作者：B.A.Dubrovin/等 日期：2021-03-03 12:09:36
片断：
CHAPTER1
GeometryinRegionsofaSpace.
BasicConcepts
?Co-ordinateSystems
Webeginbydiscussingsomeoftheconceptsfundamentaltogeometry.In
schoolgeometry-theso-called"elementaryEuclidean"geometryofthe
ancientGreeks-themainobjectsofstudyarevariousmetricalproperties
ofthesimplestgeometricalfigures.Thebasicgoalofthatgeometryisto
findrelationshipsbetweenlengthsandanglesintrianglesandotherpolygons.
Knowledgeofsuchrelationshipsthenprovidesabasisforthecalculation
ofthesurfaceareasandvolumesofcertainsolids.Thecentralconcepts
underlyingschoolgeometryarethefollowing:thelenethofastraightline
segment(orofacirculararc);andtheanglebetweentwointersectingstraight
lines(orcirculararcs).
Thechiefaimofanalytic(orco-ordinate)geometryistodescribegeo-
metricalfiguresbymeansofalgebraicformulaereferredtoaCartesian
systemofco-ordinatesoftheplaneor3-dimensionalspace.Theobjects
studiedarethesameasinelementaryEuclideangeometry:thesoledifference
liesinthemethodology.Again,differentialgeometryisthesameoldsubject,
exceptthatherethesubtlertechniquesofthedifferentialcalculusandlinear
algebraarebroughtintofullplay.Beingapplicabletogeneral"smooth"
geometricalobjects,thesetechniquesprovideaccesstoawiderclassofsuch
objects.
1.1.CartesianCo-ordinatesinaSpace
Ourmostbasicconceptionofgeometryissetoutinthefollowingtwopara-
graphs:
(i)WedoourgeometryinacertainspaceconsistingofpointsP,Q,....
(ii)Asinanalyticgeometry,weintroduceasystemofco-ordinatesforthe
space.Thisisdonebysimplyassociatingwitheachpointofthespace
anorderedn-tuple(x,...,x)ofrealnumbers-theco-ordinatesofthe
point-insuchawayastosatisfythefollowingtwoconditions:
(a)Distinctpointsareassigneddistinctn-tuples.Inotherwords,points
PandQwithco-ordinates(xl....,x)and(y,...,y)areoneand
thesamepointifandonlyifx'=y,i=1,...,n.
(b)Everypossibien-tuple(x....,x)isused,i.e.isassignedtosome
pointofthespace.
1.1.1.Definition.AspacefurnishedwithasystemofCartesianco-ordinates
satisfyingconditions(a)and(b)iscalledann-dimensionalCartesianspace.
andisdenotedbyR".Theintegerniscalledthedimensionofthespace.
Weshalloftenrefersomewhatlooselytothen-tuples(x,....x)them-
selvesasthepointsofthespace.ThesimplestexampleofaCartesianspace
istherealnumberline.Hereeachpointhasjustoneco-ordinatex,sothat
n=1,i.e.itisal-dimensionalCartesianspace.Otherexamples,familiar
fromanalyticgeometry,areprovidedbyCartesianco-ordinatizationsof
theplane(whichisthena2-dimensionalCartesianspace).andofordinary
(i.e.3-dimensional)space(Figure1).TheseCartesianspacesarecompletely
adequateforsolvingtheproblemsofschoolgeometry.