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1、Introductiontodi erentialformsDonuArapuraThisisasupplementformyMath362class.Thecalculusofdi erentialformsgivesanalternativetovectorcalculuswhich,althoughquitecompelling,israrelytaughtatthislevel.11-formsAdi erential1-form(orsimplyadi erentialora1-form)onanopensubset
2、ofR2isanexpressionF(x;y)dx+G(x;y)dywhereF;GareR-valuedfunctionsontheopenset.Averyimportantexampleofadi erentialisgivenasfollows:Iff(x;y)isC1R-valuedfunctiononanopensetU,thenitstotaldi erential(orexteriorderivative)is@f@fdf=dx+dy@x@gItisadi erentialonU.Inasimilarfash
3、ion,adi erential1-formonanopensubsetofR3isanexpressionF(x;y;z)dx+G(x;y;z)dy+H(x;y;z)dzwhereF;G;HareR-valuedfunctionsontheopenset.Iff(x;y;z)isaC1functiononthisset,thenitstotaldi erentialis@f@f@fdf=dx+dy+dz@x@y@zAtthisstage,itisworthpointingoutthatadi erentialformisve
4、rysimilartoavector eld.Infact,wecansetupacorrespondence:Fi+Gj+Hk$Fdx+Gdy+HdzUnderthissetup,thegradientrfcorrespondstodf.Thusitmightseemthatallwearedoingiswritingthepreviousconceptsinafunnynotation.However,thenotationisverysuggestiveandultimatelyquitepowerful.Suppose
5、thatthatx;y;zdependonsomeparametert,andfdependsonx;y;z,thenthechainrulesaysdf@fdx@fdy@fdz=++dt@xdt@ydt@zdtThustheformulafordfcanbeobtainedbycancelingdt.12ExactnessinR2SupposethatFdx+Gdyisadi erentialonR2withC1coecients.Wewillsaythatitisexactifonecan ndaC2functionf(
6、x;y)withdf=Fdx+GdyMostdi erentialformsarenotexact.Toseewhy,notethattheaboveequationisequivalentto@f@fF=;G=:@x@yThereforeiffexiststhen@F@2f@2f@G===@y@y@x@x@y@xButthisequationwouldfailformostexamplessuchasydx.Wewillcalladi erentialclosedif@Fand@Gareequal.Sowehavejusts
7、hownthatifa@y@xdi erentialistobeexact,thenithadbetterbeclosed.Exactnessisaveryimportantconcept.You'veprobablyalreadyencountereditinthecontextofdi erentialequations.Givenanequationdy=F(x;y)dxwecanrewriteitasFdx dy=0Ifthedi erentialontheleftisexactandequaltosay,df,the
8、nthecurvesf(x;y)=cgivesolutionstothisequation.Theseconceptsariseinphysics.Forexamplegivenavector eldF=F1i+F2jrepresentingaforce,onewouldli
