanopen subset UofRn.Letc: ca,b de Ubeaparametrized curve. Thenf c ag gc gb h_h`b g gc ga h#h. PROOF.ByTheor em3.11 wehave c i ac idg adc ig.Writing h gt hjac ig gt hja g gc gt h#hwehave c i adh,sof c a fQk a,b lc i a fb adh ah gb hmbh ga h, wher eweused the(ordinary) fundamental theor emofcalculus, formula (B.1) . Hence nc ag gc gb h#hb g gc ga h#h. QED The physical interpr etation ofthis result isthat when aparticle moves ina conservative forceeld, itspotential ener gydecr eases bytheamount ofwork done bytheeld. This claries what itmeans foraeld tobeconservative: itmeans that thework done isentir elyconverted into mechanical ener gyand that none is dissipated byfriction into heat, radiation, etc. Thus thefundamental theor emof calculus “explains” thelawofconservation ofener gy. 50 4.INTEGRA TION OF1-FORMS Italso yields anecessary and sufcient criterion fora1-form onUtobeexact. Acurve c: oa,b pq Uiscalled closed ifc ra sGtc rb s. 4.5.THEOREM.Letbea1-form onanopen subset UofRn.Then thefollowing statements areequivalent. (i)isexact. (ii) uc t0forallclosed curves c. (iii)ucdepends only ontheendpoints ofcforevery curve cinU. PROOF.(i) tmv (ii):if tdgand cisclosed, then uc tg rc rb s#sIw g rc ra s_sxt 0bythefundamental theor emofcalculus, Theor em4.4. (ii) tv (iii): assume uc t0forallclosed curves c.Let c1: oa1,b1 pq U and c2: oa2,b2 pq U betwo curves with thesame endpoints, i.e.c1 ra1 sDt c2 ra2 sand c1 rb1 s0t c2 rb2 s. Weneed toshow thatuc1tyuc2.After reparametrizing c1and c2wemay assume that a1 ta2 t0and b1 tb2 t1.Dene anew curve cby c rt szt|{c1 rt s for0 }t }1, c2 r2wtsfor1}t}2. (First traverse c1,then traverse c2backwar ds.) Then cisclosed, so uc t0.But Theor em4.3implies uc t)uc1 w~uc2,so uc1 t)uc2. (iii) tmv (i):assume that, forallc,ucdepends only ontheendpoints ofc. Wemust dene afunction gsuch that tdg.Fixapoint x0inU.Foreach point xinUchoose acurve cx: o0,1 p`q Uwhich joins x0tox.Dene g rx sCt) cx. Weassert that dgiswell-dened and equal to.Writetån i €1fidxi.Wemust show that ¶g¶xi tfi.Fromthedenition ofpartial differentiation, ¶g ¶xi rx sjtlim h ‚0grxƒhei s>wgrxs h tlim h ‚01 h „  cx …hei w~ cx †. Now consider acurve ˜ccomposed oftwo pieces: for0 }t }1travel fromx0 toxalong thecurve cxand then for1 }t }2travel fromxtox ƒheialong the straight line given byl rt s‡t x ƒ)rt w1 shei.Then ˜chas thesame endpoints as cxˆhei.Ther efore ucx…hei t)u˜c,and hence ¶g ¶xi rx sjtlim h ‚01 h „  ˜c w‰ cx †Št lim h ‚01 h „  cx ƒ‹ l w‰ cx †tlim h ‚01 h  l tlim h ‚01 h  Œ 1,2 l Ž.(4.2) 4.3.THE GLOBAL ANGLE FUNCTION AND THE WINDING NUMBER 51 Leti,jbetheKronecker delta ,which isdened byi,i 1andi,j 0ifi j.Then wecanwrite lj ‘t ’xj “i,j ‘t ”1 ’h,and hence l •j ‘t ’i,jh.This shows that l –n å j —1fj ‘x“Š‘t ”1 ’hei ’dlj n å j —1fj ‘x“˜‘t ”1 ’hei ’l •j ‘t ’dtn å j —1fj ‘x“Š‘t ”1 ’hei ’i,jhdthfi ‘x“˜‘t ”1 ’hei ’dt.(4.3) Taking equations (4.2) and (4.3) together wend ¶g ¶xi ‘x’lim h™01 h š2 1hfi ‘x“Š‘t”1’hei ’dtlim h™0 š1 0fi ‘x“shei ’dsš1 0lim h™0fi ‘x“shei ’dsš1 0fi ‘x’dsfi ‘x’. This proves that gissmooth and that dg. QED This theor em,and itsproof, canbeused inmany differentways. Forexample, ittells usthat once weknow a1-formtobeexact wecannd an“antiderivative” g‘x ’byintegratingalong anarbitrary path running fromaxed point x0tox. (See Exer cises 4.3–4.5 foranapplication.) Ontheother hand, thetheor emalso enables ustodetect closed 1-forms that arenotexact. 4.6.EXAMPLE.The angle form ›œ1‘R2”ž0 Ÿ ’ofExample 2.10 isclosed, butnotexact. Indeed, itsintegral around thecircleis20.Mark thecon- trast with closed 1-forms onRn,which arealways exact! (See Exer cise 2.6.) This phenomenon underlines theimportance ofbeing careful about thedomain ofdef- inition ofaform. 4.3. The global angle function and thewinding number Inthis section wewill have acloser look attheangle form and seethat it carries inter esting information ofa“topological” natur e.Throughout this section Uwill bethepunctur edplane R2”ž0Ÿ,will denote theangle form,  ”ydx“xdy x2“y2, andandwill denote thefunctions x¡ x2“y2,y¡ x2“y2. Thenisaclosed 1-form andandaresmooth functions onU.Infact,and arejust thecomponents ofx ¢\£x £,theunit vector pointing inthedirection ofx. These functions satisfy d ”d. (4.4) (Youwill beasked tocheck this formula inExer cise 4.6.) Now let:U¤¦¥0,2’ betheangle between apoint and thepositive x-axis, chosen tolieintheinterval¥0,2’.Thencosandsin,sobyequation (4.4) cosdsin ”sindcoscos2d“Š‘ ”sin ’2dd. This equation isnotvalid onallofU(itcannot bebecause wesaw inExample 4.6 thatisnotexact), butonly wher eisdifferentiable, i.e.onthecomplement ofthe 52 4.INTEGRA TION OF1-FORMS positive x-axis. Hence thenonexactness ofisclosely related totheimpossibility ofdening aglobal differentiable angle function onU.(The precise meaning of thisassertion will become clear inExer cise 4.6.) However ,along acurve c: §a,b ¨ © Uwecandene acontinuous angle func- tion, and thefact that ªdalmost everywher esuggests how: byintegrating along c!Forsimplicity assume that aª0and bª1.Start byxing any#0such that cos#0 ª«c«0¬#¬and sin#0 ª«c«0¬#¬and then dene # «t ¬jª#0 ­®°¯ 0,t±c ². The following result says that# «t ¬measur estheangle between c «t ¬and theposi- tive x-axis (uptoaninteger multiple of2)and that thefunction#: §0,1 ¨ © Ris smooth. Inthissense#isa“differentiable choice ofangle” along thecurve c. 4.7.THEOREM.Thefunction#issmooth andsatises #«0¬jª#0,cos#«t¬jª«c«t¬#¬and sin#«t¬zª«c«t¬#¬. PROOF.Toseethat# «0 ¬*ª#0,plug t ª0into thedenition of#.Toprove theother assertions werescale thecurve c «t ¬toanew curve c «t ¬#³\´c «t ¬µ´moving ontheunit circle.Letf«t¬and g«t¬bethex-and y-components ofthisnew curve. Then f«t¬jª«c«t¬_¬and g«t¬zª«c«t¬#¬and f«t¬2­g«t¬2ª1 forallt.Inother wordsf ªc²and g ªc²,soitfollows fromformula (4.4) that c² ªfdg ¶gdf.Ther efore #«t¬jª#0­ ®t 0·f«s¬g¸¹«s¬>¶g«s¬f¸6«s¬»ºds. Bythefundamental theor emofcalculus, formula (B.2) ,#isdifferentiable and #¸ªfg¸¶gf¸. (4.5) Since theright-hand side issmooth,#issmooth aswell. Toprovethat cos# «t ¬@ª f«t¬and sin#«t¬jªg«t¬foralltitisenough toshow that thedifference vector¼ f «t ¬ g «t ¬¾½ ¶ ¼ cos# «t ¬ sin# «t ¬_½ haslength 0.Itslength isequal to«f ¶cos# º2­·g ¶sin# ¬2ªf2­g2¶2 «fcos#­gsin# ¬­cos2#­sin2#ª2¶2«fcos#­gsin#¬. Hence weneed toshow that thefunction u ªfcos#­gsin#isaconstant equal to1.Fort ª0wehave u «0 ¬jª f «0 ¬cos#0 ­g «0 ¬sin#0 ªcos2#0 ­sin2#0 ª1. Furthermor e,thederivative ofuis u¸ªf¸cos#¶f#¸sin#­g¸sin#­g#¸cos#ª¿«f¸¶g2f¸­fgg¸¬cos#­ «g¸¶f2g¸­fgf¸¬sin# byformula (4.5)ª¿«f2f ¸­fgg ¸¬cos#­ «g2g ¸­fgf ¸¬sin# since f2­g2ª1ªf«ff ¸­gg ¸¬cos#­g«gg ¸­ff ¸¬sin#. EXERCISES 53 Now f2 Àg2 Á1implies ff  Àgg  Á0,sou ÂÄÃt Å Á0forallt.Hence uisaconstant function, sou Ãt Å Á1forallt. QED Itisuseful tothink ofthevectorÃfÃtÅ,gÃtÅ#ÅTÁcÃtÅ#Æ\ÇcÃtÅTÇasadial that points inthesame direction asthevector cÃtÅ. 0 0 Astincreases from0to1,thedial starts attheangle#Ã0Å Á#0,itmoves around themeter ,and ends upatthenal angle# Ã1 Å.The difference# Ã1 ÅIÈ# Ã0 Åmeasur es thetotal angle swept outbythedial. 4.8.COROLLARY.Ifc: É0,1 ÊÌË Uisaclosed curve, then# Ã1 ÅzÈ# Ã0 Å Á2k, wher ekisaninteger . PROOF.ByTheor em4.7,c Ã0 Å Ác Ã1 ÅimpliesÍ cos# Ã0 Å,sin# Ã0 ŻΠÁ Í  Ãc Ã0 Å#Å, Ãc Ã0 Å#Å#Î Á Í  Ãc Ã1 Å#Å, Ãc Ã1 Å#Å#ÎÁ Í cos# Ã1 Å,sin# Ã1 Å Î. Inother wordscos#Ã0Å Ácos#Ã1Åand sin#Ã0Å Ásin#Ã1Å,so#Ã0Åand#Ã1Ådiffer byaninteger multiple of2. QED The integer k ÁÃ2Å_Ï1Ð ciscalled thewinding number oftheclosed curve c about theorigin. Itmeasur eshow many times thecurve loops around theorigin. winding number ofaclosed curve about origin Á1 2Ñc. (4.6) 4.9.EXAMPLE.ByExample 4.1,thewinding number ofthecirclec Ãt Å ÁÃcost,sint ÅT (0ÒtÒ2)isequal to1. Exercises 4.1.Consider thecurve c: Ó0, Ô2 Õ\Ö R2dened byc ×t ØGÙ)× acost,bsint ØT,wher ea and barepositive constants. Let Ùxydx Úx2ydy. (i)Sketch thecurve cfora Ù2and b Ù1. (ii)Find Ûc(forarbitrary aand b). 4.2.Restate Theor em4.5interms offorceelds, potentials and ener gy.Explain why theresult isplausible onphysical grounds. 54 4.INTEGRA TION OF1-FORMS 4.3.Consider the1-form Ü)Ýx Ýaån i Þ1xidxionRnßJà0 á,wher eaisarealconstant. Forevery x â Ü0letcxbetheline segment starting attheorigin and ending atx. (i)Show thatisclosed forany value ofa. (ii)Determine forwhich values ofathefunction gãxä9Ü¿åcxiswell-dened and compute it. (iii) Forthevalues ofayou found inpart (ii)check that dg Ü. 4.4.Letæèç1ãRnßéà0áêäbethe1-form ofExer cise 4.3. Now letcxbethehaline pointing fromxradially outwar dtoinnity .Parametrize cxbytravelling frominnity inwar dtox.(Youcandothis byusing aninnite time interval ã ßUë,0 ìinsuch away that cx ã0 äÜx.) (i)Determine forwhich values ofathefunction g ãx ä9Ü¿åcxiswell-dened and compute it. (ii)Forthevalues ofayou found inpart (i)check that dgÜ. (iii) Show how torecover fromthis computation thepotential ener gyforNewton's gravitational force.(See Exer ciseB.4.) 4.5.Letæ‡ç1ãRn ߇à0áêäbeasinExer cise 4.3. Ther eisone value ofawhich isnot cover edbyExer cises 4.3and 4.4. Forthis value ofand asmooth function gonRn ßJà0 á such that dg Ü. 4.6. (i)Verify equation (4.4). (ii)LetU ÜR2ßéà0 á.Prove that theredoes notexist asmooth function:U í Rsatisfying cos ãx,y äîÜ x ïñðx2òy2and sin ãx,y äîÜ y ïñðx2òy2forallãx,yäóæ U.(Argue bycontradiction, bylettingÜôã ßydx òxdyäöõ÷ãx2òy2ä and showing that Üdifwas such afunction.) 4.7.Calculate directly fromthedenition thewinding number about theorigin ofthe curve c: ø0,2 ì°í R2given byc ãt ämܞã coskt,sinkt äT. 4.8.Letx0beapoint inR2and caclosed curve which does notpass through x0. How would you dene thewinding number ofcaround x0?Trytoformulate two different denitions: a“geometric” denition and adenition interms ofanintegral over cofa certain 1-form analogous toformula (4.6). 4.9.Letc: ø0,1 ìQí R2ßBà0 ábeaclosed curve with winding number k.Determine the winding numbers ofthefollowing curves ˜c:ø0,1ìmí R2߇à0ábyusing theformula, and then explain theanswer byappealing togeometric intuition. (i)˜cãtäÜcã1 ßtä; (ii) ˜c ãt äÜ ãt äc ãt ä,wher e: ø0,1 ìQíùã 0, ëäisafunction satisfying ã0 äÜ ã1 ä; (iii) ˜c ãt äÜŠÝ c ãt äúÝ_û1c ãt ä; (iv) ˜c ãt äÜ ãc ãt äöä,wher e ãx,y ä>Üüã y,x äT; (v) ˜c ãt äÜ ãc ãt äöä,wher e ãx,y ä>Ü1 x2òy2 ãx, ßy äT . 4.10.Foreach ofthefollowing closed curves c: ø0,2 ìQí R2ßýà0 ásetuptheintegral dening thewinding number about theorigin. Evaluate theintegral ifyou can(but don't give uptoosoon). Ifnot, sketch thecurve (the useofsoftwar eisallowed) and obtain the answer geometrically . (i)c ãt äÜüã acost,bsint äT,wher ea þ0and b þ0; (ii)c ãt äÜüã cost ß2,sint äT; (iii) cãtämÜüã cos3t,sin3täT; (iv) cãtäÜ)ÿöã acost òbäcost òãb ßaäöõ2,ãacost òbäsintT,wher e0
b
a. EXERCISES 55 4.11.Letb 0and a  0beconstants with a  b.Dene aplanar curve c: 0,2  R2  0by c t  a b cost acosa b at, a b sint asina b at T . (i)Sketch thecurve cfora b 3. (ii)Forwhat values ofaand bisthecurve closed? (iii) Assume cisclosed. Setuptheintegral dening thewinding number ofcaround theorigin and evaluate it.Ifyou getstuck, nd theanswer geometrically . 4.12.LetUbeanopen subset ofR2and letF F1e1 F2e2:U R2beasmooth vector eld. The differential form  F1dF2 F2dF1 F2 1 F2 2 iswell-dened atallpoints xofUwher eF x  0.Letcbeaparametrized circlecontained inU,traversed once inthecounter clockwise direction. Assume that Fx 0forallxc. The index ofFrelative tocis indexF,c 1 2c. Provethefollowing assertions. (i) F,wher eistheangle form ydxxdy x2y2 !; (ii)isclosed; (iii) index F,c isthewinding number ofthecurve F "cabout theorigin; (iv) index F,c isaninteger . 4.13. (i)Find theindices ofthefollowing vector elds around theindicated circles. 56 4.INTEGRA TION OF1-FORMS (ii)Draw diagrams ofthreevector elds intheplane with respective indices 0,2 and 4around suitable circles. CHAPTER 5 Integration and Stokes' theorem 5.1. Integration offorms over chains Inthischapter wegeneralize thetheory ofChapter 4tohigher dimensions. In thesame way that 1-forms areintegrated over parametrized curves, k-forms can beintegrated over k-dimensional parametrized regions. LetUbeanopen subset ofRnand letbeak-form onU.The simplest k-dimensional analogue ofan interval isarectangular block inRkwhose edges areparallel tothecoor dinate axes. This isasetoftheform R #%$a1,b1 &(' $a2,b2 &('*)+)+)' $ak,bk & #,t -Rk.ai/ti/bifor1/i/k 0, wher eai 1bi.The k-dimensional analogue ofaparametrized path isasmooth map c:R 2U.Although theimage c 3R 4may look very differentfromtheblock R,wethink ofthemap casaparametrization ofthesubset c3R4ofU:each choice ofapoint tinRgives rise toapoint c3t4inc3R4.The pullback c5isak-form onRand thereforelooks like g3t4dt1dt2)+)+)dtkforsome function g:R2R.The integral ofover cisdened as6 c# 6 Rc 5# 6bk ak )+)+) 6b2 a2 6b1 a1g3t4dt1dt2)+)+)dtk. Fork #1thisreproduces thedenition given inChapter 4.(The denition makes sense ifwereplace therectangular block Rbymoregeneral shapes inRk,such as skew blocks, k-dimensional balls, cylinders, etc. Infact any compact subset ofRk will do.) The case k#0isalso worth examining. Azero-dimensional “block” Rin R0#7,00isjust thepoint 0.Wecan thereforethink ofamap c:R2Uasa collection,x0consisting ofasingle point x#c304inU.The integral ofa0-form (function) fover cisbydenition thevalue offatx,6 cf #f 3x 4. Asintheone-dimensional case, integrals ofk-forms arealmost wholly unaf- fected byachange ofvariables. Let ¯R #8$¯a1,¯b1 & ' $¯a2,¯b2 & '*)+)+)9' $¯ak,¯bk & beasecond rectangular block. Areparametrization isamap p:¯R2Rsatisfying thefollowing conditions: pisbijective (i.e. one-to-one and onto) and thek'k- matrix Dp 3s 4isinvertible foralls -¯R.Then detDp 3s 4;: #0foralls -¯R,soeither detDp 3s 4=< 0forallsordetDp 3s 410foralls.Inthese cases wesaythat the reparametrizion preserves ,respectively reverses theorientation ofc. 57 58 5.INTEGRA TION AND STOKES' THEOREM 5.1.THEOREM.Letbeak-form onUandc:R >Uasmooth map. Letp:¯R >R beareparametrization. Then? c @p ACB Dcifppreserves theorientation ,EDcifpreverses theorientation . PROOF.Almost verbatim thesame proofasfork A1(Theor em4.3). Itfollows fromthedenition oftheintegral and fromthenaturality ofpullbacks, Proposition 3.10(iii) ,that? c @p A ? ¯R Fc Gp HJI A ? ¯Rp IFc I H. Now letuswrite cI Agdt1dt2 K+K+Kdtkand t ApFs H.Then p IFc I HLAp IFgdt1dt2 K+K+Kdtk HLAFp Ig HdetDpds1ds2 K+K+Kdsk byTheor em3.13, so? c @p A ? ¯RgFpFs HJHdetDpFs Hds1ds2 K+K+Kdsk. Ontheother hand,DcADRgFtHdt1dt2K+K+Kdtk,sobythesubstitution formula, Theor emB.7, wehaveDc @pANMDc,wher etheOoccurs ifdetDpP0and theEifdetDpQ0. QED 5.2.EXAMPLE.The unit interval istheinterval R0,1 Sintherealline. Any curve c: Ra,b ST> Ucan bereparametrized toacurve c Gp: R0,1 SU> Ubymeans of thereparametrization pFs HVAFb Ea Hs Oa.Similarly ,theunit cube inRkisthe rectangular blockR0,1SkAXWtYRk Zti Y[R0,1Sfor1\i\k]. LetRbeany other block, given byai \ti \bi.Dene p:R0,1Sk>RbypFsH;A AsOa,wher e A A_^ ```ab1 Ea1 0 ... 0 0 b2 Ea2... 0 ............ 0 0 ...bk Eak bdccceand a A7^ ```aa1 a2 ... ak bdccce. (“Squeeze theunit cube until ithasthesame edgelengths asRand then move itto theposition ofR.”)Then pisone-to-one and onto and DpFsHfAA,sodetDpFsHLA detA AvolR P0foralls,sopisanorientation-pr eserving reparametrization. HenceDc @p ADcforany k-formonU. 5.3.REMARK.Auseful fact you learned incalculus isthat one may inter - change theorderofintegration inamultiple integral, asintheformula?b1 a1 ?b2 a2fFt1,t2 Hdt1dt2 A ?b2 a2 ?b1 a1fFt1,t2 Hdt2dt1. (5.1) (This follows forinstance fromthesubstitution formula, Theor emB.7.) Onthe other hand, wehave also learned that fFt1,t2 Hdt2dt1 A EfFt1,t2 Hdt1dt2.How canthis besquar edwith formula (5.1) ?The explanation isasfollows. Let A fFt1,t2 Hdt1dt2.Then theleft-hand side offormula (5.1) istheintegral ofover 5.2.THE BOUNDAR YOFACHAIN 59 c: ga1,b1 h i ga2,b2 hkj R2,theparametrization oftherectangle given byc lt1,t2 monlt1,t2 m.The right-hand side istheintegral of pnotover c,butover c qp,wher e p: ga2,b2 h(i ga1,b1 h j ga1,b1 h i ga2,b2 h isthereparametrization pls1,s2mLn ls2,s1m.Since preverses theorientation, Theo- rem5.1says thatrc spn ptrc;inother wordsrcn rc sp lupm,which isexactly formula (5.1) .Analogously wehavevxw 0,1 ykflt1,t2,...,tk mdt1dt2z+z+zdtk n v w 0,1 ykflt1,t2,...,tk mdtidt1dt2z+z+z|{dti z+z+zdtk forany i. WeseefromExample 5.2that anintegral over any rectangular block canbe written asanintegral over theunit cube. Forthis reason, fromnow onweshall usually take Rtobetheunit cube. Asmooth map c:g0,1hkjUiscalled ak-cube inU(orsometimes asingular k-cube, thewordsingular meaning that themap cis notassumed tobeone-to-one, sothat theimage canhave self-intersections.) Itisoften necessary tointegrate over regions that aremade upofseveral pieces. Ak-chain inUisaformal linear combination ofk-cubes, cna1c1}a2c2}~z+z+z|} apcp, wher ea1,a2,...,aparerealcoefcients and c1,c2,...,cparek-cubes. Forany k-formwethen denev cnp å i1ai v ci. (Inthelanguage oflinear algebra, thek-chains form anabstract vector space with abasis consisting ofthek-cubes. Integration, which isapriori only dened on cubes, isextended tochains insuch away astobelinear .) Recall that a0-cube isnothing butasingleton€xconsisting ofasingle point xinU.Thus a0-chain isaformal linear combination ofpoints, cnåp i 1ai €xi . Agood way tothink ofcisasacollection ofppoint char ges, with anelectric char geaiplaced atthepoint xi.(Youmust carefully distinguish between theformal linear combination åp i1ai €xi ,which represents adistribution ofpoint char ges, and thelinear combination ofvectors åp i1aixi,which represents avector inRn.) The integral ofafunction fover the0-chain isbydenitionv cfnp å i1ai vƒ‚ xi„fnp å i1aiflxi m. Likewise, ak-chain åp i1aicicanbepictur edasachar gedistribution, with anelec- tricchar geaispreadalong thek-dimensional “patch” ci. 5.2. The boundary ofachain Consider acurve (“1-cube”) c: g0,1h…j U.Itsboundary isbydenition the 0-chain dened by¶cn €c l1m †p[€ c l0m .‡‰ˆc Š0 ‹dŒ  ˆcŠ1‹dŒ c 60 5.INTEGRA TION AND STOKES' THEOREM The boundary ofa2-cube c: Ž0,1 2 Uconsists offour pieces corresponding totheedges oftheunit squar e:c1 ‘t ’”“ c‘t,0 ’,c2 ‘t ’•“ c‘1,t ’,c3 ‘t ’”“ c‘t,1 ’and c4 ‘t ’”“ c‘0,t ’.The pictur ebelow suggests that weshould dene ¶c “c1 –c2 — c3 —c4. c˜ ˜ ™™ (Alternatively wecould dene ¶c“c1–c2–¯c3–¯c4,with ¯c3‘t’“c‘1—t,1’and ¯c4‘t’o“c‘0,1—t’,which corresponds tothefollowing pictur e: c˜ ˜ ˜˜ This would work equally well, butistechnically lessconvenient.) Ak-cube c: Ž0,1 k Uhas2kfaces ofdimension k—1,which aredescribed asfollows. Lett “‘t1,t2,...,tkš1 ’†›[Ž 0,1 k š1and fori “1,2,...,kput ci,0 ‘t’f“c‘t1,t2,...,ti š1,0,ti,...,tk š1 ’, ci,1 ‘t ’f“c‘t1,t2,...,tiš1,1,ti,...,tkš1 ’. (“Insert 0,resp. 1inthei-thslot”.) Now dene ¶c “k å i œ1 ‘u—1 ’i‘ci,0 —ci,1 ’f“k å i œ1å œ0,1 ‘u—1 ’i ci,. Foranarbitrary k-chain c“åiaiciweput¶c“åiai¶ci.Then ¶isalinear map fromk-chains tok—1-chains. Youshould check that fork“0and k“1this denition isconsistent with theone- and two-dimensional cases consider edabove. Ther eareanumber ofcurious similarities between theboundary operator ¶ and theexterior derivative d,themost important ofwhich isthefollowing. (Ther e arealso many differences, such asthefact that draises thedegr eeofaform by1, wher eas¶lowers thedimension ofachain by1.) 5.4.PROPOSITION.¶‘¶c ’L“0forevery k-chain cinU.Inshort, ¶2“0. PROOF.Bylinearity of¶itsufces toprovethisfork-cubes c:Ž0,1kU.Let t “‘t1,t2,...,tkš2 ’›žŽ 0,1 k š2and letandbe0or1.Then for1 Ÿi Ÿj Ÿk—1 5.3.CYCLES AND BOUNDARIES 61 wehave   ci,¡j,   t1,t2,...,tk ¢2 ¡f£ci,   t1,t2,...,tj¢1,,tj,...,tk ¢2 ¡£c   t1,t2,...,ti ¢1,,ti,...,tj ¢1,,tj,...,tk ¢2 ¡. Ontheother hand,  cj ¤1, ¡i,   t1,t2,...,tk¢2 ¡L£cj ¤1,   t1,t2,...,ti ¢1,,ti,...,tk¢2 ¡£c   t1,t2,...,ti¢1,,ti,...,tj¢1,,tj,...,tk ¢2¡, because inthevector   t1,t2,...,ti¢1,,ti,...,tk¢2¡theentry tjoccupies thej ¥1st slot! Weconclude that   ci,¡j,£   cj¤1,¡i,for1¦i¦j¦k§1.Ther eforethe k §2-chain ¶   ¶c¡isgiven by ¶   ¶c¡o£¶k å i ¨1å  ¨0,1  §1¡i ¤ci,£k å i ¨1å  ¨0,1  §1¡i ¤¶ci,£k å i ¨1k ¢1 å j ¨1å ,¨0,1  §1¡i ¤j ¤ ¤   ci,¡j,. The double sum over iand jcanberearranged inasum over i ¦jand asum over i ©jtogive ¶   ¶c¡L£ å 1ªiªjªk¢1å , ¨0,1  §1¡i¤j¤¤   ci,¡j,¥å 1 ªj «i ªkå ,¨0,1  §1¡i ¤j ¤ ¤   ci,¡j,.(5.2) Intherst term ontheright in(5.2) wesubstitute   ci,¡j,£   cj¤1,¡i,and then r£j¥1,s£i,£,£toget å 1 ªi ªj ªk ¢1å ,¨0,1  §1¡i ¤j ¤ ¤   ci,¡j,£å 1 ªi ªj ªk ¢1å ,¨0,1  §1¡i ¤j ¤ ¤   cj ¤1,¡i,£å 1ªs«rªkå , ¨0,1  §1¡s¤r¢1¤¤   cr,¡s,£ §å 1ªs«rªkå , ¨0,1  §1¡s¤r¤¤   cr,¡s,. Thus thetwo terms ontheright in(5.2) cancel out. QED 5.3. Cycles and boundaries Ak-cube cisdegenerate ifc   t1,...,tk ¡isindependent oftiforsome i.Ak-chain cisdegenerate ifitisalinear combination ofdegenerate cubes. Inparticular ,ade- generate 1-cube isaconstant curve. The work done byaforceeld onamotionless particle is0.Mor egenerally wehave thefollowing. 5.5.LEMMA.Letbeak-form andcadegenerate k-chain. Then ¬c£0. PROOF.Bylinearity wemay assume that cisadegenerate cube. Suppose cis constant asafunction ofti.Then c   t1,...,ti,...,tk¡f£c   t1,...,0,...,tk¡f£g ­f   t1,...,ti,...,tk¡J®, 62 5.INTEGRA TION AND STOKES' THEOREM wher ef: ¯0,1 °k ±¯0,1 °k ²1and g: ¯0,1 °k ²1 ±Uaregiven respectively by f³t1,...,ti,...,tk ´fµ ³t1,...,ˆti,...,tk ´, g ³s1,...,sk²1´fµc ³s1,...,si²1,0,si¶1,...,tk²1´. Now g ·isak-form on ¯0,1 °k ²1and hence equal to0,and soc ·µf ·¸³g ·´µ0. Weconclude ¹cµ ¹+º0,1 »kc ·µ0. QED Sodegenerate chains areirrelevant wher eintegration isconcerned. This mo- tivates thefollowing denition. Ak-chain cisclosed ,oracycle ,if¶cisadegenerate k ¼1-chain. Ak-chain cisaboundary ifcµ¶b ½c ¾forsome k ½1-chain band some degenerate k-chain c¾. 5.6.EXAMPLE.Ifc1and c2arecurves arranged head totailasinthepictur e below ,then c1 ½c2isa1-cycle. Likewise, theclosed curve cisa1-cycle. c1c2 c 5.7.LEMMA.Theboundary ofadegenerate k-chain isadegenerate k ¼1-chain. PROOF.Bylinearity itsufces toconsider thecase ofadegenerate k-cube c. Suppose cisconstant asafunction ofti.Then ci,0µci,1,so ¶cµå j ¿ Ài ³J¼1´j³cj,0 ¼cj,1´. Lettµ ³t1,t2,...,tk²1´.For j Áithecubes cj,0 ³t´and cj,1 ³t´areindependent of tiand forj Âithey areindependent ofti ²1.So¶cisacombination ofdegenerate k ¼1-cubes and hence isdegenerate. QED 5.8.COROLLARY.Every boundary isacycle. PROOF.Suppose cµ¶b ½c ¾with c ¾degenerate. Then byLemma 5.5¶cµ ¶ ³¶b´ ½¶c ¾µ¶c ¾,wher eweused Proposition 5.4. Lemma 5.7says that ¶c ¾is degenerate, and thereforesois¶c. QED 5.9.EXAMPLE.Consider theunit circleintheplane c ³t´;µ ³cos2t,sin2t´ with 0ÃtÃ1.This isaclosed 1-cube. The circleistheboundary ofthedisc ofradius 1and thereforeitisreasonable toexpect that cisaboundary ofa2- cube. This isindeed trueinthesense dened above, that cµ¶b½c ¾wher ec ¾ isaconstant 1-chain. (Itisactually notpossible tond absuch that cµ¶b; seeExer cise 5.2.) The 2-cube bisdened by“shrinking ctoapoint”, b³t1,t2´Äµ³1¼t2´c³t1´for³t1,t2´intheunit squar e.Then b³t1,0´oµc³t1´, b³0,t2´Lµb³1,t2´Lµ ³1¼t2,0´, b³t1,1´Åµ ³0,0´, sothat ¶bµc¼c¾,wher ec¾istheconstant curve located attheorigin. Ther efore cµ¶b½c¾,aboundary plus adegenerate 1-cube. Inthesame way that aclosed form isnotnecessarily exact, itmay happen that a1-cycle isnotaboundary .SeeExample 5.11. 5.4.STOKES' THEOREM 63 5.4. Stokes' theorem Inthelanguage ofchains and boundaries wecan rewrite thefundamental theor emofcalculus, Theor em4.4,asfollows:Æ cdgÇgÈcÈ1ÉJÉkÊ gÈcÈ0ÉdÉËÇ ÆÍÌ cÎ1ÏÑÐgÊ Æ¸Ì cÎ0ÏÑÐgÇ Æ¸Ì cÎ1ÏÑÐJÒ Ì cÎ0ÏÑÐgÇ Æ ¶cg, i.e. Ócdg ÇÔÓ¶cg.This istheform inwhich thefundamental theor emofcalculus generalizes tohigher dimensions. This generalization isperhaps theneatest rela- tionship between theexterior derivative and theboundary operator .Itcontains asspecial cases theclassical integration formulas ofvector calculus (Green, Gauß and Stokes) and forthat reason hasStokes' name attached toit,although itwould perhaps bebetter tocallitthe“fundamental theor emofmultivariable calculus”. 5.10.THEOREM(Stokes' theor em).Letbeak Ê1-form onanopen subset Uof Rnandletcbeak-chain inU.ThenÆ cdÇ Æ ¶c. PROOF.Bythedenition oftheintegral and byTheor em3.11 wehaveÆ cdÇ ÆxÕ 0,1Ökc×dÇ ÆÕ 0,1Ökdc×. Since c×isakÊ1-form onØ0,1Ùk,itcanbewritten as c×Çk å iÚ1gidt1dt2Û+Û+Û|Üdti Û+Û+Ûdtk forcertain functions g1,g2,...,gkdened on Ø0,1 Ùk.Ther eforeÆ cd Çk å i Ú1 Æ Õ 0,1 Ökd Ýgidt1dt2 Û+Û+Û|Üdti Û+Û+ÛdtkÞ Çk å i Ú1 ÈuÊ1 Éiß1 ÆxÕ 0,1 Ök¶gi ¶tidt1dt2 Û+Û+Ûdtk. Changing theorderofintegration (cf.Remark 5.3)and subsequently applying the fundamental theor emofcalculus inonevariable, formula (B.1) ,givesÆxÕ 0,1Ök¶gi ¶tidt1dt2Û+Û+Ûdtk Ç ÆxÕ 0,1Ök¶gi ¶tidtidt1dt2Û+Û+Û|Üdti Û+Û+ÛdtkÇ Æ Õ 0,1Ök à1 Ýgi Èt1,...,tiÒ1,1,tiß1,...,tk ÉÊgi Èt1,...,ti Ò1,0,ti ß1,...,tk ÉÞdt1dt2 Û+Û+Û|Üdti Û+Û+Ûdtk. The forms gi Èt1,...,tiÒ1,1,tiß1,...,tk Édt1dt2Û+Û+ÛÜdti Û+Û+Ûdtkand gi Èt1,...,tiÒ1,0,tiß1,...,tk Édt1dt2Û+Û+Û|Üdti Û+Û+Ûdtk 64 5.INTEGRA TION AND STOKES' THEOREM arenothing butc ái,1,resp. c ái,0.Accor dingly ,â cdãk å iä1åuæ1çiè1 âxé 0,1êk¶gi ¶tidtidt1dt2ë+ë+ëíìdti ë+ë+ëdtkãk å i ä1 åuæ1çiè1 â é 0,1êk î1åc ái,1æc ái,0çãk å i ä1å  ä0,1 åuæ1 çiè âé 0,1êkî1c ái,ãk å i ä1å  ä0,1 åuæ1 çi è â ci, ã â ¶c, which proves theresult. QED 5.11.EXAMPLE.The unit circlecåt çïãåcos2t,sin2t çisa1-cycle inthe punctur edplane UãR2æžð0ñ.Consider edasachain inR2itisalso aboundary , aswesaw inExample 5.9. However ,weclaim that itisnotaboundary inUin thesense that thereexist no2-chain band nodegenerate 1-chain còboth contained inUsuch that c ã¶b ócò.Indeed, suppose that c ã¶b ócò.Let ãåuæydx ó xdyçdôåx2óy2çbetheangle form. Thenõcã2byExample 4.1.On theother hand,â c ã â ¶b èc ö ã â bd ã0, wher ewehave used Stokes' theor em,Lemma 5.5and thefactthatisclosed. This isacontradiction. The moral ofthis example isthat thepresence ofthepunctur e inUisresponsible both fortheexistence ofthenon-exact closed 1-form(see Example 4.6) and fortheclosed 1-chain cwhich isnotaboundary .Wedetected both phenomena byusing Stokes' theor em. Exercises 5.1.LetUbeanopen subset ofRn,Vanopen subset ofRmand:U ÷Vasmooth map. Letcbeak-cube inUandak-form onV.Provethatøcùúûø üc. 5.2.LetUbeanopen subset ofRn.Itsboundary isalinear combination ofký1-cubes, ¶cúåiaici. (i)Letbbeak þ1-chain inU.Itsboundary isalinear combination ofk-cubes, ¶b úåiaici.Provethat åiai ú0. (ii)Letcbeak-cube inU.Conclude that thereexists nok þ1-chain binUsatisfying ¶búc. 5.3.Dene a2-cube c:ÿ0,12÷R3byc
t1,t2  út2 1,t1t2,t2 2,and letúx1dx2 þ x1dx3 þx2dx3. (i)Sketch theimage ofc. (ii)Calculate bothøcdandø¶cand check that they areequal. 5.4.Dene a3-cube c:ÿ0,13÷R3byc
t1,t2,t3  ú
t2t3,t1t3,t1t2 ,and letú x1dx2dx3.Calculate bothøcdandø¶cand check that they areequal. 5.5.Using polar coor dinates inndimensions (cf.Exer cise3.18) write thený1-dimen- sional unit spher eSn 1inRnastheimage ofanný1-cube c.For nú2,3,4,calculate theboundary ¶cofthis cube. (The domain ofcwill notbetheunit cube inRn1,buta EXERCISES 65 rectangular block Rdictated bytheformula inExer cise3.18. Choose Rinsuch away asto cover thespher easeconomically aspossible.) 5.6.Deduce thefollowing classical integration formulas fromthegeneralized version ofStokes' theor em. Allfunctions, vector elds, chains etc.aresmooth and aredened inan open subset UofRn.(Some formulas hold only forspecial values ofn,asindicated.) (i) cgradg dx gc1  gc0 forany function gand any curve c. (ii)Green's formula:  c ¶g ¶x ¶f ¶ydxdy  ¶c fdx gdy forany functions f,g and any 2-chain c.(Her en 2.) (iii) Gauß' formula:  cdivFdx1dx2   dxn  ¶cF dxforany vector eld Fand any n-chain c. (iv) Stokes' formula: ccurlF dx  ¶cF dxforanyvector eld Fand any 2-chain c.(Her en 3.) Inparts (iii) and (iv)weusethenotations dxand dxexplained inSection 2.5. Weshall give ageometric interpr etation oftheentity dxinterms ofvolume forms later on. (See Corollary 8.15.) CHAPTER 6 Manifolds 6.1. The denition Intuitively ,ann-dimensional manifold intheEuclidean space RNisasubset that intheneighbour hood ofevery point “looks like” Rnupto“smooth distor - tions”. The formal denition isgiven below and isunfortunately abitlong. Itwill help toconsider rst thebasic example ofthesurface oftheearth, which isatwo- dimensional spher eplaced inthree-dimensional space. Auseful way torepresent theearth isbymeans ofaworld atlas, which isacollection ofmaps. Each map depicts aportion oftheworld, such asacountry oranocean. The correspondence between points onamap and points ontheearth's surface isnotentir elyfaithful, because charting acurved surface onaatpiece ofpaper inevitably distorts the distances between points. Butthedistortions arecontinuous, indeed differentiable (inmost traditional cartographic projections). Maps ofneighbouring areasover - lapnear their edges and thetotality ofallmaps inaworld atlas covers thewhole world. Anarbitrary manifold isdened similarly ,asann-dimensional “world” rep- resented byan“atlas” consisting of“maps”. These maps areaspecial kind of parametrizations known asembeddings. 6.1.DEFINITION.LetUbeanopen subset ofRn.Anembedding ofUinto RN isaC map :U RNsatisfying thefollowing conditions: (i) isone-to-one (i.e.if t1 t2,then t1t2); (ii)D tisone-to-one forallt U; (iii) theinverse of ,which isamap 1: U U,iscontinuous. The image oftheembedding istheset U! t#"tU$consisting ofallpoints oftheform twith tU.The inverse map 1iscalled achart orcoordinate map.Youshould think of Uasann-dimensional “patch” inRN parametrized bythemap .Condition (i)means that todistinct values ofthe “parameter ”tmust correspond distinct points tinthepatch U.Thus the patch Uhasnoself-intersections. Condition (ii)means that foreach tinUall ncolumns oftheJacobi matrix D tmust beindependent. This isimposed to prevent theoccurr ence ofcusps and other singularities intheimage U.Since D thas Nrows, this condition also implies that N %n:thetargetspace RN must have dimension greater than orequal tothat ofthesour cespace U,orelse cannot beanembedding. The column space ofD tiscalled thetangent space to thepatch atthepoint x tand isdenoted byTx U, Tx U&D t Rn. 67 68 6.MANIFOLDS Thetangent space ateach point isann-dimensional subspace ofRNbecause D 't ( hasnindependent columns. Condition (iii)canberestated astherequir ement that iftiisanysequence ofpoints inUsuch that limi )+* 'ti (exists and isequal to 't ( forsome t ,U,then limi )+*ti -t.This isintended toavoid situations wher ethe image 'U (doubles back onitself “atinnity”. (See Exer cise6.4foranexample.) 6.2.EXAMPLE.The pictur ebelow shows anembedding ofanopen rectangle intheplane into three-space, theimage ofwhich isaportion ofatorus. Tryto write aformula forsuch anembedding! (Ifwechose Utoobig, theimage would self-intersect and themap would notbeanembedding.) Forone particular value oftthecolumn vectors oftheJacobi matrix arealso shown. Asyou cansee, they span thetangent plane attheimage point. .t /D .t/e1 D .t /e2 .U /tU e1e2e1e2e3 6.3.EXAMPLE.LetUbeanopen subset ofRnand letf:U 0Rmbeasmooth map. The graph offisthecollection graphf-2143t f 't (6587777t ,U 9. Since tisann-vector and f't(anm-vector ,thegraph isasubset ofRNwith N- n :m.Weclaim that thegraph istheimage ofanembedding :U 0RN.Dene 't (-;3t f 't ( 5. Then bydenition graphf- 'U(.Furthermor e isanembedding. Indeed, 't1 (- 't2 (implies t1-t2,so isone-to-one. Also, D 't (-;3In Df 't (<5, 6.1.THE DEFINITION 69 soD =t >hasnindependent columns. Finally theinverse of isgiven by ?1@t f =t >6ACBt, which iscontinuous. Hence isanembedding. Amanifold isanobject patched together outoftheimages ofseveral embed- dings. Mor eprecisely , 6.4.DEFINITION.Ann-dimensional manifold1(orn-manifold forshort) inRNis asubset MofRNsuch that forallx DMthereexistEanopen subset VFRNcontaining x,Eanopen subset UFRn,Eand anembedding :U GRNsatisfying =U >BV HM. The codimension ofMinRNisNIn.Choose tDUsuch that =t>Bx.Then the tangent space toMatxisthecolumn space ofD =t>, TxMBD =t>J=Rn>. (Using thechain ruleone canshow that TxMisindependent ofthechoice ofthe embedding .)The elements ofTxMaretangent vectors toMatx.Acollection of embeddings i:Ui GRNwith Uiopen inRnand such that Mistheunion ofall thesets i =Ui >isanatlas forM. One-dimensional manifolds arecalled (smooth) curves ,two-dimensional man- ifolds (smooth) surfaces ,and n-manifolds inRn K1(smooth) hypersurfaces .Inthese cases thetangent spaces areusually called tangent lines ,tangent planes ,and tangent hyperplanes ,respectively . The following pictur eillustrates thedenition. HereMisacurve intheplane, sowehave NB2and nB1.Uisanopen interval inRand Visanopen disc inR2.The map sends ttoxand parametrizes theportion ofthecurve inside V. Since nB1,theJacobi matrix D =t >consists ofasingle column vector ,which is tangent tothecurve atxB =t >.The tangent line TxMistheline spanned bythis vector . tU VM xTxM Sometimes amanifold hasanatlas consisting ofonesingle chart. Inthat event wecantake VBRN,and choose oneopen U FRnand anembedding :U GRN such that MB =U >.However ,usually one needs morethan one chart tocover amanifold. (For instance, one chart isnotenough forthecurve Minthepictur e above.) 1Intheliteratur ethis isusually called asubmanifold ofEuclidean space. Itispossible todene manifolds moreabstractly ,without reference toasurrounding vector space. However ,itturns out that practically allabstract manifolds canbeembedded into avector space ofsufciently high dimen- sion. Hence theabstract notion ofamanifold isnotsubstantially moregeneral than thenotion ofa submanifold ofavector space. 70 6.MANIFOLDS 6.5.EXAMPLE.Anopen subset UofRncanberegar ded asamanifold ofdi- mension n(hence ofcodimension 0).Indeed, Uistheimage ofthemap :U L Rngiven by Mx NOx,theidentity map. The tangent space toUatany point isRn itself. 6.6.EXAMPLE.LetNPnand dene :RnLRNby Mx1,x2,...,xn NQOMx1,x2,...,xn,0,0,...,0N.Itiseasy tocheck that isanembedding. Hence the image MRnNisann-manifold inRN.(Note that MRnNisjust alinear subspace isomorphic toRn;e.g.ifNO3and nO2itisjustthexy-plane. Weshall usually identify Rnwith itsimage inRN.)Combining this example with theprevious one, weseethat ifUisany open subset ofRn,then MUNisamanifold inRNof codimension NRn.Itstangent space atany point isRn. 6.7.EXAMPLE.LetM Ographf,wher ef:U LRmisasmooth map. As shown inExample 6.3, Mistheimage ofasingle embedding :U LRn Sm,so Misann-dimensional manifold inRnSm,cover edbyasingle chart. AtapointMx,fMxNNinthegraph thetangent space isspanned bythecolumns ofD .For instance, ifnOmO1,Misone-dimensional and thetangent linetoMatMx,fMxNN isspanned bythevectorM1,fTUMxNN.This isequivalent tothewell-known fact that theslope ofthetangent line tothegraph atxisfTVMxN. xfWxX Y1 f ZVWx X6[graphf Forn O2and m O1,Misasurface inR3.The tangent plane toMatapointMx,y,f Mx,y N\Nisspanned bythecolumns ofD Mx,y N,namely]^1 ¶f ¶x Mx,y N 0 _` and ]a^ 1 0 ¶f ¶y Mx,y N _\b` . The diagram below shows thegraph off Mx,y NcOx3 dy3R3xyfromtwo different angles, together with afew points and tangent vectors. (Toimpr ovethescale the 6.1.THE DEFINITION 71 z-coor dinate and thetangent vectors have been compr essed byafactor of2.) e1e2e3 e1e2e3 6.8.EXAMPLE.Consider thepath :ReR2given by ftgihetfcost,sintg. Letuscheck that isanembedding. Observe rst that j ft gkjh et.Ther efore ft1 glh ft2 gimplies et1het2.The exponential function isone-to-one, sot1 ht2. This shows that isone-to-one. The velocity vector is mnft glhetocost psint cost qsint r. Ther efore m ftgsh 0ifand only ifcosthsinth0,which isimpossible because cos2t qsin2t h1.So m ft g+t h0forallt.Mor eover wehave t hlnethln j ft gkj. Hence theinverse of isgiven by u1fx glhln jx jforx v fR gand soiscontinu- ous. Ther efore isanembedding and fR gisa1-manifold. The image fR gisa spiral, which fortewpyx conver gestotheorigin. Itwinds innitely many times around theorigin, although that ishardtoseeinthepictur e. xy Even though fRgisamanifold, theset fRg{z}| 0~isnot: ithasavery nasty singularity attheorigin! 72 6.MANIFOLDS 6.9.EXAMPLE.Anexample ofamanifold which cannot becover edbyasingle chart istheunit spher eMSn€1inRn.LetURn €1and let :URnbethe map ‚t ƒ„1…t …2 †1 ‡2t †‚ …t …2ˆ1 ƒen‰ given inExer cise B.7. Aswesaw inthat exercise, theimage of isthepunctur ed spher eM ˆ‹ŠenŒ,soifweletVbetheopen setRn ˆŠenŒ,then ‚U ƒŽ M V. Also wesaw that hasatwo-sided inverse: ‚U ƒ& U,thestereographic pro- jection fromthenorth pole. Ther efore isone-to-one and itsinverse iscontinuous (indeed, differentiable). Mor eover ,  ‚t ƒ‘timplies D ‚ ‚t ƒƒD ‚t ƒv vfor allvinRn €1bythechain rule. Ther efore,ifvisinthenullspace ofD ‚t ƒ, v D ‚ ‚t ƒƒD ‚t ƒv D ‚ ‚t ƒƒ0 0. Thus weseethat isanembedding. Tocover allofMweneed asecond map, for example theinverse ofthestereographic projection fromthesouth pole. This is also anembedding and itsimage isM ˆŠ’ˆenŒ MV,wher eVRn ˆ‹ŠenŒ. This nishes theproofthat Misann ˆ1-manifold inRn. Asthis example shows, thedenition ofamanifold canbealittle awkwar d towork with inpractice, even foravery simple manifold. Aside fromtheabove examples, inpractice itcanberather hardtodecide whether agiven subset isa manifold using thedenition alone. Fortunately thereexists amoremanageable criterion forasettobeamanifold. 6.2. The regular value theorem Denition 6.4isbased onthenotion ofanembedding, which canberegar ded asan“explicit” way ofdescribing amanifold. However ,embeddings canbehard nd inpractice. Instead, manifolds areoften given “implicitly”, byasystem ofm equations inNunknowns, 1 ‚x1,...,xN ƒ„c1, 2 ‚x1,...,xN ƒ„c2, ... m ‚x1,...,xN ƒ„cm. Herethei'saresmooth functions presumed tobedened onsome common open subset UofRN.Writing intheusual way x ”“ •••–x1 x2 ... xN —\˜˜˜™, ‚x ƒš“ •••–1 ‚x ƒ 2 ‚x ƒ ... m ‚x ƒ —\˜˜˜™,c !“ •••–c1 c2 ... cm —\˜˜˜™, wecanabbr eviate this system toasingle equation ‚xƒc. Foraxed vector c›Rmwedenote thesolution setby  €1‚c ƒl Šx ›U œ ‚x ƒcŒ 6.2.THE REGULAR VALUE THEOREM 73 and callitthelevel setorthebreofatc.(The notation 1 žc Ÿforthesolution set isstandar d,butabitunfortunate because itsuggests falsely thatisinvertible, which itisusually not.) Ifisalinear map, thesystem ofequations isinho- mogeneous linear and bylinear algebra thesolution setisanafne subspace of RN.The dimension ofthis afne subspace isN  m,provided thathasrank m (i.e. hasmindependent columns). Wecangeneralize this idea tononlinear equa- tions asfollows. Wesaythat c ¡Rmisaregular value ofiftheJacobi matrix D žxŸ:RN ¢Rmhasrank mforallx¡1žcŸ.Avector that isnotaregular value iscalled asingular value. (Asanextreme, though slightly silly,special case, if1žcŸisempty ,then cisautomatically aregular value.) The following result isthemost useful criterion forasettobeamanifold. (Don't getcarried away though, because itdoes notapply toevery possible mani- fold. Inother words,itisasufcient butnotanecessary criterion.) The proofuses thefollowing important factfromlinear algebra, nullityA £rankA ¤l, valid forany k¥l-matrix A.Heretherank isthenumber ofindependent columns ofA(inother wordsthedimension ofthecolumn space A žRlŸ)and thenullity isthenumber ofindependent solutions ofthehomogeneous equation Ax ¤0(in other wordsthedimension ofthenullspace kerA). 6.10.THEOREM(regular value theor em).LetUbeopen inRNandlet:U ¢ Rmbeasmooth map. Suppose that cisaregular value ofandthat M ¤ 1žc Ÿis nonempty .Then Misamanifold inRNofcodimension m.Itstangent space atxisthe nullspace ofD žxŸ, TxM ¤kerD žx Ÿ. PROOF.Letx ¡M.Then D žx Ÿhas rank mand sohas mindependent columns. After relabelling thecoor dinates onRNwemay assume thelast m columns areindependent and thereforeconstitute aninvertible m ¥m-submatrix AofD žxŸ.Letusputn¤N m.Identify RNwith Rn¥Rmand correspond- ingly write anN-vector asapair žu,vŸwith uan-vector and vanm-vector .Also write x¤ žu0,v0 Ÿ.Now refer toAppendix B.4and observe that thesubmatrix Aisnothing butthe“partial” Jacobian Dv žu0,v0 Ÿ.This matrix being invertible, bytheimplicit function theor em, Theor emB.4, thereexist open neighbour hoods Uofu0inRnand Vofv0inRmsuch that foreach u ¡Uthereexists aunique v ¤f žu Ÿ‘¡Vsatisfying žu,f žu ŸŸl¤ c.The map f:U ¢VisC ¦.Inother words M § žU ¥V Ÿ‘¤graphfisthegraph ofasmooth map. Weconclude fromExam- ple6.7that M § žU ¥V Ÿisann-manifold, namely theimage oftheembedding :U ¢RNgiven by žuŸ¤ žu,f žuŸŸ.Since U¥Visopen inRNand theabove argument isvalid forevery x¡M,weseethat Misann-manifold. Tocompute TxMnote that ž žu ŸŸ„¤ c,aconstant, forallu ¡U.Hence D ž žu ŸŸD žu Ÿ¤0 bythechain rule. Plugging inu ¤u0gives D žx ŸD žu0 Ÿl¤0. The tangent space TxMisbydenition thecolumn space ofD žu0 Ÿ,soevery tangent vector vtoMatxisoftheform v ¤D žu0 Ÿaforsome a ¡Rn.Ther efore D žx Ÿv ¤D žx ŸD žu0 Ÿa ¤0,i.e.TxM ¨kerD žx Ÿ.The tangent space TxM 74 6.MANIFOLDS isn-dimensional (because thencolumns ofD ©u0 ªareindependent) and sois thenullspace ofD ©xª(because nullityD ©xª¬« N ­m«n).Hence TxM« kerD ©xª. QED The case ofone single equation (m«1)isespecially important. Then Dis asingle rowvector and itstranspose isthegradient of:DT«grad.Ithas rank 1atxifand only ifitisnonzer o,i.e.atleast one ofthepartials ofdoes notvanish atx.The solution setofascalar equation ©xªŽ« cisknown asalevel hypersurface .Level hypersurfaces, especially level curves, occur frequently inall kinds ofapplications. Forexample, isotherms inweather charts and contour lines intopographical maps aretypes oflevel curves. 6.11.COROLLARY(level hypersurfaces) .LetUbeopen inRNandlet:U ®R beasmooth function. Suppose that M« ¯1©cªisnonempty andthat grad ©xª±° «0 forallxinM.Then Misamanifold inRNofcodimension 1.Itstangent space atxisthe orthogonal complement ofgrad©xª, TxM« ©grad ©xª\ª². 6.12.EXAMPLE.LetU«R2and ©x,yª³« xy.The level curves ofare hyperbolas intheplane and thegradient isgrad ©xª´« ©y,xªT.The diagram below shows afew level curves aswell asthegradient vector eld, which asyou canseeisperpendicular tothelevel curves. xy The gradient vanishes only attheorigin, so©0ªi«0istheonly singular value of .ByCorollary 6.11 this means that¯1©cªisa1-manifold forc° «0.The bre  ¯1©0ªistheunion ofthetwo coor dinate axes, which hasaself-intersection and soisnotamanifold. However ,theset¯1©0ª ­‹µ0¶isa1-manifold since thegra- dient isnonzer ooutside theorigin. Think ofthis diagram asatopographical map representing thesurface z«©x,yªshown below .The level curves ofarethe contour lines ofthesurface, obtained byintersecting thesurface with horizontal planes atdifferentheights. Asexplained inAppendix B.2, thegradient points in thedirection ofsteepest ascent. Wher ethecontour lines self-intersect thesurface 6.2.THE REGULAR VALUE THEOREM 75 hasa“mountain pass” orsaddle point. e1e2e3 6.13.EXAMPLE.Amoreinter esting example ofanequation intwo variables is ·x,y ¸º¹ x3 »y3 ¼3xy ¹c.Heregrad ·x ¸y¹ 3 ·x2 ¼y,y2 ¼x ¸T,sograd vanishes attheorigin and at ·1,1 ¸T.The corresponding values ofare0,resp.¼1,which arethesingular values of. xy The level “curve” ½1· ¼1 ¸isnotacurve atall,butconsists ofthesingle point·1,1 ¸T.Herehasaminimum and thesurface z ¹ ·x,y ¸hasa“valley”. The level curve ½1·0¸has aself-intersection attheorigin, which corresponds toa saddle point onthesurface. These featur esarealso clearly visible inthesurface itself, which isshown inExample 6.7. 6.14.EXAMPLE.LetU¹RNand·x¸i¹¿¾ x¾2.Then grad·x¸¹2x,soasin Example 6.12 gradvanishes only attheorigin 0,which iscontained in½1·0¸. Soagain any c À ¹0isaregular value of.Clearly , ½1·c ¸isempty forc Á0.For c Â0, ½1·c ¸isanN ¼1-manifold, thespher eofradius ÃcinRN.The tangent 76 6.MANIFOLDS space tothespher eatxisthesetofallvectors perpendicular tograd Äx ÅºÆ 2x. Inother words, TxMÆxÇ}ÆÉÈ yÊRN ËyÌxÆ0Í. Finally ,0isasingular value (the absolute minimum) ofand Î1Ä0 ńÆÉÈ 0 Íisnot anN Ï1-manifold. (Ithappens tobea0-manifold, though, just like thesingular bre Î1ÄÏ1 ÅinExample 6.13. Soifcisasingular value, you cannot becertain that Î1Äc Åisnotamanifold. However ,even ifasingular brehappens tobea manifold, itisoften ofthe“wrong” dimension.) Hereisanexample ofamanifold given bytwo equations (mÆ2). 6.15.EXAMPLE.Dene:R4 ÐR2by  Äx ÅlÆ;Ñx2 1 Òx2 2 x1x3Òx2x4Ó. Then DÄxÅlÆ Ñ2x12x20 0 x3 x4x1x2Ó. Ifx1 Ô Æ0therst and thirdcolumns ofD Äx Åareindependent, and ifx2 Ô Æ0the second and fourth columns areindependent. Ontheother hand, ifx1 Æx2 Æ0, D Äx Åhasrank 1and Äx ÅÕÆ 0.This shows that theorigin 0inR2istheonly singular value of.Ther efore,bytheregular value theor em, forevery nonzer o vector ctheset Î1Äc Åisatwo-manifold inR4.Forinstance, M Æ Î1 Ö1 0 ×isa two-manifold. Note that Mcontains thepoint x Æ¿Ä1,0,0,0 ÅT.Letusnd abasis ofthetangent space TxM.Again bytheregular value theor em, this tangent space isequal tothenullspace of DÄxńÆ;Ñ2000 0010Ó, which isequal thesetofallvectors ysatisfying y1 Æy3 Æ0.Abasis ofTxMis thereforegiven bythestandar dbasis vectors e2and e4. Wenow come toamoresophisticated example ofamanifold determined bya largesystem ofequations. 6.16.EXAMPLE.Recall that ann Øn-matrix Aisorthogonal ifATA ÆI.This means that thecolumns (and also therows) ofAareperpendicular tooneanother and have length 1.(Inother words,they form anortho normal basis ofRn—note theregrettable inconsistency intheterminology .)The collection oforthogonal ma- trices form agroup under matrix multiplication, which isusually called theorthog- onal groupand denoted byOÄnÅ.Letusproveusing theregular value theor emthat OÄnÅisamanifold. First observe that thatÄATAÅTÆATA,soATAisasymmetric matrix. Inother words,ifV ÆRnÙnisthevector space ofalln Øn-matrices and W ÆÚÈC ÊV ËC ÆCTÍthelinear subspace ofallsymmetric matrices, then  ÄA ÅÆATA denes amap:V ÐW.Clearly O Än űÆ Î1ÄIn Å,sotoprove that O Än Åisa manifold itsufces toshow that Iisaregular value of.The derivative ofcan EXERCISES 77 becomputed byusing theformula derived inExer ciseB.3: D ÛA ÜB Ýlim h Þ01 h Û ÛA ßhB Üáà ÛA ÜÜÝlim hÞ01 h ÛATA ßhATB ßhBTA ßh2BTB àATA ÜÝBATßABT. Weneed toshow that forA âO Ûn Üthelinear map D ÛA Ü:V ãWhasrank equal tothedimension ofW.Bylinear algebra this amounts toshowing that the equation BATßABTÝC (6.1) issolvable forB,given any orthogonal Aand any symmetric C.Hereisaway of guessing asolution: observe that CÝ1 2 ÛCßCTÜand rst trytosolve BATÝ1 2C. Left multiplying both sides byAand using ATA ÝIgives B Ý1 2CA.Itisnow easy tocheck that B Ý1 2CAisasolution ofequation (6.1) . Exercises 6.1.This isacontinuation ofExer cise 1.1.Dene :R äR2by åt æ{çCå t èsint,1 è costæT.Show that isone-to-one. Determine alltforwhich éUåtæáç0.Provethat åRæis notamanifold atthese points. 6.2.Leta ê³å0,1 æbeaconstant. Provethat themap :R äR2given by åt æ{çëå t è asint,1èacostæTisanembedding. (This becomes easier ifyou rst show that tèasint isanincreasing function oft.)Graph thecurve dened by . 6.3.Provethat themap :R äR2given by åt æìç1 2 åetíe ît,etèe îtæTisanembed- ding. Conclude that Mç åRæisa1-manifold. Graph thecurve M.Compute thetangent line toMat å1,0 æand trytond anequation forM. 6.4.LetIbetheopen interval åïè1, ð³æand let :I äR2bethemap åt æñçòå 3at óôå1 í t3,3at2óõå1 ít3æïæT,wher eaisanonzer oconstant. Show that isone-to-one and that éöåtæ±÷ ç0foralltêI.Is anembedding and is åIæamanifold? (Observe that åIæ isaportion ofthecurve studied inExer cise1.2.) 6.5.Dene :R äR2by åt æñçCøúù èf åt æ,f åt æüûTift ý0,ùf åt æ,f åt æïûTift þ0, wher efisthefunction given inExer cise B.6. Show that issmooth, one-to-one and that itsinverse î1: åRæÿä Riscontinuous. Sketch theimage of .Is åRæamanifold? 6.6.Dene amap :R2äR4by t1 t2
ç t3 1 t2 1t2 t1t2 2 t3 2  . (i)Show that isone-to-one. (ii)Show that D åtæisone-to-one forallt÷ ç0. (iii) LetUbethepunctur edplane R2è 0.Show that :UäR4isanembedding. Conclude that åU æisatwo-manifold inR4. 78 6.MANIFOLDS (iv) Find abasis ofthetangent plane to U atthepoint 1,1 . 6.7.Let:Rn 0  Rbeahomogeneous function ofdegr eepasdened inEx- ercise B.5. Assume thatissmooth and that p  0.Show that 0istheonly possible singular value of.(Use theresult ofExer cise B.5.) Conclude that, ifnonempty ,1 cis ann 1-manifold forc 0. 6.8.Let x a1x2 1a2x2 2anx2n,wher etheaiarenonzer oconstants. Deter - mine theregular and singular values of.Forn 3sketch thelevel surface1 cfora regular value c.(Youhave todistinguish between afew differentcases.) 6.9.Show that thetrajectories oftheLotka-V olterra system ofExer cise 1.10 areone- dimensional manifolds. 6.10.Compute thedimension oftheorthogonal group O n and show that itstangent space attheidentity matrix Iisthesetofallantisymmetric n n-matrices. 6.11.LetVbethevector space ofn n-matrices and dene:V Rby A   detA. (i)Show that D AB n å i 1det a1,a2...,ai1,bi,ai 1,...,an , wher ea1,a2,...,anand b1,b2,...,bndenote thecolumn vectors ofA,resp. B. (Apply theformula ofExer cise B.3forthederivative and usethemultilinearity ofthedeterminant.) (ii)The special linear groupisthesubset ofVdened by SL n A V!detA 1. Show that SL n isamanifold. What isitsdimension? (iii) Show that forA I,theidentity matrix, wehave D A B ån i 1bi,i trB, thetrace ofB.Conclude that thetangent space toSL natIisthesetoftraceless matrices, i.e.matrices Asatisfying trA 0. 6.12. (i)LetWbepunctur ed4-space R4 "0 and dene:W Rby  x x1x4 x2x3. Show that 0isaregular value of. (ii)LetAbeareal2 2-matrix. Show that rankA 1ifand only ifdetA 0and A 0. (iii) LetMbethesetof2 2-matrices ofrank 1.Show that Misathree-dimensional manifold. (iv) Compute TAM,wher eA $# 11 00%. 6.13.Dene:R4R2by  x  '&x1x2x3x4 x1x2x3x4 (. (i)Show that D xhasrank 2unless xisoftheform t2,t2,t,t3forsome t  0.(Compute all2 2-subdeterminants ofDand setthem equal to0.) (ii)Show that M 1 0isa2-manifold (wher e0istheorigin inR2). (iii) Find abasis ofthetangent space TxMforallx Mwith x3 0.(The answer depends onx.) EXERCISES 79 6.14.LetUbeanopen subset ofRnand let:U )Rmbeasmooth map. LetMbe themanifold*1 +c,,wher ecisaregular value of.Letf:U)Rbeasmooth function. Apoint x-Miscalled acritical point fortherestricted function f.MifDf +x,v/0forall tangent vectors v -TxM.Prove that x -Miscritical forf .Mifand only ifthereexist numbers1,2,...,msuch that gradf +x ,0/1grad1 +x ,212grad2 +x ,31544461mgradm +x ,. (Use thecharacterization ofTxMgiven bytheregular value theor em.) 6.15.Find thecritical points ofthefunction f +x,y,z ,7/98 x 12y 13zover thecircleC given by x21y21z2/1, x1z/0. Wher earethemaxima and minima off.C? 6.16 (eigenvectors viacalculus) .LetA /ATbeasymmetric n :n-matrix and dene f:Rn)Rbyf +x ,0/x 4Ax.LetMbetheunit spher e ;x -Rn.=<x <>/1 ?. (i)Calculate gradf +x ,. (ii)Show that x -Misacritical point off .Mifand only ifxisaneigenvector forA oflength 1. (iii) Given aneigenvector xoflength 1,show that f +x ,isthecorresponding eigen- value ofx. CHAPTER 7 Differential forms onmanifolds 7.1. First denition Ther eareseveral differentways todene differential forms onmanifolds. In this section wepresent apractical, workaday denition. Amoretheor etical ap- proach istaken inSection 7.2. LetMbeann-manifold inRNand letusrst consider what wemight mean by a0-form orsmooth function onM.Afunction f:M@Rissimply anassignment ofaunique number f Ax Btoeach point xinM.Forinstance, Mcould bethesurface oftheearth and fcould represent temperatur eatagiven time, orheight above sea level. Buthow would wedene such afunction tobedifferentiable? The difculty hereisthat ifxisinMand ejisone ofthestandar dbasis vectors, thestraight line xChejmay notbecontained inM,sowecannot form thelimit ¶fD¶xjE limh F0 Af Ax Chej BHGf Ax BBDh. Hereisoneway outofthisdifculty .Because Misamanifold thereexist open sets UiinRnand embeddings i:Ui @RNsuch that theimages i AUi Bcover M: ME$I i i AUi B.(Her eiranges over some unspecied, possibly innite, index set.) Foreach iwedene afunction fi:Ui @Rbyfi AtBEfA i AtBB,i.e.fiE Jif.We call fithelocal representative offrelative totheembedding i.(For instance, ifM istheearth's surface, fistemperatur e,and iisamap ofNew YorkState, then fi represents atemperatur echart ofNY.)Since fiisdened ontheopen subset Uiof Rn,itmakes sense toaskwhether itspartial derivatives exist. Wesaythat fisCk ifeach ofthelocal representatives fiisCk.Now suppose that xisintheoverlap oftwo charts. Then wehave two indices iand jand vectors t KUiand u KUj such that xE i AtBE j AuB.Then wemust have fAxBEfA i AtBBEfA j AuBB,so fi AtBEfj AuB.Also i AtBE j AuBimplies tE L1 i M j AuBand thereforefj AuBE fiN L1 i M j Au BO.This identity must hold forallu KUjsuch that j Au BPK i AUi B, i.e.foralluin L1 j A i AUi BB.Wecanabbr eviate thisbysaying that fjE A L1 i M j B Jfi on L1 j A i AUi BQB.This isaconsistency condition onthefunctions fiimposed bythe fact that they arepullbacks ofasingle function fdened everywher eonM.The map L1 i M jisoften called achange ofcoordinates and theconsistency condition isalso known asthetransformation lawforthelocal representatives fi.(Pursuing theweather chart analogy ,itexpr esses nothing buttheobvious factthat wher ethe maps ofNew Yorkand Pennsylvania overlap, thecorresponding two temperatur e charts must show thesame temperatur es.) Conversely ,thecollection ofalllocal representatives fidetermines f,because wehave f Ax BEfi A L1 i Ax BBifx K i AUi B. 81 82 7.DIFFERENTIAL FORMS ON MANIFOLDS (That istosay,ifwehave acomplete setofweather charts forthewhole world, we know thetemperatur eeverywher e.) Following thiscueweformulate thefollowing denition. 7.1.DEFINITION.Adiffer ential form ofdegreek,orsimply ak-form ,onMisa collection ofk-formsionUisatisfying thetransformation law jRTS U1 i V jWXi (7.1) on U1 j S i SUi WW.Wecallithelocal representative ofrelative totheembedding iand denote itbyiR Xi.The collection ofallk-forms onMisdenoted byYkSMW. This denition israther indir ect,butitworks really well ifaspecic atlas for themanifold Misknown. Denition 7.1isparticularly tractible ifMistheimage ofasingle embedding :U ZRN.Inthat case thecompatibility relation (7.1) is vacuous and ak-formonMisdetermined byonesingle representative, ak-form XonU. Sometimes itisuseful towrite thetransformation law(7.1) incomponents. We candothisbyappealing toTheor em3.12. If iRå IfIdtI andjRå JgJdtJ aretwo local representatives for,then gJRå I S U1 iV jW XfIdetDS U1 iV jWI,J. on U1 j S iSUiWW. Just likeforms onRn,forms onamanifold canbeadded, multiplied, differen- tiated and integrated. Forexample, supposeisak-form andanl-form onM. Supposei,resp.i,isthelocal representative of,resp.,relative toanembed- ding i:Ui ZM.Then wedene theproduct Rbysetting iRii.Tosee that thisdenition makes sense, wecheck that theforms isatisfy thetransforma- tion law (7.1) : jRjjR[S U1 i V jW XiS U1 i V jW XiR\S U1 i V jW XSiiW]RTS U1 i V jW X i. Herewehave used themultiplicative property ofpullbacks, Proposition 3.10(ii) . Similarly ,theexterior derivative ofisdened bysettingSdWiRdi.Asbefor e, letuscheck that theformsSdWisatisfy thetransformation law(7.1) :SdWjRdjRdS U1 i V jW XiRTS U1 i V jW XdiRTS U1 i V jW XSdWi, wher eweused Theor em3.11. 7.2. Second denition This section presents some ofthealgebraic underpinnings ofthetheory of differential forms. This branch ofalgebra, now called exterior oralternating algebra was invented byGraßmann inthemid-nineteenth century and isaprerequisite formuch ofthemoreadvanced literatur eonthesubject. 7.2.SECOND DEFINITION 83 Covectors. Befor egiving arigor ousdenition ofdifferential forms onmani- folds weneed tobemoreprecise about thedenition ofadifferential form onRn. Recall that Rnisthecollection ofallcolumn vectors x ^ _```ax1 x2 ... xn bQcccd. LetUbeanopen subset ofRn.The denition ofa0-form onUrequir esnofurther clarication: itissimply afunction onU.Formally ,a1-formonUcanbedened asarowvector ^Tef1,f2,...,fn f whose entries arefunctions onU.The form iscalled constant iftheentries f1,..., fnareconstant. The setofconstant rowvectors isdenoted byeRnfgand iscalled thedual ofRn.Constant 1-forms arealso known ascovariant vectors orcovectors and arbitrary 1-forms ascovariant vector elds orcovector elds .Bydenition dxiis theconstant 1-form dxi ^eT i ^Te0,...,0,1,0,...,0f, thetranspose ofei,thei-thstandar dbasis vector ofRn.Every 1-form canthus be written as ^\ef1,f2,...,fn f ^n å i h1fidxi. Using thisformalism wecanwrite forany smooth function gonU dg^n å i h1¶g ¶xidxi ^i¶g ¶x1,¶g ¶x2,...,¶g ¶xn j, sodgissimply theJacobi matrix Dgofg!(This isthereason that many authors usethenotation dgfortheJacobi matrix.) Wewould liketoextend thenotions ofcovectors and 1-forms tovector spaces other than Rn.Toseehow,letusstart byobserving that arowvector yisnothing buta1 kn-matrix. Wecanmultiply itbyacolumn vector xtoobtain anumber , yx ^Tey1,y2,...,yn f _``` ax1 x2 ... xn bcccd ^n å ih1yixi. Obviously wehave yec1x1lc2x2 f ^c1yx1lc2yx2.Thus arowvector can be viewed asalinear map which sends column vectors inRntoone-dimensional vectors (scalars) inR1^R. This motivates thefollowing denition. IfVisany vector space over thereal numbers (for example Rnorasubspace ofRn),then Vg,thedual ofV,istheset oflinear maps fromVtoR.Elements ofVgarecalled dual vectors orcovectors or linear functionals .The dual isavector space initsown right: if1and2areinVg wedene1 l2and c1bysetting e1 l2 f evf ^1 evfl2 evfforallv mV and ec1 f evf ^c1 evf. 84 7.DIFFERENTIAL FORMS ON MANIFOLDS 7.2.EXAMPLE.LetV nC0 oqpa,b r,R s,thecollection ofallcontinuous real- valued functions onaclosed and bounded interval pa,b r.Alinear combination ofcontinuous functions iscontinuous, soVisavector space. Dene of stnub af ox sdx.Then oc1f1 vc2f2 sn c1 of1 svc2 of2 s,soisalinear functional onV. 7.3.EXAMPLE.LetV nRnand xv wV.Dene ox sxnv yx,wher e“ y”isthe standar dinner product onRn.Thenisalinear functional onV. Now suppose that Visavector space ofnite dimension nand choose abasis v1,v2,...,vnofV.Then every vector vwVcanbewritten inaunique way as alinear combination åjcjvj.Dene acovectori wVzbyi ovs{n ci.Inother words,iisdetermined bytherule i ovj s]ni,j n}|1ifinj, 0ifi ~ nj. Wecallithei-thcoordinate function . 7.4.LEMMA.Thecoordinate functions1,2,...,nconstitute abasis ofV z.Hence dimV znn ndimV. PROOF.Let wV z.Weneed towriteasalinear combination  nån i €1cii. Assuming forthemoment that this ispossible, wecan apply both sides tothe vector vjtoobtain  ovj sxnn å i€1cii ovj s]nn å i€1cii,j ncj. (7.2) Socj n ovj sistheonly possible choice forthecoefcient cj.Toshow that this choice ofcoefcients works, letusdene ‚n ån i €1 ovi si.Then byequation (7.2) , ovj s]n ovj sforallj,soƒn,i.e.nån i €1 ovi si.Wehave proved that every wV zcanbewritten uniquely asalinear combination ofthei. QED The basis „1,2,...,n …ofV zissaid tobedual tothebasis „v1,v2,...,vn … ofV. 7.5.EXAMPLE.Consider Rnwith standar dbasis„e1,...,en ….Then dxi oej s†n eT iej ni,j,sothedual basis of oRnszis „dx1,dx2,...,dxn …. Dual bases come inhandy when writing thematrix ofalinear map. Let L:V ‡Wbealinear map between abstract vector spaces Vand W.Towrite thematrix ofLweneed tostart bypicking abasis v1,v2,...,vnofVand abasis w1,w2,...,wmofW.Then foreach j n1,2,...,nthevector Lvjcanbeexpanded uniquely interms ofthew's:Lvj nåm i €1li,jwi.The m ˆnnumbers li,jmake up thematrix ofLrelative tothetwo bases ofVand W. 7.6.LEMMA.Let1,2,...,n wV zbethedual basis ofv1,v2,...,vnand1, 2,...,n wW zthedual basis ofw1,w2,...,wn.Then thei,j-thmatrix element ofa linear map L:V ‡Wisequal toli,j ni oLvj s. PROOF.Wehave Lvj nåm k€1lk,jwk,so i oLvj s]nm å k €1lk,ji owk s‰nm å k €1lk,jik nli,j, that isli,j ni oLvj s. QED 7.2.SECOND DEFINITION 85 Multilinear algebra. LetVbeavector space and letVkdenote theCarte- sian product VŠ‹Œ‹Œ‹Š V(ktimes). Thus anelement ofVkisanorderedk-tupleŽv1,v2,...,vkofvectors inV.Ak-multilinear function onVisafunction:Vk  Rwhich islinear ineach vector ,i.e.  Žv1,v2,...,cvi ‘c ’v ’i,...,vkŒ“ c Žv1,v2,...,vk ‘c’ Žv1,v2,...,v’i,...,vk forallscalars c,c’and allvectors v1,v2,...,vi,v’i,...,vk. 7.7.EXAMPLE.LetV “ Rnand let Žx,y “ x ‹y,theinner product ofxand y.Thenisbilinear (i.e.2-multilinear). 7.8.EXAMPLE.LetV “ R4,k “ 2.The function Žv,w “ v1w2 ”v2w1‘ v3w4”v4w3isbilinear onR4. 7.9.EXAMPLE.LetV “ Rn,k “ n.The determinant det Žv1v2,,...,vnisan n-multilinear function onRn. Ak-multilinear function isalternating orantisymmetric ifithasthealternating property ,  Žv1,...,vj,...,vi,...,vk “” Žv1,...,vi,...,vj,...,vk. Mor egenerally ,ifisalternating, then forany permutation  •Skwehave  Žv –1 —,...,v –k —  “ sign Ž Žv1,...,vk. 7.10.EXAMPLE.The inner product ofExample 7.7isbilinear ,butitisnotal- ternating. Indeed itissymmetric :y ‹x “ x ‹y.The bilinear function ofExample 7.8 isalternating, and soisthedeterminant function ofExample 7.9. Hereisauseful trick togenerate alternating k-multilinear functions starting fromkcovectors1,2,...,k •V˜.The (wedge) product isthefunction 12 ‹Œ‹Œ‹k:VkR dened by 12 ‹Œ‹Œ‹k Žv1,v2,...,vk  “ det™i Žvjš1 ›i,j ›k. (The determinant ontheright isakŠk-determinant.) Itfollows fromthemulti- linearity and thealternating property ofthedeterminant that12 ‹Œ‹Œ‹kisanal- ternating k-multilinear function. The wedge product isoften denoted by1œ2œ‹Œ‹Œ‹œktodistinguish itfromother products, such asthetensor product dened inExer cise7.6. The collection ofallalternating k-multilinear functions isdenoted byAkV. Fork “ 1thealternating property isvacuous, soanalternating 1-multilinear function isnothing butalinear function. Thus A1V “ V ˜. For k “ 0ak-multilinear function isdened tobeasingle number .Thus A0V “ R. Forany k,k-multilinear functions canbeadded and scalar -multiplied justlike ordinary linear functions, sothesetAkVforms avector space. Ther eisanice way toconstr uctabasis ofthevector space AkVstarting from abasis v1,...,vnžofV.The idea istotake wedge products ofdual basis vectors. 86 7.DIFFERENTIAL FORMS ON MANIFOLDS Let Ÿ1,...,n bethecorresponding dual basis ofV ¡.LetI ¢¤£i1,i2,...,ik¥be anincreasing multi-index, i.e.1 ¦i1 §i2 §©¨Œ¨Œ¨§ ik ¦n.Write I ¢i1i2 ¨Œ¨Œ¨ik ªAkV, vI ¢T£vi1,vi2,...,vik ¥ªVk. 7.11.EXAMPLE.LetV¢R3with standar dbasisŸe1,e2,e3  .The dual basis of£R3¥ ¡isŸdx1,dx2,dx3  .Letk¢2and I¢[£1,2¥,J¢T£2,3¥.Then dxI £eI¥ ¢}««««dx1 £e1¥dx1 £e2¥ dx2 £e1¥dx2 £e2¥ «««« ¢¬««««10 01 «««« ¢1, dxI £eJ¥ ¢ ««««dx1 £e2¥dx1 £e3¥ dx2 £e2¥dx2 £e3¥ «««« ¢ ««««00 10 «««« ¢0, dxJ £eI¥ ¢}««««dx2 £e1¥dx2 £e2¥ dx3 £e1¥dx3 £e2¥ «««« ¢¬««««01 00 «««« ¢0, dxJ £eJ¥ ¢}««««dx2 £e2¥dx2 £e3¥ dx3 £e2¥dx3 £e3¥ «««« ¢¬««««10 01 «««« ¢1. This example generalizes asfollows. 7.12.LEMMA.LetIandJbeincreasing multi-indices ofdegreek.Then I £vJ¥ ¢I,J ¢}­1ifI¢J, 0ifI ® ¢J. PROOF.LetI¢T£i1,...,ik ¥and J¢T£j1,...,jk ¥.Then I £vJ¥ ¢det ¯ir £vjs ¥°1±r,s±k ¢ ««««««««««««i1,j1...i1,jk...... il,j1...il,jk...... ik,j1...ik,jk «««««««««««« ¢1ifI ¢J. IfI ® ¢J,then il ® ¢jlforsome l.Choose lassmall aspossible, sothat im ¢jmfor m§l.Ther earetwo cases: il §jland il²jl.Ifil §jl,then il §jl §jl³1 §¨Œ¨Œ¨7§ jkbecause Jisincreasing, soallentriesil,jminthedeterminant with m ´l are0.Form§lwehave jm ¢im§ilbecause Iisincreasing, soil,jm ¢0for m§l.Inother wordsthel-throwinthedeterminant is0and henceI £vJ¥ ¢0. Ifil ²jlwend that thel-thcolumn inthedeterminant is0and thereforeagain I £vJ¥ ¢0. QED Weneed one further technical result befor eshowing that thefunctionsIare abasis ofAkV. 7.13.LEMMA.LetªAkV.Suppose £vI¥ ¢0forallincreasing multi-indices I ofdegreek.Then ¢0. PROOF.The assumption implies  £vi1,...,vik ¥ ¢0 (7.3) 7.2.SECOND DEFINITION 87 forallmulti-indices µi1,...,ik¶,because ofthealternating property .Weneed to show that µw1,...,wk¶¸·0forarbitrary vectors w1,...,wk.Wecanexpand the wiusing thebasis: w1·a11v1¹»ºŒºŒºŒ¹ a1kvk, ... wk·ak1v1 ¹¼ºŒºŒº½¹ akkvk. Ther eforebymultilinearity  µw1,...,wk ¶x·k å i1¾1 ºŒºŒºk å ik ¾1a1i1 ºŒºŒºakik µvi1,...,vik ¶. Each term intheright-hand side is0byequation (7.3) . QED 7.14.THEOREM.LetVbeann-dimensional vector space with basis ¿v1,...,vn À. Let ¿1,...,n Àbethecorresponding dual basis ofV Á.Then thealternating k-multilinear functionsI·i1 ºŒºŒºik,wher eIranges over thesetofallincreasing multi-indices of degreek,form abasis ofAkV.Hence dimAkV·¬Ân kÃ. PROOF.The proofisclosely analogous tothat ofLemma 7.4. LetÄAkV. Weneed towriteasalinear combination ·åIcII.Assuming forthemoment that this ispossible, wecanapply both sides tothek-tuple ofvectors vJ.Using Lemma 7.12 weobtain  µvJ¶‰·å IcII µvJ¶]·å IcII,J·cJ. SocJ· µvJ¶istheonly possible choice forthecoefcient cJ.Toshow that this choice ofcoefcients works, letusdeneÅ·åIµvI¶I.Then forallincreasing multi-indices IwehaveµvI¶>Æ ŵvI¶·µvI¶>ƵvI¶{· 0.Applying Lemma 7.13 toÆ ÅwendÆ Å·0.Inother words,·åI µvI¶I.Wehave proved that every ÄV Ácanbewritten uniquely asalinear combination ofthei.QED 7.15.EXAMPLE.LetV·Rnwith standar dbasis ¿e1,...,enÀ.The dual basis of µRn¶ Áis ¿dx1,...,dxnÀ.Ther eforeAkVhasabasis consisting ofallk-multilinear functions oftheform dxI·dxi1dxi2 ºŒºŒºdxik, with 1 Çi1 È\ºŒºŒºHÈ ik Çn.Hence ageneral alternating k-multilinear function onRnlooks like ·å IaIdxI, with aIconstant. ByLemma 7.12, µeJ¶É· åIaIdxI µeJ¶Ê· åIaII,J·aJ,soaIis equal toµeI¶. Anarbitary k-formonaregion UinRnisnow dened asachoice ofan alternating k-multilinear functionxforeach x ÄU;hence itlooks likex· åIfI µx¶dxI,wher ethecoefcients fIarefunctions onU.Weshall abbr eviate this to ·å IfIdxI, 88 7.DIFFERENTIAL FORMS ON MANIFOLDS and weshall always assume thecoefcients fItobesmooth functions. ByExample 7.15 wecanexpr essthecoefcients asfI Ë ÌeI Í(which istobeinterpr eted as fI ÌxÍ]Ëx ÌeI Íforallx). Pullbacks re-examined. Inthelight ofthisnew denition wecangive afresh interpr etation ofapullback. This will beuseful inourstudy offorms onmanifolds. LetUand Vbeopen subsets ofRn,resp. Rm,and:UÎVasmooth map. Fora k-form ÏÐkÌVÍdene thepullback Ñ ÏÐkÌUÍbyÌ ÑÍx Ìv1,v2,...,vk Í]Ë Òx Ó ÌDÌxÍv1,DÌxÍv2,...,DÌxÍvk Í. Letuscheck that thisformula agreeswith theolddenition. SupposeËåIfIdyI and ÑËåJgJdxJ.What istherelationship between gJand fI?WeusegJ Ë  Ñ ÌeJ Í,ournew denition ofpullback and thedenition ofthewedge product toget gJ ÌxÍ]Ë Ì ÑÍx ÌeJ ÍxË Òx Ó ÌD ÌxÍej1,D ÌxÍej2,...,D ÌxÍejk ÍËå IfI ÌÌxÍÍdyI ÌDÌxÍej1,DÌxÍej2,...,DÌxÍejk ÍËå I ÑfI ÌxÍdet Ôdyir ÌD ÌxÍejs ÍÕ1 Ör,s Ök. ByLemma 7.6thenumber dyir ÌDÌxÍejs Íistheirjs-matrix entry oftheJacobi ma- trixD ÌxÍ(with respect tothestandar dbasis e1,e2,...,enofRnand thestandar d basis e1,e2,...,emofRm).Inother words,gJ ÌxÍ×Ë åI ÑfI ÌxÍdetDI,J ÌxÍ.This formula isidentical totheone inTheor em3.12 and thereforeournew denition agreeswith theold! Forms onmanifolds. LetMbeann-dimensional manifold inRN.Foreach point xinMthetangent space TxMisann-dimensional linear subspace ofRN. Adiffer ential form ofdegreekorak-formonMisachoice ofanalternating k- multilinear mapxonthevector space TxM,oneforeach x ÏM.This alternating mapxisrequir edtodepend smoothly onxinthefollowing sense. Accor ding to thedenition ofamanifold, foreach x ÏMthereexists anembedding :U ÎRN such that ÌUÍØË MÙVforsome open setVinRNcontaining x.The tangent space atxisthen TxMËD ÌtÍ ÌRnÍ,wher etÏUischosen such that ÌtÍÚË x. The pullback ofunder thelocal parametrization isdened byÌ ÑÍt Ìv1,v2,...,vk ͉Ë Òt Ó ÌD ÌtÍv1,D ÌtÍv2,...,D ÌtÍvk Í. Then Ñisak-form onU,anopen subset ofRn,so ÑËåIfIdtIforcertain functions fIdened onU.Wewill requir ethefunctions fItobesmooth. (The form ÑËåIfIdtIisthelocal representative ofrelative totheembedding ,as introduced inSection 7.1.) Torecapitulate: 7.16.DEFINITION.Ak-formonMisachoice, foreach xÏM,ofanalternat- ingk-multilinear mapxonTxM,which depends smoothly onx. The book [BT82 ]describes ak-form asan“animal” that inhabits a“world” M, eats orderedk-tuples oftangent vectors, and spits outnumbers. 7.17.EXAMPLE.LetMbeaone-dimensional manifold inRN.Letuschoose an orientation (“dir ection”) onM.Atangent vector toMispositive ifitpoints inthe EXERCISES 89 same direction astheorientation and negative ifitpoints intheopposite direction. Dene a1-formonMasfollows. Forx ÛMand atangent vector v ÛTxMput xÜv Ý‰Þ¬ß àvàifvispositive ,áàvàifvisnegative. This form istheelement ofarclength ofM.Weshall seeinChapter 8how togener - alize ittohigher -dimensional manifolds and inChapter 9how touseittocalculate arclengths and volumes. Wecancalculate thelocal representative âofak-formforanyembedding :UãRNparametrizing aportion ofM.Suppose wehad two differentsuch embeddings i:Ui ãMand j:Uj ãM,such that xiscontained inboth Wi Þ i ÜUi Ýand Wj Þ j ÜUj Ý.How dothelocal expr essionsi Þ âiandj Þ âjforcompar e?Toanswer thisquestion, consider thecoor dinate change map ä1 j å i,which maps ä1 i ÜWiæWj Ýto ä1 j ÜWiæWj Ý.Fromi Þ âiandj Þ âjwerecover thetransformation law (7.1) j ÞÜ ä1 iå j Ý âi. This shows that Denitions 7.1and 7.16 ofdifferential forms onamanifold are equivalent. Exercises 7.1.The vectors e1 çe2and e1 èe2form abasis ofR2.What isthedual basis of éR2 êìë? 7.2.Letív1,v2,...,vnîbeabasis ofRnand letí1,2,...,nîbethedual basis oféRnê ë.LetAbeaninvertible n ïn-matrix. Then byelementary linear algebra thesetof vectors íAv1,...,Avnîisalso abasis ofRn.Show that thecorresponding dual basis isthe setofrowvectorsí1Að1,2Að1,...,nAð1î. 7.3.Suppose thatisabilinear function onavector space Vsatisfying év,v ê0ñ0for allvectors vòV.Provethatisalternating. Generalize this observation tok-multilinear functions. 7.4.Show that thebilinear functionofExample 7.8isequal todx1dx2çdx3dx4. 7.5.Thewedge product isageneralization ofthecrossproduct toarbitrary dimensions inthesense that x ïy ñ$óõôéxT öyTêì÷T forallx,y òR3.Provethisformula. (Interpr etation: xand yarecolumn vectors, xTand yT arerowvectors, xT öyTisa2-form onR3, ôéxT öyT êisa1-form, i.e.arowvector .Soboth sides oftheformula represent column vectors.) 7.6.LetVbeavector space and let1,2,...,k òV ëbecovectors. Their tensor product isthefunction 1ø2øúùùQù6øk:Vk ûR dened by 1 ø2 ø5ùùùüøk év1,v2,...,vk ê0ñ1 év1 ê2 év2 êùùùk évk ê. Show that1ø2øýùQùù6økisak-multilinear function. 90 7.DIFFERENTIAL FORMS ON MANIFOLDS 7.7.Let:VkþRbeak-multilinear function. Dene anew function Alt:VkþR by Altÿv1,v2,...,vk 
1 k!å  Sksignÿÿv1,v2,...,vk . Provethefollowing. (i)Altisanalternating k-multilinear function. (ii)Alt
ifisalternating. (iii) AltAlt
Altforallk-multilinear. (iv) Let1,2,...,k V.Then 12  k
1 k!Altÿ1 2   k. 7.8.Show that detÿv1,v2,...,vn
dx1dx2 dxn ÿv1,v2,...,vnforallvectors v1, v2,...,vnRn.Inshort, det
dx1dx2 dxn. 7.9.LetVand Wbevector spaces and L:V þWalinear map. Show that L ÿ
ÿL  ÿL forallcovectors,W . CHAPTER 8 Volume forms 8.1. n-Dimensional volume inRN Leta1,a2,...,anbevectors inRN.The block orparallelepiped spanned bythese vectors isthesetofallvectors oftheform ån i1ciai,wher ethecoefcients cirange over theunit interval 0,1 .Forn 1thisisalso called alinesegment and forn 2 aparallelogram .Wewill need aformula forthevolume ofablock. Ifn Nthere isnocoher entway ofdening anorientation onalln-blocks inRN,sothisvolume will benotanoriented butanabsolute volume. Weappr oach this problem ina similar way astheproblem ofdening thedeterminant, namely byimposing a few reasonable axioms. 8.1.DEFINITION.Anabsolute n-dimensional Euclidean volume function isafunc- tion voln:RN RN RN   ntimes R with thefollowing properties: (i)homogeneity: volna1,a2,...,cai,...,an ! c volna1,a2,...,an forallscalars cand allvectors a1,a2,...,an; (ii)invariance under shear transformations: voln a1,...,ai "caj,...,aj,...,an  voln a1,...,aj,...,ai,...,an  forallscalars cand any i # j; (iii) invariance under Euclidean motions: voln Qa1,...,Qan  voln a1,...,an  forallorthogonal matrices Q; (iv) normalization: volne1,e2,...,en 1. Weshall shortly seethat these axioms uniquely determine then-dimensional volume function. 8.2.LEMMA. (i)volna1,a2,...,an %$a1 $&$a2 $ $an $ifa1,a2,..., anareorthogonal vectors. (ii)volna1,a2,...,an 0ifthevectors a1,a2,...,anaredependent. PROOF.Suppose a1,a2,...,anareorthogonal. First assume they arenonzer o. Then wecan dene qi '$ai $(1ai.The vectors q1,q2,...,qnareortho normal . Complete them toanorthonormal basis q1,q2...,qn,qn )1,...,qNofRN.Let 91 92 8.VOLUME FORMS Qbethematrix whose i-thcolumn isqi.Then Qisorthogonal and Qei *qi. Ther efore voln+a1,a2,...,an,*.-a1-&-a2-0///1- an-voln+q1,q2,...,qn , byAxiom (i)*.-a1-&-a2-0///1- an-voln+Qe1,Qe2,...,Qen,*.-a1-&-a2-0///1- an-voln +e1,e2,...,en , byAxiom (iii)*.-a1-&-a2-0///1- an- byAxiom (iv), which proves part (i)ifallaiarenonzer o.Ifoneoftheaiis0,thevectors a1,a2,..., anaredependent, sothestatement follows frompart (ii),which weprovenext. Assume a1,a2,...,anaredependent. For simplicity suppose a1isalinear combination oftheother vectors, a1*ån i 22ciai.Byrepeatedly applying Axiom (ii)weget voln+a1,a2,...,an ,*voln 3n å i 22ciai,a2,...,an 4*voln 3n å i 23ciai,a2,...,an4*5///6* voln+0,a2,...,an,. Now byAxiom (i), voln +0,a2,...,an ,*voln +00,a2,...,an ,*0voln +0,a2,...,an ,*0, which proves property (ii). QED This brings ustothevolume formula. Wecan form amatrix Aoutofthe column vectors a1,a2,...,an.Itdoes notmake sense totake detAbecause Ais notsquar e,unless n*N.However ,theproduct ATAissquar eand wecantake itsdeterminant. 8.3.THEOREM.Ther eexists aunique n-dimensional volume function onRN.Let a1,a2,...,an 7RNandletAbetheN 8n-matrix whose i-thcolumn isai.Then voln+a1,a2,...,an,*59 det+ATA,. PROOF.Weleave ittothereader tocheck that thefunction:det+ATA,sat- ises theaxioms foran-dimensional volume function onRN.(See Exer cise 8.2.) Hereweproveonly theuniqueness part ofthetheor em. Case 1.First assume that a1,a2,...,anareorthogonal. Then ATAisadiagonal matrix. Itsi-thdiagonal entry is-ai-2,so:det+ATA,*.-a1-&-a2-0///;- an-,which isequal tovoln+a1,a2,...,an,byLemma 8.2(i) . Case 2.Next assume that a1,a2,...,anaredependent. Then thematrix A hasanontrivial nullspace, i.e.thereexists anonzer on-vector vsuch that Av* 0.But then ATAv*0,sothecolumns ofATAaredependent aswell. Since ATAissquar e,this implies detATA*0,so:det+ATA,*0,which isequal to voln +a1,a2,...,an ,byLemma 8.2(ii) . Case 3.Finally consider anarbitrary sequence ofindependent vectors a1, a2,...,an.This sequence can betransformed into anorthogonal sequence v1, v2,...,vnbytheGram-Schmidt process. This works asfollows: letb1*0and for i <1letbibetheorthogonal projection ofaionto thespan ofa1,a2,...,ai=1;then 8.1.n-DIMENSIONAL VOLUME INRN93 vi >ai ?bi.(See illustration below .)LetVbetheN @n-matrix whose i-thcolumn isvi.Then byrepeated applications ofAxiom (ii), voln Aa1,a2,...,an B>voln Av1,a2,...,an B>voln Av1,v2,...,an B>5CCC>volnAv1,v2,...,vnB>5D detAVTVB,(8.1) wher ethelastequality follows fromCase 1.Since vi>ai?bi,wher ebiisalinear combination ofa1,a2,...,aiE1,wehave V>AU,wher eUisan @n-matrix ofthe form U> FGGGGGH1 I ICCC I 01 ICCC I 00 1CCC I .........I 00CCC 0 1 JKKKKKL. Note that Uhasdeterminant 1.This implies that VTV>UTATAUand detAATAB>detUTdetAATABdetU>detAUTATAUB>detAVTVB. Using formula (8.1) wegetvolnAa1,a2,...,anB>NM detAATAB. QED The Gram-Schmidt process transforms asequence ofnindependent vectors a1,a2,...,aninto anorthogonal sequence v1,v2,...,vn.(The horizontal “oor ” represents theplane spanned bya1and a2.)The block spanned bythea'shasthe same volume astherectangular block spanned bythev's. a1a2a3 a1Ov1a2a3 b2 b3v2v3 a1a2a3 v1v2v3 Forn>NTheor em8.3gives thefollowing result. 8.4.COROLLARY.Leta1,a2,...,anbevectors inRnandletAbethen @n-matrix whose i-thcolumn isai.Then voln Aa1,a2,...,an B>.PdetAP. 94 8.VOLUME FORMS PROOF.Aissquar e,sodet QATA RTS detATdetA SUQdetA R2byTheor em 3.7(ii) and thereforevoln Qa1,a2,...,an RVSXW QdetA R2SZYdetA YbyTheor em8.3. QED 8.2. Orientations Oriented vector spaces. Youareprobably familiar with orientations onvec- torspaces ofdimension [3.Anorientation ofaline isanassignment ofadi- rection. Anorientation ofaplane isachoice ofadirection ofrotation, clockwise versus counter clockwise. Anorientation ofathree-dimensional space isachoice of“handedness”, i.e.achoice ofaright-hand ruleversus aleft-hand rule. These notions canbegeneralized asfollows. LetVbeann-dimensional vec- torspace over therealnumbers. Suppose that \]S^Q v1,v2,...,vn Rand \`_aSQv_1,v_2,...,v_n Raretwo orderedbases ofV.Then wecanwrite v_i Såjai,jvjand vi Såjbi,jv _jforsuitable coefcients ai,jand bi,j.The n bn-matrices A SZQai,j R and BScQbi,j Rsatisfy ABSBASIand arethereforeinvertible. Wesaythat the bases \and \_dene thesame orientation ofVifdetA d0.IfdetA e0,thetwo bases dene opposite orientations . Forinstance, if Qv _1,v _2,...,v _n R&S5Q v2,v1,...,vn R,then AS fgggggh010...0 100...0 001...0 ............... 000...1 ijjjjjk, sodetA Sml 1.Hence theorderedbases Qv2,v1,...,vn Rand Qv1,v2,...,vn Rdene opposite orientations. Weknow now what itmeans fortwo bases tohave thesame orientation, but how dowedene theconcept ofanorientation itself? Intypical mathematician's fashion wedene theorientation ofVdetermined bythebasis\tobethecollec- tion ofallorderedbases that have thesame orientation as\.(Ther eisananal- ogous denition ofthenumber 1,namely asthecollection ofallsets that contain one element.) The orientation determined by \^SnQ v1,v2,...,vn Risdenoted byo\qpor ov1,v2,...,vn p.Soif \and \_dene thesame orientation then o\qprS o\_ p. Ifthey dene opposite orientations wewrite o\qpsScl o\_ p.Because thedetermi- nant ofaninvertible matrix iseither positive ornegative, therearetwo possible orientations ofV.Anoriented vector space isavector space together with achoice ofanorientation. This preferr edorientation isthen called positive . Forn S0weneed tomake aspecial denition, because azero-dimensional space hasanempty basis. Inthis case wedene anorientation ofVtobeachoice ofsign,torl. 8.5.EXAMPLE.The standard orientation onRnistheorientation oe1,...,en pde- ned bythestandar dorderedbasis Qe1,...,en R.Weshall always usethis orienta- tion onRn. Maps and orientations. LetVand Wbeoriented vector spaces ofthesame dimension and letL:V uWbeaninvertible linear map. Choose apositively oriented basis Qv1,v2,...,vn RofV.Because Lisinvertible, theorderedn-tuple 8.2.ORIENT ATIONS 95vLv1,Lv2,...,Lvnwisanorderedbasis ofW.Ifthis basis ispositively ,resp. neg- atively ,oriented wesaythat Lisorientation-pr eserving ,resp. orientation-r eversing . This denition does notdepend onthechoice ofthebasis, forif vv x1,v x2,...,v xn wis another positively oriented basis ofV,then v xi yåjai,jvjwith det vai,j w{z0.Ther e- foreLv xiyL|åjai,jvj }yåjai,jLvj,and hence thetwo bases vLv1,Lv2,...,Lvn w and vLvx1,Lvx2,...,Lvxn wofWdetermine thesame orientation. Oriented manifolds. Now letMbeamanifold. Wedene anorientation of Mtobeachoice ofanorientation foreach tangent space TxMwhich varies con- tinuously over M.“Continuous” means that forevery x~Mthereexists alo- calparametrization :W M,with Wopen inRnand x ~ vWw,such that D y:RnTyMpreserves theorientation forally ~W.(Her eRnisequipped with itsstandar dorientation.) Amanifold isorientable ifitpossesses anorienta- tion; itisoriented ifaspecic orientation hasbeen chosen. Hypersurfaces. The case ofahypersurface, amanifold ofcodimension 1,is particularly instr uctive. Aunit normal vector eld onamanifold MinRnisasmooth function n:M Rnsuch that n vxw€TxMand n vxw y1forallx ~M. 8.6.PROPOSITION.Ahypersurface inRnisorientable ifandonly ifitpossesses a unit normal vector eld. PROOF.LetMbeahypersurface inRn.Suppose Mpossesses aunit normal vector eld. Let vv1,v2,...,vn ‚1 wbeanorderedbasis ofTxMforsome x ~M. Then vn vxw,v1,v2,...,vn ‚1 wisabasis ofRn,because n vxw&€viforalli.Wesaythatvv1,v2,...,vn ‚1 wispositively oriented if vn vxw,v1,v2,...,vn ‚1 wisapositively ori- ented basis ofRn.This denes anorientation onM,called theorientation induced bythenormal vector eld n. Conversely ,letussuppose that Misanoriented hypersurface inRn.Foreach x~Mthetangent space TxMisnƒ1-dimensional, soitsorthogonal complementvTxMw„isaline. Ther earethereforeprecisely two vectors oflength 1which are perpendicular toTxM.Wecanpick apreferr edunit normal vector asfollows. Letvv1,v2,...,vn ‚1 wbeapositively oriented basis ofTxM.The positive unit normal vector isthat unit normal vector n vxwthat makes vn vxw,v1,v2,...,vn ‚1 waposi- tively oriented basis ofRn.InExer cise 8.8you will beasked tocheck that n vxw depends smoothly onx.Inthis way wehave produced aunit normal vector eld onM. QED 8.7.EXAMPLE.Letusregar dRn‚1asthesubspace ofRnspanned bytherst nƒ1standar dbasis vectors e1,e2,...,en‚1.The standar dorientation onRnis…e1,e2,...,en†,and thestandar dorientation onRn ‚1is …e1,e2,...,en ‚1 †.Since…e1,e2,...,en†y vƒ1wn ‡1 …en,e1,e2,...,en ‚1 † byExer cise 8.5,thepositive unit normal toRn‚1inRnis vƒ1wn‡1en. The positive unit normal onanoriented hypersurface MinRncanberegar ded asamap nfromMinto theunit spher eSn‚1,which isoften called theGauß map of M.The unit normal enables onetodistinguish between two sides ofM:thedirec- tion ofnis“out” or“up”; theopposite direction is“in” or“down”. Forthisreason orientable hypersurfaces areoften called two-sided ,wher easthenonorientable ones arecalled one-sided .Letusshow that ahypersurface given byasingle equation is always orientable. 96 8.VOLUME FORMS 8.8.PROPOSITION.LetUbeopen inRnandlet:U ˆRbeasmooth function. Letcbearegular value of.Then themanifold ‰1 Šc ‹hasaunit normal vector eld given byn Šx ‹&Œgrad Šx ‹ Žgrad Šx ‹;Žandisthereforeorientable. PROOF.The regular value theor emtells usthat MŒ‰1Šc‹isahypersurface inRn(ifnonempty), and also that TxMŒkerDx Œcgrad Šx‹‘’.The function n Šx‹{Œgrad Šx‹ Žgrad Šx‹;Žthereforedenes aunit normal vector eld onM. Appealing toProposition 8.6weconclude that Misorientable. QED 8.9.EXAMPLE.Taking Šx‹“Œ%Ž xŽ2and cŒr2weobtain that the Šn”1‹- spher eofradius rabout theorigin isorientable. The unit normal is n Šx ‹•Œgrad Šx ‹ Žgrad Šx ‹;Ž–Œ x Žx Ž. 8.3. Volume forms Now letMbeanoriented n-manifold inRN.Choose acollection ofembed- dings i:Ui ˆRNwith Uiopen inRnsuch that M Œ˜—i i ŠUi ‹and such that D i Št ‹:RnˆTxMisorientation-pr eserving forallt ™Ui.The volume formM, also denoted by,isthen-form onMwhose local representative relative tothe embedding iisdened by i Œ ši Œ.› det ŠD i Št ‹TD i Št ‹ ‹dt1dt2 œœœdtn. ByTheor em8.3thesquar e-rootfactor measur esthevolume ofthen-dimensional block inthetangent space TxMspanned bythecolumns ofD i Št ‹,theJacobi ma- trixof iatt.Hence you should think ofasmeasuring thevolume ofinnitesi- mal blocks inside M. 8.10.THEOREM.Foranyoriented n-manifold MinRNthevolume formMisa well-dened n-form. PROOF.Toshow thatiswell-dened weneed tocheck that itslocal repre- sentatives satisfy thetransformation law(7.1) .SoletusputŒ ‰1 i  jand sub- stitute tŒ Šu‹intoi.Since each oftheembeddings iisorientation-pr eserving, wehave detDž0.Hence byTheor em3.13 wehave  š Šdt1dt2œœœdtn ‹&ŒdetD Šu‹du1du2œœœdun Œ!ŸdetD Šu‹;Ÿdu1du2œœœdun. Ther efore  ši Œ›detD i Š Šu ‹ ‹TD i Š Šu ‹ ‹‘ ŸdetD Šu ‹;Ÿdu1du2 œœœdunŒ›detD Šu ‹TdetD i Š Šu ‹ ‹TD i Š Šu ‹‹‘detD Šu ‹du1du2 œœœdunŒ›det ŠD i Š Šu‹ ‹D Šu‹ ‹TD i Š Šu‹ ‹D Šu‹‘du1du2œœœdunŒ›det Š ŠD j Šu ‹ ‹TD j Šu ‹ ‹du1du2 œœœdun Œj, wher einthesecond tolastidentity weapplied thechain rule. QED FornŒ1thevolume form isusually called theelement ofarclength ,fornŒ2, theelement ofsurface area,and forn Œ3,thevolume element .Traditionally these are denoted byds,dA,and dV,respectively .Don't bemisled bythisold-fashioned no- tation: volume forms areseldom exact! The volume formMishighly dependent 8.3.VOLUME FORMS 97 ontheembedding ofMinto RN.Itchanges ifwedilate orshrink orotherwise deform M. 8.11.EXAMPLE.LetUbeanopen subset ofRn.Recall fromExample 6.5 that Uisamanifold cover edbyasingle embedding, namely theidentity map :U  U, ¡x ¢•£x.Then det ¡D TD ¢•£1,sothevolume form onUissimply dt1dt2 ¤¤¤dtn,theordinary volume form onRn. 8.12.EXAMPLE.LetIbeaninterval intherealline and f:I Rasmooth function. LetM ¥R2bethegraph off.ByExample 6.7Misa1-manifold inR2. Indeed, Mistheimage oftheembedding :I  R2given by ¡t ¢•£5¡ t,f ¡t ¢ ¢.Let usgive Mtheorientation induced bytheembedding ,i.e.“fromlefttoright”. What istheelement ofarclength ofM?Letuscompute thepullback ¦,a1-form onI.Wehave D ¡t¢§£©¨1 f ª«¡t ¢­¬, D ¡t¢TD ¡t¢0£¯® 1f ª«¡t ¢±°¨1 f ª«¡t ¢­¬ £1²f ª¡t¢2, so ¦£5³ det¡D ¡t¢TD ¡t¢ ¢dt£³1²fª ¡t¢2dt. The next result canberegar ded asanalternative denition ofM.Itisperhaps moreintuitive, butitrequir esfamiliarity with Section 7.2. 8.13.PROPOSITION.LetMbeanoriented n-manifold inRN.Letx´Mandv1, v2,...,vn ´TxM.Then thevolume form ofMisgiven by M,x ¡v1,v2,...,vn ¢£¶µ · ¸·¹voln ¡v1,v2,...,vn ¢ifv1,v2,...,vnarepositively oriented ,ºvoln ¡v1,v2,...,vn ¢ifv1,v2,...,vnarenegatively oriented , 0 ifv1,v2,...,vnarelinearly dependent , i.e.M,x ¡v1,v2,...,vn ¢istheoriented volume ofthen-dimensional parallelepiped in TxMspanned byv1,v2,...,vn. PROOF.Foreach xinMand n-tuple oftangent vectors v1,v2,...,vnatxlet !x ¡v1,v2,...,vn ¢betheoriented volume oftheblock spanned bythese nvectors. This denes ann-form!onMand wemust show that! £M.LetUbean open subset ofRnand :U  RNanorientation-pr eserving embedding with ¡U¢»¥ Mand ¡t¢»£ xforsome tinU.Letuscalculate then-form ¦!onU. Wehave ¦!£gdt1dt2¤¤¤dtnforsome function g.ByLemma 7.12 this function isgiven by g ¡t ¢•£ ¦!x ¡e1,e2,...,en ¢&£!x ¡D ¡t ¢e1,D ¡t ¢e2,...,D ¡t ¢en ¢, wher einthesecond equality weused thedenition ofpullback. The vectors D ¡t¢e1,D ¡t¢e2,...,D ¡t¢enareapositively oriented basis ofTxMand, more- over ,arethecolumns ofthematrix D ¡t¢,sobyTheor em8.3they span apositive volume ofmagnitude³det¡D ¡t¢TD ¡t¢ ¢.This shows that g£ ³det¡D TD ¢ and therefore ¦! £5¼ det ¡D TD ¢dt1dt2 ¤¤¤dtn. Thus ¦!isequal tothelocal representative ofMwith respect totheembedding .Since this holds forallembeddings ,wehave!£M. QED 98 8.VOLUME FORMS Volume form ofahypersurface. Fororiented hypersurfaces MinRnthereis amoreconvenient expr ession forthevolume formM.Recall thevector -valued forms dx½U¾ ¿Àdx1 ... dxn ÁÂà andÄdx½Å¾ ¿À Ädx1 ...Ädxn ÁÂà introduced inSection 2.5. Letnbethepositive unit normal vector eld onM and letFbeany vector eld onM,i.e.asmooth map F:MÆRn.Then the inner product FÇnisafunction dened onM.Itmeasur esthecomponent ofF orthogonal toM.The productÈFÇnÉMisannÊ1-form onM.Ontheother hand wehave thenÊ1-formÄËÈFÇdxÉ&½FÇÌÄdx. 8.14.THEOREM.Onthehypersurface Mwehave FÇÌÄdx½5ÈFÇnÉM. FIRSTPROOF.This proofisshort butrequir esfamiliarity with thematerial in Section 7.2.Letx ÍM.Letuschange thecoor dinates onRninsuch away that the rst n Ê1standar dbasis vectors Èe1,e2...,en Î1 Éform apositively oriented basis ofTxM.Then, accor ding toExample 8.7,thepositive unit normal atxisgiven by n Èx ÉϽ¯ÈÊ 1 Én Ð1enand thevolume form satisesM,x Èe1,...,en Î1 ɖ½1.Writing F ½ån iÑ1Fiei,wehave F Èx ÉÒÇn Èx É&½NÈÊ 1 Én Ð1Fn Èx É.Ontheother hand FÇÄdx½å i ÈÊ1ÉiÐ1Fidx1 ÇÇÇÓdxi ÇÇÇdxn, and therefore ÈF ÇÌÄdx ÉÌÈe1,...,en Î1 ɕ½5ÈÊ 1 Én Ð1Fn.This proves thatÈF ÇÌÄdx Éx Èe1,...,en Î1 É&½NÈ F Èx ÉÔÇn Èx ÉÉM Èe1,...,en Î1 É, which impliesÈFÇÕÄdxÉx ½UÈFÈxÉÇnÈxÉ ÉM.Since this equality holds forevery xÍM,wend FÇÌÄdx½5ÈFÇnÉM. QED SECONDPROOF.Choose anembedding :U ÆRn,wher eUisopen inRn Î1, such that ÈU ÉVÖ M,x Í ÈU É.Lett ÍUbethepoint satisfying Èt ÉV½ x.As apreliminary step intheproofwearegoing toreplace theembedding with a new one enjoying aparticularly nice property .Letuschange thecoor dinates on Rninsuch away that therst n Ê1standar dbasis vectors Èe1,e2...,en Î1 Éform apositively oriented basis ofTxM.Then atxthepositive unit normal isgiven by n Èx ɕ½NÈÊ 1 Én Ð1en.Since thecolumns oftheJacobi matrix D Èt Éareindependent, thereexist unique vectors a1,a2,...,an Î1inRn Î1such that D Èt Éai ½eifori ½1, 2,...,n Ê1.These vectors aiareindependent, because theeiareindependent. Ther eforethe Èn Ê1 ÉÏ×ØÈ n Ê1 É-matrix Awith i-thcolumn vector equal toaiis invertible. Put ˜U½A Î1ÈUÉ,˜t½A Î1tand ˜ ½ ÙA.Then ˜Uisopen inRn Î1, ˜ Șt É&½x,˜ :˜U ÆRnisanembedding with ˜ ȘU É&½ ÈU É,and D˜ Ștɕ½D ÈtÉÚÙDAȘt½D ÈtÉÚÙA bythechain rule. Ther eforethei-thcolumn vector ofD˜ ȘtÉis D˜ Șt Éei ½D Èt ÉAei ½D Èt Éai ½ei (8.2) 8.3.VOLUME FORMS 99 fori Û1,2,...,n Ü1.(On thelefteidenotes thei-thstandar dbasis vector in Rn Ý1,ontheright itdenotes thei-thstandar dbasis vector inRn.)Inother words, theJacobi matrix of˜ at˜tisthe Þn Ü1 ߕàn-matrix D˜ ޘtß&Û©áIn Ý1 0 â, wher eIn Ý1isthe Þn Ü1 ßàãÞ n Ü1 ßidentity matrix and 0denotes arowconsisting ofn Ü1zeros. Letusnow calculate ˜ äËåÞF æn ßM çand ˜ äèÞF æèédx ßatthepoint ˜t.Writing FænÛån i ê1Finiand using thedenition ofMweget ˜ äåÞFænßMç Û©án å iê1˜ äÞFini ßâaëdetÞD˜ TD˜ ßd˜t1d˜t2 æææd˜tnÝ1. Fromformula (8.2) wehave det ÞD˜ ޘt ßTD˜ ޘt ß ß•Û 1.Soevaluating thisexpr ession atthepoint ˜tand using n Þx ß&ÛNÞÜ 1 ßn ì1enwegetå˜ äÞFænßMç˜t ÛNÞÜ 1ßnì1Fn Þxßd˜t1d˜t2 æææd˜tnÝ1. FromF æédx Ûån iê1 ÞÜ1 ßi ì1Fidx1dx2 æææ6ídxi ææædxnweget ˜ äÞF æÌédx ߕÛn å i ê1 Þ Ü1 ßi ì1˜ äFid˜ 1d˜ 2 æææîd˜ i æææd˜ n. Fromformula (8.2) wesee¶˜ i ޘt ß ï¶˜tj Ûi,jfor1 ði,j ðn Ü1and ¶˜ n ޘt ß ï¶˜tj Û0 for1ðjðnÜ1.Ther eforeå˜ äÞF æédx ßç˜t Û.ÞÜ 1 ßn ì1Fn Þx ßd˜t1d˜t2 æææd˜tn Ý1. Weconclude thatå˜ ä1ÞFænßMç˜t Ûcå˜ äÕÞFæédxßç˜t,inother wordsåÞFænßMçx ÛÞF æÌédx ßx.Since thisholds forallx ñMwehave F æédx Û5ÞF æn ßM. QED This theor emgives insight into thephysical interpr etation ofnÜ1-forms. Think ofthevector eld Fasrepresenting theow ofauid orgas. The direc- tion ofthevector Findicates thedirection oftheow and itsmagnitude measur es thestrength oftheow .Then Theor em8.14 says that then Ü1-form F æ;édxmea- sures,forany unit vector ninRn,theamount ofuid perunit oftime passing through ahyperplane ofunit volume perpendicular ton.WecallF æédxtheux ofthevector eld F. Another application ofthetheor emisthefollowing formula forthevolume form onahypersurface. The formula provides aheuristic interpr etation ofthe vector -valued formédx:ifnisaunit vector inRn,then thescalar -valued nÜ1- form næèédxmeasur esthevolume ofaninnitesimal nÜ1-dimensional paral- lelepiped perpendicular ton. 8.15.COROLLARY.Letnbetheunit normal vector eld andMthevolume form of anoriented hypersurface MinRn.Then M ÛnæÌédx. PROOF.SetFÛninProposition 8.14. Then FænÛ1becauseònò»Û1.QED 100 8.VOLUME FORMS 8.16.EXAMPLE.Suppose thehypersurface Misgiven byanequation óx ôsõ c,wher ecisaregular value ofafunction:U öR,with Uopen inRn.Then by Proposition 8.8Mhasaunit normal n õgrad ÷øgrad ø.The volume form is therefore õnøgrad ø;ù1grad ú;ûdx.Inparticular ,ifMisthespher eofradius Rabout theorigin inRn,then n óx ô&õx ÷R,soM õR ù1x úûdx. Exercises 8.1.Deduce fromTheor em8.3that theareaoftheparallelogram spanned byapair of vectors a,binRnisgiven byüaüübüsin,wher eistheangle between aand b(which is taken toliebetween 0and).Show that üa ürüb üsin ýþüa ÿb üinR3. 8.2.Check that thefunction volna1,a2,...,an
ý detATA
satises theaxioms of Denition 8.1. 8.3.Letu1,u2,...,ukand v1,v2,...,vlbevectors inRNsatisfying uivj ý0foriý1, 2,...,kand j ý1,2,...,l.(“The u'sareperpendicular tothev's.”) Provethat volk l u1,u2,...,uk,v1,v2,...,vl
ývolk u1,u2,...,uk
voll v1,v2,...,vl
. 8.4.Leta1,a2,...,anberealnumbers, letcý 1ån i 1a2 iand let u1 ý 1 0 ... 0 a1 ,u2 ý 0 1 ... 0 a2 ,...,un ý 0 0 ... 1 an ,un1 ý1 c  a1a2 ...an 1  bevectors inRn1. (i)Deduce fromExer cise8.3that volnu1,u2,...,un
ývoln 1 u1,u2,...,un,un 1
. (ii)Provethat1 a2 1a1a2 a1a3 ...a1an a2a1 1 a2 2a2a3 ...a2an a3a1 a3a2 1 a2 3...a3an ............... ana1 ana2 ana3 ...1 a2n  ý1n å i 1a2 i. 8.5.Justify thefollowing identities concerning orientations ofavector space V.Here thev'sform abasis ofV(which inpart (i)isn-dimensional and inparts (ii)–(iii) two- dimensional). (i)If Snisany permutation, then v 1 ,v 2 ,...,v n  ýsign
v1,v2,...,vn. (ii) v1,v2  ý v1,v2 . (iii)3v1,5v2  ýv1,v2 . 8.6.LetUbeopen inRnand letf:URbeasmooth function. Let :URn 1be theembedding x
ýx,fx

and letM ý U
,thegraph off.Dene anorientation on Mbyrequiring tobeorientation-pr eserving. Deduce fromExer cise 8.4that thevolume form ofMisgiven by M ý1 ügradfx
ü2dx1dx2 dxn. 8.7.LetM ýgraphfbetheoriented hypersurface ofExer cise8.6. EXERCISES 101 (i)Show that thepositive unit normal vector eld onMisgiven by n ! "#1$n %1& 1')(gradf"x$(2 *+++++,¶f -¶x1 ¶f-¶x2 ... ¶f -¶xn 1 ./////0. (ii)Derive theformula 1M ! & 1 '2(gradf"x $3(2dx1dx2 4 44dxnfromCorollary 8.15 bysubstituting xn %1 !f"x1,x2,...,xn $.(Caution: forconsistency you must replace nwith n'1inCorollary 8.15.) 8.8.Show that theunit normal vector eld n:M 5Rndened intheproofofPropo- sition 8.6issmooth. (Compute ninterms ofanorientation-pr eserving parametrization :U5Mofanopen subset ofM.) 8.9.Let :"a,b $65 Rnbeanembedding. Letbetheelement ofarclength onthe embedded curve M ! "a,b $.Show that 1isthe1-form on"a,b $given by ( 7"t $3(dt !8 71"t$2' 72"t$2'4 44 ' 7n"t$2dt. CHAPTER 9 Integration and Stokes' theorem onmanifolds Inthis chapter wewill seehow tointegrate ann-form over anoriented n- manifold. Inparticular ,byintegrating thevolume form wend thevolume ofthe manifold. Wewill also discuss aversion ofStokes' theor emformanifolds. This requir estheslightly moregeneral notion ofamanifold with boundary . 9.1. Manifolds with boundary The notion ofaspherical earth developed inclassical Greece around thetime ofPlato and Aristotle. Older cultur es(and also Western cultur euntil theredis- covery ofGreekastronomy inthelate Middle Ages) visualized theearth asaat disc surrounded byanocean oravoid. Aclosed disc isnotamanifold, because noneighbour hood ofapoint ontheedge istheimage ofanopen subset ofR2un- deranembedding. Rather ,itisamanifold with boundary ,anotion which canbe dened asfollows. The n-dimensional halfspace is Hn 9;:x<Rn =xn >0?. The boundary ofHnis¶Hn 9@:x <Rn =xn 90 ? 9Rn A1and itsinterior is intHn 9B:x <Rn =xnC0 ?. 9.1.DEFINITION.Ann-dimensional manifold with boundary (orn-manifold with boundary )inRNisasubset MofRNsuch that forallx <MthereexistDanopen subset V ERNcontaining x,Danopen subset UERn,Dand anembedding :U FRNsatisfying GU HHn I 9V HM. Youshould compar ethis denition carefully with Denition 6.4ofamanifold. If x 9 Gt Iwith t <¶Hn,then xisaboundary point ofM.The boundary ofMisthe setofallboundary points and isdenoted by¶M.Itscomplement M J¶Misthe interior ofMand isdenoted byintM. Somewhat confusingly ,theboundary ofamanifold with boundary isallowed tobeempty .Ifnonempty ,theboundary ¶Misann J1-dimensional manifold. Likewise theinterior intMisann-manifold. The most obvious example ofann-manifold with boundary isthehalfspace Hnitself, which hasboundary ¶Hn 9Rn A1and interior theopen halfspace :x < Rn =xnC0 ?.Hereisamoreinter esting type ofexample, which generalizes the graph ofafunction. 9.2.EXAMPLE.LetU Kbeanopen subset ofRn A1and letf:U KLF Rbea smooth function. PutU 9U KNMRand write elements ofUas Oxy Pwith xinU Kand 103 104 9.INTEGRA TION AND STOKES' THEOREM ON MANIFOLDS yinR.The region below thegraph offisthesetconsisting ofall Qxy RinUsuch that y Sf Tx U. ¶M Vgraphf M xy Weassert that theregion below thegraph isann-manifold whose boundary is exactly thegraph off.Wewill provethis bydescribing itastheimage ofasingle embedding. Dene :UWRnby Xt uY[Z Xt fTtU]\uY. AsinExample 6.3oneveries that isanembedding, using thefactthat D Xt uY[Z XIn^1 0 DfTtU_\ 1Y, wher e0istheorigin inRn ^1.Bydenition therefore,thesetMZ TU `HnUis ann-manifold inRnwith boundary ¶MZ TU `¶HnU.What areMand ¶M?A point Qxy RisinMifand only ifitisoftheformXx yY[Z Xt uY[Z Xt fTtU]\uY forsome Qtu RinU `Hn.Since Hnisgiven byu a0,this isequivalent tox bU c and ySfTxU.Thus Misexactly theregion below thegraph. On¶Hnwehave uZ0,so¶Misgiven bytheequality yZfTxU,i.e.¶Misthegraph. 9.3.EXAMPLE.Iff:U cdWRmisavector -valued map onecannot speak about theregion “below” thegraph, butonecandothefollowing. Again putUZU cfe R.LetNZngm\1and think ofRNasthesetofvectors Qxy Rwith xinRn ^1and yinRm.Dene :U WRNby Xt uY[Z Xt fTtU]\uem Y This time wehave D Tt UZ XIn ^1 0 Df Tt Uh\ em Y and again isanembedding. Ther eforeMZ TU `HnUisann-manifold inRN with boundary ¶MZ TU `¶HnU.This time Misthesetofpoints Qxy RoftheformXx yY Z Xt f Tt U]\uem Y 9.1.MANIFOLDS WITH BOUNDAR Y 105 with t iU jand u k0.Hence Misthesetofpoints lxy mwher exisinU jand wher e ysatises m n1equalities and oneinequality: y1 of1 px q,y2 of2 px q,...,ym r1 ofm r1 px q,ymsfmpx q. Again ¶Misgiven byyofpx q,so¶Misthegraph off. Hereisanextension oftheregular value theor em,Theor em6.10, tomanifolds with boundary .The proof, which wewill notspell out, issimilar tothat ofTheo- rem6.10. 9.4.THEOREM(regular value theor emformanifolds with boundary) .LetUbe open inRNandlet:UtRmbeasmooth map. LetMbethesetofxinRNsatisfying 1pxqoc1,2pxqoc2,...,mr1pxqocmr1,mpxqscm. Suppose that coupc1,c2,...,cm qisaregular value ofandthat Misnonempty .Then Misamanifold inRNofcodimension m n1andwith boundary ¶Mo r1pc q. 9.5.EXAMPLE.LetUoRn,mo1andpxqowvxv2.The setgiven bythe inequalitypxqs1isthen theclosed unit ballxxiRn yvxv s1z.Since gradpx qo2x,any nonzer ovalue isaregular value of.Hence theball isan n-manifold inRn,whose boundary is r1p1q,theunit spher eSnr1. Ifmorethan one inequality isinvolved, singularities often arise. Asimple example istheclosed quadrant inR2given bythepair ofinequalities xk0and y k0.This isnotamanifold with boundary because itsedge hasasharp angle at theorigin. Similarly ,aclosed squar eisnotamanifold with boundary . However ,one canshow that asetgiven byapair ofinequalities oftheform asfpx qsb,wher eaand bareboth regular values ofafunction f,isamanifold with boundary .Forinstance, thespherical shellxx iRnyR1 svxv sR2 z isann-manifold whose boundary isaunion oftwo concentric spher es. Other examples ofmanifolds with boundary arethepairofpants ,a2-manifold whose boundary consists ofthreeclosed curves, and theMöbius band shown inChapter 1.The Möbius band isanonorientable manifold with boundary .Wewill notgive aproofofthisfact, butyou canconvince 106 9.INTEGRA TION AND STOKES' THEOREM ON MANIFOLDS yourself that itistruebytrying topaint thetwo sides ofaMöbius band indifferent colours. Ann-manifold with boundary contained inRn(i.e.ofcodimension 0)isoften called adomain .Forinstance, aclosed ball isadomain inRn. Todene thetangent space toamanifold with boundary Matapoint xchoose Uand asinthedenition and put TxM{D |t}~|Rn}. Asinthecase ofamanifold, thisdoes notdepend onthechoice oftheembedding .Now suppose xisaboundary point ofMand letv TxMbeatangent vector . Then v {D |t }uforsome u Rn.Wesaythat vpoints inwards ifun€0and outwards ifun0.Ifun {0,then vistangent totheboundary .Inother words, Tx¶M {D |t }~|Rn ‚1}. The above pictur eofthepair ofpants shows some tangent vectors atboundary points that aretangent totheboundary oroutwar d-pointing. Orienting theboundary .LetMbeanoriented manifold with boundary .The orientation onMinduces anorientation on¶Mbyamethod very similar tothe proofofProposition 8.6.Namely ,forx ¶Mdene n |x }ƒTxMtobetheunique outwar d-pointing tangent vector oflength 1which isorthogonal toTx¶M.This de- nes theunit outward-pointing normal vector eld on¶M.Abasis |v1,v2,...,vn ‚1 } ofTx¶Miscalled positively oriented if |n |x },v1,v2,...,vn ‚1 }isapositively ori- ented basis ofTxM.This denes anorientation of¶M,called theinduced orientation . Forinstance, letM {Hnwith thestandar dorientation „e1,...,en ….Ateach point of¶M {Rn ‚1theoutwar dpointing normal is †en.This implies that induced orientation on¶Mis |†1 }n„e1,e2,...,en ‚1 …,because„‡†en,e1,e2,...,en ‚1 … {ˆ|† 1 }n„e1,e2,...,en ‚1,en …. 9.2. Integration over orientable manifolds Aswesaw inChapter 5,aform ofdegr eencan beintegrated over achain ofdimension n.The integral does notchange ifwereparametrize thechain in anorientation-pr eserving manner .This opens upthepossibility ofintegrating an n-form over anoriented n-manifold. LetMbeann-dimensional oriented manifold (possibly with boundary) inRN and letbeann-form onM.Todene theintegral ofover Mletusassume that Miscompact. (Asubset ofRNiscalled compact ifitisclosed and bounded; seeAppendix A.2. This assumption ismade toensur ethat theintegral isaproper integral and thereforeconver ges.) Forastart, letusalso make theassumption that thereexists asmooth map c: „0,1…n ‰RNsuch thatŠc ‹ „0,1…n Œ{MandŠtherestriction ofcto |0,1 }nisanorientation-pr eserving embedding. Forinstance, thisassumption issatised forthen-spher eSn(see Exer cise5.5)and thetorusS1 S1(see Exer cise9.4). The pullback c Žisthen ann-form onthecube„0,1…n.Wedene  M{ ‘ 0,1’nc Ž. 9.2.INTEGRA TION OVER ORIENT ABLE MANIFOLDS 107 Suppose ¯c: “0,1 ”n •RNisasmooth map with thesame properties asc.Toensur e that –Miswell-dened weneed tocheck thefollowing equality . 9.6.LEMMA.—™˜ 0,1šnc›œ—‘˜ 0,1šn¯c›. SKETCHOFPROOF.Letusdenote theclosed cube“0,1”nbyRand letUbe theopen cubež0,1Ÿn.LetVœU c¡1ž¯cžUŸ Ÿand ¯VœU ¯c¡1žcžUŸ Ÿ.The complement ofVand of¯VinRarenegligeable inthesense that— Rc › œ¢— Vc › and — R¯c › œ— ¯V¯c ›. (9.1) Byassumption therestriction ofctoUisanembedding. This implies that c:V • Misabijection onto itsimage, and soweseethat c¡1£¯cisabijection from ¯V onto V.Itisorientation-pr eserving, because cand ¯careorientation-pr eserving. Ther efore,byTheor em5.1,— Vc › œ — ¯V žc ¡1£¯c Ÿ ›žc › Ÿ¤œ — ¯V žc £c ¡1£¯c Ÿ › œ — ¯V¯c ›. Combining thiswith theequalities (9.1) wegettheresult. QED Not every manifold canbecover edwith one single n-cube. However ,itcan beshown that therealways exists anite collection ofn-cubes ci:“0,1”n•Mfor iœ1,2,...,k,such that (i) ¥k i ¦1ci§ “0,1 ”n ¨œM, (ii)ci§ ž0,1Ÿn¨ cj § ž0,1Ÿn¨isempty fori© œj, (iii) foreach itherestriction ofcito ž0,1 Ÿnisanorientation-pr eserving em- bedding. Wecanthen dene— M œk å i ¦1 —˜ 0,1 šnc ›i, and check asinLemma 9.6that theresult does notdepend onthemaps ci.(The condition (ii)onthemaps isimposed toavoid “double counting” intheintegral.) 9.7.DEFINITION.LetMacompact oriented manifold inRN.The volume ofM isvolM œ –M,wher eisthevolume form onM.(IfdimM œ1,resp. 2,we speak ofthearclength ,resp. surface areaofM.)The integral ofafunction fonM isdened as –Mf.The mean oraverage offisthenumber ¯f œªžvolM Ÿ ¡1–Mf. The centr oidorbarycentr eofMisthepoint ¯xinRnwhose i-thcoor dinate isthe mean value ofxiover M,i.e. ¯xi œ1 volM — Mxi. The volume form depends ontheembedding ofMinto RN,sothenotions dened above depend ontheembedding aswell. The most important property oftheintegral isthefollowing version ofStokes' theor em, which canbeviewed asaparametrization-independent version ofThe- orem5.10 and isproved inasimilar way. 108 9.INTEGRA TION AND STOKES' THEOREM ON MANIFOLDS 9.8.THEOREM(Stokes' theor emformanifolds) .Letbeann «1-form ona compact oriented n-manifold with boundary M.Give theboundary ¶Mtheinduced ori- entation. Then ¬ Md ­ ¬ ¶M. 9.3. Gauß and Stokes Stokes' theor em, Theor em9.8,contains asspecial cases theintegral theor ems ofvector calculus. These classical results involve avector eld F­ån i ®1Fieide- ned onanopen subset UofRn.Asdiscussed inSection 2.5,tothis vector eld corresponds a1-form­F¯dx­ån i®1Fidxi,which wecanthink ofasthework done bytheforceFalong aninnitesimal line segment dx.Wewill now derive theclassical integral theor ems byapplying Theor em9.8toone-dimensional, resp. n-dimensional, resp. two-dimensional manifolds Mcontained inU. Fundamental theorem ofcalculus. IfFisconservative, F­gradgforafunc- tion g,then­gradg¯dx­dg.IfMisacompact oriented 1-manifold with boundary inRn,then°Mdg­±°¶MgbyTheor em9.8. The boundary consists of two points aand b(ifMisconnected). Iftheorientation ofMis“fromatob”, then aacquir esaminus and baplus. Stokes' theor emthereforegives thefunda- mental theor emofcalculus inRn,¬ MF ¯dx ­g ²b ³´«g ²a ³. Ifweinterpr etFasaforceacting onaparticle travelling along M,then«gstands forthepotential ener gyoftheparticle intheforceeld. Thus thepotential ener gy oftheparticle decr eases bytheamount ofwork done. Gauß' divergence theorem. Wehaveµ­F¯ µdx and d µ­divFdx1dx2 ¯¶¯¶¯dxn. IfNisaoriented hypersurface inRnwith positive unit normal n,then µ­·²F¯ n ³NonNbyTheor em8.14. Inthis situation itisbest tothink ofFastheow vector eld ofauid, wher ethedirection ofF ²x ³gives thedirection oftheow at apoint xand themagnitude ¸F ²x ³¹¸gives themass oftheamount ofuid passing perunit time through ahypersurface ofunit areaplaced atxperpendicular tothe vector F ²x ³.Then µdescribes theamount ofuid passing perunit time and per unit areathrough thehypersurface N.Forthis reason then «1-form µisalso called theuxofF,and itsintegral over Nthetotal uxthrough N. Applying Stokes' theor emtoacompact domain MinRnweget °Md µ ­°¶M.Written interms ofthevector eld Fthis isGauß' diver gence theor em,¬ MdivFdx1dx2 ¯¶¯¶¯dxn ­ ¬ ¶M ²F¯n³¶M. Thus thetotal ux outofthehypersurface ¶Mistheintegral ofdivFover M.If theuid isincompr essible (e.g. most liquids) then this formula leads totheinter - pretation ofthediver gence ofF(orequivalently d µ)asameasur eofthesour ces orsinks oftheow .Thus divF ­0foranincompr essible uid without sour ces EXERCISES 109 orsinks. Iftheuid isagasand iftherearenosour cesorsinks then divF ºx »ƒ¼0 (resp. ½0)indicates that thegasisexpanding (resp. being compr essed) atx. Classical version ofStokes' theorem. Now letMbeacompact two-dimen- sional oriented surface with boundary and letusrewrite Stokes' theor em¾Md¿¾¶Minterms ofthevector eld F.The right-hand side represents thework of Fdone around theboundary curve(s) ofM,which isnotnecessarily 0ifFisnot conservative. The left-hand side hasanice interpr etation ifn¿3.ThenÀd¿ curlFÁdx,sod¿curlFÁ3Àdx.Hence ifnisthepositive unit normal ofthesurface MinR3,then d¿uºcurlFÁn»MonM.Inthis way wegettheclassical formula ofStokes, M ºcurlF Án »M ¿  ¶MF Ádx. Inother words,thetotal ux ofcurlFthrough thesurface Misequal tothework done byFaround theboundary curves ofM.This formula shows that curlF,or equivalentlyÀd,canberegar ded asameasur eofthevorticity ofthevector eld. Exercises 9.1.LetUbeanopen subset ofRnand letf,g:U ÃRbetwo smooth functions satisfying fÄxÅÇÆgÄxÅforallxinU.LetMbethesetofallpairsÄx,yÅsuch that xinUand f Äx Å´Èy Èg Äx Å. (i)Draw apictur eofMifUistheopen unit disc given byx2 Éy2Æ1and f Äx,y ÅËỄÍ1 Ìx2 Ìy2and gÄx,yÅÎÊ2 Ìx2Ìy2. (ii)Show directly fromthedenition that Misamanifold with boundary .(Use two embeddings tocover M.)What isthedimension ofMand what arethe boundary and theinterior? (iii) Give anexample showing that Misnotnecessarily amanifold with boundary if thecondition f Äx Å´Èy Èg Äx Åfails. 9.2. (i)Let Êxdy Ìydxand letMbeacompact domain intheplane R2. Show that ϶Mistwice thesurface areaofM. (ii)Apply theobservation ofpart (i)tond theareaenclosed bytheastroidxÊ cos3t,y Êsin3t. (iii) LetÊxÐÒÑdxand letMbeacompact domain inRn.Show that϶Misa constant times thevolume ofM.What isthevalue oftheconstant? 9.3.Writethediver gence theor emforthevector eld F Ê ÌcxnenonRn,wher ecis apositive constant. Deduce Archimedes' Law: thebuoyant forceexerted onasubmer ged body isequal totheweight ofthedisplaced uid. E´! 9.4.LetR1 ÓR2 Ó0beconstants. Dene a3-cube c: Ô0,R2 ÕNÖ Ô0,2Õ‘Ö Ô0,2Õ ÃR3 by c×Ør 1 2 ÙÚÊÛ× Ø ÄR1 Ércos2 Åcos1ÄR1 Ércos2 Åsin1 rsin2 ÙÚ . (i)Sketch theimage ofc. (ii)Letx1,x2,x3bethestandar dcoor dinates onR3.Compute c Üdx1,c Üdx2,c Üdx3 and cÜ Ädx1dx2dx3 Å. (iii) Find thevolume ofthesolid parametrized byc. (iv) Find thesurface areaoftheboundary ofthissolid. 110 9.INTEGRA TION AND STOKES' THEOREM ON MANIFOLDS 9.5.LetMbeacompact domain inRn.Letfand gbesmooth functions onM.The Dirichlet integral offand gisD Ýf,g Þ]ßàM Ýgradf ágradg Þ,wher e ßdx1dx2 áá ádxnis thevolume form onM. (i)Show that df Ýãâdg ÞäßÝ gradf ágradg Þ. (ii)Show that d âdg ßåÝçæ g Þ,wher e æg ßån iè1¶2g é¶x2 i. (iii) Deduce fromparts (i)–(ii) that d Ýf Ýãâdg ÞÞËßåÝ gradf ágradg êf æg Þ. (iv) Letnbetheoutwar d-pointing unit normal vector eld on¶M.Write¶g é¶nfor thedirectional derivativeÝDgÞnßgradgán.Show thatë ¶Mf Ýãâdg Þìß ë ¶Mf¶g ¶n¶M. (v)Deduce fromparts (iii)and (iv)Green's formula,ë ¶Mf¶g ¶n¶M ßD Ýf,g ޙê ë M Ýf æg Þ. (vi) Deduce Green's symmetric formula,ë ¶M íf¶g ¶nîg¶f ¶nï¶M ß ë M ÝfægîgæfÞ. 9.6.Inthis problem wewill calculate thevolume ofaball and aspher einEuclidean space. LetB ÝR Þbetheclosed ball ofradius Rabout theorigin inRn.Then itsboundary S ÝR Þä߶B ÝR Þisthespher eofradius R.PutVn ÝR ÞäßvolnB ÝR Þand An ÝR ÞËßvoln ð1S ÝR Þ. Also putVn ßVn Ý1Þand An ßAn Ý1Þ. (i)Deduce fromCorollary 8.15 that thevolume form onS ÝR Þistherestriction of toS ÝR Þ,wher eisasinExer cise2.16. Conclude that An ÝR ÞËßñàSòRó. (ii)Show that Vn ÝRÞôß RnVnand An ÝRÞôß Rn ð1An.(Substitute yßRxinthe volume forms ofB ÝR Þand S ÝR Þ.) (iii) Let f: õ0, ö÷Þùø Rbeacontinuous function. Dene g:RnøRbyg Ýx Þúß fÝüûxû Þ.Use Exer cise2.16(ii) toprovethatë B òR ógdx1dx2 ááádxn ß ëR 0f Ýr ÞAn Ýr Þdr ßAn ëR 0f Ýr Þrn ð1dr. (iv) Show that ýëÿþð þe ðr2dr nßAn ëôþ 0e ðr2rn ð1dr. (Take fÝrÞËße ðr2inpart (iii)and letRø ö.) (v)Using Exer cises B.10 and B.11 conclude that An ß2n 2
n 2 ,whence A2m ß2mÝmî1 Þ!and A2m1 ß2m 1m 1 á3 á5 áá áÝ2mî1 Þ. (vi) Bytaking fÝrÞËß1inpart (iii)show that An ßnVnand An ÝRÞË߶Vn ÝRÞ é¶R. (vii) Deduce that Vn ßn 2
n 2 ê1,whence V2m ßm m!and V2m 1 ß2m 1m 1á3á5ááá Ý2mê1Þ. (viii) Complete thefollowing table. (Conventions: aspace ofnegative dimension is empty; thevolume ofazero-dimensional manifold isitsnumber ofpoints.) n 0 1 2 3 4 5 Vn ÝRÞ R2 4 3R3 An ÝR Þ 2R EXERCISES 111 (ix) Find limn An,limn Vnand limn  An 1 An .Use Stirling's formula, limx  x 1 ex xx 1 2 2. CHAPTER 10 Applications totopology 10.1. Brouwer 'sxed point theorem LetMbeamanifold, possibly with boundary .Aretraction ofMonto asubset Aisasmooth map:M Asuch that x xforallxinA.Forinstance, let Mbethepunctur edunit ball inn-space, MxRn0x 1!. Then thenormalization map x "x #$x isaretraction ofMonto itsboundary A¶M,theunit spher e.The following theor emsays that aretraction onto the boundary isimpossible ifMiscompact and orientable. 10.1.THEOREM.LetMbeacompact orientable manifold with nonempty boundary . Then theredoes notexist aretraction fromMonto ¶M. PROOF.Suppose:M¶Mwas aretraction. Letuschoose anorientation ofMand equip ¶Mwith theinduced orientation. Let¶Mbethevolume form ontheboundary (relative tosome embedding ofMinto RN).Let  %beits pullback toM.Letndenote thedimension ofM.Note thatisann&1-form onthen&1-manifold ¶M,sod0.Ther eforedd%%d0and hence byStokes' theor em0('Md)'¶M.Butisaretraction onto ¶M,sothe restriction ofto¶Mistheidentity map and thereforeon¶M.Thus 0 )* ¶M )* ¶M vol¶M + 0, which isacontradiction. Ther eforedoes notexist. QED This brings ustooneoftheoldest results intopology .Suppose fisamap from asetXinto itself. Anelement xofXisaxed point offiffx,x. 10.2.THEOREM(Brouwer 'sxed point theor em).Every smooth map fromthe closed unit ballintoitself hasatleast onexed point. PROOF.LetM- xRn x. 1!betheclosed unit ball. Suppose f:M Mwas asmooth map without xed points. Then f x /+ xforallx.For each xintheballconsider thehaline starting atf x and pointing inthedirection ofx.This haline intersects theunit spher e¶Minaunique point that weshall call 113 114 10.APPLICA TIONS TOTOPOLOGY  0x 1,asinthefollowing pictur e. xf 2x 3 2x3y f 2y 3 2y 3 This denes asmooth map:M4¶M.Ifxisintheunit spher e,then0x1"5x, soisaretraction oftheball onto itsboundary ,which contradicts Theor em10.1. Ther eforefmust have axed point. QED This theor emcanbestated impr ecisely assaying that after you stiracup of coffee, atleast one molecule must return toitsoriginal position. Brouwer origi- nally stated hisresult forarbitrary continuous maps. This moregeneral statement can bederived fromTheor em10.2 byanargument fromanalysis which shows that every continuous map ishomotopic toasmooth map. (See Section 10.2 for thedenition ofhomotopy .)The theor emalso remains valid iftheclosed ball is replaced byaclosed cube orasimilar shape. 10.2. Homotopy Denition and rst examples. Suppose that0and1aretwo maps froma manifold Mtoamanifold Nand thatisaform onN.What istherelationship between thepullbacks60and61?Ther eisareasonable answer tothisquestion if0canbesmoothly deformed into1.Mor eformally ,wesaythat0and1 arehomotopic ifthereexists asmooth map:M780,19:4 Nsuch that0x,01"5 0 0x 1and 0x,1 1;51 0x 1forallxinM.The mapiscalled ahomotopy .Instead of  0x,t 1weoften writet 0x 1.Then eachtisamap fromMtoNand wecanthink oftasafamily ofmaps parametrized bytintheunit interval that interpolates between0and1,orasaone-second “movie” that attime 0starts at0and at time 1ends upat1. 10.3.EXAMPLE.LetM 5N 5Rnand0 0x 1<5x(identity map) and1 0x 1,50 (constant map). Then0and1arehomotopic. Ahomotopy isgiven by 0x,t 1,501 =t 1x.This homotopy collapses Euclidean space onto theorigin bymoving each point radially inwar d.Ther eareother ways toaccomplish this. Forinstance01 =t 12xand 01 =t21xaretwo other homotopies between thesame maps. Wecan also inter change0and1:if0 0x 1<50and1 0x 1,5x,then wend ahomotopy byreversing time (playing themovie backwar ds), 0x,t 1,5tx. 10.4.EXAMPLE.LetM 5Nbethepunctur edEuclidean space Rn=?>0 @and let0 0x 1A5 x(identity map) and1 0x 1B5 x C$Dx D(normalization map). Then0 and1arehomotopic. Ahomotopy isgiven forinstance by0x,t1E5 xC$DxDtor by0x,t1/5F0 1=t1xGtxC$DxD.Either ofthese homotopies collapses punctur ed Euclidean space onto theunit spher eabout theorigin bysmoothly stretching or shrinking each vector until ithaslength 1. 10.2. HOMOT OPY 115 10.5.EXAMPLE.Amanifold Missaid tobecontractible ifthereexists apoint x0inMsuch that theconstant map0 Hx IKJ x0ishomotopic totheidentity map Hx I<Jx.Aspecic homotopy:M LNM0,1 O$P Mfrom0to1isacontraction of Monto x0.(Perhaps “expansion” would beamoreaccurate term, a“contraction” being theresult ofreplacing twith 1 Qt.)Example 10.3 shows that Rniscon- tractible onto theorigin. (Infactitiscontractible onto any point x0.Can you write acontraction ofRnonto x0?)The same formula shows that anopen orclosed ball around theorigin iscontractible. Weshall seeinTheor em10.19 that punctur ed n-space RnQ?R0Sisnotcontractible. Homotopy ofcurves. IfMisaninterval Ma,b Oand Nanymanifold, then maps fromMtoNarenothing butparametrized curves inN.Ahomotopy ofcurves can bevisualized asapiece ofstring moving through themanifold N. a b N Homotopy ofloops. Aloop inamanifold Nisasmooth map fromtheunit circleS1into N.This can bevisualized asathin rubber band sitting inN.A homotopy ofloops:S1LM0,1 O;P Ncanbepictur edasarubber band oating through Nfromtime 0until time 1. S1N 10.6.EXAMPLE.Consider thetwo loops0,1:S1PR2intheplane given by0HxITJ xand1HxIUJ xVXW2 0Y.Ahomotopy ofloops isgiven byshifting 0totheright,tHx IZJ x V[W2t 0 Y.What ifweregar d0and1asloops inthe punctur edplane R2Q.R0 S?Clearly thehomotopydoes notwork, because it moves theloop through theforbidden point 0.(E.g.tHx IEJ 0forx JW]\1 0 Yand t J1 ^2.)Infact, however you trytomove0to1you getstuck attheorigin, so 116 10.APPLICA TIONS TOTOPOLOGY itseems intuitively clear that thereexists nohomotopy ofloops from0to1in thepunctur edplane. This isindeed thecase, asweshall seeinExample 10.13. The homotopy formula. The product M_a`0,1bisoften called thecylinder with base M.The two maps dened by0cxd,ecx,0dand1cxd<ecx,1dsend Mto thebottom, resp. thetopofthecylinder .Ahomotopy:M_`0,1b,f M_`0,1b between these maps isgiven bytheidentity mapcx,td,ecx,td.(“Slide thebottom tothetopatspeed 1.”) 04 g101 base cylinder IfMisanopen subset ofRn,ak h1-form onthecylinder canbewritten as eå IfIcx,tddxI hå JgJcx,tddtdxJ, with Irunning over multi-indices ofdegr eekh1and Jover multi-indices ofde- greek.(Her ewewrite thedtinfrontofthedx'sbecause that ismoreconvenient in what follows.) The cylinder operator turns forms onthecylinder into forms onthe base lowering thedegr eeby1, :ikj1cM_k`0,1bldmfnikcMd, bytaking thepiece of involving dtand integrating itover theunit interval,  eå J op1 0gJcx,tddtqdxJ. (Inparticular e0forany that does notinvolve dt.)Forageneral manifold Mwecanwrite akh1-form onthecylinder as ehdt ,wher eand are forms onM_`0,1b(ofdegr eekh1and krespectively) that donotinvolve dt.We then dene e(r1 0 dt. The following result will enable ustocompar epullbacks offorms under ho- motopic maps. Itcanberegar ded asanapplication ofStokes' theor em, butwe shall give adirectproof. 10.7.LEMMA(cylinder formula) .LetMbeamanifold. Thens1 ts0 ed h d forallk h1-forms onM _k`0,1 b.Inshort,  s1 t s0 ed hd. PROOF.Wewrite outtheproofforanopen subset ofRn.The proofforar- bitrary manifolds issimilar .Itsufces toconsider two cases: efdxIand egdtdxJ. 10.2. HOMOT OPY 117 Case 1.If ufdxI,then u0and d u0.Also d u¶f ¶tdtdxI vå i¶f ¶xidxidxI u¶f ¶tdtdxI vterms notinvolving dt, so d vd ud uxwy1 0¶f ¶tzx,t {dt |dxIu~}fzx,1 {€fzx,0 {‚dxI u ƒ1  ƒ0 . Case 2.If ugdtdxJ,thenƒ0 uƒ1 u0and d uå i¶g ¶xidxidtdxJ u„å i¶g ¶xidtdxidxJ, so d u„n å i …1 wy1 0¶g ¶xi zx,t{dt|dxidxJ. Also u} y1 0gzx,t {dtdxJ,so d un å i…1¶ ¶xi wy1 0gzx,t {dt|dxidxJ un å i…1 wy1 0¶g ¶xi zx,t {dt|dxidxJ. Hence d vd u0uƒ1 ƒ0 . QED Now suppose wehave apair ofmaps0and1going fromamanifold M toamanifold Nand that:M †ˆ‡0,1 ‰<Š Nisahomotopy between0and1. ForxinMwehave ‹0zx {Tuzx,0 {Tu0zx {,inother words0 u ‹0. Similarly1 u ‹1.Hence forany kv1-formonNwehaveƒ0ƒ uƒ0and ƒ1ƒ uƒ1.Applying thecylinder formula to uƒweseethat thepullbacks ƒ0andƒ1arerelated inthefollowing manner . 10.8.THEOREM(homotopy formula) .Let0,1:MŠNbesmooth maps from amanifold Mtoamanifold Nandlet:M†‡0,1‰ŒŠ Nbeahomotopy from0to1. Thenƒ1ƒ0uƒdvdƒforallkv1-formsonN.Inshort, ƒ1 ƒ0 uƒdvdƒ. Inparticular ,ifdu0wegetƒ1uƒ0vdƒ. 10.9.COROLLARY.If0,1:M ŠNarehomotopic maps between manifolds and isaclosed form onN,thenƒ0andƒ1differ byanexact form. This implies that ifthedegr eeofisequal tothedimension ofM,ƒ0and ƒ1have thesame integral. 10.10 .THEOREM.LetMand Nbemanifolds andletbeaclosed n-form onN, wher enudimM.Suppose Miscompact andoriented andhasnoboundary .Let0and 1behomotopic maps fromMtoN.Theny Mƒ0 u)y Mƒ1. 118 10.APPLICA TIONS TOTOPOLOGY PROOF.ByCorollary 10.9, 1 Ž 0 dforann Ž1-formonM.Hence byStokes' theor em  M ‘ 1 Ž 0 ’;  Md   ¶M 0, because ¶Misempty . QED ALTERNATIVEPROOF.Hereisaproofbased onStokes' theor emforthemani- fold with boundary M “”0,1 •.The boundary ofM “”0,1 •consists oftwo copies ofM,namely M “–1 —and M “–0 —,therst ofwhich iscounted with aplus sign and thesecond with aminus. Ther efore,if:M “”0,1 •€˜ Nisahomotopy between0and1, 0  M™Œš0,1› d  M™Œš0,1›d   ¶œM™Œš0,1›ž   M 1Ž  M 0. QED IfMisthecircleS1,Nthepunctur edplane R2Ž.–0 —andtheangle form‘ Žydx Ÿxdy ’¡ ‘x2Ÿy2’ofExample 3.8,then amap fromMtoNisaloop inN and theintegral ofis2times thewinding number oftheloop. Thus Theor em 10.10 gives thefollowing result. 10.11 .COROLLARY.Homotopic loops inR2Ža–0—have thesame winding number about theorigin. 10.12 .EXAMPLE.Unfolding thethreeself-intersections inthecurve pictur ed below does notaffectitswinding number . 0 0 10.13 .EXAMPLE.The two circles0and1ofExample 10.6 have winding number 1,resp. 0and thereforearenothomotopic (asloops inthepunctur ed plane). 10.3. Closed and exact forms re-examined The homotopy formula throws light onouroldproblem ofwhen aclosed form isexact, which welooked into inSection 2.3. The answer turns outtodepend on the“shape” ofthemanifold onwhich theforms aredened. Onsome manifolds allclosed forms (ofpositive degr ee)areexact, onothers thisistrueonly incertain degr ees. Failur eofexactness istypically detected byintegrating over asubmani- fold ofthecorrectdimension and nding anonzer oanswer .Inacertain sense all obstr uctions toexactness areofthis natur e.Weshall notattempt tosaythelast 10.3. CLOSED AND EXACT FORMS RE-EXAMINED 119 wordonthis problem, butstudy afew representative special cases. The matter is explor edin[Fla89 ]and atamoreadvanced level in[BT82 ]. 0-forms. Aclosed 0-form onamanifold isasmooth function fsatisfying df ¢0.This means that fisconstant (oneach connected component ofM).If this constant isnonzer o,then fisnotexact (because forms ofdegr ee £1areby denition 0).Soaclosed 0-form isnever exact (unless itis0)forarather uninter - esting reason. 1-forms and simple connectivity. Letusnow consider 1-forms onamanifold M.Theor em4.5says that theintegral ofanexact 1-form along aloop is0.With astronger assumption ontheloop thesame istrueforarbitrary closed 1-forms. A loop c:S1 ¤Misnull-homotopic ifitishomotopic toaconstant loop. The integral ofa1-form along aconstant loop is0,sofromTheor em10.10 (wher ewesettheM ofthetheor emequal toS1)wegetthefollowing. 10.14 .PROPOSITION.Letcbeanull-homotopic loop inM.Then ¥c ¢0forall closed formsonM. Amanifold issimply connected ifevery loop initisnull-homotopic. 10.15 .THEOREM.Allclosed 1-forms onasimply connected manifold areexact. PROOF.Letbeaclosed 1-form and caloop inM.Then cisnull-homotopic, so¥c¢0byProposition 10.14. The result now follows fromTheor em4.5. QED 10.16 .EXAMPLE.The punctur edplane R2£?¦0§isnotsimply connected, be- cause itpossesses anonexact closed 1-form. (See Example 4.6.) Incontrast itcanbe proved that forn¨3thespher eSn©1and punctur edn-space Rn£¦0§aresimply connected. Intuitively ,thereason isthat intwo dimensions aloop that encloses thepunctur eattheorigin cannot becrumpled uptoapoint without getting stuck atthepunctur e,wher easinhigher dimensions thereisenough room toslide any loop away fromthepunctur eand then squeeze ittoapoint. The Poincaré lemma. Onacontractible manifold allclosed forms ofpositive degr eeareexact. 10.17 .THEOREM(Poincaré lemma) .Allclosed k-forms onacontractible manifold areexact fork ¨1. PROOF.LetMbeamanifold and let:M ªk«0,1 ¬ ¤Mbeacontraction onto apoint x0inM,i.e.asmooth map satisfying ­x,0 ®m¢x0and ­x,1 ®m¢xforallx. Letbeaclosed k-form onMwith k¨1.Then¯1¢and¯0¢0,soputting ¢¯weget d ¢d ¯ ¢ ¯1 £ ¯0 £d ¯ ¢. Hereweused thehomotopy formula, Theor em10.8, and theassumption that d ¢ 0.Hence d ¢. QED The proofprovides uswith aformula forthe“antiderivative”, namely ¢  ¯,which canbemade quite explicit incertain cases. 120 10.APPLICA TIONS TOTOPOLOGY 10.18 .EXAMPLE.LetMbeRnand let °x,t ±³² txbetheradial contraction. Let ²åigidxibea1-form. Then ´²å igi °tx±d°txi ±;²å igi °tx±µ°xidt¶tdxi ±, so ²´²å ixi ·1 0gi °tx±dt. Accor ding totheproofofthePoincaré lemma, thefunctionsatises d²pro- vided that d ²0.Itisinstr uctive tocompar ewith thefunction fconstr ucted intheproofofTheor em4.5.(See Exer cise10.5.) Another typical application ofthePoincaré lemma isshowing that amanifold isnotcontractible byexhibiting aclosed form that isnotexact. Forexample, the punctur edplane R2¸º¹0 »isnotcontractible because itpossesses anonexact closed 1-form, namely theangle form. (See Example 4.6.) The angle form generalizes to ann ¸1-form onpunctur edn-space Rn¸ˆ¹0 », ²x ¼¾½dx¿x ¿n. 10.19 .THEOREM.isaclosed butnon-exact n ¸1-form onpunctur edn-space. Hence punctur edn-space isnotcontractible. PROOF.d ²0follows fromExer cise 2.1(ii) .The n ¸1-spher eM ²Sn À1has unit normal vector eld x,sobyCorollary 8.15 onMwehave ²,thevolume form. Hence ÁM ²volM  ²0.Ontheother hand, supposewas exact, ²d forann ¸1-form.Then·M²·Md²·¶M²0 byStokes' theor em, Theor em9.8.This isacontradiction, soisnotexact. Itnow follows fromthePoincaré lemma, Theor em10.17, that Rn ¸N¹0»isnotcontractible. QED Using thesame form,butrestricting ittotheunit spher eSn À1,weseethat Sn À1isnotcontractible. Buthow about forms ofdegr eenotequal ton ¸1?With- outproofwestate thefollowing fact. 10.20 .THEOREM.OnRn¸a¹0 »andonSn À1every closed form ofdegreek  ²1, n ¸1isexact. For acompact oriented hypersurface without boundary Mcontained inin Rn ¸?¹0 »theintegral 1 voln À1SnÀ1 ·Mx ¼µ½dx¿x ¿n isthewinding number ofMabout theorigin. Itgeneralizes thewinding number ofaclosed curve inR2 ¸ˆ¹0»around theorigin. Itcanbeshown that thewinding number inany dimension isalways aninteger .Itprovides ameasur eofhow many times thehypersurface wraps around theorigin. Forinstance, theproof ofTheor em10.19 shows that thewinding number ofthen ¸1-spher eabout the origin is1. 10.3. CLOSED AND EXACT FORMS RE-EXAMINED 121 Contractibility versus simple connectivity .Theor ems 10.15 and 10.17 sug- gest that thenotions ofcontractibility and simple connectivity arenotindepen- dent. 10.21 .PROPOSITION.Acontractible manifold issimply connected. PROOF.Use acontraction tocollapse any loop onto apoint. Mx0 c1 Mx0 c1 Formally ,letc1:S1 ÃMbealoop,:MÄÆÅ0,1Ç ÃMacontraction ofMonto x0. Putc Ès,t ÉKÊ Èc1 Ès É,t É.Then cisahomotopy between c1and theconstant loop c0 Èt É;Ê Èc1 Ès É,0 ÉmÊx0positioned atx0. QED Asmentioned inExample 10.16, thespher eSn Ë1and punctur edn-space Rn ÌÍ0 Îaresimply connected forn Ï3,although itfollows fromTheor em10.19 that they arenotcontractible. Thus simple connectivity isweaker than contractibility . The Poincaré conjecture. Not long after inventing thefundamental group Poincaré posed thefollowing question. LetMbeacompact three-dimensional manifold without boundary .Suppose Missimply connected. IsMhomeomor - phic tothethree-dimensional spher e?(This means: does thereexist abijective map M ÃS3which iscontinuous and hasacontinuous inverse?) This question became (inaccurately) known asthePoincaré conjectur e.Itisfamously difcult and was theforcethat drovemany ofthedevelopments intwentieth-century topology . Ithasann-dimensional analogue, called thegeneralized Poincaré conjectur e,which asks whether every compact n-dimensional manifold without boundary which is homotopy equivalent toSnishomeomorphic toSn.Wecannot heregointo this fascinating problem inany serious way,other than toreport that ithasnow been completely solved. Strangely ,thecase nÏ5ofthegeneralized Poincaré conjec- tureconjectur ewas theeasiest and was conrmed byS.Smale in1960. The case nÊ4was done byM.Freedman in1982. The case nÊ3,theoriginal version of theconjectur e,turned outtobethehardest, butwas nally conrmed byG.Perel- man in2002-03. Foradiscussion and references, seethepaper Towards thePoincaré conjectur eandtheclassication of3-manifolds byJ.Milnor ,which appear edinthe November 2003 issue oftheNotices oftheAmerican Mathematical Society and canbereadonline at ÐÒÑÓÑÕÔ<ÖØ×Ó×ÚÙÛÙÓÙ<Ü]ݵތß:ÜáàÕâÕãä×ÚåÒàÕÑçæÛèÚéêßÕ× . 122 10.APPLICA TIONS TOTOPOLOGY Exercises 10.1.Writeaformula forthemapguring intheproofofBrouwer 'sxed point theor emand provethat itissmooth. 10.2.Letx0beany point inRn.Byanalogy with theradial contraction onto theorigin, write aformula forradial contraction onto thepoint x0.Deduce that any open orclosed ballcentr edatx0iscontractible. 10.3.Asubset MofRnisstar-shaped relative toapoint x0 ëMifforallxëMthe straight line segment joining x0toxisentir elycontained inM.Show that ifMisstar- shaped relative tox0,then itiscontractible onto x0.Give anexample ofacontractible set that isnotstar-shaped. 10.4.Asubset MofRnisconvex ifforallxand yinMthestraight linesegment joining xtoyisentir elycontained inM.Provethefollowing assertions. (i)Misconvex ifand only ifitisstar-shaped relative toeach ofitspoints. Give an example ofastar-shaped setthat isnotconvex. (ii)The closed ball Bì",xíofradius"centr edatxisconvex. (iii) Same fortheopen ball B î¾ì",x í. 10.5.LetxëRnand letcxbethestraight line connecting theorigin tox.Letbea 1-form onRnand letbethefunction dened inExample 10.18. Show that ìx í€ï?ðcx. 10.6.Letbethek-form fdxI ïfdxi1dxi2ñòñòñdxikonRnand let:Rn óõô0,1 ö÷ Rn betheradial contractionìx,tí€ïtx.Verify that øïk å mù1 ìáú1ímû1üêý1 0fìtxítkþ1dtÿximdxi1dxi2 ñòñòñ
dxim ñòñ¡ñdxik, and check directly that dø dø ïfork 1. 10.7.Let ïfdxdy gdzdx hdydzbea2-form onR3and let ìx,y,z,t í ï tìx,y,zíbetheradial contraction ofR3onto theorigin. Verify that øï ü ý1 0fìtx,ty,tzítdtÿ"ìxdyúydxí ü ý1 0gìtx,ty,tzítdtÿ ìzdxúxdzíüÒý1 0h ìtx,ty,tz ítdt ÿ"ìydz úzdy í. 10.8.LetïåIfIdxIbeaclosed k-form whose coefcients fIaresmooth functions dened onRnú0 that areallhomogeneous ofthesame degr eep  ïˆú k.Let  ï1 p kå Ik å lù1 ì]ú1 íl û1xilfIdxi1dxi2 ñ‚ñòñdxil ñ¡ñòñdxik. Show that d ï.(Use d ï0and apply theidentity proved inExer ciseB.5toeach fI;see also Exer cise2.7.) 10.9.LetMand Nbemanifolds and0,1:M ÷Nhomotopic maps. Show thatðcø0ïðcø1forallclosed k-chains cinMand allclosed k-formsonN. 10.10 .Prove that any two maps0and1fromMtoNarehomotopic ifMorNis contractible. (First show that every map M÷Nishomotopic toaconstant mapìxí ï y0.) 10.11 .Letx0 ï ì2,0íand letMbethetwice-punctur edplane R2ú 0,x0 .Letc1,c2, c3: ô0,2ö÷ Mbetheloops dened byc1 ìtí ï ìcost,sintí,c2 ìtí ï)ì2 cost,sintíand c3 ìt í ï ì1 2cost,2sint í.Show that c1,c2and c3arenothomotopic. (Constr ucta1-form onMsuch that theintegrals ðc1, ðc2and ðc3aredistinct.) EXERCISES 123 10.12 .Afunction g:R Ris2-periodic ifg x 2 g x forallx. (i)Letg:R Rbeasmooth 2-periodic function and let gdt,wher etis thecoor dinate onR.Provethat thereisaunique number ksuch that kdt  dhforsome smooth 2-periodic function h.(Tond k,integrate theequation kdtdhover0,2.Then check that thisvalue ofkworks.) (ii)Letbeany 1-form ontheunit circleS1and letbetheelement ofarclength ofS1.(Youcanthink ofastherestriction toS1oftheangle form.) Provethat thereisaunique number ksuch that kisexact. (Use theparametrization ct  cost,sint and apply theresult ofpart (i).) APPENDIX A Sets and functions A.1. Glossary Westart with alistofset-theor etical notations that arefrequently used inthe text. LetXand Ybesets. xX:xisanelement ofX.a,b,c :thesetcontaining theelements a,band c. X Y:Xisasubset ofY,i.e.every element ofXisanelement ofY. X Y:theintersection ofXand Y.This isdened asthesetofallxsuch that x Xandx Y. X Y:theunion ofXand Y.This isdened asthesetofallxsuch that x Xorx Y. X Y:thecomplement ofYinX.This isdened asthesetofxinXsuch that xisnotinY. XY:theCartesian product ofXand Y.This isbydenition thesetof allorderedpairsx,ywith xXand yY.Examples: RRis theEuclidean plane, usually written R2;S1 0,1 !isacylinder wall of height 1;S1S1isatorus. R2 S1 "$#0,1 % S1 "S1xX&Px:thesetofallxXwhich have theproperty Px.Exam- ples:x R &1 'x (3 istheinterval 1,3 ,x&xXand xYistheintersection XY,x &x Xorx Y istheunion X Y,x X &x ) Y isthecomplement X Y. f:X *Y:fisafunction (also called amap) fromXtoY.This means that fassigns toeach x Xaunique element f x +Y.The setXiscalled thedomain orsourceoff,and Yiscalled thecodomain ortargetoff. 125 126 A.SETS AND FUNCTIONS f ,A -:theimage ofaAunder themap f.IfAisasubset ofX,then itsimage under fisbydenition theset f,A-/.10 y2Y3y.f,x-forsome x2A4. f51,B-:thepreimage ofBunder themap f.IfBisasubset ofY,this isby denition theset f 51,B -6.70 x 2X 3f ,x -82B 4. (This isasomewhat confusing notation. Itisnotmeant toimply that fis requir edtohave aninverse.) f51,c-:anabbr eviation forf51,0c4
-,i.e.theset0x2X3f,x-9. c4.This isoften called thebreorlevel setoffatc. f3A:therestriction offtoA.IfAisasubset ofX,f3Aisthefunction dened by,f3A-:,x-;. < f ,x - ifx 2A, notdened ifx = 2A. Inother words,f 3Aisequal tofonA,but“forgets” thevalues offat points outside A. g >f:thecomposition offand g.Iff:X ?Yand g:Y ?Zarefunctions, then g >f:X ?Zisdened by ,g >f ,x -@. g ,f ,x -A-.Weoften saythat thefunction g >fisobtained by“substituting y .f ,x -into g ,y -”. Afunction f:X ?Yisinjective orone-to-one ifx1 = .x2implies f ,x1 -B= .f ,x2 -. (Equivalently ,fisinjective iff ,x1 -8. f ,x2 -implies x1 .x2.)Itiscalled surjective oronto iff ,X - . Y,i.e.ify 2Ythen y .f ,x -forsome x 2X.Itiscalled bijective ifitisboth injective and surjective. The function fisbijective ifand only ifithasatwo-sided inverse f51:Y ?Xsatisfying f51,f ,x -A-B. xforallx 2X and f ,f51,y -A-6. yforally 2Y. IfXisanite setand f:X ?Rareal-valued function, thesum ofallthe numbers f ,x -,wher exranges through X,isdenoted byåxCXf ,x -.The setXis called theindex setforthesum. This notation isoften abbr eviated orabused in various ways. Forinstance, ifXisthecollection 01,2,...,n 4,oneuses thefamiliar notation ån i D1f,i-.Inthese notes wewill often deal with indices which arepairs ork-tuples ofintegers, also known asmulti-indices .Asasimple example, letnbe axed nonnegative integer ,letXbethesetofallpairs ofintegers,i,j-satisfying 0EiEjEn,and letf,i,j-6.iFj.Forn.3wecandisplay Xand finatableau asfollows. ij 0123 234 45 6 EXERCISES 127 The sum åxGXf Hx Iofallthese numbers iswritten as å 0JiJjJn HiKjI. Youwill beasked toevaluate itexplicitly inExer cise A.2. A.2. General topology ofEuclidean space Letxbeapoint inEuclidean space Rn.The open ballofradius"about apoint xisthecollection ofallpoints ywhose distance toxislessthan", B LMH",x I6NPO y QRn RTSy Ux SWV" X. x" Asubset OofRnisopen ifforevery x QOthereexists an" Y0such that BL H",x I iscontained inO.Intuitively thismeans that atevery point inOthereisalittle bit ofroom inside Otomove around inany direction you like. Anopen neighbour hood ofxisany open setcontaining x. Asubset CofRnisclosed ifitscomplement RnUCisopen. This denition isequivalent tothefollowing: Cisclosed ifand only ifforevery sequence of points x1,x2,...,xn,...that conver gestoapoint xinRn,thelimit xiscontained inC.Loosely speaking, closed means “closed under taking limits”. Anexample ofaclosed setistheclosed ballofradius"about apoint x,which isdened asthe collection ofallpoints ywhose distance toxislessthan orequal to", B H",x I/N7O y QRnRTSy Ux SBZ" X. x" Closed isnottheopposite ofopen! Ther eexist lots ofsubsets ofRnthat are neither open norclosed, forexample theinterval [0,1 IinR.(On theother hand, therearenotsomany subsets that areboth open and closed, namely justtheempty setand Rnitself.) Asubset AofRnisbounded ifthereexists some R Y0such that Sx S ZR forallxinA.(That is,Aiscontained intheball BHR,0Iforsome value ofR.)A compact subset ofRnisonethat isboth closed and bounded. The importance ofthe notion ofcompactness, asfarasthese notes areconcerned, isthat theintegral of acontinuous function over acompact subset ofRnisalways awell-dened, nite number . Exercises A.1.Parts (iii)and (iv)ofthis problem requir etheuseofanatlas (ortheWeb;seee.g.\^]_]a`bdc_cAegfa]ghjiAegf_kjfj]^kjf:lnmpo^iaq).LetXbethesurface oftheearth, letYbetherealline and let 128 A.SETS AND FUNCTIONS f:X rYbethefunction which assigns toeach x sXitsgeographical latitude measur ed indegr ees. (i)Find f tX u. (ii)Find f v1t0 u,f v1t90 u,f v1txw90 u. (iii) LetAbethecontiguous United States. Find ftAu.Round thenumbers towhole degr ees. (iv) LetB yf tA u,wher eAisasinpart (iii).Find (a)acountry other than Athat iscontained infv1tBu;(b)acountry that intersects fv1tBubutisnotcontained infv1tBu;and (c)acountry inthenorthern hemispher ethat does notintersect f v1tB u. A.2.LetStnuzyå0{i{j{n ti|ju.Provethefollowing assertions. (i)S t0 uy0and S tn |1 uyS tn u}|3 2 tn |1 uAtn |2 u. (ii)S tn uy1 2n tn |1 u~tn |2 u.(Use induction onn.) A.3.Prove that theopen ball B:t",xuisopen. (This isnotatautology! State your reasons asprecisely asyou can, using thedenition ofopenness stated inthetext. Youwill need thetriangle inequality €y wx €‚ƒ€ y wz €„| € z wx €.) A.4.Provethat theclosed ballisBt",xuisclosed. (Same comments asforExer ciseA.3.) A.5.Show that thetwo denitions ofclosedness given inthetext areequivalent. A.6.Complete thefollowing table. HereSnv1denotes theunit spher eabout theorigin inRn,that isthesetofvectors oflength 1. closed? bounded? compact?…w3,5 † yes yes yes…w3,5u…w3,‡ utxw3, ‡ u B t",x u B_t",xu Snv1 xy-plane inR3 unit cube …0,1†n APPENDIX B Calculus review This appendix isabrief review ofsome single- and multi-variable calculus needed inthestudy ofmanifolds. Refer ences forthismaterial are[Edw94 ],[HH02 ] and [MT03 ]. B.1. The fundamental theorem ofcalculus Suppose that Fisadifferentiable function ofasingle variable xand that the derivative f ˆF ‰iscontinuous. Let Ša,b ‹beaninterval contained inthedomain ofF.The fundamental theor emofcalculus says thatŒb aftŽdtˆFbŽFaŽ. (B.1) Ther earetwo useful alternative ways ofwriting this theor em. Replacing bwith x and differentiating with respect toxwend d dx Œx aftŽdtˆfxŽ. (B.2) Writing ginstead ofFand g‰instead offand adding gaŽtoboth sides informula (B.1) weget gxŽ6ˆgaŽz‘ Œx ag ‰tŽdt. (B.3) Formulas (B.1) –(B.3) areequivalent, butthey emphasize differentaspects ofthe fundamental theor emofcalculus. Formula (B.1) isaformula foradenite integral: ittells you how tond the(signed) surface areabetween thegraph ofthefunction fand thex-axis. Formula (B.2) says that theintegral ofacontinuous function is adifferentiable function oftheupper limit; and thederivative istheintegrand. Formula (B.3) isan“integral formula”, which expr esses thefunction ginterms of thevalue gaŽand thederivative g‰.(See Exer ciseB.1foranapplication.) B.2. Derivatives Let1,2,...,mbefunctions ofnvariables x1,x2,...,xn.Asusual wewrite x ˆ“’ ”””•x1 x2 ... xn –˜———™, x Ž/ˆ“’ ”””•1 xŽ 2 xŽ ... m xŽ –˜———™, and view x Žasasingle map fromRntoRm.(Incalculus theword“map” isoften used forvector -valued functions, while theword“function” isgenerally reserved 129 130 B.CALCULUS REVIEW forreal-valued functions.) Wesaythatiscontinuously differ entiable ifthepartial derivatives ¶i ¶xj šx›‚œlim h0išx žhej ›Ÿišx › h(B.4) arewell-dened and continuous functions ofxforalli œ1,2,...,nand j œ1, 2,...,m.Here e1 œ  ¡¡¡¡¡¡¡¢1 0 0 ... 0 0 £˜¤¤¤¤¤¤¤¥,e2 œ  ¡¡¡¡¡¡¡¢0 1 0 ... 0 0 £˜¤¤¤¤¤¤¤¥,...,en œ  ¡¡¡¡¡¡¡¢0 0 0 ... 0 1 £˜¤¤¤¤¤¤¤¥ arethestandar dbasis vectors ofRn.The (total) derivative orJacobi matrix ofatx isthen them ¦n-matrix Dšx ›‚œ  ¡¡¢¶1 ¶x1šx›...¶1 ¶xnšx› ...... ¶m ¶x1šx›...¶m ¶xnšx› £˜¤¤¥. Ifvisany vector inRn,thedirectional derivative ofalong visdened tobethe vector Dšx›vinRm,obtained bymultiplying thematrix Dšx›bythevector v. Fornœ1isavector -valued function ofone variable x,often called apath or(parametrized) curve inRm.Inthis case thematrix Dšx›consists ofasingle column vector ,called thevelocity vector ,and isusually denoted simply by §šx ›. Form œ1isascalar -valued function ofnvariables and Dšx ›isasingle rowvector .The transpose matrix ofDšx ›isthereforeacolumn vector ,usually called thegradient of: Dšx ›Tœgradšx ›. The directional derivative ofalong vcanthen bewritten asaninner product, Dšx ›v œgradšx ›;¨v.Ther eisanimportant characterization ofthegradient, which isbased ontheidentity a ¨b œª©a ©‚©b ©cos.Here0 « «istheangle subtended byaand b.Ifvisaunit vector (©v©9œ1),then Dšx ›v œgradšx ›¬¨v œ­©gradšx ›^©cos, wher eistheangle between gradšx ›and v.SoDšx ›vtakes onitsmaximal value ifcos œ1,i.e. œ0.This means that vpoints inthesame direction as gradšx ›.Thus thedirection ofthevector gradšx ›isthedirection ofsteepest ascent ,i.e.inwhichincreases fastest, and themagnitude ofgradšx ›isequal tothedirectional derivative Dšx ›v,wher evistheunit vector pointing along gradšx›. Frequently afunction isnotdened onallofRn,butonly onasubset U.We must bealittle careful indening thederivative ofsuch afunction. Letusassume that Uisanopen set. Let:U®Rmbeafunction dened onUand letx¯U. Because Uisopen, thereexists"°0such that thepoints xžtejarecontained inUfor Ÿ" ±t ±".Ther eforeišx žtej ›iswell-dened for Ÿ" ±t ±"and thus itmakes sense toaskwhether thepartial derivatives (B.4) exist. Ifthey do, forallx ¯Uand alliand j,and ifthey arecontinuous, thefunctioniscalled continuously differ entiable orC1. B.3. THE CHAIN RULE 131 Ifthesecond partial derivatives ¶2i ¶xj¶xk ²x³ exist and arecontinuous forallx ´Uand foralli µ1,2,...,nand j,k µ1,2,..., m,theniscalled twice continuously differ entiable orC2.Likewise, ifallr-fold partial derivatives ¶ri ¶xj1¶xj2 ¶:¶:¶¶xjr ²x ³ exist and arecontinuous, thenisrtimes continuously differ entiable orCr.IfisCr forallr ·1,then wesaythatisinnitely many times differ entiable ,C ¸,orsmooth . This means thatcanbedifferentiated arbitrarily many times with respect toany ofthevariables. Letusnow review some ofthemost important facts concerning derivatives. B.3. The chain rule Recall that ifA,Band Caresets and:A¹Band :B¹Carefunctions, wecanapply aftertoobtain thecomposite function² º³²x³/µ ²²x³˜³. B.1.THEOREM(chain rule).LetU»RnandV»Rmbeopen andlet:U¹V and :V ¹RkbeCr.Then ºisCrand D² º³²x³/µD ²²x³A³D²x³ forallx ´U. HereD ²²x ³A³D²x ³denotes thecomposition ortheproduct ofthetwo ma- trices D ²²x ³˜³and D²x ³. B.2.EXAMPLE.Intheone-variable case n µm µk µ1thederivatives Dand D are1 ¼1-matrices² ½²x ³A³and² ½²y ³A³,and matrix multiplication isordinary multiplication, sowegettheusual chain rule² º³ ½²x³/µ ½²²x³A³ ½²x³. B.3.EXAMPLE.Ifnµkµ1,then ºisareal-valued function ofone vari- able x,soD² º³isa1¼1-matrix containing thesingle entry² º³ ½.Mor eover , D²x³;µ¿¾ ÀÁd1 dx²x³ ... dm dx²x³ ˜ÃÄ and D ²y³‚µÆŶ ¶y1²y ³...¶ ¶ym²y ³Ç, sobythechain rule d² º ³ dx²x ³/µD ²²x ³A³D²x ³µm å i È1¶ ¶yi²²x ³A³di dx²x ³. (B.5) This isperhaps themost important special case ofthechain rule. Sometimes we aresloppy and abbr eviate this identity to d² º ³ dx µm å i È1¶ ¶yidi dx. 132 B.CALCULUS REVIEW Aneven sloppier ,butnevertheless quite common, notation is d dx Ém å i Ê1¶ ¶yidi dx. Inthese notes wefrequently usetheso-called “pullback” notation. Instead of Ë weoften write Ì ,sothat Ì Íx Îstands for Í Íx ÎAÎ.Similarly , Ì
Ͷ ϶yi Î_Íx Î stands for¶ ϶yi Í Íx ÎAÎ.Inthis notation wehave d Ì dxÉm å i Ê1 ̬ж ¶yi Ñdi dx. (B.6) B.4. The implicit function theorem Let:WÒRmbeacontinuously differentiable function dened onanopen subset WofRnÓm.Letusthink ofavector inRnÓmasanorderedpair ofvectorsÍu,vÎwith uÔRnand vÔRm.Consider theequation  Íu,v ÎÉ0. Under what circumstances isitpossible tosolve forvasafunction ofu?The answer isgiven bytheimplicit function theor em. Weform theJacobi matrices of with respect totheu-and v-variables separately , DuÉ¿Õ ÖÖ׶1 ¶u1...¶1 ¶un...... ¶m ¶u1...¶m ¶un ؘÙÙÚ, DvÉ“Õ ÖÖ׶1 ¶v1...¶1 ¶vm...... ¶m ¶v1...¶m ¶vm ؘÙÙÚ. Observe that thematrix Dvissquar e.Weareinbusiness ifwehave apointÍu0,v0 Îatwhichis0and Dvisinvertible. B.4.THEOREM(implicit function theor em).Let:W ÒRmbeCr,wher eW isopen inRn Óm.Suppose thatÍu0,v0 ÎÛÔ Wisapoint such thatÍu0,v0 ÎÉ0and DvÍu0,v0 Îisinvertible. Then thereareopen neighbour hoods UÜRnofu0and V ÜRmofv0such that foreach u ÔUthereexists aunique vÉf Íu Î Ô Vsatis- fying Íu,f Íu ÎAÎÉ0.Thefunction f:U ÒVisCrwith derivative given byimplicit differ entiation: Df Íu ÎÉ7ÝDv Íu,v ÎAÞ1Du Íu,v Îàßßv Êf áu â forallu ÔU. This iswell-known formÉnÉ1,whenisafunction oftwo realvariablesÍu,vÎ.If¶϶vãÉ0atacertain pointÍu0,v0 Î,then foruclose tou0and vclose to v0wecansolve theequationÍu,vÎÉ0forvasafunction vÉfÍuÎofu,and fäÉPݶ ϶u ¶ ϶v. Now letustaketobeoftheformÍu,vÎÉgÍvÎÝu,wher eg:WÒRnisa given function with Wopen inRn.Solving Íu,v ÎÉ0hereamounts toinverting thefunction g.Mor eover ,DvÉDg,sotheimplicit function theor emyields the following result. B.5. THE SUBSTITUTION FORMULA FOR INTEGRALS 133 B.5.THEOREM(inverse function theor em).Letg:W åRnbecontinuously differ entiable, wher eWisopen inRn.Suppose that v0 æWisapoint such that Dg çv0 è isinvertible. Then thereisanopen neighbour hood U éRnofv0such that g çUèisan open neighbour hood ofg çv0 èandthefunction g:U åg çUèisinvertible. Theinverse g ê1:V åUiscontinuously differ entiable with derivative given by Dg ê1çuè‚ëDg çvè ê1ììv íg î1ïu ð forallvæV. Again letusspell outtheone-variable case në1.Invertibility ofDg çv0 è simply means that g ñòçv0 èôó ë0.This implies that near v0thefunction gisstrictly monotone increasing (ifg ñdçv0 èöõ0)ordecr easing (ifg ñdçv0 èö÷0).Ther eforeifIis asufciently small open interval around u0,then g çIèisanopen interval around g çu0 èand therestricted function g:I åg çIèisinvertible. The inverse function hasderivativeçg ê1è ñçuè/ë1 gñ çvè, with vëgê1çuè. B.6.EXAMPLE(squar eroots) .Letg çvè‚ëv2.Then g ñçv0 èøó ë0whenever v0 ó ë 0.Forv0 õ0wecantake Ië ç0, ùè.Then g çIèúë ç0, ùè,g ê1çuèûëýü u,andçg ê1è ñòçuè9ë1 þnç2üuè.Forv0 ÷0wecantake Ië ç~ÿWù ,0è.Then g çIè ë ç0, ùè, g ê1çuè ë ÿüu,and çg ê1è ñ çuè ë ÿ1 þnç2üuè.Inaneighbour hood of0itisnot possible toinvert g. B.5. The substitution formula forintegrals LetVbeanopen subset ofRnand letf:V åRbeafunction. Suppose we want tochange thevariables intheintegral
Vf çyèdy.(This isshorthand foran n-fold integral over y1,y2,...,yn.)This means wesubstitute yëp çxè,wher e p:UåVisamap fromanopen UéRntoV.Under asuitable hypothesis we canchange theintegral over ytoanintegral over x. B.7.THEOREM(change ofvariables formula) .LetUandVbeopen subsets ofRn andletp:U åVbeamap. Suppose that pisbijective andthat panditsinverse are continuously differ entiable. Then foranyintegrable function fwehave Vfçyèdyë  UfçpçxèAèdetDpçxèdx. Again thisshould look familiar fromone-variable calculus: ifp:ça,bè å çc,dè isC1and hasaC1inverse, thend cf çyèdyë
b afçpçxèAèpñ çxèdxifpisincreasing,ÿ
b afçpçxèAèpñ çxèdxifpisdecr easing . This canbewritten succinctly as
d cf çyèdyë
b af çp çxè˜èp ñ çxèdx,which looks moresimilar tothemultidimensional case. 134 B.CALCULUS REVIEW Exercises B.1.Letg: a,b  RbeaCn 1-function, wher en 0.Suppose a x band put hxa. (i)Bychanging variables inthefundamental theor emofcalculus (B.3) show that g x g a h 1 0g a th dt. (ii)Show that g x g a h t 1 g a th 1 0 h21 0 1 t g  a th dtgahgah21 0 1tg athdt. (Integrate theformula inpart (i)byparts and don't forgettousethechain rule.) (iii) Byinduction onndeduce frompart (ii)that gxn å k 0gk a k!hkhn 1 n! 1 0 1tng n1 athdt. This isTaylor 'sformula with integral remainder term . B.2.Letxand vbeconstant vectors inRn.Dene c t x tv.Find c t . B.3.Deduce fromthechain rulethat Dxvlim t !0xtv"x t. B.4.Accor ding toNewton's law ofgravitation, aparticle ofmass m1placed atthe origin inR3exerts aforceonaparticle ofmass m2placed atx #R3%$0 &equal to F 'Gm1m2(x (3x, wher eGisaconstant ofnatur e.Show that Fisthegradient offxGm1m2 ) (x (. B.5.Afunction f:Rn*$0 &+ Rishomogeneous ofdegr eepiff tx , tpf x forall x #Rn-$0 &and t .0.Herepisarealconstant. (i)Show that thefunctions fx,y/0 x2xy) x2y2,fx,y/01 x3y3, f x,y,z 2 x2z63x4y2z243652arehomogeneous. What aretheir degr ees? (ii)Assume that fisdened at0and continuous everywher e.Show that p 0. Show that fisconstant ifp0. (iii) Show that iffishomogeneous ofdegr eepand smooth, then n å i1xi¶f ¶xi x pf x . (Differentiate therelation ftxtpfxwith respect tot.) B.6.Dene afunction f:R Rbyf 0 0and f x e317x2forx 8 0. (i)Show that fisdifferentiable at0and that f 0 0. (ii)Show that fissmooth and that fn 0 0foralln. (iii) Plot thefunction fover theinterval5x5.Using softwar eoragraphing calculator isne, butpay special attention tothebehaviour near x0. B.7.Dene amap fromRn31toRnby t 1(t (2192t : (t (21 en ;. (i)Show that t liesontheunit spher eSn31about theorigin. EXERCISES 135 (ii)Show that <t =istheintersection point ofthespher eand theline through the points enand t.(Her eweregar dt >?<t1,t2,...,tn @1 =asapoint inRnbyidenti- fying itwith<t1,t2,...,tn@1,0=.) (iii) Compute D <t =. (iv) LetXbethespher epunctur edatthe“north pole”, X >Sn @1 ACBenD.Stereo- graphic projection fromthenorth pole isthemap:XERn @1given by<x=F><xn A1 = @1<x1,x2,...,xn@1 =.Show thatisatwo-sided inverse of . (v)Draw diagrams illustrating themapsand forn >2and n >3. (vi) Now letybeany point onthespher eand letPthehyperplane which passes through theorigin and isperpendicular toy.The stereographic projection fromy ofany point xinthespher edistinct fromyisdened astheunique intersec- tion point oftheline joining ytoxand thehyperplane P.This denes amap :Sn @1ACByD EP.The point yiscalled thecentr eoftheprojection. Writea formula forthestereographic projectionfromthesouth pole Aenand forits inverse :Rn@1ESn @1. B.8.Amap:RnERmiscalled even if< Ax=G><x=forallxinRn.Find D<0=if iseven and C1. B.9.Leta0,a1,a2,...,anbevectors inRn.Alinear combination ån i H0ciaiisconvex if ån iH0ci >1.The simplexIspanned bytheai'sisthecollection ofalltheir convex linear combinations,IC>KJn å iH0ciaiLLLLn å iH0ci >1 M. The standard simplex inRnisthesimplex spanned bythevectors 0,e1,e2,...,en. (i)Forn>1,2,3draw pictur esofthestandar dn-simplex aswell asanonstandar d n-simplex. (ii)The volume ofaregion RinRnisdened as N Rdx1dx2 OPOPOdxn.Show that vol IC>1 n! QdetAQ, wher eAisthen Rn-matrix with columns a1 Aa0,a2 Aa0,...,an Aa0.(First compute thevolume ofthestandar dsimplex byrepeated integration. Then mapItothestandar dsimplex byanappr opriate substitution and apply thesubsti- tution formula forintegrals.) The following two calculus problems arenot review problems, but theresults are needed inChapter 9. B.10 .ForxS0deneT<x =>NVU 0e @ttx @1dt and provethefollowing assertions. (i) T<x W1 =>x T<x =forallx S0. (ii) T<n =>X< n A1 =!forpositive integers n. (iii)NVU 0e @u2uadu>1 2 TZY aW1 2 [. B.11 .Calculate T<n W1 2 =(wher e T isthefunction dened inExer ciseB.10) byestablish- ingthefollowing identities. Forbrevity write > T<1 2 =. (i) >NU@Ue @s2ds. (ii) 2>NU@U NU@Ue @x2@y2dxdy. 136 B.CALCULUS REVIEW (iii) 2\^]2 0 ]`_ 0rear2drd. (iv) \^b. (v) cedn f1 2 g \1 h3 h5 hPhPhji2n k1 l 2n bforn m1. Bibliography [Bac06] D.Bachman, Ageometric approach todiffer ential forms ,Birkhäuser ,Boston, MA, 2006. Arecent text, aversion ofwhich isavailable through the author 'swebsitenPojoqprsjstpPuPvxw4vqyz{p}|~ojuPq€zqyq‚ƒsP„tyq…ƒvxw†n4‡v4ˆ‰s. [BS91] P.Bamber gand S.Sternber g,Acourse inmathematics forstudents ofphysics ,Cambridge Uni- versity Press,Cambridge, 1991. [BT82] R.Bott and L.Tu,Differ ential forms inalgebraic topology ,Springer -Verlag, New York, 1982. Masterly exposition atbeginning graduate level oftheuses ofdifferential forms intopology and deRham cohomology . [Bre91] D.Bressoud, Second year calculus ,Under graduate Texts inMathematics, Springer -Verlag, New York, 1991. Multivariable calculus fromthepoint ofview ofNewton's law and special relativity with coverage ofdifferential forms. [Bre05] O.Bretscher ,Linear algebra with applications ,thirded., Pearson Prentice Hall, Upper Saddle River ,New Jersey ,2005. Good reference forthebasic linear algebra requir edinthese notes. [Car94] M.doCarmo, Differ ential forms andapplications ,Springer -Verlag, Berlin, 1994, translated from the1971 Portuguese original. Slightly moreadvanced than these notes, with some coverage ofRiemannian geometry ,in- cluding theGauß-Bonnet theor em. [Dar94] R.Darling, Differ ential forms andconnections ,Cambridge University Press,Cambridge, 1994. Advanced under graduate text onmanifolds and differential forms, including expositions of Maxwell theory and itsmodern generalization, gauge theory . [Edw94] H.Edwar ds,Advanced calculus: Adiffer ential forms approach,Birkhäuser Boston, Boston, Mas- sachusetts, 1994, corrected reprint ofthe1969 original. One oftheearliest under graduate textbooks covering differential forms. Still recommended asanalternative orsupplementary sour ce. [Fla89] H.Flanders, Differ ential forms with applications tothephysical sciences ,second ed.,Dover Publi- cations, New York, 1989. Written for1960s engineering graduate students. Concise butlucid (and cheap!). [Gra98] A.Gray ,Modern differ ential geometry ofcurves andsurfaces with Mathematica ,second ed., CRC Press,Boca Raton, FL,1998. Leisur elyand thorough exposition atintermediate under graduate level with plenty ofcom- puter graphics. [GP74] V.Guillemin and A.Pollack, Differ ential topology ,Prentice-Hall, Englewood Cliffs,N.J., 1974. Written forbeginning graduate students, butvery intuitive and with lotsofinter esting appli- cations totopology . [HH02] J.Hubbar dand B.Hubbar d,Vector calculus, linear algebra, anddiffer ential forms: Aunied ap- proach,second ed.,Prentice Hall, Upper Saddle River ,New Jersey ,2002. [MT03] J.Marsden and A.Tromba, Vector calculus ,fth ed.,W.H.Freeman, New York, 2003. Standar dmultivariable calculus reference. [Opr03] J.Oprea,Differ ential geometry anditsapplications ,second ed.,Prentice Hall, 2003. Curves, surfaces, geodesics and calculus ofvariations with plenty ofMAPLE programming. Accessible tointermediate under graduates. [Sin01] S.Singer ,Symmetry inmechanics. Agentle, modern introduction ,Birkhäuser ,Boston, MA, 2001. [Spi65] M.Spivak, Calculus onmanifolds. Amodern approach toclassical theor emsofadvanced calculus ,W. A.Benjamin, New York-Amster dam, 1965. Efcient and rigor oustreatment ofmany ofthetopics inthese notes. 137 138 BIBLIOGRAPHY [Spi99] ,Acompr ehensive introduction todiffer ential geometry ,thirded.,Publish orPerish, Hous- ton, TX,1999. Differential geometry textbook atadvanced under graduate level inve massive butfunto readvolumes. [Wei97] S.Weintraub, Differ ential forms. Acomplement tovector calculus ,Academic Press, San Diego, CA, 1997. Written asacompanion tomultivariable calculus texts. Contains careful and intuitive expla- nations ofseveral oftheideas cover edinthese notes, aswell asanumber ofstraightforwar d exercises. The Greek alphabet upper case lower case name A  alpha B betaŠ gamma‹ delta E," epsilon Z  zeta H  etaŒ,# theta I iota K  kappa lambda M  mu N  nuŽ xi O o omicr on,$ pi P  rho sigma T  tau‘ upsilon’,' phi X  chi“ psi”! omega 139 Notation Index•,Hodge star operator ,24 relativistic, 29–0,1—k,unit cube inRk,58– ˜—,orientation dened byabasis ˜,94™,Euclidean inner product (dot product), 8š,composition ofmaps, 37,126, 131›,integral ofaform over achain, 57 over amanifold, 106›,integral ofaform over achain, 47œVœ,Euclidean norm (length), 8,tensor multiplication, 89 ¶ ¶xi,partial derivative, 130ž,orthogonal complement, 74Ÿ,exterior multiplication, 17,85 AT,transpose ofamatrix A,35 AkV,setofalternating k-multilinear functions onV,85 A,permutation matrix, 43 AI,J,I,J-submatrix ofA,42•,Hodge star of,24 relativistic, 29› M,integral ofover amanifold M,106› c,integral ofover achain c,47,57 Alt,alternating form associated to,90 B ",x¡,closed ball inRn,127 B ¢4 ",x ¡,open ball inRn,127– ˜—,orientation dened byabasis ˜,94 Cr,rtimes continuously differentiable, 131 curl,curl ofavector eld, 27 D,Jacobi matrix of,130 ¶,boundary ofachain, 59 ofamanifold, 103 d,exterior derivative, 20£,Laplacian ofafunction, 28 I,J,Kronecker delta, 86 i,j,Kronecker delta, 51 detA,determinant ofamatrix A,31 div,diver gence ofavector eld, 26 •dx,innitesimal hypersurface, 26 dx,innitesimal displacement, 25 dxI,short fordxi1dxi2 ™~™†™dxik,17 dxi,covector (“innitesimal increment”), 17, 83¤ dxi,omit dxi,18 ei,i-thstandar dbasis vector ofRn,130 f ¥A,restriction offtoA,126 g šf,composition offand g,37,126, 131¦,Gamma function, 110, 135 grad,gradient ofafunction, 26 graph,graph ofafunction, 68 Hn,upper halfspace inRn,103 I,multi-index i1,i2,...,ik ¡(usually increas- ing), 17 intM,interior ofamanifold with boundary , 103 kerA,kernel (nullspace) ofamatrix A,73 l   ¡,length ofapermutation ,34 M,volume form ofamanifold M,96§n k ¨,binomial coefcient, 19,23 n,unit normal vector eld, 95 nullityA,dimension ofthekernel ofA,73 O  n ¡,orthogonal group, 76©k M¡,vector space ofk-forms onM,19,82  ª,pullback ofaform, 37,88 ofafunction, 37,132 Rn,Euclidean n-space, 1 rankA,dimension ofthecolumn space ofA, 73 Sn,unit spher eabout theorigin inRn «1,8,64 Sn,permutation group, 33 sign   ¡,sign ofapermutation ,34 141 142 NOT ATION INDEX SL ¬n ­,special linear group, 78 TxM,tangent space toMatx,8,69,73,74 V®,dual ofavector space V,83 Vk,k-fold Cartesian product ofavector space V,85 voln,n-dimensional Euclidean volume, 91¯x ¯,Euclidean norm (length) ofavector x,8 x °y,Euclidean inner product (dot product) of vectors xand y,8 xT,transpose ofavector x,1 Index Page numbers inboldface refer todenitions ortheor ems; italic page numbers refer toexamples orapplications. Inafew cases italic boldface isinorder. afne space, 7,8,73 alternating algebra, 82 multilinear function, 32,85,87–90 property ,18,19,21 Ampèr e,André Marie (1775–1836), 29 Ampèr e'sLaw,29 angle form, 23,36,47,51,64,118, 120, 123 function along acurve, 52–53 anticommutativity ,18 antisymmetric matrix, 78 multilinear function, seealternating multi- linear function arclength, 89,96,101, 107 Archimedes ofSyracuse (287–212 BC), 3,109 Archimedes' Law,109 atlas, 67,69,82 average ofafunction, 107 ball, seeclosed ball, open ball barycentr e,107 bilinear ,85,89 block, 31,42,91,93,96 rectangular ,seerectangular block Bonnet, Pierr e(1819–1892), 137 boundary ofachain, 60,62–65 ofamanifold, 2–7,103,104–111 ,118 bounded, 47,106, 127 Brouwer ,Luitzen Egbertus Jan(1881–1966), 113, 122 Brouwer 'sxed point theor em,113, 122 Cartan, Elie (1869–1951), 17 Cartesian product, 10,85,116, 125 Cartesius, Renatus, seeDescartes, René centr oid, 107 chain, 59,60–65 chart, 67,69,72circle,9,47,48,51–53, 55,62,64,72,115, 118, 123 closed ball, 8,105, 106, 110, 115, 122, 127 chain, 62,64,122 curve, 1,50,53,62 form, 22,27–29 ,40,51,54,55,117–123 set,4,36,47,84,103, 106, 107, 127,128 codimension, 69,73,74,105 cohomology ,137 column operation, 32,42 vector ,1,31,45,69,83,89,92,130 compact, 57,106–110, 117, 120, 127 complementary ,24 conguration space, 9,14 connected component, 12,119 conservative, 49,108, 109 constant form, 19,28,83 continuously differentiable, 130 contractible, 115,119–122 contraction, 115,120,121–122 contravariance, 38 convex, 122 linear combination, 135 coor dinate map, seechart covariant vector ,seecovector covector ,83,84,85,89,90 eld, 83 Coxeter ,HaroldScott MacDonald (1907–2003), 43 Coxeter relations, 43 critical point, 79 cross-cap, 7 cube inanopen set,59,60–62, 64–65 curl, 27,29,65,109 curvatur e,17 curve, 1,69 cycle, 62,64 cylinder formula, 116 143 144 INDEX with base M,116 d'Alembert, Jean LeRond (1717–1783), 29 d'Alembertian, 29 deRham, Geor ges(1903–1990), 137 degenerate chain, 61,64 degr ee ofaform, 17 ofahomogeneous function, 134 ofamulti-index, 17 degr eesoffreedom, 9,14 Descartes, René (1596–1650), 10,85,125 determinant, 31,32–35, 40–42, 43,78,85,86, 91–94 differential equation, 15 form, seeform dimension, 1–11 ,69 Dirichlet, Lejeune (1805–1859), 110 Dirichlet integral, 110 disconnected, 12,108 diver gence, 26,29,65,108, 109 domain, 106,108, 109 ofafunction, 125 dotproduct, seeinner product dual basis, 84,86,87,89 vector ,seecovector space, 83 electr omagnetic wave, 29 electr omagnetism, 29 element ofarclength, 89,96,101, 123 ofsurface area,96 embedding, 67,68,70,71,77–78 ,81,88,96–98, 100, 101, 103, 104,106, 107 Euclid ofAlexandria (ca.325–265 BC), 1,67, 69,91,110, 114, 125, 127 Euclidean motion, 91 plane, 125 space, 1,67,69,110, 114, 127 volume, 91 "´,109 even map, 135 permutation, 34,43 exact form, 22,28,49–52, 64,117–120, 123 exterior algebra, 82 derivative, 20,21,25,27,60,63 onamanifold, 82 differential calculus, 17 product, seeproduct offorms Faraday ,Michael (1791–1867), 29 Faraday's Law,29 bre,seelevel setxed point, 113 ux, 26,99,108, 109 form asavector -eating animal, 88 closed, seeclosed form exact, seeexact form onamanifold, 82,88 onEuclidean space, 17,18–30, 88 volume, seevolume form freespace, 29 Freedman, Michael (1951–), 121 function, 125,130 functional, seecovector fundamental theor emofcalculus, 27,47,49, 52,64,129,134 inRn,49,63,108 Gamma function, 110–111, 135 Gauß, Carl Friedrich (1777–1855), v,29,63,65, 95,108, 137 Gauß map, 95 Gauß' Law,29 graded commutativity ,18,19 gradient, 26,28,49,65,74–79, 96,100, 101, 108, 110, 130 Gram, Jorgen (1850–1916), 92 Gram-Schmidt process, 92 graph, 68,70,73,97,100, 103, 129 Graßmann, Hermann (1809–1877), 82 gravitation, 54,134 Greekalphabet, 139 Green, Geor ge(1793–1841), v,63,65,110 Green's theor em,65,110 halfspace, 103 Hodge, William (1903–1975), 23,25,28,29,38 Hodge star operator ,23,25,28,38,seealsorel- ativity homogeneous function, 28,78,122, 134 homotopy ,114 formula, 117 ofcurves, 115 ofloops, 115, 119, 121 hypersurface, 26,69,95–101, 106, 108, 120 increasing multi-index, 19,24,28,41,86,87, seealsocomplementary index ofavector eld, 55 inner product, 8,84,85,98,130 offorms, 28 integrability condition, 22 integral ofa1-form over acurve, 47,48,49,51 ofaform over achain, 57,58–59, 63–65, 106 over amanifold, 106,107, 108, 117, 119, 120, 122 inversion, 34 inwar dpointing, 106 INDEX 145 k-chain, seechain k-cube, seecube k-form, seeform k-multilinear function, seemultilinear function Klein, Felix (1849–1925), 5 Klein bottle, 5 Kronecker ,Leopold (1823–1891), 51 Kronecker delta, 51 Laplace, Pierr e-Simon (1749–1827), 28 Laplacian, 28 Leibniz, Gottfried Wilhelm von (1646–1716), 20, 21,40 Leibniz rule forforms, 21,40 forfunctions, 20 length ofapermutation, 34,42–43 ofavector ,8,76,79,95,106, 114, 128 level curve, 74 hypersurface, 74 set,73,126 surface, 74 lightlike, 29 line segment, 91 local representative, 82,88,89,96 Lotka, Alfred(1880–1949), 15,78 Lotka-V olterra model, 15,78 manifold, 1–15 ,69,70–79 abstract, 9–13, 69 given explicitly ,9,72 given implicitly ,7,72 with boundary ,2–7,103,104–111 ,118 map, 125,130 Maxwell, James Clerk (1831–1879), 29 mean ofafunction, 107 measurable set,36 Milnor ,John (1931–), 121 minimum, 75 Minkowski, Hermann (1864–1909), 29 Minkowski inner product, 29 space, 29 Möbius, August (1790–1868), 4,105 Möbius band, 4,105 multi-index, 17,126, seealsoincreasing multi- index multilinear algebra, 17 function, 85,89 n-manifold, seemanifold naturality ofpullbacks, 38,48,58 Newton, Isaac (1643–1727), 54,134, 137 norm ofavector ,8 normal vector eld, seeunit normal vector eldodd permutation, 34,43 open ball, 127 neighbour hood, 127 set,127,128 isamanifold, 70 orientation ofaboundary ,106,108 ofahypersurface, 95,100 ofamanifold, 88,95,108 ofavector space, 91,94,100 preserving, 48,57,95–97, 101, 106, 107 reversing, 48,57,95 orthogonal complement, 74,95 group, 76,78 matrix, 76,91,92 operator ,28 projection, 93 orthonormal, 28,91 outwar dpointing, 106 outwar d-pointing, 110 pair ofpants, 105 paraboloid, 4 parallelepiped, seeblock parallelogram, 13,14,91,100 parametrization, 9,57,67 parametrized curve, 13,47,49,53–55 ,77–78 , 130 partial differential equation, 22 operator ,20 path, seeparametrized curve pentagon, 14 Perelman, Grigori (1966–), 121 periodic function, 123 permutation, 33,34,35,41,42–43 ,85,100 group, 33,43 matrix, 43 pinch point, 7 plane curve, 1,13,69 Poincaré, Jules Henri (1854–1912), 119, 121 Poincaré conjectur e,121 lemma, 119 potential, 49,53,54,108 predator ,15 prey,15 product offorms, 19,27 onamanifold, 82 ofpermutations, 34,43 ofsets, seeCartesian product product rule, seeLeibniz rule projective plane, 5 pullback ofaform 146 INDEX onamanifold, 88,97,106, 114, 116, 117 onEuclidean space, 37,40–42, 47,48,57, 58,82,88 ofafunction, 132 punctur ed Euclidean space, 78,114,119–121 plane, 47,51,64,77,115, 119, 120 quadrilateral, 11 rectangular block, 18,57,65 regular value, 73,74–79 ,96,100, 105 relativity ,10,29,137 reparametrization ofacurve, 47,48,50,58 ofarectangular block, 57,58 restriction ofamap, 47,106, 107, 110, 126 retraction, 113 Riemann, Bernhar d(1826–1866), 137 rigid body ,10 row operation, 42 vector ,1,33,45,74,83,89,130 saddle point, 75 Schmidt, Erhard(1876–1959), 92 sign ofapermutation, 34,42–43 simple permutation, 43 simplex, 135 simply connected, 119,121 singular cube, seecube value, 73,74–76 ,78 singularity ,2,8,13,14,67,71,105 Smale, Stephen (1930–), 121 smooth curve, 69 function ormap, 131,134 hypersurface, 69 manifold, 4 point, 2 surface, 69 solution curve, 9,15 space curve, 1 space-time, 29 spacelike, 29 special linear group, 78 spher e,4,8,10,11,14,64,67,75,79,96,100, 105, 106, 110, 113, 114, 119–121, 128, 134 spherical coor dinates, 44 pendulum, 9 standar d basis ofRn,130 orientation, 94 simplex, 135 star-shaped, 122 state space, seeconguration space steepest ascent, 75,130stereographic projection, 72,135 Stirling, James (1692–1770), 111 Stirling's formula, 111 Stokes, Geor ge(1819–1903), v,47,63,65,107, 109 Stokes' theor em classical version, 65,109 forchains, 47,63,64 formanifolds, 103, 108,116, 118,120 submanifold, 69 surface, 4,14,69,81,109 area,9,17,96,107,109,129 symmetric bilinear function, 85 matrix, 76,79 tangent hyperplane, 69 line, 1,13,69,70,77 plane, 14,69,78 space, 1,8,14,69,70,73–76 ,78,88,95,96, 106 vector ,48,69,79,88,106 Taylor ,Brook (1685–1731), 134 Taylor 'sformula, 134 tensor product, 17,85,89 timelike, 29 topological manifold, 4 torus,4,68,106 trace, 78 trajectory ,15,78 transformation law,82 transpose ofamatrix oravector ,1,26,35,74, 83,130 transposition, 42 unit circle,seecircle cube, 35,58,65,128 interval, 35,58,91,114, 116 normal vector eld, 95,98,99,101,106, 108– 110, 120 spher e,seespher e squar e,35,60,62 vector ,51,130 vector ,seealso column, length, row,unit, tan- gent eld, 24,28,29,48,55,65,98,108, 109, see alsoconservative, curl, diver gence, gradi- ent, index, potential, unit normal Volterra, Vito(1860–1940), 15,78 volume change, 36,42 element, 17,96 Euclidean, seeEuclidean volume form, 65,96,97,103 ofahypersurface, 99 onRn,18,24,42 INDEX 147 onahypersurface, 100,110,120 ofablock, 18,31,91,92,93 ofamanifold, 89,107,109 ofasimplex, 135 wave operator ,29 wedge product, 17,85,89 winding number ofclosed curve, 53,54–55 ,118, 120 ofhypersurface, 120 work, 17,25,48,49,61,108, 109 zeroofavector eld, 25