数学专业英语5.docx

数学专业英语5.docx

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数学专业英语5

MathematicalEnglish

Dr.XiaominZhang

Email:

zhangxiaomin@

§2.5BasicConceptsofCartesianGeometry

TEXTAThecoordinatesystemofCartesiangeometry

Asmentionedearlier,oneoftheapplicationsoftheintegralisthecalculationofarea.Ordinarilywedonottalkaboutareabyitself,instead,wetalkabouttheareaofsomething.Thismeansthatwehavecertainobjects（polygonalregions,circularregions,parabolicsegmentsetc.）whoseareaswewishtomeasure.Ifwehopetoarriveatatreatmentofareathatwillenableustodealwithmanydifferentkindsofobjects,wemustfirstfindaneffectivewaytodescribetheseobjects.

Themostprimitivewayofdoingthisisbydrawingfigures,aswasdonebytheancientGreeks.AmuchbetterwaywassuggestedbyRenaDescartes（1596-1650）,whointroducedthesubjectofanalyticgeometry（alsoknownasCartesiangeometry）.Descartes’ideawastorepresentgeometricpointsbynumbers.Theprocedureforpointsinaplaneisthis:

Twoperpendicularreferencelines（calledcoordinateaxes）arechosen,onehorizontal（calledthe“x-axis”）,theothervertical（the“y-axis”）.Theirpointofintersection,denotedbyO,iscalledtheorigin.Onthex-axisaconvenientpointischosentotherightofOanditsdistanceformOiscalledtheunitdistance.Verticaldistancesalongthey-axisareusuallymeasuredwiththesameunitdistance,althoughsometimesitisconvenienttouseadifferentscaleonthey-axis.Noweachpointintheplane（sometimescalledthexy-plane）isassignedapairofnumbers,calleditscoordinates.Thesenumberstellushowtolocatethepoint.

Figure2-5-1illustratessomeexamples.Thepointwithcoordinates（3,2）liesthreeunitstotherightofthey-axisandtwounitsabovethex-axis.Thenumber3iscalledthex-coordinateofthepoint,2itsy-coordinate.Pointstotheleftofthey-axishaveanegativex-coordinate;thosebelowthex-axishaveanegativey-coordinate.Thex-coordinateofapointissometimescalleditsabscissaandthey-coordinateiscalleditsordinate.

Whenwewriteapairofnumberssuchas（a,b）torepresentapoint,weagreethattheabscissaorx-coordinate,a,iswrittenfirst.Forthisreason,thepair（a,b）isoftenreferredtoasanorderedpair.Itisclearthattwoorderedpairs（a,b）and（c,d）representthesamepointifandonlyifwehavea=candb=d.Points（a,b）withbothaandbpositivearesaidtolieinthefirstquadrant,thosewitha<0andb>0areinthesecondquadrant;thosewitha<0andb<0areinthethirdquadrant;andthosewitha>0andb<0areinthefourthquadrant.Figure2-5-1showsonepointineachquadrant.

Theprocedureforpointsinspaceissimilar.Wetakethreemutuallyperpendicularlinesinspaceintersectingatapoint（theorigin）.Theselinesdeterminethreemutuallyperpendicularplanes,andeachpointinspacecanbecompletelydescribedbyspecifying,withappropriateregardforsigns,itsdistancesfromtheseplanes.Weshalldiscussthree-dimensionalCartesiangeometryinmoredetaillateron;forthepresentweconfineourattentiontoplaneanalyticgeometry.

Notations

RenaDescartes（1596-1650）Frenchscientificphilosopherwhodevelopedatheoryknownasthemechanicalphilosophy.ThisphilosophywashighlyinfluentialuntilsupersededbyNewton'smethodology,andmaintained,forexample,thattheuniversewasaplenuminwhichnovacuumcouldexist.Descarteswasthefirsttomakeagraph,allowingageometricinterpretationofamathematicalfunctionandgivinghisnametoCartesiancoordinates（originatedfromPappus’problem）.

Descartesbelievedthatasystemofknowledgeshouldstartfromfirstprinciplesandproceedmathematicallytoaseriesofdeductions,reducingphysicstomathematics.InDiscoursdelaMéthode（1637）,headvocatedthesystematicdoubtingofknowledge,believingasPlatothatsenseperceptionandreasondeceiveusandthatmancannothaverealknowledgeofnature.Theonlythingthathebelievedhecouldbecertainofwasthathewasdoubting,leadingtohisfamousphrase"Cogitoergosum,"（Ithink,thereforeIam）.Fromthisonephrase,hederivedtherestofphilosophy,includingtheexistenceofGod.

Pappus’problemThefullenunciationoftheproblemisratherinvolved,butthemostimportantcaseistofindthelocusofapointsuchthattheproductoftheperpendicularsonmgivenstraightlinesshallbeinaconstantratiototheproductoftheperpendicularsonnothergivenstraightlines.Theancientshadsolvedthisgeometricallyforthecasem=1,n=1,andthecasem=1,n=2.Pappushadfurtherstatedthat,ifm=n=2,thelocusisaconic,buthegavenoproof;Descartesalsofailedtoprovethisbypuregeometry,butheshowedthatthecurveisrepresentedbyanequationoftheseconddegree,thatis,aconic;subsequentlyNewtongaveanelegantsolutionoftheproblembypuregeometry.

TEXTBGeometricfigures

Ageometricfigure,suchasacurveintheplane,isacollectionofpointssatisfyingoneofmorespecialconditions.Bytranslatingtheseconditionsintoexpressions,involvingthecoordinatesxandy,weobtainoneormoreequationswhichcharacterizethefigureinquestion.Forexample,consideracircleofradiusrwithitscenterattheorigin,asshowninFigure2-5-2.LetPbeanarbitrarypointonthiscircle,andsupposePhascoordinates（x,y）.ThenthelinesegmentOPisthehypotenuseofarighttrianglewhoselegshavelengths|x|and|y|andhence,bythetheoremofPythagoras,

x2+y2=r2.

Theequation,calledaCartesianequationofthecircle,issatisfiedbyallpoints（x,y）onthecircleandbynoothers,sotheequationcompletelycharacterizesthecircle.Thisexampleillustrateshowanalyticgeometryisusedtoreducegeometricalstatementsaboutpointstoanalyticalstatementsaboutrealnumbers.

Throughouttheirhistoricaldevelopment,calculusandanalyticgeometryhavebeenintimatelyintertwined.Newdiscoveriesinonesubjectledtoimprovementsintheother.Thedevelopmentofcalculusandanalyticgeometryinthisbookissimilartothehistoricaldevelopment,inthatthetwosubjectsaretreatedtogether.However,ourprimarypurposeistodiscusscalculus,Conceptsfromanalyticgeometrythatarerequiredforthispurposewillbediscussedasneeded.Actually,onlyafewveryelementaryconceptsofplaneanalyticgeometryarerequiredtounderstandtherudimentsofcalculus.Adeeperstudyofanalyticgeometryisneededtoextendthescopeandapplicationsofcalculus,andthisstudywillbecarriedoutinlaterchaptersusingvectormethodsaswellasthemethodsofcalculus.Untilthen,allthatisrequiredformanalyticgeometryisalittlefamiliaritywithdrawinggraphoffunction.

TEXTCSetsofpointsintheplane

Wehavealreadyshownthatthereisaone-to-onecorrespondencebetweenpointsinaplaneandpairsofnumbers（x,y）.Certainsetsofpointsintheplanemaybeofspecialinterest.Forexample,wemaywishtoexaminethesetofpointscomprisingthecircumferenceofacertaincircle,orthesetofpointsconstitutingtheinteriorofacertaintriangle.Onemaywonderifsuchsetsofpointmaybesuccinctlydescribedinacompactmathematicalnotation.

Wemaywrite

{|（x,y）|y=2x}

（1）

todescribethesetoforderedpairs（x,y）,orcorrespondingpoints,suchthattheordinateisequaltotwicetheabscissas.Ineffect,then,theverticallinein

（1）isread“suchthat“.By“thegraphofthesetoforderedpairs”ismeantthesetofallpointsoftheplanecorrespondingtothesetoforderedpairs.Thestudentwillreadilyinferthatthesetofpointsconstitutingthegraphliesonastraightline.

Considertheset

{（x,y）|y=x2}.

Consistentwithourpreviousinterpretation,thissymbolrepresentsthesetoforderedpairs（x,y）suchthattheordinateisequaltothesquareoftheabscissa.Here,thetotalgraphcomprisesasimplerecognizablegeometricalfigure,acurveknownasaparabola.

Onthebasisofthesetwoexamples,onemaybetemptedtobelievethatanyarbitrarilydrawncurve,whichofcoursedeterminesasetofpointsororderedpairs,couldbedescribedsuccinctlybyasimpleequation.Unfortunately,thisisnotthecase.Forexample,thebrokenlineinfigure2-2-3isoneofsuchcurves.

Considernowtheset

{（x,y）|y>2x}

todescribethesetofpoints（x,y）whoseordinateisgreaterthantwiceitsabscissa.Inthecase,oursetofpointsconstitutesnotacurve,butaregionofthecoordinateplane.

SUPPLEMENTConicSection

Theconicsectionsarethenondegeneratecurvesgeneratedbytheintersectionsofaplanewithoneortwonappesofacone.Foraplaneperpendiculartotheaxisofthecone,acircleisproduced.Foraplanethatisnotperpendiculartotheaxisandthatintersectsonlyasinglenappe,thecurveproducediseitheranellipseoraparabola.Thecurveproducedbyaplaneintersectingbothnappesisahyperbola.

Becauseofthissimplegeometricinterpretation,theconicsectionswerestudiedbytheGreekslongbeforetheirapplicationtoinversesquarelaworbitswasknown.ApolloniuswrotetheclassicancientworkonthesubjectentitledOnConics.Keplerwasthefirsttonoticethatplanetaryorbitswereellipses,andNewton wasthenabletoderivetheshapeoforbitsmathematicallyusingcalculus,undertheassumptionthatgravitationalforcegoesastheinversesquareofdistance.Dependingontheenergyoftheorbitingbody,orbitshapesthatareanyofthefourtypesofconicsectionsarepossible.

AconicsectionmaymoreformallybedefinedasthelocusofapointPthatmovesintheplaneofafixedpointFcalledthefocusandafixedlinedcalledtheconicsectiondirectrix（withFnotond）suchthattheratioofthedistanceof