数学专业英语 第二版 (吴炯圻 著) 高等教育出版社 课后答案_NoRestriction 144页

'数学专业英语Lesson1MathematicsasaLanguageofScienceassertvt.断言；坚持主张；维护表明qualitativeadj.性质的；定性的quantitativeadj.量的；数量的；定量的；与数量有关的astronomyn.天文学postulaten.假定,基本条件,基本原理vt.要求,假定vi.要求hypotheticaladj.假设的,假定的,爱猜想的Lesson2deductionn.减除,扣除,减除额,推论,演绎inductionn.归纳；归纳法；归纳所得之结论verificationn.验证；证实correlatekhdaw.comvt.使相互关联vi.和...相关discardvt.丢弃,抛弃v.放弃discreditn.不信任；失信consistentadj.一致的,调和的,坚固的,[数、统]相容的inadequacyn.不充分,不适当,不适合,不足额conic,conicaladj圆锥的；圆锥形的ellipsen.椭圆,椭圆形ellipt(n.)hyperbolicadj.双曲线的hyperbola(n.)parabolicadj.用寓言表达的:抛物线的，像抛物线的parabola(n.)algebraicadj.代数的,关于代数学的mineralogyn.矿物学refractionn.折光,折射stimulusn.刺激物,促进因素,刺激,刺激impetusn.冲力推动力；刺激课后答案网Lesson3Axioms,definitionsandTheoremsaxiomn.[数]公理www.hackshp.cndefinitionn.阐明；确定定义；界说extravagantadj.奢侈的,浪费的,过分的,放纵的collinearadj.在同一直线上的,同线的convexadj.凸出的；凸面的segmentn.部分；片段；节,弓形；圆缺；弧形,线段conswquentlyadv.从而,因此intermsofadv.根据,按照,用...的话,在...方面pretensen.主张,要求,伪称,借口,自称Lesson4GeometryandGeometricaltermstermn.学期,期限,期间,条款,条件,术语khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com trianglen.[数]三角形,三人一组,三角关系parallelogramn.平行四边形straightanglen.[数]平角rightanglen.直角acuteanglen.锐角obtuseanglen.钝角reflexanglen.优角rectilinearadj直线的；由直线组成的；循直线进行的isoscelestrianglen.等腰三角形equilateraltrianglen.等边三角形righttrianglen.直角三角形obtusetrianglen.钝角三角形acutetrianglen.锐角三角形equiangulartrianglekhdaw.comn.正三角形,等角三角形hypotenusen.（直角三角形的）斜边circle圆center中心；中央；圆心diametern.直径radiusn.半径,范围,辐射光线,有效航程,范围,界限circumferencen.圆周,周围Lesson5TheMethodofLimitslimitn.限度,极限,极点infiniteadj.无限的；无穷的infinitesimaladj.无穷小的,极小的,无限小的calculusn.微积分学,结石课后答案网exemplifyvt.例证,例示,作为...例子inscribev.记下polygonn.[数]多角形,多边形diminishv.(使)减少,(使)变小www.hackshp.cncurvilinearadj曲线的,由曲线组成的intuitionn.直觉,直觉的知识integraln.[数学]积分,完整,部分defectiveadj.有缺陷的,(智商或行为有)欠缺的differentialcoefficient微分系数arithmeticaladj.算术的,算术上的convergencen.集中,收敛criterionn.(批评判断的)标准,准据,规范sequencen.次序,顺序,序列irrationalnumbersn.[数]无理数domain，定义域contradiction矛盾reversaln.颠倒,反转,反向,逆转,撤销khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Lesson6Functioncontinuousvariable连续变量;［连续变数］variation变分,变化interval区间independentvariable自变量dependentvariable应变量rectangularcoordinate直角坐标abscissan.〈数〉横坐标ordinaten.[数]纵线,纵座标gradientadj.倾斜的n.梯度,倾斜度,坡度slopekhdaw.comn.斜坡,斜面,倾斜v.(使)顺斜Lesson7DifferentialandIntegralcalculusdifferentialadj.微分的n.微分(differentiation)Integraln.[数学]积分,完整,部分(integration)calculusn.微积分学,结石interrelationn.相互关系trigonometryn.三角法exponentialadj.指数的,幂数的logarithmn.[数]对数derivativen.导数；微商tangentn.切线,[数]正切counterclockwiseadj.反时针方向的课后答案网adv.反时针方向(clockwise)definiteintegral定积分approximationn.接近,走近,[数]近似值culminatev.达到顶点meann.平均数,中间,中庸www.hackshp.cndifferentialequation微分方程extremevaluen.极值multipleintegral多重积分doubleintegrallineintegralfunctionalanalysis泛函分析Lesson8TheConceptofCardinalNumber(I)cardinalnumbern.基数（如：1,2,3,...有别于序数）denumerableadj.可数的aggregaten.合计,总计,集合体adj.合计的,集合的,聚合的v.聚集,集合,合计khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com purportn.主旨v.声称fanciern.空想家,培育动物(或植物)的行家,爱好者sniffv.用力吸,嗅,闻到,发觉,轻视,用力吸气n.吸,闻,吸气声,嗤之以鼻schemen.安排,配置,计划,阴谋,方案,图解,摘要v.计划,设计,图谋,策划,*n.（计算数学）方法，格式superiorn.长者,高手,上级adj.较高的,上级的,上好的,出众的,高傲的cumbersomeadj.讨厌的,麻烦的,笨重的instructionn.指示,用法说明(书),教育,指导,指令drasticallyadv.激烈地,彻底地conservation守衡律quadraturekhdaw.comn.求积,求积分interpolationn.插值extrapolationn.[数]外推法,推断internalpoint内点identicaladj.同一的,同样的generalizedsolution广义解functional泛函hydrodynamics流体力学，水动力学divergence发散（性），梯度，发散量playanimportant(fundamental...)role起着重要的（...）作用integro-interpolationmethod积分插值法Variationalmethod变分方法comparativelyadv.比较地,相当地deficiencyn.缺乏,不足课后答案网fictiveadj.虚构的,想象上的,虚伪的self-adjoint(nonself-adjoint)自治的，自伴的，自共轭的finiteelementmethod有限元法splineapproximation样条逼近www.hackshp.cnParticles-in-the-Cell网格质点法heraldn.使者,传令官,通报者,先驱,预兆vt.预报,宣布,传达,欢呼advectionn.水平对流phenomenologicaladj.现象学的,现象的fluctuationn.波动,起伏optimismn.乐观,乐观主义pessimismn.悲观,悲观主义unjustifiedadj未被证明其正确的mean-square均方dispersionn.[数]离差,差量Polynomialnadj.[数]多项式的interpolation插值khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com arithmeticn.算术,算法roundingerrors舍入误差multiplen.倍数,若干subjectiveadj.主观的,个人的objectiveadj.客观的,outcomen.结果,成果patternn.样品tossv.投,掷exhaustvt.用尽,耗尽,抽完,使精疲力尽divisibleadj.可分的dice,dien.骰子assignvt.分配,指派attachvt.缚上,系上,贴上v.配属,隶属于khdaw.compitfalln.缺陷chairperson主席mechanicsn.(用作单数)机械学、力学,(用作复数)技巧,结构staticsn.[物]静力学dynamicsn.动力学adequatelyadv.充分地celestialadj.天上的macroscopicadj.肉眼可见的,巨观的classicalfieldtheory经典场理论rigitadj.刚硬的,刚性的,严格的elasticadj.弹性的课后答案网plasticn.可塑的,塑性的,塑料的quantumn.量,额,[物]量子,量子论inceptionn.起初,获得学位pertainv.适合,属于www.hackshp.cngravitationn.地心吸力,引力作用tiden.潮,潮汐,潮流,趋势monumentaladj.纪念碑的,纪念物的,不朽的,非常的encompassv.包围,环绕,包含或包括某事物ingredientn.成分,因素acquaintedadj.有知识的,知晓的synonymousadj.同义的configurationn.构造,结构,配置,外形referencen.提及,涉及,参考,参考书目inertian.惯性,惯量attribute特性momentumn.动量khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com proportionaladj.比例的,成比例的,相称的,均衡的designate指明negligibleadj.可以忽略的,不予重视的projectilen.射弹adj.发射的ballisticsn.弹道学,发射学intractableadj.难处理的{MechanicsofaParticleinconsequenceofadv.由于的...缘故exertvt.尽(力),施加(压力等),努力v.发挥,竭尽全力,尽galaxyn.星系,银河,一群显赫的人,一系列光彩夺目的东furnishvt.供应,提供,装备,布置v.供给torquekhdaw.comn.扭矩,转矩moment力矩的friction摩擦dissipationn.消散,分散,挥霍,浪费,消遣,放荡,狂饮inferv.推断HookesLawandItsConsequenceselasticityn.弹力,弹性constitutiveadj.构成的,制定的atomisticadj.原子论的crackn.裂缝,噼啪声v.(使)破裂,裂纹,(使)爆裂continuummechanicsn.连续介质力学课后答案网superpositionn.重叠,重合,叠合strainn.过度的疲劳,紧张,张力,应变vt.扭伤,损伤v.拉紧,扯紧,(使)紧张,尽力thermodynamicsn.[物]热力学www.hackshp.cnreckonvt.计算,总计,估计,猜想vi.数,计算,估计,依赖,料想lesson20strength强度load载荷empirical以经验为依据的member构件isolated孤立的segment部分、段、节stress应力khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com strain应变tension拉伸shear剪切bend弯曲torsion扭转、扭力insofar在……范围cohesive内聚性的tensile拉力、张力stiffness硬度furnish供给Lesson23FluidMechanicseruption喷发、爆发khdaw.comturbulent湍流laminar层流isothermal等温isotropic各向同性prevalent普遍的、流行的tornado旋风、飓风eddy旋涡viscosity粘性、粘度nonviscous无粘性的rotation旋转adiabatic绝热的reversible可逆的课后答案网isentropic等熵的instant瞬时的streamline流线streamtube流管www.hackshp.cntangential切线的incompressible不可压缩的resultant合成的,组合的downstream下游的,顺流的elbow弯管,肘similitude相似性hydraulic水力的,水力学的predominante占主导地位spillway(河或水坝的)放水道,泄洪道prototype原型,样板Lesson24MechanicalVibrationrepetitive重复的,反复的khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com periodic周期的,定期的tidal潮的,像潮的stationary固定的,不动的vibratory振动的,摆动的propagation传播couplev.连接,连合acoustic听觉的,声学的annoyance烦恼,困惑adjacent接近的,邻近的damp阻尼,衰减restore复职,归还neutral平衡excitingforce激励力resonantadj.共振的,谐振的khdaw.comstiffness刚度,刚性proportionality成比例地inclusion包含,包括magnitude数值,大小substantiallyadv.实质上的perturb干扰,扰乱resonancen.共振vibratoryadj.振动的,可知的perceptible可见的,可知的adudible听得见的,可闻的foregoing前述的impulsive冲击的shock冲击课后答案网Fourierseries傅里叶级数excitation激发,激励discrete分离,离散的contendwith向…作斗争www.hackshp.cncompressor压气机fatigue疲劳perceptible可见的,可知觉的shredder切菜器disposal处理urban都市的metropolitan大都市的at-grade在同一水平面上elevated高架的guideway导轨Lesson25AprefecttotheContinuumMechanicspreface序言khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com continuum连续pl.continuuarigidbody刚体contemporary当代的,同时期的widespread分布广的,普及的accommodate容纳,使适应medium介质plasticity塑性residual剩余的,残留的creep蠕变,爬行,塑性变形aging老化polymeric聚合(物)的sandy沙的,沙质的aubterranean地下的,隐藏的essence精髓,本质khdaw.comthermodynamics热力学self-similar自相似expedient方便的sonsolidate把…联合为一体,统一justify证明…有理radically根本地,本质上deliberate从容不迫的,深思熟虑Lesson33whatisacomputerAttributev.赋予medieval中世纪的astronomer天文学家Mars火星课后答案网resemblevt.像,相似tediousadj.冗长乏味的pulp浆状物,果肉filtervt.过滤www.hackshp.cnunderlyingadj.潜在的,基本的oren.矿沙,矿石perceivev.察觉,看见interventionn.干涉,插入intelligentadj.有智力的,聪明的Lesson34Acomputersystemmanipulatevt.操纵,使用chipn.芯片etchvt.蚀刻,蚀镂fingernail指甲mountvt.安装,安置assemblevt.集合,聚集khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com cabinet橱柜executevt.执行,实现paycheckn.支付薪金的支票barchart直方图joystick游戏杆encountervt.遇到,遇上MathematicalModelingindustryn.工业,产业,行业,勤奋commercen.商业complexityn.复杂(性),复杂的事物,复杂性careern.(原意:道路,轨道)事业,生涯,速度outsetn.开端,开始essencekhdaw.comn.基本,[哲]本质,香精advocationn.(=advocacy)拥护支持provisionn.供应,(一批)供应品,预备,防备,规定publicizev.宣扬roundaboutadj.迂回的,转弯抹角的n.道路交叉处的环形路,迂回路线,兜圈子的话trial-errorvt.n.试制,试生产maneuverabilityn.可操作性,机动性vehiclen.交通工具,车辆,媒介物,传达手段junctionn.连接,接合,交叉点,汇合处ponderv.沉思,考虑contrivev.发明,设计,图谋snookern.(=snookerpool)彩色台球,桌球contextn.上下文,文章的前后关系课后答案网deviationn.背离www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－(a)(a)(a)HowHowHowtototodefineddefineaefiefineneaamathematicalmathematicmathematicalmathematicalterm?alterm?terterm?m?数学术语的定义和数学定理的叙述，其基本格式可归纳为似“if…then…”的格式，其他的格式一般地说可视为这一格式的延伸或变形。如果一定语短语或定语从句，以界定被定义的词，所得定义表面上看虽不是“If……then……”的句型，而实际上是用“定语部分”代替了“If”句，因此我们可以把“定语部分”写成If句，从而又回到“If……then……”的句型。至于下面将要叙述的“Let…if…then”，“Letandassume…,If…then…”等句型，其实质也是基本句型“If……then……”的延伸。有时，在定义或定理中，需要附加说明某些成份，我们还可在“if…then…”句中插入如“where…”等的句子，加以延伸（见后面例子）。khdaw.com总之，绝大部分（如果不是全部的话）数学术语的定义和定理的叙述均可采用本附录中各种格式之。（a）HowHowtototodefineddefineaefiefineneaamathematicalmathematicmathematicalmathematicalterm?alterm?terterm?m?isdefinedas1.SomethingsomethingiscalledTheunionofAandBisdefinedasthesetofthoseelementswhichareinA,inBorinboth.Themapping,ad-bc0,iscalledaMobiustransformation.isdefinedtobe2.Something课后答案网something(oradjective)issaidtobeThedifferenceA-BisdefinedtobethesetofallelementsofAwhicharenotinB.Arealnumberthatcannotbeexpressedastheratiooftwointegersissaidtobewww.hackshp.cnanirrationalnumber.Realnumberswhicharegreaterthanzeroaresaidtobepositive.define3.Wesomethingtobesomething.callWedefinetheintersectionofAandBtobethesetofthoseelementscommontobothAandB.Wecallrealnumbersthatarelessthanzero(tobe)negativenumbers.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 4.如果在定义某一术语之前，需要事先交代某些东西（前提），可用如下形式：iscalledLet…,then…issaidtobeisdefinedasLetx=()beann-tupleofrealnumbers.Thenthesetofallsuchn-tuplesisdefinedastheEuclideann-spaceR.isdefinedtobeLetd(x,y)denotethedistancebetweentwopointsxandyofasetA.ThenthenumberD=iscalledkhdaw.comthediameterofA.5．如果被定义术语，需要满足某些条件，则可用如下形式：iscalledIf…,then…issaidtobeisdefinedasIfthenumberofrowsofamatrixAequalsthenumberofitscolumns,thenAiscalledasquarematrix.isdefinedtobeIfafunctionfisdifferentiableateverypointofadomainD,thenitissaidtobeanalyticinD.课后答案网6.如果需要说明被定义术语应在什么前提下，满足什么条件，则可用下面形式：iscalledissaidtobewww.hackshp.cnLetSuppose….If…then……Letf(z)beananalyticfunctiondefinedonadomainD(前提条件).Ifforeverypairorpoints,andinDwith,wehavef()f()(直接条件)，thenf(z)iscalledaschlichtfunctionorissaidtobeschlichtinD.7.如果被定义术语需要满足几个条件（大前提，小前提，直接条件）则可用如下形式：khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com supposeassumeLet…and….If…then…iscalled…LetDbeadomainandsupposethatf(z)isanalyticinD.IfforeverypairofpointsandinDwith,wehavef()f(),thenf(z)iscalledaschlichtfunction.Notes:(a)一种形式往往可写成另一种形式。Let{}beasequenceofsets.Ifforalln,then{}iscalledanascendingoranon-decreasingsequence.khdaw.com我们可用一定语短语来代替“If”句，使其变为“Let……then”句Let{}beasequenceofsetswithforalln,then{}iscalledanascendingoranon-decreasingsequence.(b)注意“Let”，“suppose”（“assume”），“if”的使用次序，一般来说，前面的可用后面的替换，但后面的用前面的替换就不好了，如上面句子可改写为：Suppose{}isasequenceofsets.If,then{}iscalledanascendingsequence.Let{}beasequenceofsetsandsupposethatthen{}iscalledanascendingsequence.但下面的句子是错误的（至少是不好的句子）；If{}isasequenceofsets,andlet,then{}iscalledanascendingsequence.课后答案网(c)在定义一些术语后，往往需要用符号来表达，或者需要对句中某些成份作附加说明，这时我们需要把定义句扩充，扩充的办法是在定义的原有结构中，插入一个由连接词引导的句子，这类连接词或短语经常是“and”，“where”，“inthis(that)casewww.hackshp.cn”…请参看PARTIA第一课注1和第二课注4、5、6。IfeveryelementofasetAalsobelongstoanothersetB,thenAissaidtobethesubsetofB,andwewriteArealnumberissaidtobearationalifitcanbeexpressedastheratiooftwointegers,wherethedenominatorisnotzero.(d)在定义中，“if”句是关键句，且往往比较复杂，要特别注意在一些定义中，“if”句又有它自己的表达格式，读者对这类句子的结构也要掌握，下面我们以函数极限定义中的“if”句的结构作为例子加以说明：Ifforevery＞0,thereis(thereexists)a＞0,suchthat＜whenever0＜＜,thenwesayf(x)hasalimitAatthepointa.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 上面是函数极限的定义，其中的“if”句是它的典型结构，凡与极限相关的概念，如连续，收敛，一致连续，一致收敛等定义均有类似结构。例：Asequenceoffunctions{}issaidtohavetheCauchypropertyuniformlyonasetEifforany＞0,thereisanNsuchthat＜whenevern,m＞N.当然，极限定义还有其他表达形式但基本结构是一样的，只不过对句中某些部分用等价的语法结构互作替换而已。下面是函数极限定义中“if”句的另一些表达式，读者可把这些句子和原来的句子作比较。If,givenany＞0,thereexistsa＞0,suchthat＜whenever(if,for)0＜＜,…khdaw.comIf,correspondingtoany＞0,a＞0canbefoundsuchthat＜whenever0＜＜,…If,forevery＞0,thereisa＞0,suchthat0＜＜implies＜.数学专业英语－（b）HowHowtototostatesstatestatetateaaatheorem?ththeorem?eorem?定理叙述的格式，基本上与数学术语的定义一样，只不过在术语的定义中，“then”句有比较固定的格式，而定理的“then”句则随其结果而变吧了。1．某些定理可用简单句叙述。Theunionofafinitenumberofclosedsetsisstillaclosedset.Thespace(E,f)iscomplete.课后答案网2.如果定理的结论是在一定前提下得到的，则可用下面形式：“Suppose…Then…”or“Let….Then…”Letf(x)beacontinuousfunctiondefinedon[a,b].Thenf(x)attainsitsmaximumandminwww.hackshp.cnimumon[a,b].Supposethatf(z)isanalyticinasimplyconnecteddomainD,thenforanyclosedsimplecurveClyingwithinD,wehave3.如果定理的结论在一定假设条件下成立，则可用下面的形式“If…,then…”IfP(z)isanon-constantpolynomialthenthereisacomplexnumbercwithP(c)=0khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 4.如果定理的结论除了在一定条件下，还需在一定前提下才成立，这时可用如下形式“Let….If…,then…”or“Suppose….If…,then…”Let,,,befourdistinctpoints.Ifallthesefourpointslieonacircle,thenthecross-ratio(,,,)isreal.5.如果定理的结论在不同层次的几种条件下面成立，可用如下形式：“Let…,andassume….If…then…”Letf(x)bedefinedonopenintervalI,andassumethatf(x)hasarelativemaximumorarelativeminimumataninteriorpointcofI.Ifthederivativef’(c)exists,thenf’(c)=khdaw.com0.数学专业英语－（c）HowHowtototowritewwrwriteanriiteteananabstract?abstract?ababstract?stract?论文摘要的写法不像数学术语的定义和数学定理的叙述那样。有一定的格式可循，但对于初学者来说仍有一些常见的句子可加以摹仿。现略举一些这样的句子，并附上一些论文摘要作为例子，供读者参考。需要指出的是，我们这里所举的例句对普遍的文章均适合，比较抽象，具体的论文摘要除了可用上下面某些句子外，必须有具体内容，更确切地说摘要中要包括一些keywords以说明该文涉及的内容，但一般不要在摘要中引用文献。1．开门见山，说明文章内容，可用下面的句子起句：prove课后答案网showpresentwww.hackshp.cndevelopgeneralizeinvestigatepapernoteaimobjectpurposekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Theofthisisto…proveshowpresentdevelopgeneralizeinvestigateItisthepurposeofthispapertokhdaw.com…isconcerneddealsThispaperwith…provepresentproposetoshowInthispaperwe课后答案网…2.如果需要简略回顾历史，然后再说明自己文章的内容，则可参考采用下面句子。Theproblem…wasfirsttreatedbywww.hackshp.cn…andlater…improvedby…Thepurposeofthispaperistoprovethatitholdsinamoregeneralcase.…firstraisedtheproblemwhichwaslaterpartlysolvedby…Wenowsolvethisprobleminthecaseof…3.如果文章推广了别人的结果，或减弱了别人结果中的条件，则可参考采用下面句子：Thepurposeofthispaperistogeneralizetheresultsobtainedby…toamoregeneralcase,i.e.,…Inthispaperweshallproveseveraltheoremswhicharegeneralizationstotheresultsgivenby…Thispaperintendstoremovesomeunnecessaryassumptions(e.g.,regularity)fromthepaperon…Thispaperdealswithgeneralizationsofthefollowingproblem…khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thispaperimprovestheresultof…on…byweakeningtheconditions…例：Itisthepurposeofthepresentpapertopointoutthatcertainbasicaspectsofinformation-processingsystemspossessdynamicalanalogy,andtoshowthattheseanalogiescanbeexploitedtoobtaindeeperinsightsintothebehaviorofcomplexsystems.Wepresentageneralcomparisionprincipleforsystemsofboundaryvalueproblemsandemploythisresultforprovingexistenceanduniquenessofsolutions,stabilityandexistenceofperiodicsolutionsfornon-linearboundaryvalueproblems.Weprovedatheoremforgeneralizednon-expansivemappingsinlocallyconvexspacesandextendtheresultsofKirkandKaun.WealsoobtainatheoremwhichgeneralizestheresultsofBrouder.khdaw.comThispaperisconcernedwiththeexistenceofmultiplesolutionsofboundaryproblemsforthenon-lineardifferentialequationoftheform….ThispaperisconcernedwiththequestionoflocaluniquenessofsolutionsofCauchyProblemforellipticpartialdifferentialequationswithcharacteristicsofmultiplicitynotgreaterthan2.Theobjectofthispaperistoinvestigatethebehaviorattheboundaryofsolutionstotheuniformlysemi-linearequation…Theaimofthispaperistotrytominimizethefunctional课后答案网overtheclassofallabsolutelycontinuousfunctionsf(x)whichsatisfytheboundaryconditionsf()=,f()=.www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－BasicBasiBasicConceptscConcepConceptsConceptsoftsofofthethetheTheoryThTheoryofeoryofofSetsSetsSetsIndiscussinganybranchofmathematics,beitanalysis,algebra,orgeometry,itishelpfultousethenotationandterminologyofsettheory.Thissubject,whichwasdevelopedbyBooleandCantorinthelatterpartofthe19thcentury,hashadaprofoundinfluenceonthedevelopmentofmathematicsinthe20thcentury.Ithasunifiedmanyseeminglydisconnectedideasandhashelpedtoreducemanymathematicalconceptstotheirlogicalfoundationsinanelegantandsystematicway.Athoroughtreatmentoftheoryofsetswouldrequirealengthydiscussionwhichweregardasoutsidethescopeofthisbook.Fortunately,thebasicnoticnsarefewinnumber,anditispossibletodevelopaworkingknowledgeofthemethodsandideasofsettheorythroughaninformaldiscussion.Actually,weshalldiscussnotsomuchanewtheoryasanagreementaboutthepreciseterminologythatwewishtoapplytomoreorlessfamiliaridekhdaw.comas.Inmathematics,theword“set”isusedtorepresentacollectionofobjectsviewedasasingleentityThecollectionscalledtomindbysuchnounsas“flock”,“tribe”,‘crowd”,“team’,areallexamplesofsets,Theindividualobjectsinthecollectionarecalledelementsormembersoftheset,andtheyaresaidtobelongtoortobecontainedintheset.Thesetinturn,issaidtocontainorbecomposedofitselements.Weshallbeinterestedprimarilyinsetsofmathematicalobjects:setsofnumbers,setsofcurves,setsofgeometricfigures,andsoon.Inmanyapplications课后答案网itisconvenienttodealwithsetsinwhichnothingspecialisassumedaboutthenatureoftheindividualobjectsinthecollection.Thesearecalledabstractsets.Abstractsettheoryhasbeendevelopedtodealwithsuchcollectionsofarbitraryobjects,andfromthisgeneralitythetheoryderivesitspower.www.hackshp.cnNOTATIONS.Setsusuallyaredenotedbycapitalletters:A,B,C,….X,Y,Z;elementsaredesignatedbylower-caseletters:a,b,c,….x,y,z.WeusethespecialnotationX∈STomeanthat“xisanelementofS“or”xbelongstoS”.IfxdoesnotbelongtoS,wewritex∈S.Whenconvenient,weshalldesignatesetsbydisplayingtheelementsinbraces;forexample，thesetofpositiveevenintegerslessthan10isdenotedbythesymbol{2,4,6,8}whereasthesetofallpositiveevenintegersisdisplayedas{2,4,6,…},thedotstakingtheplaceof“andsoon”.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thefirstbasicconceptthatrelatesonesettoanotherisequalityofsets:DEFINITIONDEFINDEFINITIONDEFINITIONOFITIONOFOFSETSESETSETEQUALITYTEQUALEQUALITYITYTwosetsAandBaresaidtobeequal(oridentical)iftheyconsistofexactlythesameelements,inwhichcasewewriteA=B.Ifoneofthesetscontainsanelementnotintheother,wesaythesetsareunequalandwewriteA≠B.SUBSETS.FromagivensetSwemayformnewsets,calledsubsetsofS.Forexample,thesetconsistingofthosepositiveintegerslessthan10whicharedivisibleby4(theset{4,8})isasubsetofthesetofallevenintegerslessthan10.Ingeneral,wehavethefollowingdefinition.DEFINITIONDEFINDEFINITIONDEFINITIONOFITIONOFOFAAASUBSET.SSUBSET.UBSET.AsetAissaidtobeasubsetofasetB,andkhdaw.comwewriteABWhenevereveryelementofAalsobelongstoB.WealsosaythatAiscontainedinBorBcontainsA.Therelationisreferredtoassetinclusion.ThestatementABdoesnotruleoutthepossibilitythatBA.Infact,wemayhavebothABandBA,butthishappensonlyifAandBhavethesameelements.Inotherwords,A=BifandonlyifABandBA.Thistheoremisanimmediateconsequenceoftheforegoingdefinitionsofequalityandinclusion.IfABbutA≠B,thenwesaythatAisapropersubsetofB:weindicatethisbywritingAB.课后答案网Inallourapplicationsofsettheory,wehaveafixedsetSgiveninadvance,andweareconcernedonlywithsubsetsofthisgivenset.TheunderlyingsetSmayvaryfromoneapplicationtoanother;itwillbereferredtoastheuniversalsetofeachparticulardiscourse.www.hackshp.cnThenotation{X∣X∈S.andXsatisfiesP}willdesignatethesetofallelementsXinSwhichsatisfythepropertyP.Whentheuniversalsettowhichwearereferringidunderstood,weomitthereferencetoSandwesimplywrite{X∣XsatisfiesP}.Thisisread“thesetofallxsuchthatxsatisfiesp.”SetsdesignatedinthiswayaresaidtobedescribedbyadefiningpropertyForexample,thesetofallpositiverealnumberscouldbedesignatedas{X∣X>0};theuniversalsetSinthiscaseisunderstoodtobethesetofallrealnumbers.Ofcourse,theletterxisadummyandmaybereplacedbyanyotherconvenientsymbol.Thuswemaywritekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com {x∣x>0}={y∣y>0}={t∣t>0}andsoon.Itispossibleforasettocontainnoelementswhatever.Thissetiscalledtheemptysetorthevoidset,andwillbedenotedbythesymbolφ.Wewillconsiderφtobeasubsetofeveryset.Somepeoplefindithelpfultothinkofasetasanalogoustoacontainer(suchasabagorabox)containingcertainobjects,itselements.Theemptysetisthenanalogoustoanemptycontainer.Toavoidlogicaldifficulties,wemustdistinguishbetweentheelementxandtheset{x}whoseonlyelementisx,(Aboxwithahatinitisconceptuallydistinctfromthehatitself.)Inparticular,theemptysetφisnotthesameastheset{khdaw.comφ}.Infact,theemptysetφcontainsnoelementswhereastheset{φ}hasoneelementφ(Aboxwhichcontainsanemptyboxisnotempty).Setsconsistingofexactlyoneelementaresometimescalledone-elementsets.UNIONS,INTERSECTIONSUNIONUNIONS,INTERSECTIONSS,INTERSECTIONS,COMPLEMENTS.FromtwogivensetsAandB,wecanformanewsetcalledtheunionofAandB.ThisnewsetisdenotedbythesymbolA∪B(read:“AunionB”)AndisdefinedasthesetofthoseelementswhichareinA,inB,orinboth.Thatistosay,A∪BisthesetofallelementswhichbelongtoatleastoneofthesetsA,B.Similarly,theintersectionofAandB,denotedby课后答案网A∩B(read:“AintersectionB”)IsdefinedasthesetofthoseelementscommontobothAandB.TwosetsAwww.hackshp.cnandBaresaidtobedisjointifA∩B=φ.IfAandBaresets,thedifferenceA-B(alsocalledthecomplementofBrelativetoA)isdefinedtobethesetofallelementsofAwhicharenotinB.Thus,bydefinition,A-B={X|X∈AandXB}Theoperationsofunionandintersectionhavemanyformalsimilaritieswith(aswellasdifferencesfrom)ordinaryadditionandmultiplicationsofunionandintersection,itfollowsthatA∪B=B∪AandA∩B=B∩A.Thatistosay,unionandintersectionarecommutativeoperations.Thedefinitionsarealsophrasedinsuchawaythattheoperationsareassociative:khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com (A∪B)∪C=A∪(B∪C)and(A∩B)∩C=A=∩(B∩C).Theoperationsofunionandintersectioncanbeextendedtofiniteorinfinitecollectionsofsets.VocabularyVocabulVocabularyarySet集合propersubset真子集Settheory集合论universalset泛集Branch分支emptyset空集Analysiskhdaw.com分析voidset空集Geometry几何学union并，并集Notation记号，记法intersection交，交集Terminology术语，名词表complement余，余集Logic逻辑relativeto相对于Logical逻辑的finite有限的Systematic系统的disjoint不相交Informal非正式的课后答案网infinite无限的Formal正式的cardinalnumber基数，纯数Entity实在物www.hackshp.cnordinalnumber序数Element元素generality一般性，通性Abstractset抽象集subset子集Designate指定，divisible可除的Notion概念setinclusion集的包含Braces大括号immediateconsequence直接结果Identical恒同的，恒等的NotesNoteskhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 1.Indiscussinganybranchofmathematics,beitanalysis,algebra,orgeometry,itishelpfultousethenotationandterminologyofsettheory.意思是：在讨论数学的任何分支时，无论是分析，代数或分析，利用集合论的记号和术语是有帮助的。这一句中beitanalysis,algebra,orgeometry是以be开头的状语从句，用倒装形式。类似的句子还有：peoplewillusethetoolsinfurtherinvestigations,beitinmathematic,hysics,orwhathaveyou.2.Actually,weshalldiscussnotsomuchanewtheoryasanagreementaboutthepreciseterminologythatwewishtomoreorlessfamiliarideas.意思是：事实上，我恩将讨论的与其说是一种新理论，不如说是关于精确术语的一种约定，我们希望将它们应用到或多或少熟悉的思想上去。khdaw.com注意：notsomuchAasB在这里解释为“与其说A不如说B。”类似的用法如：Thisisnotsomuchalectureasafriendlychat.(与其说这是演讲不如说是朋友间的交谈。)3．TwosetsAandBaresaidtobeequaliftheyconsistofexactlythesameelements,inwhichcasewewriteA=B.数学上常常在给定了定义后，就用符号来表达。上面句子是常见句型。类似的表达法有：AsetAissaidtobeasubsetofasetB,andwewriteA=BwhenevereveryelementofAalsobelongstoB.课后答案网Thissetiscalledtheemptysetorthevoidset,andwillbedenotedbythesymbolΦ.www.hackshp.cnExerciseExerciExerciseseⅰ.TurnthefollowingmathematicalexpressionsinEnglish:ⅰ)x∈A∪Bⅱ)A∩B=φⅲ)A={Φ}ⅳ)A={X:a0.Thepositivenumberriscalledtheradiusoftheneighborhood.WedesignateN(p)byN(p;r)ifwewishtospecifyitsradius.Theinequalitiesp-r0,thereisaδ>0suchthat∣f(x)–A∣<εwhenever0<∣x–p∣<δ“One-sided”limitsmaybedefinedinasimilarway.Forexample,iff(x)→Aasx→pthroughvaluesgreaterthan课后答案网p,wesaythatAisright-handlimitoffatp,andweindicatethisbywritingf(x)=Awww.hackshp.cnInneighborhoodterminologythismeansthatforeveryneighborhoodN1(A),thereissomeneighborhoodN2(p)suchthatf(x)∈N1(A)wheneverx∈N1(A)andx>pLeft-handlimits,denotedbywritingx→p-,aresimilarlydefinedbyrestrictingxtovalueslessthanp.IffhasalimitAatp,thenitalsohasaright-handlimitandaleft-handlimitatp,bothofthesebeingequaltoA.Butafunctioncanhavearight-handlimitatpdifferentfromtheleft-handlimit.Thedefinitionofcontinuityofafunction.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Inthedefinitionoflimitwemadenoassertionaboutthebehaviouroffatthepointpitself.Moreover,eveniffisdefinedatp,itsvaluethereneednotbeequaltothelimitA.However,ifithappensthatfisdefinedatpandifitalsohappensthatf(p)=A,thenwesaythefunctionfiscontinuousatp.Inotherwords,wehavethefollowingdefinition.Definitionofcontinuityofafunctionatapoint.Afunctionfissaidtobecontinuousatapointpif(a)fisdefinedatp,and(b)f(x)=f(p)Thisdefinitioncanalsobeformulatedintermofneighborhoods.Afunctionfiscontinuousatpifforeveryneighborhoodkhdaw.comN1(f(p))thereisaneighborhoodN2(p)suchthatf(x)∈N1(f(p))wheneverx∈N2(p).Intheε-δterminology,wherewespecifytheradiioftheneighborhoods,thedefinitionofcontinuitycanberestatedadfollows:Functionfiscontinuousatpifforeveryε>0,thereisaδ>0suchthat∣f(x)–f(p)∣<εwhenever∣x–p∣<δIntherestofthislessonweshalllistcertainspecialpropertiesofcontinuousfunctionsthatareusedquitefrequently.Mostofthesepropertiesappearobviouswheninterpretedgeometrically;consequentlymanypeopleareinclinedto课后答案网acceptthemadself-evident.However,itisimportanttorealizethatthesestatementsarenomoreself-evidentthanthedefinitionofcontinuityitself,andthereforetheyrequireproofiftheyaretobeusedwithanydegreeofgenerality.Theproofsofmostofthesepropertiesmakeuseoftheleast-upperboundaxiowww.hackshp.cnmfortherealnumbersystem.THEOREM1.(Bolzano’stheorem)Letfbecontinuousateachpointofaclosedinterval[a,b]andassumethatf(a)anf(b)haveoppositesigns.Thenthereisatleastonecintheopeninterval(a,b)suchthatf(c)=0.THEOREM2.Sign-preservingpropertyofcontinuousfunctions.Letfbecontinuiousatcandsupposethatf(c)≠0.Thenthereisaninterval(c-δ,c+δ)aboutcinwhichfhasthesamesignasf(c).THEOREM3.Letfbecontinuousateachpointofaclosedinterval[a,b].Choosetwoarbitrarypointsx10,suchthat∣f(x)∣≤Mforallxin[a,b].THEOREM5.(extremevaluetheorem)Assumefiscontinuousonaclosedinterval[a,b].Thenthereexistpointscanddin[a,b]suchthatf(c)=supfandf(d)=inff.Note.Thistheoremshowsthatiffiscontinuouson[a,b],thensupfisitsabsolutemaximum,andinffisitsabsoluteminimum.VocabularyVocabulVocabularyarycontinuitykhdaw.com连续性assume假定,取continuous连续的specify指定,详细说明continuousfunction连续函数statement陈述,语句intuitive直观的right-handlimit右极限corresponding对应的left-handlimit左极限correspondence对应restrict限制于graph图形assertion断定approach趋近,探索,入门课后答案网consequently因而,所以tendto趋向prove证明regardless不管,不顾proofwww.hackshp.cn证明discontinuous不连续的bound限界jumpdiscontinuity限跳跃不连续leastupperbound上确界mathematician科学家greatestlowerbound下确界formulate用公式表示,阐述boundedness有界性limit极限maximum最大值Interval区间minimum最小值openinterval开区间extremevalue极值khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com equation方程extremum极值neighborhood邻域increasingfunction增函数midpoint中点decreasingfunction减函数symmetric对称的strict严格的radius半径(单数)uniformlycontinuous一致连续radii半径(复数)monotonic单调的inequality不等式monotonicfunction单调函数equivalentkhdaw.com等价的NotesNotes1.Itwadnotuntillateinthe18thcenturythatdiscontinuousfunctionsbeganappearinginconnectionwithvariouskindsofphysicalproblems.意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.这里Itwasnotuntil…that译为“直到……才”2.Thesymbolf(x)=Ameansthatforeveryε>0,thereisaδ>0,suchthat|f(x)-A|<εwhenever0<|x课后答案网–p|<δ注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.有时句中thereis可换为thereexists;suchthatwww.hackshp.cn可换为satisfying;whenever换成if或for.3.Let…andassume(suppose)…Then…这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课Let假设条件的大前提,assume(suppose)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.4.Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:f(x)approachesAasxapproachesp.Approach有时可代以tendto.如f(x)tendstoAasxtendstop.值得留意的是approach后不加to而tend之后应加to.5.asclosetoAasweplease=arbitrarilyclosetoA..Exercisekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com I.Fillineachblankwithasuitablewordtobechosenfromthewordsgivenbelow:independentdomaincorrespondenceassociatesvariablerange(a)Lety=f(x)beafunctiondefinedon[a,b].Then(i)xiscalledthe____________variable.(ii)yiscalledthedependent___________.(iii)Theinterval[a,b]iscalledthe___________ofthefunction.(b)khdaw.comInsetterminology,thedefinitionofafunctionmaybegivenasfollows:GiventwosetsXandY,afunctionf:X→Yisa__________which___________witheachelementofXoneandonlyoneelementofY.II.a)Whichfunction,theexponentialfunctionorthelogarithmicfunction,hasthepropertythatitsatisfiesthefunctionalequationf(xy)=f(x)+f(v)b)Givethefunctionalequationwhichwillbesatisfiedbythefunctionwhichyoudonotchoosein(a).III.Letfbeareal-valuedfunctiondefinedonasetSofrealnumbers.Thenwehavethefollowingtwodefinitions:课后答案网i)fissaidtobeincreasingonthesetSiff(x)0(ε不依赖于E上的点)存在一个正数δ使得当p和q属于E且|p–q|<δ时有|f(p)–f(q)|<ε,则称f在E上一致连续.khdaw.com课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－DifferentialCalculusHistoricalHistoriHistoricalHistoricalIntroductioncalIntroduIntroductionctionNewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyduetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.Thecentralideaofdifferentialcalculusisthenotionofderivative.Liketheintegral,thederivativeoriginatedfromaproblemingeometry—theproblemoffindingthetangentlineatapointofacurve.Unliletheintegral,however,thederivakhdaw.comtiveevolvedverylateinthehistoryofmathematics.Theconceptwasnotformulateduntilearlyinthe17thcenturywhentheFrenchmathematicianPierredeFermat,attemptedtodeterminethemaximaandminimaofcertainspecialfunctions.Fermat’sidea,basicallyverysimple,canbeunderstoodifwerefertoacurveandassumethatateachofitspointsthiscurvehasadefinitedirectionthatcanbedescribedbyatangentline.Fermatnoticedthatatcertainpointswherethecurvehasamaximumorminimum,thetangentlinemustbehorizontal.Thustheproblemoflocatingsuchextremevaluesisseentodependonthesolutionofanotherproblem,thatoflocatingthehorizontaltangents.Thisraisesthemoregeneralquestionofdeterminingthedirectionofthetange课后答案网ntlineatanarbitrarypointofthecurve.ItwastheattempttosolvethisgeneralproblemthatledFermattodiscoversomeoftherudimentaryideasunderlyingthenotionofderivative.Atfirstsightthereseemstobenoconnectionwhateverbetweentheproblemwww.hackshp.cnoffindingtheareaofaregionlyingunderacurveandtheproblemoffindingthetangentlineatapointofacurve.Thefirstpersontorealizethatthesetwoseeminglyremoteideasare,infact,ratherintimatelyrelatedappearstohavebeenNewton’steacher,IsaacBarrow(1630-1677).However,NewtonandLeibnizwerethefirsttounderstandtherealimportanceofthisrelationandtheyexploitedittothefullest,thusinauguratinganunprecedentederainthedevelopmentofmathematics.Althoughthederivativewasoriginallyformulatedtostudytheproblemoftangents,itwassoonfoundthatitalsoprovidesawaytocalculatevelocityand,moregenerally,therateofchangeofafunction.Inthenextsectionweshallconsideraspecialprobleminvolvingthecalculationofavelocity.Thesolutionofthkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com isproblemcontainsalltheessentialfcaturesofthederivativeconceptandmayhelptomotivatethegeneraldefinitionofderivativewhichisgivenbelow.AAProblemProbleProblemProblemInvolvingmInvolvinInvolvingInvolvingVelocitygVelociVelocitytySupposeaprojectileisfiredstraightupfromthegroundwithinitialvelocityof144feetpersecond.Neglectfriction,andassumetheprojectileisinfluencedonlybygravitysothatitmovesupandbackalongastraightline.Letf(t)denotetheheightinfeetthattheprojectileattainstsecondsafterfiring.Iftheforceofgravitywerenotactingonit,theprojectilewouldcontinuetomoveupwardwithaconstantvelocity,travelingadistanceof144feeteverysecond,andattimetwewoulehavef(t)=144t.Inactualpractice,gravitycausestheprojectiletoslowdownuntilitsvelocitydecreasestozeroandthenitdropsbacktoearth.Physicalexperimentssuggestthatastheprojectileisaloft,itsheightf(t)isgikhdaw.comvenbytheformula(1)f(t)=144t–16t2Theterm–16t2isduetotheinfluenceofgravity.Notethatf(t)=0whent=0andwhent=9.Thismeansthattheprojectilereturnstoearthafter9secondsanditistobeunderstoodthatformula(1)isvalidonlyfor00，根据连续函数的保号性质，存在c点的一个领域，在此领域里，Q（x）是正的；（4）因此在此领域内，对所有x≠c，Q（x）的分子和分母同号；（5）即是说，当x>c时，f（x）>f(c),而当x0不可能，同理可证课后答案网Q（c）<0也不真。www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－FirstOrderDifferentialEquationsAdifferentialequationisanequationbetweenspecifiedderivativesofafunction,itsvalves,andknownquantities.Manylawsofphysicsaremostsimplyandnaturallyformu-latedasdifferentialequations(orDE’s,asweshallwriteforshort).Forthisreason,DE’shavebeenstudiesbythegreatestmathematiciansandmathematicalphysicistssincethetimeofNewton..khdaw.comOrdinarydifferentialequationsareDE’swhoseunknownsarefunctionsofasingleva-riable;theyarisemostcommonlyinthestudyofdynamicsystemsandelectricnetworks.Theyaremucheasiertotreatthanpartialdifferentialequations,whoseunknownfunctionsdependontwoormoreindependentvariables.OrdinaryDE’sareclassifiedaccordingtotheirorder.TheorderofaDEisd课后答案网efinedasthelargestpositiveinteger,n,forwhichann-thderivativeoccursintheequation.Thiswww.hackshp.cnchapterwillberestrictedtorealfirstorderDE’softheformΦ(x,y,y′)=0(1)GiventhefunctionΦofthreerealvariables,theproblemistodetermineallrealfunctionsy=f(x)whichsatisfytheDE,thatis,allsolutionsof(1)inthefollowingsense.DEFINITIONDEFINDEFINITIONITIONAsolutionof(1)isadifferentiablefunctionf(x)suchthatΦ(x.f(x),f′(x))=0forallxintheintervalwheref(x)isdefined.EXAMPLEEXAMPEXAMPLEEXAMPLE1.LE1.Inthefirst-otherDEkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com x+yy′=0(2)thefunctionΦisapolynomialfunctionΦ(x,y,z)=x+yzofthreevariablesin-volved.Thesolutionsof(2)canbefoundbyconsideringtheidentityd(x²+y²)/dx=2(x+yyˊ).Fromthisidentity,oneseesthatx²+y²isacon-stantify=f(x)isanysolutionof(2).Theequationx²+y²=cdefinesyimplicitlyasatwo-valuedfunctionofx,foranypositiveconstantc.Solvingfory,wegettwosolutions,the(single-valued)khdaw.comfunctionsy=±(c-x²)0.5,foreachpositiveconstantc.Thegraphsoftheseso-lutions,theso-calledsolutioncurves,formtwofamiliesofscmicircles,whichfilltheupperhalf-planey>0andthelowerhalf-planey>0,respectively.Onthex-axis,wherey=0,theDE(2)impliesthatx=0.HencetheDEhasnosolutionswhichcrossthex-axis,exceptpossiblyattheorigin.Thisfactiseasilyoverlooked,becausethesolutioncurvesappeartocrossthex-axis;henceyˊdoesnotexist,andtheDE(2)isnotsatisfiedthere.课后答案网TheprecedingdifficultyalsoarisesifonetriestosolvetheDE(2)foryˊ.Dividingthroughbyy,onegetsyˊ=-x/y,anequationwhichcannotbesatisfiedify=0.TheprecedingdifficultyisthusavoidedifonerestrictsattentiontoregionswheretheDE(1)isnormal,inthefollowingsense.www.hackshp.cnDEFINITION.DEFINDEFINITION.ITION.Anormalfirst-orderDEisoneoftheformyˊ=F(x,y)(3)Inthenormalformyˊ=-x/yoftheDE(2),thefunctionF(x,y)iscontinuousintheupperhalf-planey>0andinthelowerhalf-planewherey<0;itisundefinedonthex-axis.FundamentalFundamFundamentalFundamentalTheorementalTheoreTheoremTheoremofmofofthethetheCalculus.CalculCalculus.us.Themostfamiliarclassofdifferentialequationsconsistsofthefirst-orderDE’softheformkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com yˊ=g(x)(4)SuchDE’sarenormalandtheirsolutionsaredescriedbythefundamentalthoremofthecalculus,whichreadsasfollows.FUNDAMENTALFUNDAFUNDAMENTALFUNDAMENTALTHEOREMMENTALTHEORTHEOREMTHEOREMOFEMOFOFTHETHTHETHECALCULUSECALCUCALCULUSLUS.Letthefunctiong(x)inDE(4)becontinuousintheintervala0)Σg(tk)Δtk,Δtk=tk-tk-1(5ˊ)isdefinedforeachfixedxasalimitofRicmannsums;itisnotnecessarytofindaformalexpressionfortheindefiniteintegral∫g(x)dxtogivemeaningtothedefiniteintegral∫xg(t)dt,providedonlythatg(t)iscontinuous.Suchfuanctionsastheerrorfunctioncrfx=(2/(π)0.5)∫xe-t²dtandthesineintegralfu0∞nctionSI(x)=∫x[(sint)/t]dtareindeedcommonlydefinedasdefiniteintegrals.课后答案网SolutionsSolutionSolutionsSolutionsandsandandIntegralsInIntegralstegralsAccordingtothedefinitiongivenaboveasolutionofaDEisalwaysafunctiwww.hackshp.cnon.Forexample,thesolutionsoftheDEx+yyˊ=0inExampleIarethefunctionsy=±(c-x²)0.5,whosegraphsaresemicirclesofarbitrarydiameter,centeredattheorigin.Thegraphofthesolutioncurvesare,however,moreeasilydescribedbytheequationx²+y²=c,describingafamilyofcirclescenteredattheorigin.InwhatsensecansuchafamilyofcurvesbeconsideredasasolutionoftheDE?Toanswerthisquestion,werequireanewnotion.DEFINITION.DEFINDEFINITION.ITION.AnintegralofDE(1)isafunctionoftwovariables,u(x,y),whichassumesaconstantvaluewheneverthevariableyisreplacedbyasolutiony=f(x)oftheDE.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Intheaboveexample,thefunctionu(x,y)=x²+y²isanintegraloftheDEx+yyˊ=0,because,uponreplacingthevariableybyanyfunction±(c-x²)0.5,weobtainu(x,y)=c.Thesecond-orderDEd²x/dt²=-x(2ˊ)becomesafirst-orderDEequivalentto(2)aftersettingdx/dx=y:y(dy/dx)=-x(2)Aswehaveseen,thecurvesu(x,y)=x²+y²=careintegralsofthisDE.WhentheDE(2khdaw.comˊ)isinterpretedasequationofmotionunderNewton’ssecondlaw,theintegralsc=x²+y²representcurvesofconstantenergyc.Thisillustratesanimportantprinciple:anintegralofaDErepresentingsomekindofmotionisaquantitythatremainsunchangedthroughthemotion.VocabularyVocabulVocabularyarydifferentialequation微分方程errorfunction误差函数ordinarydifferentialequation常微分方程sineintegralfunction正弦积分函数order阶,序diameter课后答案网直径derivative导数curve曲线knownquantities已知量www.hackshp.cnreplace替代unknown未知量substitute代入singlevariable单变量strip带形dynamicsystem动力系统exactdifferential恰当微分electricnetwork电子网络lineintegral线积分partialdifferentialequation偏微分方程pathofintegral积分路径classify分类endpoints端点polynomial多项式generalsolution通解khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com severalvariables多变量parameter参数family族rigorous严格的semicircle半圆existence存在性half-plane半平面initialcondition初始条件region区域uniqueness唯一性normal正规,正常Riemannsum犁曼加identity恒等(式)khdaw.comNotesNotes1.TheorderofaDEisdefinedasthelargestpositiveintegraln,forwhichannthderivativeoccursinthequestion.这是另一种定义句型,请参看附录IV.此外要注意nthderivative之前用an不用a.2.ThischapterwillberestrictedtorealfirstorderdifferentialequationsoftheformΦ(x,y,yˊ)=0意思是;文章限于讨论形如Φ(x,y,yˊ)=0的实一阶微分方程.有时可以用ofthetype代替oftheform课后答案网的用法.Theequationcanberewrittenintheformyˊ=F(x,y).3.Dividingthroughbyy,onegetsywww.hackshp.cnˊ=-x/y,…划线短语意思是:全式除以y4.Aswehaveseen,thecurvesu(x,y)=x²+y²=careintegralsofthisDE这里x²+y²=c因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.5.Theirsolutionsaredescribedbythefundamentaltheoremofthecalculus,whichreadsasfollows.意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下.句中readsasfollows就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.ExerciseExerciExercisesekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Ⅰ.TranslatethefollowingpassagesintoChinese:1.AdifferentialM(x,y)dx+N(x,y)dy,whereM,Narerealfunctionsoftwovariablesxandy,iscalledexactinadomainDwhenthelineintegral∫cM(x,y)dx+N(x,y)dyisthesameforallpathsofintegrationcinD,whichhavethesameendpoints.Mdx+Ndyisexactifandonlyifthereexistsacontinuouslydifferentiablefunctionu(x,y)suchthatM=u/x,N=u/y.2.ForanynormalfirstorderDEyˊ=F(x,y)andanyinitialx0,theinitialvalveproblemconsistsoffindingthesolutionorsolutionsoftheDE,forx>x0whichassumesagiveninitialvalvef(x0)=c.3.Toshowthattheinitialvalveproblemiswell-setrequiresprovingtheoremsofexistence(thereisasolution),uniqueness(thereisonlyonesolution)andcontinuity(thesolutiondependscontinuouslyontkhdaw.comheinitialvalue).Ⅱ.TranslatethefollowingsentencesintoEnglish:1)因为y=ч(x)是微分方程dy/dx=f(x,y)的解,故有dч(x)/dx=f(x,ч(x))2)两边从x0到x取定积分得xf(x,ч(x))dxxч(x)-ч(x0)=∫x000)isa___________.6.theniscalleda_________________sequence.7.isa_________oftwoequationswiththree_______.8.Numberssuchasandπarecalled________numbers.9.Therelationbetweenthecelementsofasetofrealnumbersdenotedby<(or<;>;>)iscalledan_________relation.10.Therelationbetweensets,denotedbyiscalledan_________relation.Ⅱkhdaw.com.Eachofthefollowingsentencesisgrammaticallywrong.Correctthesesentences.1.Letisacontinuousfunctiondefinedon[a,b].2.Differentiatingbothsidesofwithrespecttox,theequationbecomesy’=—x/y3.Takethederivativesofbothsidesoftheequation,wegetx+yy’=0.4.TheprimtiveofhereCisaconstant.5.WesaythathasalimitAatifapproachestoAwhenXtendsto.Ⅲ.TranslatethefollowingsentencesintoChinese(payattentiontothephrasesunderlined):1.Wearenowinapositionto课后答案网provethemaintheorem.2.Ananalogousargumentgivesaproofofthecorrespondingtheoremfordecreasingfunctions.3.Animmediateconsequenceofwww.hackshp.cnBolzano’stheoremistheintermediate-valuetheoremforcontinuousfunctions.4.Weclaimthathasnorealsolution,Infactifisarealsolution,thenwehavewhichisimpossible.5.Itisclearthatthemethoddescribedabovealsoappliestothegeneralcase.6.Itiseasytoshowthathasderivativesuptoordernatthepointx=0,wheren>1.Ⅳ.TranslatethefollowingpassageintoChinese:1.Itishelpfultointroducethewords”local”and“global”tocontrasttwotypesofsituationsthatfrequentlyarise.IfweareconsideringagivensetD,thenwesaythatanyspecificpropertyholds“locally”atofDifitistreatifandatallpointsnear；thustherewillbeanopenballBapoutandtheproperkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com tywillholdforall.Ontheotherhand,apropertythatholdsatallpointsinDissaidtohold“globally”inD.2.Thestudyofsequencesisconcernedprimarilywiththefollowingtypeofquestion:ifeachtermofasequencehasacertainproperty,suchascontinuity,differentiabilityorintegrability,towhatextendisthispropertytransferredtothelimitfunction?khdaw.com课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－LinearAlgebraForthedefinitionthatfollowsweassumethatwearegivenaparticularfieldK.ThescalarstobeusedaretobeelementsofK.DEFINITION.AvectorspaceisasetVofelementscalledvectorssatisfyingthefollowingaxioms.(A)Toeverypair,xandy,ofvectorsinVcorrespondsavectorx+y,calledthesumofxandy,insuchawaythat.(1)additioniscommutative,x+y=y+x.(2)additionisassociative,x+(y+z)=(x+y)+z.khdaw.com(3)thereexistsinVauniquevector0(calledtheorigin)suchthatx+0=xforeveryvectorx,and(4)toeveryvectorxinVtherecorrespondsauniquevector-xsuchthatx+(-x)=0.(B)Toeverypair,αandx,whereαisascalarandxisavectorinV,therecorrespondsavectorαxinV,calledtheproductofαandx,insuchawaythat(1)multiplicationbyscalarsisassociative,α(βx)=(αβ)x(2)1x=xforeveryvectorx.课后答案网(C)(1)multiplicationbyscalarsisdistributivewithrespecttovectoraddition,α(x+y)=αx+βy,and(2)multiplicationbyvectorsisdistributivewithrespecttoscalaraddition,(www.hackshp.cnα+β)x=αx+βx.TherelationbetweenavectorspaceVandtheunderlyingfieldKisusuallydescribedbysayingthatVisavectorspaceoverK.TheassociatedfieldofscalarsisusuallyeithertherealnumbersRorthecomplexnumbersC.IfVislinearspaceandM真包含于V,andifαu-vbelongtoMforeveryuandvinMandeveryα∈K,thenMislinearsubspaceofV.IfU={u1,u2,…}isacollectionofpointsinalinearspaceV,thenthe(linear)spanofthesetUisthesetofallpointsotheform∑ciui,whereci∈K,andallbutafinitenumberofthescalarsciare0.ThespanofUisalwaysalinearsubspaceofV.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Akeyconceptinlinearalgebraisindependence.Afiniteset{u1,u2,…,uk}issaidtobelinearlyindependentinViftheonlywaytowrite0=∑ciuiisbychoosingalltheci=0.Aninfinitesetislinearlyindependentifeveryfinitesetisindependent.Ifasetisnotindependent,itislinearlydependent,andinthiscase,somepointinthesetcanbewrittenasalinearcombinationofotherpointsintheset.AbasisforalinearspaceMisanindependentsetthatspansM.AspaceMisfinite-dimensionalifitcanbespannedbyafiniteset;itcanthenbeshownthateveryspanningsetcontainsabasis,andeverybasisforMhasthesamenumberofpointsinit.ThiscommonnumberiscalledthedimensionofM.Anotherkeyconceptisthatoflineartransformation.IfVandWarelinearspaceswiththesamescalarfieldK,amappingLfromVintoWiscalledlinearifL(u+v)=L(u)+L(v)andL(khdaw.comαu)=αL(u)foreveryuandvinVandαinK.WithanyI,areassociatedtwospeciallinearspaces:ker(L)=nullspaceofL=L-1(0)={allx∈VsuchthatL(X)=0}Im(L)=imageofL=L(V)={allL(x)forx∈V}.Thenr=dimensionofIm(L)iscalledtherankofL.IfWalsohasdimensionn,thenthefollowingusefulcriterionresults:Lis1-to-1ifandonlyifLisonto.Inparticular,ifLisalinearmapofVintoitself,andtheonlysolutionofL(x)=0is0,thenLISontoandisthereforeanisomorphismofVontoV,andhasaninverseL课后答案网-1.SuchatransformationVisalsosaidtobenonsingular.SupposenowthatLisalineartransformationfromVintoWwheredim(V)=nanddim(W)=www.hackshp.cnm.Chooseabasis{υ1,υ2,…,υn}forVandabasis{w1,w2,…,wm}forW.ThenthesedefineisomorphismsofVontoKnandWontoKm,respectively,andtheseinturninducealineartransformationAbetweenthese.Anylineartransformation(suchasA)betweenKnandKmisdescribedbymeansofamatrix(aij),accordingtotheformulaA(x)=y,wherex={x1,x2,…,xn}y={y1,y2,…,ym}andYnj=Σj=iaijxiI=1,2,…,m.ThematrixAissaidtorepresentthetransformationLandtobetherepresentationinducedbytheparticularbasischosenforVandW.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com IfSandTarelineartransformationsofVintoitself,soisthecompositictransformationST.IfwechooseabasisinV,andusethistoobtainmatrixrepresentationsforthese,withArepresentingSandBrepresentingT,thenSTmusthaveamatrixrepresentationC.ThisisdefinedtobetheproductABofthematrixesAandB,andleadstothestandardformulaformatrixmultiplication.Theleastsatisfactoryaspectoflinearalgebraisstillthetheoryofdeterminantseventhoughthisisthemostancientportionofthetheory,datingbacktoLeibnizifnottoearlyChina.Onestandardapproachtodeterminantsistoregardann-by-nmatrixasanorderedarrayofvectors(u1,u2,…,un)andthenitsdeterminantdet(A)asafunctionF(u1,u2,…,un)ofthesenvectorswhichobeyscertainrules.khdaw.comThedeterminantofsuchanarrayAturnsouttobeaconvenientcriterionforcharacterizingthenonsingularityoftheassociatedlineartransformation,sincedet(A)=F(u1,u2,…,un)=0ifandonlyifthesetofvectorsuiarelinearlydependent.Therearemanyotherusefulandelegantpropertiesofdeterminants,mostofwhichwillbefoundinanyclassicbookonlinearalgebra.Thus,det(AB)=det(A)det(B),anddet(A)=det(A"),whereA"isthetransposeofA,obtainedbytheformulaA"=(aji),therebyrotatingthearrayaboutthemaindiagonal.Ifasquarematrixistriangular,meaningthatallitsentriesabovethemaindiagonalare0,thendet(A)turnsouttobeexactlytheproductofthediagonalentries.Anotherusefulconceptisthatofeigenvalue.AscalarissaidtobeaneigenvalueforatransformationTifthereisanonzerovector课后答案网υwithT(υ)λυ.Itisthenclearthattheeigenvalueswillbethosenumbersλ∈KsuchthatT-λIisasingulartransformation.AnyvectorinthenullspaceofT-λIiscalledaneigenvectorofTassociatedwitheigenvalueλ,andtheirspantheeigenspace,Eλ.ItisinvariantundertheactionofT,meaningthatTcarriesEλintoitself.TheeigenvaluesofTarethenexactlythesetofrootsofthewww.hackshp.cnpolynomialp(λ)=det(T-λI).IfAisamatrixrepresentingT,thenonehasp(λ)det(A-λI),whichpermitsonetofindtheeigenvaluesofTeasilyifthedimensionofVisnottoolarge,orifthematrixAissimpleenough.TheeigenvaluesandeigenspacesofTprovideameansbywhichthenatureandstructureofthelineartransformationTcanbeexaminedindetail.VocabularyVocabulVocabularyarylinearalgebra线性代数non-singular非奇异field域isomorphism同构vector向量isomorphic同构khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com scalar纯量,无向量matrix矩阵(单数)vectorspace向量空间matrices矩阵(多数)span生成,长成determinant行列式independence无关(性),独立(性)array阵列dependence有关(性)diagonal对角线linearcombination线性组合triangular三角形的basis基(单数)entry表值,元素basiskhdaw.com基(多数)eigenvalue特征值,本征值dimension维eigenvector特征向量lineartransformation线性变换invariant不变,不变量nullspace零空间row行rank秩column列singular奇异systemofequations方程组homogeneous齐次课后答案网www.hackshp.cnNotesNotes1.IfU={u1,u2,…}isacollectionofpointsinalinearspaceV,thenthe(linear)spanofthesetUisthesetofallpointsoftheform∑ciui,wwhereci∈K,andallbutafinitenumberofscalarscIare0.意思是:如果U={u1,u2,…}是线性空间V的点集,那么集U的(线性)生成是所有形如∑ciui的点集,这里ci∈K,且除了有限个ci外均为0.2.Afiniteset{u1,u2,…,uk}issaidtobelinearlyindependentiftheonlywaytowrite0=∑ciuIisbychoosingalltheci=0.这一句可以用更典型的句子表达如下:Afiniteset{u1,u2,…,uk}issaidtobelinearlyindependentinVif∑ciuiisbychoosingalltheci=0.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 这里independent是形容词,故用linearly修饰它.试比较F(x)isacontinuousperiodicfunction.这里periodic是形容词但它前面的词却用continuous而不用continuously,这是因为continuous这个词不是修饰periodic而是修饰作为整体的名词periodicfunction.3.ThenthesedefineisomorphismsofVontoKnandWontoKMrespectively,andtheseinturninducealineartransformationAbetweenthese.这里第一个these代表前句的两个基(basis);第二个these代表isomorphisms;第三个these代表什么留给读者自己分析.4.Theleastsatisfactoryaspectoflinearalgebraisstillthetheoryofdeterminants-khdaw.com意思是:线性代数最令人不满意的方面仍是有关行列式的理论.leastsatisfactory意思是:最令人不满意.5.Ifasquarematrixistriangular,meaningthatallitsentriesabovethemaindiagonalare0,thendet(A)turnsouttobeexactlytheproductofthediagonalentries.意思是:如果方阵是三角形的,即所有在主对角线上方的元素均为零,那末det(A)刚好就是对角线元素的乘积.这里meaningthat可用thatistosay代替,turnsouttobe解为”结果是”.ExerciseExerciExerciseseI.Answerthefollowingquestions:1.Howcanwedefinethelinearindependenceofaninfiniteset?2.LetTbealineartransformation(T:V→W)whoseassociatedmatrixisA.Giveacriterionforthenon-singularityofthetransformationT.课后答案网3.Whereistheentrya45ofam-by-nmatrix(m>4;n>5)located?4.LetA,Bbetworectangularmatrices.Underwhatconditionistheproductmatrixwell-defined?www.hackshp.cnII.TranslatethefollowingtwoexamplesandtheirproofsintoChinese:1.Example1.Letukk=t,k=0,1,2,...andtreal.Showthattheset｛u0,u1,u2,…｝isindependent.Proof:Bythedefinitionofindependenceofaninfiniteset,itsufficestoshowthatforeachn,then+1polynomialsunnk0,u1,...,unareindependent.Arelationoftheform∑k=0ckuk=0means∑k=0ckt=0forallt.Whent=0,thisgivescnk0=0.Differentiatingbothsidesof∑k=0ckt=0andsettingt=0,wefindthatc1=0.Repeatingtheprocess,wefindthateachcocfficientiszero2.Example2.LetVbeafinitedimensionallinearspace,TheneveryfinitebasisforVhasthesamenumberofelements.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Proof:LetSandTbetwofinitebasesforV.SupposeSconsistsofkelemntsandTconsistsofmelements.SinceSisindependentandspansV,everysetofk+1elementsinVisdependent.ThereforeeverysetofmorethankelementsinVisdependent.SinceTisanindependentset,wemusthavem(greaterthan),or≥(greaterthanorequalto)iscalledalinearinequalityintwovariablesxandy.Thuslx+my+n≤0,lx+my+n≥0arealllinearliequalities.Asolutionofalinearinequalityisanorderedpair(x,y)ofnumbersxandyforwhichtheinequalityistrue.EXAMPLE1GraphthesolutionsetofthepairofinequalitiesSOLUTIONLetAbethesolutionsetoftheinequalityx+y-7≤0andBbethatoftheinequalityx-3y+6≥0.ThenA∩Bisthesolutionsetofthegivenpairofinequalities.SetAisrepresentedbytheregionshadedwithhorizontallinesandsetBbytheregionshadedwithverticallinesinFig.1.Thereforethekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com crossed-hatchedregionrepresentsthesolutionsetofthegivenpairofinequalities.Observethatthepointofintersection(3.4)ofthetwolinesisinthesolutionset.Generallyspeaking,linearprogrammingproblemsconsistoffindingthemaximumvalueorminimumvalueofalinearfunction,calledtheobjectivefunction,subjecttosomelinearconditions,calledconstraints.Forexample,wemaywanttomaximizetheproductionorprofitofacompanyortomaximizethenumberofairplanesthatcanlandatortakeofffromanairportduringpeakhours;orwemaywanttominimizethecostofproductionoroftransportationortominimizegroceryexpenseswhilestillmeetingtherecommendednutritionalrequirements,allsubjecttocertainrestrictions.Linearprogrammingisaveryusefultoolthatcaneffectivelybeappliedtosolveproblemsofthiskind,asillustratedbythefollowingexample.khdaw.comEXAMPLE2Maximizethefunctionf(x,y)=5x+7ysubjecttotheconstraintsx≥0y≥0x+y-7≤02x-3y+6≥0SOLUTIONFirstwefindthesetofallpossiblepairs(x,y)ofnumbersthatsatisfyallfourinequalities.Suchasolutioniscalledafeasiblesulutionoftheproblem.Forexample,(0,0)isafeasiblesolutionsince(0,0)satisfiesthegivenconditions;soare(1,2)and(4,3).课后答案网Secondly,wewanttopickthefeasiblesolutionforwhichthegivenfunctionf(x,y)isamaximumorminimum(maximuminthiscase).Suchafeasiblesolutioniscalledanoptimalsolution.www.hackshp.cnSincetheconstraintsx≥0andy≥0restrictustothefirstquadrant,itfollowsfromexample1thatthegivenconstraintsdefinethepolygonalregionboundedbythelinesx=0,y=0,x+y-7=0,and2x-3y+6=0,asshowninFig.2.Fig.2.Observethatiftherearenoconditionsonthevaluesofxandy,thenthefunctionfcantakeonanydesiredvalue.Butrecallthatourgoalistodeterminethelargestvalueoff(x,y)=5x+7ywherethevaluesofxandyarerestrictedbythegivenconstraints:thatis,wemustlocatethatpoint(x,y)inthepolygonalregionOABCatwhichtheexpression5x+7yhasthemaximumpossiblevalue.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Withthisinmind,letusconsidertheequation5x+7y=C,whereCisanynumber.Thisequationrepresentsafamilyofparallellines.Severalmembersofthisfamily,correspondingtodifferentvaluesofC,areexhibitedinFig.3.Noticethatastheline5x+7y=CmovesupthroughthepolygonalregionOABC,thevalueofCincreasessteadily.Itfollowsfromthefigurethattheline5x+7y=43hasasingularpositioninthefamilyoflines5x+7y=C.Itisthelinefarthestfromtheoriginthatstillpassesthroughthesetoffeasiblesolutions.ItyieldsthelargestvalueofC:43.(Remember,wearenotinterestedinwhathappensoutsidetheregionOABC)Thusthelargestvalueofthefunctionf(x,y)=5x+7ysubjecttotheconditionthatthepoint(x,y)mustbelongtotheregionOABCis43;clearlythismaximumvalueoccursatthepointB(3,4).Fig.3.khdaw.comConsiderthepolygonalregionOABCinFig.3.ThisshadedregionhasthepropertythatthelinesegmentPQjoininganytwopointsPandQintheregionliesentirelywithintheregion.Suchasetofpointsinaplaneiscalledaconvexset.Aninterestingobservationaboutexample2isthatthemaximumvalueoftheobjectivefunctionfoccursatacornerpointofthepolygonalconvexsetOABC,thepointB(3,4).Thefollowingcelebratedtheoremindicatesthatitwasnotaccidental.THEOREM(Fundamentaltheoremoflinearprogramming)Alinearobjectivefunctionfdefinedoverapolygonalconvexsetattainsamaximum(orminimum)valueatacornerpointoftheset.Wenowsummarizetheprocedureforsolvingalinearprogrammingproblem:课后答案网1.Graphthepolygonalregiondeterminedbytheconstraints.2.Findthecoordinatesofthecornerpointsofthepolygon.www.hackshp.cn3.Evaluatetheobjectivefunctionatthecornerpoints.4.Identifythecornerpointatwhichthefunctionhasanoptimalvalue.VocabularyVocabulVocabularyarylinearprogramming线形规划quadrant象限objectivefunction目标函数convex凸的constraints限制条件，约束条件convexset凸集feaseblesolution容许解，可行解cornerpoint偶角点khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com optimalsolution最优解simplexmethod单纯形法NotesNotes1.AYaleeconomist,aStanfordmathematician这里YaleStanford是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)2.subjecttosomelincarconditions解作“在某些线形条件的限制下”。试比较下面各词组在用法上的异同：subjectto;underthecondition(s)of;satisfyingthecondition(s).3.celebratedtheorem意思为“著名定理”。4.Since…itfollows…与Notice…itfollow…都是数学中常用的句型，以表达“根据什么，可得什么”这一意思，请参看附录khdaw.comIII。5.本课多次用到recall,observe,notice,remember等词，用以提醒读者一些已知的事实或定理，读者可从这些例句中体会这些词的用法。请参看附录III。ExerciseExerciExerciseseI.TranslatethefollowingpassageintoChinesre:Althoughsatisfactoryfortwo-variablelinearprograms,thecorner-pointmethodisnotcomputationallyeffectivewhenextendedbeyondthree-variablesituations.Itsusefulnessisrestrictedtointroducingthegeneralideaoflinearprogramming,andtheneedformorepowefulsolutiontechniquesisobvious.Thesimpl课后答案网exmethod,devisedbyGeorgeDantziginthelate1940’s,was,centraltotheexplosioninlinearprogrammingapplicationsthatoccurredinthe1950’sand1960’s.Althoughmorepowerfultechniquesexistforsolvingspecialproblems,thesimplexmethodisstillthemostcomputationallyeffectivegeneralmethodavailableforsolvingthewidestvarietyoflinearprogrammingproblems.www.hackshp.cnII.Readthecontentofthislessoncarefullyandcompletethefollowingsentences:1.Linearprogrammingproblemsconsistof2.Thesetoffeasiblesolutionsofalinearprogrammingproblemis3.Apolygonalregionisaregionboundedby4.AconvexsetinaplaneisIII.TranslatethefollowingsentencesintoEnglish:1.点(3,4)是两条直线x+y-7=0和2x-3y+6=0的交点。khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 2.(3,4)是例2的最优解。3.计算目标函数在偶角点处的值，然后进行比较，求出目标函数的最大值或最小值。khdaw.com课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－MathematicalDiscoveryTogivetheflavorofPolya’sthinkingandwritinginaverybeautifulbutsubtlecase,acasethatinvolveachangeintheconceptualmode,IshallquoteatlengthfromhisMathematicalDiscovery(vol.II,pp.54ff):EXAMPLEItakethelibertyalittleexperimentwiththereader,Ishallstateasimplebutnottoocommonplacetheoremofgeometry,andthenIshalltrytoreconstructthesequenceofidoasthatledtoitsproof.Ishallproceedslowly,veryslowly,revealingoneclueaftertheother,andrevealingeachgradually.IthinkthatbeforeIhavefinishedthewholestory,thereaderwillseizethemainidea(unlessthereissomespecialhamperingcircumstance).Butthismainideaisratherunexpected,andsothereadermayexperiencethepleasureofalittlediscovery.khdaw.comA.Ifthreecircleshavingthesameradiuspassthroughapoint,thecirclethroughtheirotherthreepointsofintersectionalsohasthesameradius.Fig.1Threecirclesthroughonepoint.Thisisthetheoremthatwehavetoprove.Thestatementisshortandclear,butdoesnotshowthedetailsdistinctlyenough.Ifwedrawafigure(Fig.1)andintroducesuitablenotation,wearriveatthefollowingmoreexplicitrestatement:B.Threecirclesk,l,mhavethesameradiusrandpassthroughthesamepointO.Moreover,landmintersectinthepointA,mandkinB,kandlinC.ThenthecircleethroughA,B,Chasalsotheradius课后答案网Fig.2toocrowded.Fig.1exhibitsthefourcirclesk,l,m,andeandtheirfourpointsofinwww.hackshp.cntersectionA,B,C,andO.Thefigureapttobeunsatisfactory,however,foritisnotsimple,anditisstillincomplete;somethingseemstobemissing;wefailedtotakeintoaccountsomethingessential,itseems.Wearedialingwithcircles.Whatisacircle?Acircleisdeterminedbycenterandradius;allitspointshavethesamedistance,measuredbythelengthoftheradius,fromthecenter.Wefailedtointroducethecommonradiusr,andsowefailedtotakeintoaccountanessentialpartofthehypothesis.Letus,therefore,introducethecenters,Kofk,Lofl,andMofm.Whereshouldweexhibittheradiusr?thereseemstobenoreasontotreatanyoneofthethreegivencirclesk;l,andmoranyoneofthethreepointsofintersectionA,B,andCbetterthantheothers.Wearepromptedtocokhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com nnectallthreecenterswithallthepointsofintersectionoftherespectivecircle;KwithB,C,andO,andsoforth.Theresultingfigure(Fig.2)isdisconcertinglycrowded.Therearesomanylines,straightandcircular,thatwehavemuchtroubleold－fashionedmagazines.Thedrawingisambiguousonpurpose;itpresentsacertainfigureifyoulooktitintheusualway,butifyouturnittoacertainpositionandlookatitinacertainpeculiarway,suddenlyanotherfigureflashesonyou,suggestingsomemoreorlesswittycommentonthefirst.Canyourecognizeinourpuzzlingfigure,overladenwithstraightandcircles,asecondfigurethatmakessense?Wemayhitinaflashontherightfigurehiddeninouroverladendrawing,khdaw.comorwemayrecognizeitgradually.Wemaybeledtoitbytheefforttosolvetheproposedproblem,orbysomesecondary,unessentialcircumstance.Forinstance,whenweareabouttoredrawourunsatisfactoryfigure,wemayobservethatthewholefigureisdeterminedbyitsrectilinearpart(Fig.3).Thisobservationseemstobesignificant.Itcertainlysimplifiesthegeometricpicture,anditpossiblyimprovesthelogicalsituation.Itleadsustorestateourtheoreminthefollowingform.C.IftheninesegmentsKO,KC,KB,课后答案网LC,LO,LA,MB,MA,MO,areallequaltor,thereexistsapointEsuchthatthethreesegmentswww.hackshp.cnEA,EB,EC,arealsoequaltor.Fig.3Itremindsyou－ofwhat?ThisstatementdirectsourattentiontoFig.3.Thisfigureisattractive;itremindsusofsomethingfamiliar.(Ofwhat?)Ofcourse,certainquadrilateralsinFig.3.suchasOLAMhave,byhypothesis,fourequalsided,theyarerhombi,ArhombusIafamiliarobject;havingrecognizedit,wecan“see“thefigurebetter.(Ofwhatdoesthewholefigureremindus?)khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Oppositcsidesofarhombusareparallel.Insistingonthisremark,werealizethatthe9segmentsofFig.3.areofthreekinds;segmentsofthesamekind,suchasAL,MO,andBK,areparalleltoeachother.(Ofwhatdoesthefigureremindusnow?)Weshouldnotforgettheconclusionthatwearerequiredtoattain.Letusassumethattheconclusionistrue.IntroducingintothefigurethecenterEorthecirclee,anditsthreeradiiendinginA,B,andC,weobtain(supposedly)stillmorerhombi,stillmoreparallelsegments;seeFig.4.(Ofwhatdoesthewholefigureremindusnow?)Ofcourse,Fig.4.istheprojectionofthe12edgesofaparallelepipedhavingtheparticularitythattheprojectionofalledgesareofequallength.khdaw.comFig.4ofcourse!Fig.3.istheprojectionofa“nontransparent“parallelepiped;weseeonly3faces,7vertices,and9edges;3faces,1vertex,and3edgesareinvisibleinthisfigure.Fig.3isjustapartofFig.4.butthispartdefinesthewholefigure.Iftheparallelepipedandthedirectionofprojectionaresochosenthattheprojectionsofthe9edgesrepresentedinFig.3areallequaltor(astheyshouldbe,byhypothesis),theprojectionsofthe3remainingedgesmustbeequaltor.These3linesoflengthrareissuedfromtheprojectionofthe8th,theinvisiblevertex,andthisprojectionEisthecenterofacirclepassingthroughthepointsA,B,andC,theradiusofwhichisr.Ourtheoremisproved,andprovedbyasurprising,artisticconceptionof课后答案网aplanefigureastheprojectionofasolid.(Theproofusesnotionsofsolidgeometry.Ihopethatthisisnotatreatwrong,butifsoitiseasilyredressed.NowthatwecancharacterizethesituationofthecenterEsosimply,itiseasytoexaminethelengthsEA,EB,andECindependentlyofanysolidwww.hackshp.cngeometry.Yetweshallnotinsistonthispointhere.)Thisisverybeautiful,butonewonders.Isthisthe“lightthatbreaksforthlikethemorning.“theflashinwhichdesireisfulfilled?OrisitmerelythewisdomoftheMondaymorningquarterback?Dotheseideasworkoutintheclassroom?FollowupsofattemptstoreducePolya’sprogramtopracticalpedagogicsaredifficulttointerpret.Thereismoretoteaching,apparently,thanagoodideafromamaster.——FromFromMathematicalMathemMathematicalMathematicalExperienceaticalExperiExperienceenceVocabularyVocabulVocabularyarykhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com subtle巧妙的，精细的clue线索，端倪hamper束缚，妨碍disconcert使混乱，使狼狈ambiguous含糊的，双关的witty多智的，有启发的rhombikhdaw.com菱形（复数）rhombus菱形parallelepiped平行六面体projection射影solidgeometry立体几何pedagogics教育学，教授法commonplace老生常谈；平凡的课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－MathematicansLeonhardEulerwasbornonApril15,1707,inBasel,Switzerland,thesonofamathematicianandCaivinistpastorwhowantedhissontobecomeapastoraswell.AlthoughEulerhaddifferentideas,heenteredtheUniversityofBaseltostudyHebrewandtheology,thusobeyinghisfather.Hishardworkattheuniversityandremarkableabilitybroughthimtotheattentionofthewell-knownmathematicianJohannBernoulli(1667—1748).Bernoulli,realizingEuler’stalents,persuadedEuler’sfathertochangehismind,andEulerpursuedhisstudiesinmathematics.Attheageofnineteen,Euler’sfirstoriginalworkappeared.HispaperfailedtowintheParisAcademyPrizein1727;howeverthislosswascompensatedforlaterashewontheprizetwelvetimes.khdaw.comAttheageof28,EulercompetedforthePairsprizeforaprobleminastronomywhichseveralleadingmathematicianshadthoughtwouldtakeseveralmonthstosolve.Totheirgreatsurprise,hesolveditinthreedays!Unfortunately,theconsiderablestrainthatheunderwentinhisrelentlesseffortcausedanillnessthatresultedinthelossofthesightofhisrighteye.Attheageof62,Eulerlostthesightofhislefteyeandthusbecametotallyblind.Howeverthisdidnotendhisinterestandworkinmathematics;instead,hismathematicalproductivityincreasedconsiderably.OnSeptember18,1783,whileplayingwithhisgrandsonanddrinkingtea,Eulersufferedafatalstroke.课后答案网Eulerwasthemostprolificmathematiciantheworldhaseverseen.Hemadesignificantcontributionstoeverybranchofmathematics.Hehadphenomenalmemory:Hecouldremembereveryimportantformulaofhistime.Agenius,hecouldworkanywhereandunderanycondition.www.hackshp.cnGeorgecantor(March3,1845—June1,1918),thefounderofsettheory,wasborninSt.PetersburgintoaJewishmerchantfamilythatsettledinGermanyin1856.Hestudiedmathematics,physicsandphilosophyinZurichandattheUniversityofBerlin.Afterreceivinghisdegreein1867inBerlin,hebecamealecturerattheuniversityofHallefrom1879to1905.In1884,underthestrainofoppositiontohisideasandhiseffortstoprovethecontinuumhypothesis,hesufferedthefirstofmanyattacksofdepressionwhichcontinuedtohospitalizehimfromtimetotimeuntilhisdeath.Thethesishewroteforhisdegreeconcernedthetheoryofnumbers;however,hearrivedatsettheoryfromhisresearchconcerningtheuniquenessoftrigonometricseries.In1874,heintroducedforthefirsttimetheconceptofcardinalkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com numbers,withwhichheprovedthattherewere“more”transcendentalnumbersthanalgebraicnumbers.Thisresultcausedasensationinthemathematicalworldandbecamethesubjectofagreatdealofcontroversy.CantorwastroubledbytheoppositionofL.Kronecker,buthewassupportedbyJ.W.R.DedekindandG.Mittagleffer.Inhisnoteonthehistoryofthetheoryofprobability,herecalledtheperiodinwhichthetheorywasnotgenerallyacceptedandcriedout“theessenceofmathematicsliesinitsfreedom!”Inadditiontohisworkontheconceptofcardinalnumbers,helaidthebasisfortheconceptsofordertypes,transfiniteordinals,andthetheoryofrealnumbersbymeansoffundamentalsequences.HealsostudiedgeneralpointsetsinEuclideanspaceanddefinedtheconceptsofaccumulationpoint,closedsetandopenset.Hewasapioneerindimensiontheory,whichledtothedevelopmentoftopology.KantorovichwasbornonJanuary19,1912,inSt.Petersburg,nowcalledLenikhdaw.comngrad.HegraduatedfromtheUniversityofLeningradin1930andbecameafullprofessorattheearlyageof22.Attheageof27,hispioneeringcontributionsinlinearprogrammingappearedinapaperentitledMathematicalMethodsfortheOrganizationandplanningofproduction.In1949,hewasawardedaStalinPrizeforhiscontributionsinabranchofmathematicscalledfunctionalanalysisandin1958,hebecameamemberoftheRussianAcademyofSciences.Interestinglyenough,in1965,kantorovichwonaLeninPrizeforthesameoutstandingworkinlinearprogrammingforwhichhewasawardedtheNobelPrize.Since1971,hehasbeenthedirectoroftheInstituteofEconomicsofManagementinMoscow.PaulR.HalmosisadistinguishedprofessorofMathematicsatIndianaUniversity,andEditor-ElectoftheAmericanMathematicalMonthly.HereceivedhisP课后答案网h.D.fromtheUniversityofIllinois,andhasheldpositionsatIllinois,Syracuse,Chicago,Michigan,Hawaii,andSantaBarbara.Hehaspublishednumerousbooksandnearly100articles,andhasbeentheeditorofmanyjournalsandseveralbookseries.TheMathematicalAssociationofAmericahasgivenhimtheChauvenetPrizeand(twice)theLesterFordawardformathematicalexpositiowww.hackshp.cnn.Hismainmathematicalinterestsareinmeasureandergodictheory,algebraic,andoperatorsonHilbertspace.VitoVolterra,bornintheyear1860inAncona,showedinhisboyhoodhisexceptionalgiftsformathematicalandphysicalthinking.Attheageofthirteen,afterreadingVerne’snovelonthevoyagefromearthtomoon,hedevisedhisownmethodtocomputethetrajectoryunderthegravitationalfieldoftheearthandthemoon;themethodwasworthlaterdevelopmentintoageneralprocedureforsolvingdifferentialequations.HebecameapupilofDiniattheScuolaNormaleSuperioreinPisaandpublishedmanyimportantpaperswhilestillastudent.HereceivedhisdegreeinPhysicsattheageof22andwasmadefullprofessorofRationalMechanicsatthesameUniversityonlyoneyearlater,asasuccessorofBetti.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Volterrahadmanyinterestsoutsidepuremathematics,rangingfromhistorytopoetry,tomusic.Whenhewascalledtojoinin1900theUniversityofRomefromTurin,hewasinvitedtogivetheopeningspeechoftheacademicyear.VolterrawasPresidentoftheAccademiadeiLinceiintheyears1923-1926.HewasalsothefounderoftheItalianSocietyfortheAdvancementofScienceandoftheNationalCouncilofResearch.Formanyyearshewasoneofthemostproductivescientistsandaveryinfluentialpersonalityinpubliclife.WhenFascismtookpowerinItaly,Volterradidnotacceptanycompromiseandpreferredtoleavehispublicandacademicactivities.VocabularyVocabulVocabularyarypastorkhdaw.com牧师hospitalize住进医院theology神学thesis论文strain紧张、疲惫transcendentalnumber超越数relentless无情的sensation感觉，引起兴趣的事prolific多产的controversy争论，辩论depression抑郁；萧条，不景气essence本质，要素transfinite超限的课后答案网www.hackshp.cnNote0.本课文由几篇介绍数学家生平的短文组成，属传记式体裁。读者从中可学到如何写介绍人物（如推荐信）之类文章。1.Theconsiderablestrainthatheunderwentinhisrelentlesseffortcausedanillnessthatresultedinthelossofthesightofhisrighteye.意思是：不幸，在这无情的拼搏中，巨大的劳累使他病倒，以致右眼失明。Resultin意思是：“导致”。2.Underthestrainofoppositiontohisideas…fromtimetotimeuntilhisdeath.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 意思是：当他的思想和他对连续统假设的证明作出努力而受到强烈反对时，他第一次遭遇到许多令人沮丧的打击，致使他在去世前一次又一次地住进医院。3.NobelPrize诺贝尔奖4.ChauvenetPrize美国数学联合会所设，创立于1925年，以表彰联合会成员认为重要的，杰出的介绍性文章。LesterFord奖，也是美国数学联合会所设，创于1965年，以表彰杰出的介绍性文章。受奖者文章需发表在重要的数学杂志上。5.ScuolaNormaleSuperiorc是“高等师范学院”的意大利文。在英文文献中，不时会出现一些非英语的外文名词，如法、德、西班牙文等，有时读者可以根据其类似于英文的发音或拼写而判断其含义。khdaw.comExerciseExerciExercisese(miscellaneousexercises)Ⅰ.TranslatethefollowingsentencesofexpressionsintoEnglish:1.(i)(A∪B)∩C=(A∩C)∪(B∩C)2.实数可以用数轴上的点来表示。实数0对应于原点，正（负）实数对应于正（负）实轴上的点。3.设X是一拓扑空间，则（i）任意个开集的并集是开的；课后答案网（ii）任意有限个开集的交集是开的。4.在拓扑映照下保持不变的性质称为拓扑性质。5.在一个拓扑空间中，集A是闭集当且仅当www.hackshp.cnA的余集是开的。6.f:A→B,如果f(A)=B,则称f为A到B上的映照，如果f(a)=f(a’)=>a=a’则称f为1-1对应的映照。7.设Z是一集合，在Z上有一关系（relation）~使得对任意a,b,c∈Z满足下列条件。（i）对于所有的a∈Z有a~a（ii）a~b=>b~a（iii）a~b与b~c=>a~c则称~为一等价（equivalence）关系。khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com II.TranslatethefollowingsentencesintoEnglish(ReadAppendixIIIbeforeyoutranslate):1.从方程A和方程B中消去参数t,即得方程C.2.不难验证，此一不等式对于所有自然数n均成立。3.为了证实极限函数是连续的，我们需要去证这一连续函数级数是一致收敛的。4.如函数f(x)(定义在某一区域G上)不取两个值a和b，则不失一般性，我们可以假定f(x)不取0和1。5.假若定理不真，则我们容易证明由此将导出矛盾。6.khdaw.com若我们能证明定理B，则定理A将是它的直接结果。7.这一定理的证明，没有什么新的东西，证明的过程完全类似于上一定理的证明。8.为了推导出这一不等式，我们需要某些有关积分的知识。9.现在我们可以把上面所证明的，概括为如下的定理。10.由（a）到（b）的证明是明显的，因此我们仅需证明由（b）到（a）也成立。课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－NotationsandAbbreviations(I)LearntounderstandNsetofnaturalnumbersZsetofintegersRsetofrealnumbersCsetofcomplexnumbers+plus;positive－minus;negative×khdaw.commultipliedby;times÷dividedby＝equals;isequalto≡identicallyequalto≈,≌approximatelyequalto＞greaterthan≥greaterthanorequalto＜lessthan课后答案网≤lessthanorequalto》muchgreaterthanwww.hackshp.cn《muchlessthansquarerootcuberootnthroot│a│absolutevalueofan!nfactorialatothepowern;thenthpowerofakhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com [a]thegreatestinteger≤athereciprocalofaLetA,Bbesets∈belongsto;beamemberofnotbelongstox∈AxosamemberofA∪unionAkhdaw.com∪BAunionB∩intersectionA∩BAintersectionBABAisasubsetofB;AiscontainedinBABAcontainsBcomplementofAtheclosureofAemptyset课后答案网()i=1,2,…,rj=1,2,…,sr-by-s（r×s）matrix││I,j=1,2,…,nwww.hackshp.cndeterminantoforderndet()thedeterminantofthematrix()vectorFx=(,,…,)xisann-tupleof‖‖thenormof…‖parallelto┴perpendiculartotheexponentialfunctionofxkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com linxthelogarithmicfunctionofxsiesinecoscosinetantangentsinhhyperbolicsinecoshhyperboliccosinetheinverseoffkhdaw.comfisthecompositeorthecompositionofuandvthelimitof…asnapproaches∞(asxapproaches)xaxapproachesa,thedifferentialcoefficientofy;the1stderivativeofy,thenthderivativeofythepartialderivativeoffwithrespecttoxthepartialderivativeoffwithrespecttoytheindefiniteintegraloff课后答案网thedefiniteintegraloffbetweenaandb(fromatob)theincrementofxwww.hackshp.cndifferentialxsummationof…thesumofthetermsindicated∏theproductofthetermsindicated=>impliesisequivalentto（）roundbrackets;parantheses[]squarebracketskhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com {}braceskhdaw.com课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－OperationsResearchThestartofoperationsresearchtookplaceinamilitarycontextintheUnitedKingdomduringWorldWarⅡ,anditwasquicklytakenupunderthenameoperationsresearch(OR)intheUnitedStates.Afterthewaritevolvedinconnectionwithindustrialorganization,anditsmanytechniquesallowedforexpandingareasofapplicationintheUnitedStates,theUnitedKingdom,andinotherindustrialcountries.Itis,however,noteasytogiveaprecisedefinitionofoperationsresearch,Therearethreedifferentrepresentativedefinitions.Accordingtotheclassicaldefinition,duetoP.M.MorseandG.E.Kimball,operationsresearchisascientificmethodofprovidingexecutiveswithaquantitativebasisfordecisionsregardingoperationsundertheircontrol.khdaw.comTheseconddefinition,duetoC.W.Churchman,R.L.Ackoff,andE.L.Arnoff,isasfollows:operationsresearchinthemostgeneralsensecanbecharacterizedastheapplicationofscientificmethods,techniques,andtoolstotheoperationsofsystemssoastoprovidethoseincontrolwithoptimumsolutionstoproblems.AsthethirddefinitionwementionthesuggestionduetoS.Beer:operationsresearchistheattackofmodernscienceonproblemsoflikelihood(acceptingmischance)thatariseinthemanagementandcontrolofmen,machines,materials,andmoneyintheirnaturalenvironments.Itsspecialtechniqueistoinventastrategyofcontrolbymeasuring,comparing,andpredictingprobablebehaviourthroughascientificmodelofasituation.Thesethreedefinitionshaveseveralcommonfeatures.Inthefirstplace,operati课后答案网onsresearchservesexecutivesbyprovidingpartialobservationsandadvicewhichtheycanuseinjudgingasituation.Second,theapplicabilityofoperationsresearchislimitedtoareaswherescientificmethodscanbesuccessfullyapplied.Thisisthereasonwhyoperationsresearchwouldnotbeconsideredtoextendwww.hackshp.cnbeyondonlypartialobservationandadvice.Afundamentalrequirementforascientificapproachisthatitmusthaveamathematicalmodelwhosevaliditycanbetestedbyactualdata,Third,anyoperationshouldsatisfythreenecessaryconditionsinorderthatitmaybeanobjectofscientificapproach:(1)theoperationshouldbedefinedobjectively;(2)theresults,consequences,andeffectsofitsapplicationshouldbeobjectivelymeasurable;(3)theoperationshouldbecapableofrepetition.Fourth,operationsresearchshouldaimatfindingapracticalstrategy.Althoughoperationsresearchisbasedonscientificmethodology,itdoesnotaimatestablishinggeneralscientificassertionsthatarevalidforallsitustions.Thesefourpointsareessentialtoanyoperationsresearch,andareimplicitineachofthethreeaforementioneddefinitions.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Ontheotherhandthesethreedefinitionsemphasizedifferentlysomespecificfeaturesofoperationsresearch,accordingtotheirhistoricalpositions.Incomparionwiththefirstdefinition,thesecondmakesclearertheplacewhereoperationsresearchisappliedbypointingoutthatitisconcernedwiththeoperationsofsystems,and,insteadofthevaguementionofquantitativebasisfordecisionsinthefirstdefinition,itstatesthatoperationsresearchseeksoptimumsolutions,reflectingastagewhereoptimumsolutionsweresoughtbyapplicationsofmathematicalprogrammingtechniques.Inthethirddefinitionofoperationsresearchthenotionofsystemisdefinedexplicitly,thenotionofoperationisdefinedtobeitsspecialtechnique,andtheobjectivesofoperationsresearcharegiven.Itisclearlyassertedthatoperationsresearchbelongstothemethodologyofappliedsciences.Inoperationsresearch,operationsandsystemsaredealtwithintheirintimateinterconnection.Themethodologyofoperationsresearchthereforereliesonanoverallapproachforwhichinterdisciplinarycooperationisinkhdaw.comdispensableandinwhichtheoperationsresearchteamplaysanimportantrole.Inapplyingtheoperationsresearchapproachtothecircumstanceswithwhichweareconcerned,weconcentrateourinterestonmutualrelationshipsamonginputandoutputcharacteristics.Ablack-boxmethodbywhichtheinterrelationbetweeninputandoutputcanbeclarifiedwithoutenteringtheactualmechanismofthetransformationyieldedbythesystemorbyitssubsystemsplaysafundamentalroleinoperationsresearch.Thefollowingaremajorphasesofanoperationsresearchproject:(1)formulatingtheproblem;(2)constructingamathematicalmodeltorepresentthesystemunderstudyandderivingasolutionfromthemodel;(3)testingthemodelandthesolutionderivedformit;(4)theimplementationstageofestablishingcontrolsoverthesolutionandputtingitto课后答案网work.Itisimportanttoconstructamodelofinformationcommunicationinconnection.Withamathematicalmodelofanyprobleminoperationsresearch.Processofaliocation,competition,queuing,inventory,andproductionappearfrequentlyinthemathematicalmodelsofoperationsresearch.www.hackshp.cn——FromEncyclopedicDictionaryofMathematicsVocabularyVocabulVocabularyaryOperationsresearch(OR)运筹interdiscipline交叉学科Executive行政人员interdisciplinarycooperation交叉学科的likelihood似然合作scientificapproach科学方法black-boxmethod黑箱方法khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com methodology方法论implementationstage实现阶段aforementioned前述的queue排队mathematicalprogramming数学规划NotesNotes1.OperationsResearch运筹学.运筹学是第二次世界大战期间,为解决后勤供应问题而发展起来的一门学科,它运用最优化技术去解决管理和决策问题.2.Accordingtotheclassicaldefinition,duetoP.M.MorseandG.E.Kimball,operationsresearchisascientificmethodofprovidingexecutiveswithaquantitivebasisfordecisionsregardingoperationsundertheircontrol.khdaw.com意思是:根据P.M.Morse和G.E.Kimball提出的古典定义,运筹学是一种科学方法,它提供行政人员一种定量基础,以便他们对所控的操作进行决策,这里dueto是”归功于”“由…提出”之意,providing…with…for…是”提供…给…用于…”之意.3.…insteadofthevaguementionofquantitativebasisfordecisionsinthefirstdefinition,itstatesthat…byapplicationsofmathematicalprogrammingtechniques.意思是:代替第意个定义中对于决策的定量基础那种模糊的提法,它(第二个定义)阐明了运筹学用于寻求最优解,反映了运用数学规划方法求最优解的阶段.这里reflecting至句子结束一段,属独立分词结构,用以补充说明itstatesthat…的句子.4.Themethodologyofoperationsresearchthereforerelieson…theoperationsresearchteamplaysanimportantrole.课后答案网意思是:因而运筹学的方法论依赖于…一种全面的研究,对这种研究来说,各交叉学科的合作是不可避免的,而且,在这种研究中,运筹学小组起了重要的作用.注意:前后两个which都是approach的关系代词,很容易误认为第二个which是cooperation的关系代词www.hackshp.cn,虽然这在意思上说得过去,但从语法结构上却不然.5.Ablack-boxmethodbywhichtheinterrelationbetweeninputandoutput…playsafundamentalroleinoperationsresearch.意思是:黑箱方法不需引进由系统或它的子系统所产生的变换的确切机制而能阐明输入和输出的相互关系,这种方法在运筹学中起了重要的作用,注意这一句中的主语Ablack-boxandmethod和谓语plays相隔甚远.ExerciseExerciExerciseseⅠ.Answerthefollowingquestions:1.Whatarethenecessaryconditionsforoperationtobecomeanobjectofscientificapproach?khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 2.Pointoutthemainpointsthe2ndandthe3rddefinitionsemphasizeascomparedwiththefirstdefinition.Ⅱ.1.TranslatethethirddefinitionsofORduetoS.Beer.2.TranslatethefollowingsentencesintoChinese;ⅰ)ItwasG.Gantorwhofirstintroducedtheconceptofthesetasobjectofmathematicalstudy.ⅱ)ThedefinitionofprobabilityduetoLaplaceprovokedagreatdealofargumentwhenitwasapplied;ⅲ)Nowadays,weusuallyadoptedmeasuretheoreticfoundationsofprobabilityinitiatedbyA.N.Kolomogorov.khdaw.com课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－PhrasesorclausesfrequentlyusedinmathematiscsI.常见的涉及运算的短语或句子(句中(A),(B)表示某些表达式,如不等式,等式,方程等)DifferentiateIntegrate1.bothsidesoftheequationandweget…DifferentiatingIntegratingkhdaw.com2.bothsidesoftheequation,weget…3.Add(A)to(B)andwehave…4.Substract(B)from(A)andwehave…5.Multiplyingeachtermofequationby…,weobtain…6.Dividingtheequationthroughby…,wehave…Give课后答案网yieldimplywww.hackshp.cn7.(A)and(B)together…8.Comparing(A)with(B),itiseasytoseethat…9.Substituting(A)into(B),weobtain…10.Eliminating(theparameter)from(A)and(B),wehave…asfollowsinthefollowingform11.Byintroducinganewvariable…,wecanthenrewrite(A)khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 12.Byasimplecalculation,weobtainfrom(A)…II.定理证明过程中常见的短语和句子.1.下面的句型可用来表达”根据什么即可得到什么”的意思.definitonhypothesisassumptionstheorem(N)lemma(A)khdaw.comcorollary(B)theremarkthefactthat……………..Accordingto,itfollows…Since…,itfollows…2.如果一个论断可以通过一些简单运算或简单推理而获得课后答案网,由于这些运算或推理比较简单,读者可以自行推算,因而只需直接写出论断来,这时可用下面名句型:seewww.hackshp.cnshowproveverifycheckItiseasytothat…seenshownkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com provedverifiedcheckedItcaneasilybethat…3.如果所要提及的结论比较显浅,或是众所周知,无需作过一步的证明,这时可用下面名句型:clearobviouskhdaw.comevidentwell-knownItisthat…ClerlyObviouslyEvidently课后答案网,…4.证明一个定理有时需要引进辅助函数,这时可用下面句型:Letusfirstdefinethefuction…www.hackshp.cnLetusintroduceanewfunction…Letusconsiderthefunction…Letusfirstinvestigatethefunction…Orsimply:Let…Set…Define…khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Put…Consider…5.在一个定理中,有几个结论需要证明,其中有些结论比较明显,可不用证明,仅需证明余下结论便可,这时可用如下句型:ObvioustrivialSince(A)and(B)are,weneedonlyprove(C)khdaw.comSince(A)and(B)aretrivial,itsufficestoprove(C)6.为了证明一个定理,有时我们并不是直接去证明,而是证明一个新的论断,一旦新的论断得到证明,已给定理不难由此而得证,这时可用下面句型:TheoremresultThewillbeprovedifwecanshow…Thetheoremwillbeprovedbyshowingthat…Ifwecanprove…thenthetheoremfollowsimmediately以下各句用于新的论断被证明之后课后答案网:Thetheoremisnowadirectconsequenceofwhatwehaveproved.Thetheoremfollowsimmediatelyfromwhatwehaveproved.www.hackshp.cnThetheoremisnowevidentfromwhatwehaveproved.Itisevidenttoseethatthetheoremholds.7.在证明过程中,有时要用到一些早已学过的知识或技巧,这里可用下面句子,以提醒读者:Recallthat…Noticethat…Notethat…khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Observethat…Inordertoprovethetheorem,weneedtheknowledgeof…(e.g..theknowledgeofVariations)Inordertoobtainthefollowingequation,weneed…(e.g..Leibniz’srulefordifferentiatingunderintegralsign)8.如果需要证明的定理的假设条件是一般条件,但是,只要定理在特殊条件下成立,就不难推出定理在一般条件下也成立,这时仅需在特殊情况下去证明定理就够了,为此可用下面句型:Considerkhdaw.comassumeWithoutlossofgenerality,wemay…Withoutlossgenerality,wemayprovethetheoreminthecase…Itsufficestoprovethetheoreminthetheoreminthecase…Weneedonlyconsiderthecase…Forsimplicityconsiderthecase…(e.g..wemaytakethedomaintobethedisk)课后答案网9.如果特征的论断可用以前用过的相似的方法或步骤进行证明,则可用下面句型:thesamewayasimilarwaywww.hackshp.cntheoremstatementlemma…Thecanbeprovedinasshownbefore.Thistheoremcanbeprovedbythesamemethodasemployedinthelastsection.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thistheoremcanbecompletedbythemethodanalogoustothatusedabove.UsingthesameargumentasintheproofoftheoremN,wecaneasilycarryouttheproofofthistheorem.WenowproceedasintheproofoftheoremN.WeshalladoptthesameprocedureasintheproofoftheoremN.10.如果我们用的是反证法,则其开头及结尾可用下面句型:falsenottruekhdaw.comnotrightstatementassertionconclusionIfthewerethen…Iftheassertionwouldnothold,then…Thisiscontraryto…Thiscontradictsthefactthat..课后答案网Thisleadstoacontradiction.11.表示定理已证毕或者把前面所证的总结为一结论www.hackshp.cn.Wehavethusprovedthetheorem.Thiscompletestheproof.Theproofofthetheoremisnowcompleted.Itisnowobviousthatthetheoremholds.Thuswehavederivedthat…Consequently,weinferthat…khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thusweconcludethat…Thusweareledtotheconclusionthat…Thuswearriveattheconclusionthat…Thuswecansummarizewhatwehaveprovedasthefollowingtheorem.12.其它Thereexist(s)…suchthat…Weclaim…infact…khdaw.comWearenowinapositionto…Ifotherwise…Providedthat…课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－Polya’sCraftofDiscoveryGeorgePolyahasascientificcareerextendingmorethansevendecades.Abrilliantmathematicianwhohasmadefundamentalcontributionsinmanyfields.Polyahasalsobeenabrilliantteacher,ateacher’steacherandanexpositor.Polyabelievesthatthereisacraftofdiscovery.Hebelievesthattheabilitytodiscoverandtheabilitytoinventcanbeenchancedbyskillfulteachingwhichalertsthestudenttotheprinciplesofdiscoveryandwhichgiveshimanopportunitytopractisetheseprinciples.Inaseriesofremarkablebooksofgreatrichness,thefirstofwhichwaspublishedin1945.Polyahascrystallizedtheseprinciplesofdiscoveryandinventionoutofhisvastexperience,andhassharedthemwithusbothinpreceptandinexample.Thesebooksareatreasure-troveofstrategy,know-how,rulesofthkhdaw.comumb,goodadvice,anecdote,mathematicalhistory,togetherwithproblemafterproblematalllevelsandallofunusualmathematicalinterest.Polyaplacesaglobalplanfor“HowtoSolveIt”intheendpapersofhisbookofthatname:HOWTOSOLVEITFirst:Youhavetounderstandtheproblem.Second:Findtheconnectionbetweenthedataandtheunknown.Youmaybeobligedtoconsiderauxiliaryproblemsifanimmediateconnectioncannotbefound.Youshouldobtaineventuallyaplanofthesolution.Third:Carryoutyourplan.课后答案网Fourth:Examinethesolutionobtained.Thesepreceptsarethenbrokendownto“molecular”levelontheoppositeendpaper.There,individualstrategiesaresuggestedwhichmightbecalledintowww.hackshp.cnplayatappropriatemomentsm,suchas:Ifyoucannotsolvetheproposedproblem,lookaroundforanappropriaterelatedproblem.WorkbackwardsWorkforwardsNarrowtheconditionWidentheconditionSeekacounterexamplekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com GuessandtestDivideandconquerChangetheconceptualmodeEachoftheseheuristicprinciplesisamplifiedbynumerousappropriateexamples.SubsequentinvestigatorshavecarriedPolya’sideasforwardinanumberofways.A.H.Schoenfeldhasmadeaninterestingtabulationofthemostfrequentlyusedheuristicprinciplesincollege-levelmathematics.Wehaveappendedithere.khdaw.comFrequentlyUsedHeuristicsAnalysisAnalysiAnalysiss1)Drawadiagramifatallpossible2)Examinespecialcases:a)Choosespecialvaluestoexemplifytheproblemandgeta“feel”forit.b)Examinelimitingcasestoexploretherangeofpossibilitiesc)Setanyintegerparametersequalto1,2,3,…,insequence,andlookforaninductivepattern.课后答案网3)Trytosimplifytheproblembya)exploitingsymmetry,orwww.hackshp.cnb)“WithoutLossofGenerality”arguments(includingscaling)ExplorationExploratiExplorationon1)Consideressentiallyequivalentproblems:a)Replacingconditionsbyequivalentones.b)Re-combiningtheelementsoftheproblemindifferentways.c)Introduceauxiliaryelements.d)Re-formulatetheproblembykhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com I)changeofperspectiveornotationII)consideringargumentbycontradictionorcontrapositiveIII)assumingyouhaveasolution,anddeterminingitsproperties2)Considerslightlymodifiedproblems:a)Choosesubgoals(obtainpartialfulfillmentoftheconditions)b)Relaxaconditionandthentrytore-imposeit.c)Decomposethedomainoftheproblemandworkonitcasebycase.3)khdaw.comConsiderbroadlymodifiedproblems:a)Constructananalogousproblemwithfewervariables.b)Holdallbutonevariablefixedtodeterminethatvariable’simpact.c)TrytoexploitanyrelatedproblemswhichhavesimilarI)formII)“givens”III)conclusionsRemember:whendealingwitheasierrelatedproblems,youshouldtrytoexpl课后答案网oitboththeRESULTandtheMETHODOFSOLUTIONonthegivenproblem.www.hackshp.cnVerifyingVerifyiVerifyingVerifyingyourngyouryoursolutionsolssolutionolutionution1)Doesyoursolutionpassthesespecifictests:a)Doesituseallthepertinentdata?b)Doesitconformtoreasonableestimatesorpredictions?c)Doesitwithstandtestsofsymmetry,dimensionanalysis,orscaling?2)Doesitpassthesegeneraltests?a)Canitbeobtaineddifferently?b)Canitbesudstantiatedbyspecialcases?khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com c)Canitbereducedtoknownresults?d)Canitbeusedtogeneratesomethingyouknow?VocabularyVocabulVocabularyarycraft技巧enchance增强alert警觉，机警preceptkhdaw.com箴言，格言treasuretrove宝藏anecdote轶事，趣闻auxiliary辅助的appropriate适当的heuristic启发式的amplified扩大，详述append附加，追加exploration探查，细查课后答案网perspective透视contrapositive对换的www.hackshp.cnrelax放松decompose分解pertinent适当的substantiate证实，证明NotesNotes1．Abrilliantmathematicianwhohasmadefundamentralcontributionsinmanyfields,Polyahasalsobeenabrilliantteacher,ateacher’steacher,andanexpositor.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 意思是：Polya，一个在许多领域中都作出重要贡献的数学家，也是一位出色的教师，教师的教师和评注家。这里Polya是abrilliantmathematician的同位语2．…whichalertsthestudenttotheprinciplesofdiscoveries…这里alert的意思是：“使机警，使注意”。因此，本句意思是：这种熟练（有技巧的）的教学可使学生机敏地注意到这些发现原则……3．Polyahascrystallizedtheseprinciplesofdiscoveriesoutofhisvastexperience,…意思是：Polya从他的浩瀚的经验中，把这些发现原则提炼得更加具体而明朗。4．Rulesofthumb以经验为基础的规则，方法。5．khdaw.comThere,individualstrategiesaresuggested,whichmightbecalledintoplayatappropriatemoments,suchas…意思是：在那里，提供了许多个别的策略，它们在适当的时刻就会发挥作用，例如……这里callintoplay意思是：“发挥作用”。ExerciseExerciExerciseseI．TranslatethefollowingsentencesintoChinese(payattentiontothephrasesunderlined:1.Notethata+ib=c+idmeansa=candb=d2.Werecallthatlogz:C课后答案网－{0}Cisaninverseforwhenisrestrictedtoastrip3.Noticethatif,anglesneednotbepreserved.4.Toshowthatthetestfailswhen,observethat,byelementaryanalysis,andbutdivergeswhileconverges.www.hackshp.cn5.Toprovetheresultsofthissection,weshallusethetechniquesdevelopedinthelastsection.6.Wecandeduce,inawaysimilartothewaywededucedtheoremA,thefollowingtheorem.7.WearenowinapositiontodrawimportantconsequencesfromCauchy’stheorem.8.Wearenowinapositiontoproveeasilyanotherwisedifficulttheoremstatingthatanypolynomialofdegreenhasaroot.9.Unlessotherwisespecified(stated),curveswillalwaysbeassumedtobecontinuousandpiecewisedifferentiable.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 10.Weshallproveatheoremthatappearstobeelementaryandthatthestudenthas,inthepast,takenforgranted.11.Thesolutiontothisdifferentialequationisuniqueuptotheadditionofaconstant.12.ThefunctionthatmapsthesimplyconnecteddomainontotheunitdiscisuniqueuptoaMobiustransformation.II．TranslatethefollowingpassagesintoChinese:1．Ifwedonotsucceedinsolvingamathematicalproblem,thereasonfrequentlyconsistsinourfailuretorecognizethemoregeneralstandpointfromwhichtheproblembeforeusappearsonlyasasinglelinkinachainofrelatedproblems.Afterfindingthisstandingpoint,notonlyisthisproblemfrequentlymoreaccessibletoourinvestigation,butatthesametimewecomeintopossessionofamethodwhkhdaw.comichisapplicablealsotorelatedproblems.2．Indealingwithmathematicalproblems,specializationplays,asIbelieve,astillmoreimportantpartthangeneralization.Perhapsinmostcaseswhereweseekinvaintheanswertoaquestion,thecauseofthefailureliesinthefactthatproblemssimplerandeasierthantheoneinhandhavebeeneithernotatallorincompletelysolved.Alldependsthen,onfindingouttheseeasierproblems,andonsolvingthembymeansofmethodsasperfectaspossible.课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－ProbabilityThemathematicstowhichouryoungstersareexposedatschoolis.Withrareexceptions,basedontheclassicalyes-or-no,right-or-wrongtypeoflogic.Itnormallydoesn’tincludeonewordaboutprobabilityasamodeofreasoningorasabasisforcomparingseveralalternativeconclusions.Geometry,forinstance,isstrictlydevotedtothe“if-then”typeofreasoningandsotothenotion(idea)thatanystatementiseithercorrectorincorrect.However,ithasbeenremarkedthatlifeisanalmostcontinuousexperienceofhavingtodrawconclusionsfrominsufficientevidence,andthisiswhatwehavetodowhenwemakethetrivialdecisionastowhetherornottocarryanumbrellawhenweleavehomeforwork.Thisiswhatagreatindustryhastokhdaw.comdowhenitdecideswhetherornottoput$50000000intoanewplantabroad.Innoneofthesecaseandindeed,inpracticallynoothercasethatyoucansuggest,canoneproceedbysaying:”IknowthatA,B,C,etc.arecompletelyandreliablytrue,andthereforetheinevitableconclusionis~~”Forthereisanothermodeofreasoning,whichdoesnotsay:Thisstatementiscorrect,anditsoppositeiscompletelyfalse.”Butwhichsay:Therearevariousalternativepossibilities.Nooneoftheseiscertainlycorrectandtrue,andnoonecertainlyincorrectandfalse.Therearevaryingdegreesofplausibility—ofprobability—forallthesealternatives.Icanhelpyouunderstandhowtheseplausibility’scompare;Icanalsotellyoureliablemyadviceis.”Thisisthekindoflogic,whichisdevelopedinthetheoryofprobability.Thistheorydealswithnottwotruth-values—correctorfalse—butwithalltheinintermediatetruthvalues:almostcertainlytrue,veryprobablytrue,possiblytr课后答案网ue,unlikely,veryunlikely,etc.Beingaprecisequantitiestheory,itdoesnotusephrasessuchasthosejustgiven,butcalculatesforanyquestionunderstudythenumericalprobabilitythatitistrue.Iftheprobabilityhasthevalueof1,theanswerisanunqualified“yes”orcertainty.Ifitiszero(0),theanswerisanunqualified“no”www.hackshp.cni.e.itisfalseorimpossible.Iftheprobabilityisahalf(0.5),thenthechancesareeventhatthequestionhasanaffirmativeanswer.Iftheprobabilityistenth(0.1),thenthechancesareonly1in10thattheansweris“yes.”Itisaremarkablefactthatone’sintuitionisoftennotverygoodatcsunatinganswerstoprobabilityproblems.Forexample,howmanypersonsmustthereareatleasttwopersonsintheroomwiththesamebirthday(bornonthesamedayofthemonth)?Rememberingthatthereare356separatebirthdayspossible,somepersonsestimatethattherewouldhavetobe50,oreven100,personsintheroomtomaketheoddsbetterthaneven.Theanswer,infact,isthattheoddsarebetterthaneighttoonethatatleasttwowillhavethesamebirthday.Letusconsideronemoreexample:Everyoneisinterestedinpolls,whichinvolveestimatingtheopinionsofalargegroup(sayallthosewhovote)khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com bydeterminingtheopinionsofasample.Instatisticsthewholegroupinquestioniscalledthe“universe”or“population”.Nowsupposeyouwanttoconsultalargeenoughsampletoreflectthewholepopulationwithatleast98%precision(accuracy)in99outofahundredinstances:howlargedoesthisveryreliablesamplehavetobe?Ifthepopulationnumbers200persons,thenthesamplemustinclude105persons,ormorethanhalfthewholepopulation.Butsupposethepopulationconsistsof10,000persons,or100,000persons?Inthecaseof10,000persons,or1000,000person?Inthecaseof10,000persons,asample,tohavethestatedreliability,wouldhavetoconsistof213persons:thesampleincreasesbyonly108whenthepopulationincreasesby9800.Andifyouadd90000moretothepopulation,sothatitnownumbers100000,youhavetoaddonly4tothesample.Thelesscrediblethisseemstoyou,themorestronglyImakethepointthatitisbettertodependonthetheoryofprobabilityratherthanonintuition.khdaw.comAlthoughthesubjectstartedout(began)intheseventeenthcenturywithgamesofchancesuchasdiceandcards,itsoonbecameclearthatithadimportantapplicationstootherfieldsofactivity.IntheeighteenthcenturyLaplacelaidthefoundationsforatheoryoferrors,andGausslaterdevelopthisintoarealworkingtoolforallexperimentersandobservers.Anymeasurementorsetofmeasurementisnecessarilyisnecessarilyinexact;anditisamatterofthehighestimportancetoknowhowtotakealotofnecessarilydiscordantdata,combinetheminthebestpossibleway,andproduceinadditionsomeusefulestimateofthedependabilityoftheresults.Othermoremodernfieldsofapplicationare:inlifeinsurance;telephonetrafficproblems;informationandcommunicationtheory;gametheory,withapplicationstoallformsofcompetition,includingbusinessinternationalpoliticsandwar;modernstatisticaltheories,bothforth课后答案网eefficientdesignofexperimentsandfortheinterpretationoftheresultsofexperiments;decisiontheories,whichaidusinmakingjudgments;probabilitytheoriesfortheprocessbywhichwelearn,andmanymore.www.hackshp.cn----Weaver,W.VocabularyVocabulVocabularyaryProbability概率论permutation置换Plausibility似乎合理binomialcoefficient二次式系数Affirmative肯定的generatingfunction母函数Estimate估计even事件Discordant不一致的informationandcommunicationtheorykhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Communicationtheory通讯理论信息与通讯论Decisiontheory决策论gametheory对策论，博弈论NotesNotes1.Geometry,forexample,isstrictlydevotedtothe“if—then”typeofreasoningandsotothenotion(idea)thatanystatementiseithercorrectorincorrect.意思是：例如几何学就是严格地属于那种“如果，则”的推理类型，所以它也就属于那种对任何陈述要么khdaw.com是对的要么是不对的概念范围。Isdevotedto意思是：“奉献于”，这里可作：“属于”解，注意在andsotothenotion~~中，在前面省去isdevoted.2.However,ithasbeenremarkedthatlifeisanalmostcontinuousexperiencewhenweleavehomeforwork.意思是：然而，人们已经注意到，生活就是这样一种几乎不断地需要我们从不充分的证据中去做出结论的经历，这就是对诸如我们离家上班时是否要带雨伞做出定时，我们所需要做的。3.Iftheprobabilityhasvalueof1,theanswerisanunqualified“yes”orcertainty.这里unqualified解作：“绝对的”，“十足的”。如anunqualifiedcertainty(绝对的肯定)；Anunqualifiedsuccess(彻底胜利)。注意qualified常解作：“有资格的”，“合格的”。如aqualifiedtechnician(合格的技术员)；qualifiedexamination(课后答案网资格考试，美国高等学校研究生院的一种考试)。4．Iftheprobabilityisahalf,thenthechancesareeventhatthequestionhasanaffirmativeanswer.意思是：如果概率是一半的话，那么问题有肯定答案的机会是对等的。注意这里www.hackshp.cneven作“对等”解。5．Thelesscrediblethisseemstoyou,themorestronglyImakethepointthatitisbettertodependonthetheoryofprobabilityratherthanonintuition.意思是：这对你越不可信，我们就要强调这种论点：宁可依靠概率论而不愚信直观，这里makethepointthat意思是：“主张；强调；视~~为重要”ExerciseExerciExercisese1.TranslatethefollowingpassageintoChinesekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com TheoriginofthetheoryofprobabilitygoesBachtothemathematicalproblemsconnectedwithdicethrowingthatwerediscussesinlettersexchangedbyB.PpascalandP.deFermatinthe17thcentury.Theseproblemswereprincipallyconcernedwithconcepts,suchaspermutations,combinations,andbinomialcoefficients,whosetheorywasestablishedaboutthesametime.ThiselementarytheoryofprobabilitywaslaterenrichedbytheworkofscholarssuchasJacobBernoulli,A.deMoivre,T.Bayes,L,deBuffon,DanialBcrnoulli,A,M,Legendre,andJ.L.Lagrange.Finally,P.S.Laplacecompletedtheclassicaltheoryofprobabilityinhisbook“Throrieanalytiquedesprobabilities”(1812).Inthiswork,Laplacenotonlysystemizedalsogreatlyextendedpreviousimportantresultsbyintroducingnewmethodssuchastheuseofdifferenceequationsandgeneratingfunctions.Sincethe19thcentury,thetheoryofprobabilityhasbeenextensivelyappliedtothenaturalsciencesandeventosocialsciences.2.TranslatethefollowingsentencesintoChinese:1.khdaw.comThetermrandomprocessisusetodescribeprocessthatgivesrisetooneofanumberofadmittedpossibleoutcomesbutwhichoutcomecannotbepredictedwithanycertaintyinadvance.2.ToweventsAandBinaprobabilitymodelwithsamplespaceandprobabilityfunctionParesaidtobeindependentifP(AB)=P(A)·P(B)3.DescribebrieflythekindoflogicdevelopedinthetheoryofProbability.4.TranslatethefollowingsentencesintoEnglish(makeuseofthephraseorthephrasesinthebracket):设X=[ab],AX(AX)是一开集，又设aA令r=sup{:[a+]A},求证a+r…A.(这一部分不用翻译，仅需翻译下下面证明部分)课后答案网证明：（1）若论不成六，即是说a+rA，则由于A是开集，存在>0使得[a+r,a+r+]A,从而[(a,a+r+)A,这与r的定义矛盾。（~~~wouldnothold,or~~~werefalse,orwerenottrue;contraryto）(2)若a+rA,则由于A是开集，存在www.hackshp.cn>0使得[a+r,a+r+]A，由这推出[a,a+r+]A,这是不可能的。故a+rA.(thisimplies)(3)若论断是错的，则由于A是开集，存在>0使得[(a+r,a+r+)A,从而[a,a+r+]A,这就导至与r是的上确界这一事实相矛盾结论。（leadstocontradictiontothethat）khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－SequencesandSeriesSeriesareanaturalcontinuationofourstudyoffunctions.Inthepreviouschapterwefoundhowtoapproximateourelementaryfunctionsbypolynomials,withacertainerrorterm.Conversely,onecandefinearbitraryfunctionsbygivingaseriesforthem.Weshallseehowinthesectionsbelow.Inpractice,veryfewtestsareusedtodetermineconvergenceofseries.Essentially,thecomparisiontestisthemostfrequent.Furthermore,themostimportantseriesarethosewhichconvergeabsolutely.Thusweshallputgreateremphasisonthese.ConvergentSerieskhdaw.comSupposethatwearegivenasequcnceofnumbersa1,a2,a3…i.e.wearegivenanumberan,foreachintegerｎ＞1.WeformthesumsSn=a1+a2+…+anItwouldbemeaninglesstoformaninfinitesuma1+a2+a3+…课后答案网becausewedonotknowhowtoaddinfinitelymanynumbers.However,ifoursumsSnapproachalimitasnbecomeslarge,thenwesaythatthesumofoursequenceconverges,andwenowdefineitssumtobethatlimit.Thesymbolswww.hackshp.cn∞∑a=1anwillbecalledaseries.Weshallsaythattheseriesconvergesifthesumsapproachalimitasnbecomeslarge.Otherwise,wesaythatitdoesnotconverge,ordiverges.Iftheseriersconverges,wesaythatthevalueoftheseriesis∞∑a=1=lima→∞Sn=lima→∞(a1+a2+…+an)Inviewofthefactthatthelimitofasumisthesumofthelimits,andotherstandardpropertiesoflimits,weget:THEOREM1.Let{an}and{bn}(n=1,2,…)khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com betwosequencesandassumethattheseries∞∞∑a=1an∑a=1bn∞converge.Then∑a=1(an+bn)alsoconverges,andisequaltothesumofthetwoseries.Ifcisanumber,then∞∞∑a=1can=c∑a=1anFinally,ifsn=a1+a2+…+anandtn=b1+b2+…+bnthen∞∞∑a=1an∑a=1bn=lima→∞sntnInparticular,seriescanbeaddedtermbyterm.Ofcourse,theycannotbemultipliedtermbyterm.khdaw.comWealsoobservethatasimilartheoremholdsforthedifferenceoftwoseries.Ifaseries∑anconverges,thenthenumbersanmustapproach0asnbecomeslarge.However,thereareexamplesofsequences{an}forwhichtheseriesdoesnotconverge,andyetlima→∞an=0SerieswithPositiveTermsThroughoutthissection,weshallassumethatournumbersanare＞0.Thenthepartialsums课后答案网Sn=a1+a2+…+anareincreasing,i.e.s1＜s2＜s3＜…＜sn＜sn+1＜…www.hackshp.cnIftheyareapproachalimitatall,theycannotbecomearbitrarilylarge.ThusinthatcasethereisanumberBsuchthatSn0,wehavekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com S–ε0andletSn=a1+a2+…+anIfthesequenceofnumbers{sn}isbounded,thenitapproachesalimitS,whichisitsleastupperbound.Theorem3givesusaveryusefulcriteriontodeterminewhenaserieswithpokhdaw.comsitivetermsconverges:∞∞THEOREM3.Let∑a=1anand∑a=1bnbetwoseries,withan>0forallnandbn>0foralln.Assumethatthereisanumbercsuchthatan0andlet∑annxbeaserieswhichconvergesabsolutelyfor∣x∣1.Usethemethodsuggestedbytheproofoftheintegraltesttoshowthatkhdaw.comn-1nf(x)dx≤∑nf(k)∑k=1f(k)≤∫1k=2Takef(x)=logxanddeducetheinequalitiesc•nn•c-n0foralln.Let0a,whichisread“bisgreaterthana“.Geometrically,realnumbersareidentifiedwithpointsonastraightline.Wechooseastraightline,andaninitialpointfreferencecalledtheorigin.Totheoriginweassignthenumberzero.Bymarkingofftheunitoflengthinbothdirectionsfromtheorigin,weassignpositiveintegerstomarked-offpointsinkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com onedirection(byconvention,totherightoftheorigin)andnegativeintegerstomarked-offpointintheotherdirection.Byfollowingthroughintermsofthechosenunitoflength,arealnumberisattachedtoonepointonthenumberline,andeachpointonthenumberlinehasattachedtoitonenumber.Geometrically,intermsofournumberline,tosaythataameansthatbistotherightofa.PropertiesProperPropertiesPropertiesoftiesofofAdditionAdditiAdditionAdditionandonandandMultiplicationMuMultiplicationltiplicationAdditionandmultiplicationareprimaryoperationsonrealnumbers.Most,ifnotall,ofthebasicpropertiesoftheseoperationsarefamiliartousfromexperience.(a)khdaw.comClosurepropertyofadditionandmultiplication.Whenevertworealnumbersareaddedormultiplied,weobtainarealnumberastheresult.Thatis,performingtheoperationsofadditionandmultiplicationleavesuswithinthereal-numbersystem.(b)Commutativepropertyofadditionandmultiplication.Theorderinwhichtworealnumbersareaddedormultiplieddoesnotaffecttheresultobtained.Thatis,ifaandbareanytworealnumbers,thenwehave(i)a+b=b+aand(ii)ab=ba.Suchapropertyiscalledacommutativeproperty.Thus,additionandmultiplicationofrealnumbersarecommutativeoperations.课后答案网(c)Associativepropertyofadditionandmultiplication.Parentheses,brackets,andthelike,werecall,areusedinalgebratogrouptogetherwhatevertermsarewithinthem.Thus2+(3+4)meansthat2istobeaddwww.hackshp.cnedtothesumof3and4yielding2+7=9whereas(2+3)+4meansthesumof2and3istobeaddedto4yieldingalso9.Similarly,2•(3•4)yields2•(12)=24whereas(2•3)•4yieldsthesameendresultbytheroute6•4=24.Thatsuchisthecaseingeneralisthecontentoftheassociativepropertyofadditionandmultiplicationofrealnumbers.(d)Distributivepropertyofmultiplicationoveraddition.Weknowthat2•(3•4)=2•7=14andthat2•3+2•4=14,thus2•(3+4)=2•3+2•4.Thatsuchisthecaseingeneralforallrealnumbersisthecontentofthedistributivepropertyofmultiplicationoveraddition,moresimplycalledthedistributiveproperty.SubstractionSubstrSubstractionSubstractionandactionandandDivisionDiDivisionvisionkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thenumberszeroandone.Thefollowingarethebasicpropertiesofthenumberszeroandone.(a)Thereisauniquerealnumber,calledzeroanddenotedby0,withthepropertythata+0=0+a,whereaisanyrealnumber.Thereisauniquerealnumber,differentfromzero,calledoneanddenotedby1,withthepropertythata•1=1•a=a,whereaisanyrealnumber.(b)Ifaisanyrealnumber,thenthereisauniquerealnumberx,calledtheadditiveinverseofa,ornegativeofa,withthepropertythata+x=x+a.Ifaisanynonzerorealnumber,thenthereisauniquerealnumbery,calledthemultiplicativeinverseofa,orreciprocalofa,withthepropertythatay=ya=1khdaw.comTheconceptofthenegativeofanumbershouldnotbeconfusedwiththeconceptofanegativenumber;theyarenotthesame.”Negativeof“meansadditiveinverseof“.Ontheotherhand,a“negativenumber”isanumberthatislessthanzero.Themultiplicativeinverseofaisoftenrepresentedbythesymbol1/aora-1.Notethatsincetheproductofanynumberyand0is0,0cannothaveamultiplicativeinverse.Thus1/0doesnotexist.Nowsubstractionisdefinedintermsofadditioninthefollowingway.Ifaandbareanytworealnumbers,thenthedifferencea-bisdefinedbya-b=cwherecissuchthatb+c=aorc=a+(-b).Thatis,tosubstractbfrom课后答案网ameanstoaddthenegativeofb(additiveinverseofb)toa.Divisionisdefinedintermsofmultiplicationinthefollowingway.Ifaandbareanyrealnumbers,wherebwww.hackshp.cn≠0,thena+bisdefinedbya+b=a•(1/b)=a•b-1.Thatis,todivideabybmeanstomultiplyabythemultiplicativeinverse(reciprocal)ofb.Thequotienta+bisalsoexpressedbythefractionsymbola/b.VocabularyVocabulVocabularyaryrealnumber实数negative负的therealnumbersystem实数系khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com rationalnumber有理数collection集体,总体ratio比,比率object对象,目的denominator分母principle原理,规则numerator分子adopt采用irrationalnumberkhdaw.com无理数define定义(动词)signify表示definition定义(名词)geometrical几何的establish建立straightline直线explicit清晰的,明显的课后答案网initialpoint初始点illustrate说明pointofreference参考点www.hackshp.cnpositive正的origin原点express表达assign指定plus加unit单位sign记号,符号,正负号property性质khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com operation运算,操作closureproperty封闭性质addition加法commutative交换的multiplication乘法associative结合的substraction减法parentheses圆括号divisionkhdaw.com除法brackets括号sum和,总数algebra代数procuct乘积yield产生difference差,差分term术语,项课后答案网quotient商distributive分配的symbolism符号系统www.hackshp.cnunique唯一的minus减additiveinverse加法逆运算identify使同一multiplicativeinverse乘法逆运算count计数reciprocal倒数,互逆naturalnumber自然数khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com concept概念zero零fraction分数integer整数arithmetic算术的greaterthan大于solution解,解法lessthan小于evenkhdaw.com偶的beequalto等于odd奇的arbitrary任意的square平方absolutevalue绝对值squareroot平方根课后答案网cube立方induction归纳法www.hackshp.cnNote1.Ourworkingexperiencewithnumbershasprovidedusallwithsomefamiliaritywiththeprinciplesthatgovernthereal-numbersystem.意思是:我们对数的实际工作经验使我们大家对支配着实数系的各原则早已有了某些熟悉,这里working作”实际工作的”解,govern作”支配”解.2.Theplussign,”+”,usedherenotexpresstheoperationofaddition,butisratherpartofthesymbolismforthenumbersthemselves.意思是:这里的正符号”+”不是表示加法运算,而是数本身的符号系统的一部分.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 3.Arealnumberissaidtobearationalnumberifitcanbeexpressedastheratiooftwointegers,wherethedenominatorisnotzero.这是定义数学术语的一种形式.下面是另一种定义数学术语的形式.Amatrixiscalledasquarematrixifthenumberofitsrowsequalsthenumberofitscolumns.这里iscalled与issaidtobe可以互用,注意iscalled后面一般不加tobe而issaid后面一般要加.4.Arealnumberthatcannotbeexpressedastheratiooftwointegersissaidtobeanirrationalnumber.与注khdaw.com3比较,这里用定语从句界定术语.5.Thereisauniquerealnumber,calledzeroanddenotedby0,withthepropertythata+0=0+a,whereaisanyrealnumber.意思是:存在唯一的一个实数,叫做零并记为0,具有性质a+0=0+a,这里(其中)a是任一实数.1)这里called和denoted都是过去分词,与后面的词组成分词短语,修饰number.2)withtheproperty是前置短语,修饰number.3)注意本句和注3.中where的用法,一般当需要附加说明句子中某一对象时可用此结构.课后答案网ExerciseExerciExerciseseI.TurnthefollowingarithmeticexpressionsintoEnglish:i)3+(-2)=1ii)2+3(-4)=-10iii)=-5iv)=3www.hackshp.cnv)2/5-1/6=7/30II.Fillineachblankthemissingmathematicaltermtomarkthefollowingsentencescomplete.i)Theoftworealnumbersofunlikesignsisnegative.ii)Anintegerniscalledifn=2mforsomeintegerm.iii)Ansolutiontotheequationxn=ciscalledthenisofc.iv)Ifxisarealnumber,thentheofxisanonnegativerealnumberdenotedby|x|anddefinedasfollowskhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com x,ifx0|x|=-x,ifx<0III.TranslatethefollowingexercisesintoChinese:i)Ifxisanarbitraryrealnumber,provethatthereisexactlyoneintegernsuchthatx2,wherecisafixedpositivenumber,useinductiontoprovethatan1.iv)khdaw.comDetermineallpositiveintegersnforwhich2n0,y是任意一实数www.hackshp.cn,则存在一正整数n使得nx>y.Ⅵ.1.Trytoshowthestructureofthesetofrealnumbersgraphically.2.Listandstatethelawsthatoperationsofadditionandmultiplicationofrealnumbersobey.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－TheRoleofMathematicsinEconomicsEconomicsisamathematicaldiscipline.Thisassertionmayseemstrangetothetraditionalpoliticaleconomist,butmathematicalmethodswereintroducedatanearlystage(Cournot,1838)inthetwo-hundred-yearhistoryofoursubjectandhavebeensteadilygrowinginsignificance.Atthepresenttimeand,essentially,sincetheendofWorldWarⅡ,mathematicalmethodshavebecomepredominantinAmericaneconomics.ThemathematicalapproachwasoriginallyinspiredinEuropeandEnglandbutithasfloweredinAmerica,withnolittlestimulusfromEuropeanimmigrants.Themathematicalapproachissteadilygainingfavorthroughouttheworld,especiallybecausetheyoungergenerationindevelopingeconomicsisembracingthenewmethodsandbecausethesocialistcountrieshaveshedapreviousbiasagainsttheuseofmathematicalmethodsineconomics.Itisclearthatthefuturedevelopmentofeconomicswillseecontinuedkhdaw.comandincreasinguseofmathematics,althoughitwouldberashtoassumethatthefuturecourseofeconomicanalysiswillbepredominantlymathematicalasithasbeeninthelasttwentyyears.TheTheEconomicEcEconomicProblemonomicProbleProblemmAfavoreddefinitionofeconomics(LionelRobbins,1932)is“…thesciencewhichstudieshumanbehaviorasarelationshipbetweenendsandscarcemeanswhichhavealternativeuses.”Whetherornotweacceptthisdefinitionasbracketingallofeconomics,itisagoodstartingpointforourdiscussionoftheroleofmathematics.Imightwanttosharpenthisdefinitionbynotingthateconomiststrytoselectamongalternativeusesofscarceresourcesinsuchawayastomakethemostefficient(orleasewasteful)employmentofresourcesto课后答案网achievestatedends.Statedinthisway,weseeclearlythateconomicsinvolvesoptimization,andthisistheenginethatproducesprinciplesofeconomicanalysis.Wehaveeitheramaximumproblemoraminimumproblem,whichisacompellingreasonforwww.hackshp.cntheuseofmathematics.Anabstracteconomyisviewedasconsistingofnumerousconsumingandproducingunits,whomakeoptimaldecisionsabouttheirowneconomicbehaviour,givenmarketprices,andtheninteractwithoneanothertoclearsupplyanddemandinmarketstodetermineprices.Economictheoryusuallybeginswithananalysisoftheindividualconsumerwhoattemptstomaximizehissatisfaction,subjecttoabudgetconstraint(ortominimizebudgetoutlaysfortheattainmentofanygivenlevelofsatisfaction).Thetheorythentakesuptheanalysisofproducerswhostrivetomaximizeprofits,Subjecttoatechnologicalconstraint(orminimizecostforreachingagivenoutputlevel,subjecttoatechnologicalconstraint).Thesearethetypicaloptimizationproblemsofeconomics.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thestandardmathematicalformulationsoftheseproblemsareasfollows.Theconsumerproblemistomaximizeautilityfunctionofquantitiesofgoodsandservicesconsumed,subjecttotherequirementoflivingwithinafixedincomewherearetherespectivepricesofthegoodsandservicesconsumed.Theproducerproblemistomaximizeincomeminusproductioncostswhereareinputsoffactorsofproductionandaretheircosts,subjecttothereskhdaw.comtraintimposedbyatechnicalproductionfunctionofthequantitiesofgoodsandservicesproducedandoftheproductionfactorinputs.Thesetwoformulationsposetheeconomicproblemasthemaximizationofutility(satisfaction),subjecttoabudgetconstraint,andthemaximizationofprofit,subjecttoatechnologicsiconstraint.Wecouldalsoformulateminimumproblemsthatseekminimumproductioncostsforproducingagivencombinationofoutputsandtheleast-costbudgettoachieveagivenlevelofutility.TreatmentTreatmeTreatmentTreatmentofntofofoptimizationoptimizatioptimizationoptimizationproblemsonprobleproblemsmsTheconsequencesofthesemaximizationorminimizationproblemshavebeene课后答案网normousforeconomicsinbuildingasetofrulesofbehavior.Nearlyalleconomictruthshavesomerootintheseorcloselyrelatedpropositions.Theoriginalmathematicalattackwasquitestraightforward.Assumethatandaresmoothcontinuousfunctions(withfirstandsecondderivatives),andoptimizeaccordingwww.hackshp.cntotherulesofthedifferentialcalculus,givenmarketprices.Thenecessaryandsufficientconditionsforoptimizationdefinethewell–knowndemandandsupplyfunctionsofeconomicsofeconomicsandestablishmanypropertiesofthesefunctions.FortheproblemasIhavestatedit,thesesolutionsarewellestablishedandhavebeenintheliteratureofeconomicsformorethanfiftyyears.RefinedpointsaremadefromtimetotimebuttheramificationsofthistheoryweremadeclearinmathematicaltreatmentsbyPareto(1896),Slutsky(1915),Fisher(1892),Hotelling(1932),Frisch(1932),HicksandAllen(1943),Samuelson(1947).khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Inthe1930"s,andagainafterWorldWarⅡ,theseproblemsreceivedextendedmathematicaltreatment,Theextensionsweretooptimizeovertimeeithercontinuouslyorinfiniteincrementalperiodsandtoenlargethenumberofsideconditions.Instochasticmodels(i.e.,thosethatincorporatechance),uncertaintyaboutfutureconditionssuchaspricecanbeintroduced.Also,wecanallowfortheaccumulationoftinyneglectedfactorsthatalwaysinfluencehumandecisions.Thesubjectivenatureoftheutilityfunction,ledtoanalysisofconditionsinwhichtheresultsofoptimizationwouldbeinvariantundertransformationsofthefunctionandtostudyofthepossibilityofderivingautilityfunction,startingfromobjectivedemandfunctions.Thelatterproblembecameknownastheintegrabilityproblem.khdaw.comItmayberemarkedthattheearlydevelopmentofmathematicaleconomicsfollowedthestepsofphysicsandengineering.Therearemanyanalogiesbetweentheclassicalmethodsofmathematicaleconomicsandthelawsofmechanics,thermodynamics,andsimilarbranchesofscience.Insomecases,therewasatendencytodrawstrictanalogiesthatcouldhardlyberationalizedintermsofeconomicbehavior.AnideathatreceivedmuchencouragementfromJ.VonNeumannwasthatmathematicaleconomicsshoulddrawupondifferentbranchesofmathematicsthatweremoresuitedtothepeculiarnatureoftheeconomicproblemandeconomicvariables.Itwasevensuggestedthatnewmathematicalmethodsmightbedevelopedthatwouldbetailoredtoeconomics.Inthesensethatmathematiciansoftheeighteenthandnineteenthcenturiesdevelopedmethodsthatweresuite课后答案网dtotheproblemsofphysics,wemighthopethatmodernmathematicianswouldreceiveinspirationfromproblemsofeconomics,andsocialsciencesgenerally.Tosomeextent,thisdevelopmenthasoccurredinlinearprogrammingandoptimizationtheoryforsituationsinwhichtheordinarymethodsofdifferentialcalculusdonotapply.Itisuptothemathematiciansthemselves,however,todecwww.hackshp.cnidethesignificanceofthislineofdevelopmentinmodernmathematics.----------LawrenceR.KleinVocabularyVocabulVocabularyarypredominant主导的economicanalysis经济分析scarceresource不充足资源khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com outlay开支、费用income收入proposition命题smooth光滑increment增量incremental增量的stochastic随机的derive推出thermodynamicskhdaw.com热动力学rationalize合理化marketprice市场价格supply供应，供给demand需求budget预算budgetoutlay预算开支profit利益，利润课后答案网cost成本goods货物services服务www.hackshp.cnramification(ofthetheory)理论的）细节sidecondition附属条件NotesNotes1.象two-hundred-year(两百年)这样的复合词，year不用复数。例如：Five-year-plan(五年计划)2.ThemathematicalapproachwasoriginallyinspiredinEuropeandEnglandbutithasfloweredinAmericawithnolittlestimulusfromEuropeanimmigrants.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 意思是：这种数学方法创于欧洲大陆和英国，但是已经在美洲（美国）开花，当然少不了欧洲移民的激励。这里flower作为动词用；而且withnolittlestimulus是一种肯定语气。3.Whetherornotweacceptthisdefinitionasbracketingallofeconomics,itisagoodstartingpointforourdiscussionoftheroleofmathematics.意思是：不管我们是否接受这个定义作为概括所有经济学，它都是我们用讨论数学（用于经济学）作用的一个良好起点，这里bracketing作为“概括”解.4.Anabstracteconomyisviewedas…whomakeoptimaldecisionsabouttheirowneconomicbehaviour,givenmarketprices,andtheninteractwithoneanothertoclearsupplyanddemandinmarketstodetermineprices.意思是：抽象经济可以看成由许多个消费和生产单位所组成，这些单位（的决策者）就他们自己的经济行khdaw.com为——给出市场价格——作出最优决策，然后相互去特约市场的供需交换，以便确定价格。这里1）who可理解为units的决策人的关系代词2）givenmarketprices是economicbehaviour的同位语。3）oneanother是指units之间，而不是指marketprices之间4）clear这一词用于商业上其意思是：“卖光，买光，交换，清理”等。5.Newmathematicalmethodsmightbedevelopedthatwouldbetailoredtoeconomics.tailor是“裁缝”的意思，这里作动词用，意思是：“使其适用于经济学”课后答案网6.Upto作“取决于”解。ExerciseExerciExerciseseⅠ.GiveanexampleofatypicaloptimationproblemofEconomicssoastoshowthatEconomicsneedswww.hackshp.cnmathematics.Ⅱ.TranslatethefollowingpassageintoChinese:Economicanalysishas,inthelasttwentyyears,becomepredominantlymathematical.ThisisparticularlytrueintheUnitedStates,wheredoctoralcandidatesnowsubstitutevariouscoursesinmathematicsforatleastsomeofthetraditionalforeignlanguagerequirement.Economicproblemsinvolvingoptimaldecisionsbygovernmentandbusinessorstablegrowthofaneconomyhaveanalogiesinproblemsofphysicsandengineeringthathavelongbeensuccessfullytreatedmathematically,Buteconomicshasoutgrownthedayswhenitmerelyapedthephysicalsciencesinapplyingmathematics.Theauthorsuggeststhatinthecomingeraeconomicsmaycallforthitsownbranchofmathematicsorprovideinspirationforgreatnewmathematicaldiscoveries.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Ⅲ.TranslatethefollowingsentencesintoEnglish(makeuseofthephraseinbracketandseewhetheronecanbereplacedbytheotherornot):1.求在下列限制条件下，函数F(x,y)的最大值。（Subjectto）2.设其中是一集合，是实数集，若满足如下条件：（Ⅰ）;(Ⅱ)当且仅当x=y时，;(Ⅲ)对称性：(Ⅳ)三角不等式：其中.则称是一距离函数.（Satisfythefollowingcondition(s)）.3.设是定义在区间Ⅰ的一个连续函数，则在区间Ⅰ是有界闭的假设下，我们可以断言，在Ⅰ上一致连续（Undertheassumption(hypothesis);claim）khdaw.com课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 数学专业英语－TheTheoryofGraphsInthischapter,weshallintroducetheconceptofagraphandshowthatgraphscanbedefinedbysquarematricesandversa.1.IntroductionGraphtheoryisarapidlygrowingbranchofmathematics.Thegraphsdiscussedinthischapterarenotthesameasthegraphsoffunctionsthatwestudiedpreviously,butatotallydifferentkind.Likemanyoftheimportantdiscoveriesandnewareasoflearning,graphtheoryalsogrewoutofaninterestingphysicalproblem,theso-calledKonigsbergkhdaw.combridgeproblem.(thisproblemisdiscussedinSection2)TheoutstandingSwissmathematician,LeonhardEuler(1707-1783)solvedtheproblemin1736,thuslayingthefoundationforthisbranchofmathematics.Accordingly,Euleriscalledthefatherofgraphtheory.GustayRobertKirchoff(1824-1887),aGermanphysicist,appliedgraphtheoryinhisstudyofelectricalnetworks.In1847,heusedgraphstosolvesystemsoflinearequationsarisingfromelectricalnetworks,thusdevelopinganimportantclassofgraphscalledtrees.In1857,ArthurCaylcydiscoveredtreeswhileworkingondifferentialequations.Later,heusedgraphsinhisstudyofisomersofsaturatedhydrocarbons.CamilleJordan(1838-1922),aFrenchmathematician,WilliamRowanHamilton,andOysteinOreandFrankHarary,twoAmericanmathematicians,arealso课后答案网knownfortheiroutstandingcontributionstographtheory.Graphtheoryhasimportantapplicationsinchemistry,genetics,managementscience,Markovchains,physics,psychology,andsociology.www.hackshp.cnThroughoutthischapter,youwillfindaverycloserelationshipbetweengraphsandmatrices.2.TheKonigsbergBridgeProblemTheRussiancityofKonigsberg(nowKaliningrad,Russia)liesonthePregelRiver.(SeeFig.1)ItconsistsofbanksAandDoftheriverandthetwoislandsBandC.Therearesevenbridgeslinkingthefourpartsofthecity.Residentsofthecityusedtotakeeveningwalksfromonesectionofthecitytoanotherandgooversomeofthesebridges.This,naturally,suggestedthefollowinginterestingproblem:canonewalkthroughthecitycrossingeachbrikhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com dgeexactlyonce?Theproblemsoundssimple,doesn’tit?Youmightwanttotryafewpathsbeforegoinganyfurther.Afterall,bythefundamentalcountingprinciple,thenumberofpossiblepathscannotexceed7!=5040.Nonetheless,itwouldbetimeconsumingtolookateachofthemtofindonethatworks.Fig.1ThecityofKonigsbergIn1736,Eulerprovedthatnosuchwalkispossible.Infact,heprovedafarmoregeneraltheorem,ofwhichtheKonigsbergbridgeproblemisaspecialcase.Fig.2AmathematicalmodelfortheKonigsbergbridgeproblemkhdaw.comLetusconstructamathematicalmodelforthisproblem.rcplaceeachareaofthecitybyapointinaplane.ThepointsA,B,C,andDdenotetheareastheyrepresentandarecalledvertices.Thearcsorlinesjoiningthesepointsrepresenttherepresenttherespectivebridges.(See图２)Theyarecallededges.TheKonigsbergbridgeproblemcannowbestatedasfollows:Isitpossibletotracethisfigurewithoutliftingyourpencilfrompaperorgoingoverthesameedgetwice?Eulerprovedthatafigurelikethiscanbetracedwithoutliftingpencilandwithouttraversingthesameedgetwiceifandonlyifithasnomorethanweoverticeswithanoddnumberofedgesjoiningthem.Observethatmorethantwoverticesinthefigurehaveanoddnumberofedgesconnectingthem-----infact,theyalldo.课后答案网1.GraphsLetusreturntotheexampleFriendlyAirlinesfliestothefivecities,Boston(B),Chicago(C),Detroitwww.hackshp.cn(D),Eden(E),andFairyland(F)asfollows:ithasdirectdailyflightsfromcityBtocitiesC,D,andF,fromCtoB,D,andE;fromDtoB,C,andF,fromEtoC,andfromFtoBandD.Thisinformation,thoughitsoundscomplicated,canbeconvenientlyrepresentedgeometrically,asin图3.Eachcityisrepresentedbyaheavydotinthefigure;anarcoralinesegmentbetweentwodotsindicatesthatthereisadirectflightbetweenthesecities.Whatdoesthisfigurehaveincommonwith图２?Bothconsistofpoints(denotedbythickdots)connectedbyarcsorlinesegments.Suchafigureiscalledagraph.Thepointsaretheverticesofthegraph;thearcsandlinesegmentsareitsedges.Moregenerally,wemakethefollowingdefinition:Agraphconsistsofafinitesetofpoints,togetherwitharcsorlinesegmentsconnectingsomeofthem.Thesepointsarecalledtheverticesofthegraph;thearcsandlinesegmentsarecalledtheedgesogthekhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com graph.TheverticesofgraphareusuallydenotedbythelettersA,B,C,andsoon.AnedgejoiningtheverticesAandBisdenotedbyABorA-B.Fig.3图２and图3aregraphs.Othergraphsareshownin图4.Thegraphin图２hasfourverticesA,B,C,andD,andsevenedgesAB,AB,AC,BC,BD,CD,andBD.Forthegraphin图4b,therearefourvertices,A,B,C,andD,butonlytwoedgesADandCD.Considerthegraphin图4c,itcontainsanedgeemanatingfromandterminatingatthesamevertexA.Suchanedgeiscalledaloop.Thegraphin图4dcontainstwoedgesbetweentheverticesAandCandaloopatthevertexC.ThenumberofedgesmeetingatavertexAiscalledthevalenceordegreeofthevertex,denotedbyvkhdaw.com(A).Forthegraphin图4b,wehavev(A)=1,v(B)=0,v(C)=1,andv(D)=2.In图4b,wehavev(A)=3,v(B)=2,andv(C)=4.AgraphcanconvenientlybedescribedbyusingasquarematrixinwhichtheentrythatbelongtotherowheadedbyXandthecolumnbyYgivesthenumberofedgesfromvertexXtovertexY.Thismatrixiscalledthematrixrepresentationofthegraph;itisusuallydenotedbytheletterM.ThematrixrepresentationofthegraphfortheKonigsbergproblemisClearlythesumoftheentriesineachrowgivesthevalenceofthecorrespondingvertex.Wehavev(A)=3,v(B)=5,v(C)=3,aswewouldexpect.课后答案网Conversely,everysymmetricsquarematrixwithnonnegativeintegralentriescanbeconsideredthematrixrepresentationofsomegraph.Forexample,considerthematrixABCDwww.hackshp.cnClearly,thisisthematrixrepresentationofthegraphin图5.VocabularyVocabulVocabularyaryNetwork网络Electricalnetwork电网络khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Isomer异构体emanate出发，引出Saturatedhydrocarbon饱和炭氢化合物terminate终止，终结Geneticskhdaw.com遗传学valence度Managementsciences管理科学node结点Markovchain马尔可夫链interconnection相互连接课后答案网Psychology心理学Konigsbergbridgeproblemwww.hackshp.cn康尼格斯堡桥问题Sociology社会学Line-segment线段NotesNotes1.CamilleJordan,aFrenchmathematician,WilliamRowanHamiltonandkhdaw.com.......若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 注意：aFrenchmathematician是CamilleJordan的同位语不要误为W.R.Hamilton是aFrenchmathematician同位语这里关于W.R.Hamilton因在本文前几节已作介绍，所以这里没加说明。２．Afterall,bythefundamentalcountingprinciple,thenumberofpossiblepathscannotexceed7!=5040.Nonetheless,itwouldbetimeconsumingtolookateachofthemtofindonethatworks.意思是：毕竟，由基本的计算原理知，可能的路径的总数，不会超过5040个。然而逐一地去考察这些路径是否有一条路适合题意，那是太耗费时间了，thatworks意思是：“有效”，这里可理解为：“适合题意”。3.Itispossibletotracethefigurewithoutliftingyourpencilfrompaperorgoingthesameedgetwice?意思是：是否能够跟踪图形而使你的铅笔不离开纸且不走过同一条边两次呢？这一句在英语上等同于withoutliftingyourpencilfrompaperandwithoutgoingoverthesameedgetwice.khdaw.com1........infact,theyalldo.这里they代表顶点vertices;do代表haveanoddnumberofedgesconnectingthem.2.Aiscalledthevalenceordegreeofthevertex,denotedbyv(A).注意denoted前面的逗号，可使读者不至于误会v(A)是用来记vertex的。这里v(A)是用来记Ａ的Ｖalence.６.theentrythatbelongstotherowheadedbyXandcolumnheadedbyYgivesthenumberofedgesfromvertexXtovertexY.意思是：属于Ｘ行，Ｙ列这一项的数字给出了从顶点Ｘ到顶点Ｙ的边数。这里therowheadedbyX意是冠以Ｘ的行，可简称Ｘ行或等Ｘ行。课后答案网ExerciseExerciExerciseseⅠ.answerthefollowingquestions:www.hackshp.cn1.HowistheKonigsbergBridgeproblemstated?2.AccordingtoEuler’stheorem,whyistheansweroftheKonigsberhBridgeProblemnegative?Ⅱ.TranslatethefollowingpassagesintoChinese:Whenanumberofelectricalcomponentsareconnectedtogether,wearesaidtohaveanelectricalnetwork.Thejunctionbetweentwoormorecomponentsinanetworkarecallednodesofthenetwork,Eachpathjoiningapairofnodesandthroughinterconnectionsisbestdescribedbyadiagramwhicheliminatesalltheelectricalpropertiesofthecomponents.Thisgraphisobtainedbyredrawingthecircuitofthenetworkwithlinesreplacingtheelectricalcomponents.khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com Thegraphmakescleartheexistenceofanumberofclosedpathswhichmaybetracedalongthebranches.Suchclosedpathsarecalledloops.Ofthetotalnumberofloopsofanetwork,acertainnumberofindependentloopsmaybechosen.Onewayofchoosingasetofindependentloopsisasfollows:form,fromthenetwork,asub-networkbyremovingbranchesuntilnoloopsremain,althougheachnodeisstillconnectedbyasinglepathtoanothernode.Suchastructureiscalledatreeofthenetwork.Ⅲ.TranslatethefollowingsentencesintoEnglish(ineachsentence,makeuseofthephrasegiveninbracket):下面简写TheKonigsbergBridgeproblem为Ｋ.B.问题１．Ｋ.B.问题只不过是尤拉所证明的定理的一个特例。(aspecialcase)２．从尤拉关于图论的一个定理，即可得Ｋ.B.问题的答案。(followsimmediatelyfrom.)khdaw.com３．Ｋ.B.问题的不可能性是尤拉定理的一个直接结果。(adirectconsequenceof)课后答案网www.hackshp.cnkhdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com'