geometric approach to differential forms - d. bachman

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1、AGeometricApproachtoDifferentialFormsDavidBachmanCaliforniaPolytechnicStateUniversityE-mailaddress:dbachman@calpoly.eduarXiv:math.GT/0306194v111Jun2003FortheInstructorThepresentworkisnotmeanttocontainanynewmaterialaboutdifferentialforms.Therearemanygoodbooksouttherewhichgivenice,completetreatmentsof

2、thesubject.Rather,thegoalhereistomakethetopicofdifferentialformsaccessibletothesophomorelevelundergraduate.Thetargetaudienceforthismaterialisprimarilystudentswhohavecompletedthreesemestersofcalculus,althoughthelatersectionswillbeofinteresttoadvancedundergraduateandbeginninggraduatestudents.Atmanyin

3、stitutionsacourseinlinearalgebraisnotaprerequisiteforvectorcalculus.Consequently,thesenoteshavebeenwrittensothattheearlierchaptersdonotrequiremanyconceptsfromlinearalgebra.Whatfollowsbeganasasetoflecturenotesfromanintroductorycourseindifferentialforms,givenatPortlandStateUniversity,duringthesummero

4、f2000.ThenoteswerethenrevisedforsubsequentcoursesonmultivariablecalculusandvectorcalculusatCaliforniaPolytechnicStateUniversity.Atsomeundeterminedpointinthefuturethismayturnintoafullscaletextbook,soanyfeedbackwouldbegreatlyappreciated!Ithankseveralpeople.Firstandforemost,Iamgratefultoallthosestude

5、ntswhosurvivedtheearlierversionsofthisbook.Iwouldalsoliketothankseveralofmycolleaguesforgivingmehelpfulcomments.Mostnotably,DonHartighadseveralcommentsafterusinganearlierversionofthistextforavectorcalculuscourse.JohnEtnyreandDannyCalegarigavemefeedbackregardingChapter6.AlvinBachmanhadgoodsuggestio

6、nsregardingtheformatofthistext.Finally,theideatowritethistextcamefromconversationswithRobertGhristwhileIwasagraduatestudentattheUniversityofTexasatAustin.Healsodeservesmygratitude.Prerequisites.Mostofthetextiswrittenforstudentswhohavecompletedthreesemestersofcalculus.Inparticular,studentsareexpect

7、edtobefamiliarwithpartialderivatives,multipleintegrals,andparameterizedcurvesandsurfaces.34FORTHEINSTRUCTORConceptsfromlinearalgebraarekepttoaminimum,althoughitwillbeimportantthatstudentsknowhowtocomputethedeterm