作者	B.A.Dubrovin/等
出版社
出版时间	1999-11-01

特色：

片断：CHAPTER1GeometryinRegionsofaSpace.BasicConcepts?Co-ordinateSystemsWebeginbydiscussingsomeoftheconceptsfundamentaltogeometry.Inschoolgeometry-theso-called"elementaryEuclidean"geometryoftheancientGreeks-themainobjectsofstudyarevariousmetricalpropertiesofthesimplestgeometricalfigures.Thebasicgoalofthatgeometryistofindrelationshipsbetweenlengthsandanglesintrianglesandotherpolygons.Knowledgeofsuchrelationshipsthenprovidesabasisforthecalculationofthesurfaceareasandvolumesofcertainsolids.Thecentralconceptsunderlyingschoolgeometryarethefollowing:thelenethofastraightlinesegment(orofacirculararc);andtheanglebetweentwointersectingstraightlines(orcirculararcs).Thechiefaimofanalytic(orco-ordinate)geometryistodescribegeo-metricalfiguresbymeansofalgebraicformulaereferredtoaCartesiansystemofco-ordinatesoftheplaneor3-dimensionalspace.TheobjectsstudiedarethesameasinelementaryEuclideangeometry:thesoledifferenceliesinthemethodology.Again,differentialgeometryisthesameoldsubject,exceptthatherethesubtlertechniquesofthedifferentialcalculusandlinearalgebraarebroughtintofullplay.Beingapplicabletogeneral"smooth"geometricalobjects,thesetechniquesprovideaccesstoawiderclassofsuchobjects.1.1.CartesianCo-ordinatesinaSpaceOurmostbasicconceptionofgeometryissetoutinthefollowingtwopara-graphs:(i)WedoourgeometryinacertainspaceconsistingofpointsP,Q,....(ii)Asinanalyticgeometry,weintroduceasystemofco-ordinatesforthespace.Thisisdonebysimplyassociatingwitheachpointofthespaceanorderedn-tuple(x,...,x)ofrealnumbers-theco-ordinatesofthepoint-insuchawayastosatisfythefollowingtwoconditions:(a)Distinctpointsareassigneddistinctn-tuples.Inotherwords,pointsPandQwithco-ordinates(xl....,x)and(y,...,y)areoneandthesamepointifandonlyifx'=y,i=1,...,n.(b)Everypossibien-tuple(x....,x)isused,i.e.isassignedtosomepointofthespace.1.1.1.Definition.AspacefurnishedwithasystemofCartesianco-ordinatessatisfyingconditions(a)and(b)iscalledann-dimensionalCartesianspace.andisdenotedbyR".Theintegerniscalledthedimensionofthespace.Weshalloftenrefersomewhatlooselytothen-tuples(x,....x)them-selvesasthepointsofthespace.ThesimplestexampleofaCartesianspaceistherealnumberline.Hereeachpointhasjustoneco-ordinatex,sothatn=1,i.e.itisal-dimensionalCartesianspace.Otherexamples,familiarfromanalyticgeometry,areprovidedbyCartesianco-ordinatizationsoftheplane(whichisthena2-dimensionalCartesianspace).andofordinary(i.e.3-dimensional)space(Figure1).TheseCartesianspacesarecompletelyadequateforsolvingtheproblemsofschoolgeometry.