数学专业英语第二版的课文翻译

1、fromman1AWhatismathematicsMathematicscomesssocialpractice,forexample,industrialandagriculturalproduction,commercialactivitmilitaroperationies,ysandmathematicresearches.Andinturn,sscientificserveandtechnologicalthepracticeandplays数军事行动和科学技术研究。反过没有应用数学，任何一个现early need of man came thegreatroleinallfiel

2、ds.Nomodernscientificandtechnologicalbranchescouldberegularlydevelopedwithouttheapplicationofmathematics.学来源于人类的社会实践，比如工农业生产，商业活动，来，数学服务于实践，并在各个领域中起着非常重要的作用。在的科技的分支都不能正常发展。Fromtheconceptsofnumbersandforms.Then,geometrydevelopedoutofproblemsofmeasuringland,andtrigonometrycamefromproblemsofsurveying.e

3、stablishedTodealwithsomesolveandthendmorecomplexequatiowitnhpracticalunknownproblems,mannumbers,thusalgebraoccurred.Before17thcentury,elementarymathematics,i.e.,geometry,manconfinedtrigonometryhimselftotheandalgebra,inwhichonlytheconstantsareconsidered.形式的概念，接着，测量土地的需要形成了几何，很早的时候，人类的需要产生了数和由于测量的需要产生

4、了三角几何，为了处理更复杂的实际问题，人类建立和解决了带未知参数的方程，从而产生了代数学，17世纪前，人类局限于只考虑常数的初等数学，即几何，三角几何和代数。Therapidofdevelopmentindustryin17thcenturypromotedtheprogressofeconomicsandtechnologyandrequireddealingwithvariablequantities.Theleapfromconstantstovariablequantitiesbroughtabouttwo new branche sof mathematics-analyticbel

5、ong to the higher mathematics. higher mathematics, among whichgeometry and calculus, which therNow e are many branches inare mathematicalanalysis, higheralgebra,differentialequations,functiontheory世纪工业的快andsoon.17速发展推动了经济技术的进从而遇到需要处理变量的问题，从常数带变量的跳跃产步，生了两个新的数学分支-一解析几何和微积分，他们都属于高等数学，现在高等数学里面有很Mathemat

6、icians study多分支，其中有数学分析，高等代数，微分方程，函数论等。conceptionsproposition and s,Axioms,postulatesdefinition sandtheoremsareallpropositions.Notationsareaspecialandpowerfultoolofmathematicsandareusedtoexpressconceptionsandpropositionsmainlybylogicaldeductionsandcomputation.Foralongperiodofthevery often.FormulasSo

7、me numeralofthe best,figures knownand chartssymbolsof1,2,3,4,5,6,7,8,9,0multiplication, division and equality.定义和定理都是命题。且full are ofmathematics ofand the signs addition,differe ntaresymbols.the Arabicsubtraction数学家研究的是概念和命题，公理，公 设，符号是数学中一个特殊而有用的工常用于表达概念和命题。式，图表都是不同的符号? ? .Theconclusion sin mathemati

8、csare obtainedhistoryofmathematics,thecentricplaceofmathematicsmethodswasoccupiedbythelogicaldeductions.Now,sinceelectroniccomputersaredevelopedpromptlyandusedwidely,theroleofcomputationbecomesmoreandmoreimportant.Inourtimes,computationisnotonlyusedtodealwithalotofinformationanddata,butalsotocarryou

9、tsomeworkthat merely could be done earlier by logical deductions, for example, theproof of most of geometricaltheorems.在数学发展历史的很长时间 内，计算机的迅速发展和广泛使逻辑推理一直占据着数学方法的中心地数学结论主要由逻辑推理和计算得到,现在，由于电子 现在计算机不仅用于处理大量用，信息和数据, 作，例如，大多数几何定理的证计算机的地位越来越重要,还可以完成一些之前只能由逻辑推理来做的工明。1BEquationAnequationisastatementoftheequal

10、itybetweentwoequalnumbersornumbersymbols.Equationareoftwokinds-一identitiesandequationsofcondition.Anarithmeticoranalgebraicidentityisanequation.In such aneither theequation twomembersare alike.Orbecome alike on the performance of the indicatedoperation.个数或者数的符号相等的一种描述。等式有两种-恒等式和条件等式等式是关于两 算术或者代数恒等式是

11、等式。这种等式的两端要么一样，要么经过执行指定的运算后变成一 样。An identityinvolving letters is true for any set of numerical values ofthe letters in it. An equation which is true only for certainvalues of a letter in it, or for certain sets of relatedvalues of two or more of its letters, is an equation ofcondition, or simply an

12、equation. Thus 3x-5=7 is true for x=4only; and 2x-y=0 is true for x=6 and y=2 and for many other pairs of values for x andy.含有字母的恒等式对其中字母的任一组数值都成立。一个等式若仅仅对其中一个字母的某些值成立，或对其中两个或着多个字母的若干组相关的值成立，则它是一个条件等式，简称方程。因仅当x=4时成立，2x-y=0,当时成立，且对 x,此3x-5=7 而x=6,y=2y的其他许多对值也成立。A root of an equation is any number or

13、numbersymbolwhich satisfies the equation. There are various kinds of equation. They are linear equation, quadratic equation,etc.方程的根是满足方程的任意数或者数的符号。方程有很多种，例如：线性方程，二次方程等。To solve an equationmeansto find the value of the unknown term. To do this , we must, of course, change the terms about until the u

14、nknown term stands alone on one sideof the equation, thus making it equal to something on theother side. Wethen obtain the value of the unknown and the answer to thequestion. Tosolve the equate eherefor means to move and change equatioabout withoutmakingthenuntrue, until only theterm the sunknownqua

15、ntity is left on one side ,no matter which side.为了求未知项的值，当然必须移项,边，解决了问题。从而求得未知项的值，形，解方程意味着求未知项的值，直到未知项单独在方程的一令其等于方程的另一边，因此解方程意味着进行一系列的移项和同解变直到Equationareofverygreat未知量被单独留在方程的一边，无论那一边。use.Wecanuseequationinmanymathematicalproblems.Wemaynoticethatalmosteveryproblemgivesusoneormorestatementsthatsometh

16、ingisequaltosomething,thisgivesusequations,withwhichwemayworkif方程作用很大，可以用方程解决很多数学问题。注意到几乎每一个问题weneedit.都给由一个或多个关于一个事情与另一个事情相等的陈述，这就给由了方程，利用该方程，如果我们需要的话，可以解方程。2AWhystudygeometry?Manyleadinginstitutionsofhigherlearninghaverecognizedthatpositivebenefitscanbegainedbyallwhostudythisbranchofmathematics.Th

17、isisevidentfromthefactthattheyrequirestudyofgeometryasaprerequisitetomatriculationinthoseschools.多居于领导地位的学术机构承认，所有学习这个数学分支的人都将得到确实的受 皿校把几何的学习作为入学考试的先决条件，从这一点上可以证明。origin long ago in the measurement by the Babylonians andGeometry许多学haditsEgyptians of their landsinundatedthe floods by ofgeometry“ - me

18、asureis derivedfrom geo, meaningthe Nile River.“ earth andThe greekwordmetron, meaning.As early as 2000 B.C. we find the land surveyors of thesepeoplere-establishingvanishinglandmarksandboundariesbyutilizing几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河geo ,意思是” 土地 和metron 意思是”测量thetruthsofgeometry.洪水淹没的土地，希腊语几何来源于公元前2

19、000年之前，我们发现这些民族的土地测量者利用几何知识重新确定消失了的土地标志和边界。2 B Some geometricalterms A solid is a three-dimensionalfigure.Commonexamplesofsolidsarecube,sphere,cylinder,coneandpyramid.Acubehassixfaceswhicharesmoothandflat.Thesefacesarecalledplanesurfacesorsimplyplanes.Aplanesurfacehastwodimensions,length and width.

20、Thesurface of a blackboardor of a tabletopis anexample of a plane surface.柱体，圆锥和棱锥。立方体有 简称为平面。平面曲面是二维的, 例子。立体是一个三维图形，立体常见的例子是立方体,6个面，都是光滑的和平的，这些面被称为平面曲面或者 有长度和宽度，黑板和桌子上面的面都是平面曲面的球体,2 - C三角函数于直角三角形的解One of the most importantapplication strigonometryisthesolutionoftriangles.Letusnowtakeupthesolutionto

21、 right triangles.A triangleis composedof six partsthreesidesandthreeangles.Tosolveatriangleistofindthepartsnotgiven.Atrianglemaybesolvedifthreeparts(atleastoneoftheseisaside)aregiven.Arighttrianglehasoneangle,therightangle,alwaysgiven.Thusarighttrianglecanbesolvedwhentwosides,oronesideandanacute由6个部

22、分组成，三条边和三只角。解一个三角形就是要求由未知的部分。如果三角angle, are given.三角形最重要的应用之一是解三角形,现在我们来解直角三角 形。一个三角 形形的三个部分（其中至少有一个为边）为已知，则此三角形就可以解由。直角三角形的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一锐角为已知，则此直角三角形可解9-AIntroductionAlargevarietyofscientificproblemsariseinwhichonetriestodeterminesomethingfromitsrateofchange.Forexample,wecouldtrytoc

23、omputethepositionofamovingparticlefromaknowledgeofitsvelocityoracceleration.Orasubstancmayberadioactiveedisintegratingataknownrateandwemayberequiredtodeterminetheamountofmaterialpresentafteragiventime.大量的科学问题需要人们根据事物的变化率来确定该事物，例如，我们可以由已知速度或者加速度来计算移动粒子的位又如，某种放射性物质可能正在以已知的速度进行衰变，物质的总量unknownfunctio需要我

24、们确定在给定的时间后遗留Inexampleslikethese,wearetryingtodetermineannequationfrominvolvinprescribedatinformationexpressedthinformofanfunction.Theseleastthequationsoneofcallearedthederivativesdifferentialofunknownequations,andtheirstudyformsoneofthemostchallengingbranchesofmathematics.似的例子中，我们力求由方程的形式表示的信息来确定未知函

25、数，在类知函数的一个导数。门分这些方程称为微分方程,而这种方程至少包含了未对其研究形成了数学中最具有挑战性的一支。Thestudyofdifferentialequationsisonepartofmathematicsthat,perhapsmorethananyother,hasbeendirectlyinspiredbymechanics,astronomy,andmathematicalphysics.其他分支更多的直接受到力学，天文学和数学物理的推动。微分方程的研究是数学的一部分，也许比Itshistorybeganintheth17thsimplecenturywhenNewton

26、,differentialequationsLeibniz,arisingandfrommechanics.Theseearlydiscoveries,beginningabout1690,graduallyledtothedevelopmentofalotofkindsofdifferentialequation.“specialtricks微分方程起源于努力家族解决了一些来自几何和力学的简单的微分方程。开始于渐引起了解某些特殊类型的微分方程的大量特殊技巧的发展。tricksareapplicableinrelativelyfewcases,theydoenableustosolvediff

27、erentmanyialequationsthatariseinisofimportance.Sometheirstudypracticalofandsomeoftheproblemswhichtheyhelpussolvearediscussedneartheendofthischapter.Bernoullisproblems”forsolv17insolvedsomegeometryingcertainspecial世纪，当时牛顿，莱布尼茨,波and1690年的早期发现，逐Althoughthesespecialmechanicsthesegeometryand,specialmetho

28、ds尽管这些特殊的技巧只是用于相对较少的几种情况，但他们能够解决力学和几何中由现的许多微分方程，因此，他们的研究具有重要的实际应用。特殊的技巧和有助于我们解决的一些问题将在本章最后讨论。Experiencehasshownso这些i difficul that t is tto obtain mathematicalotheoriesfmuchgeneralityaboutsolutionofdifferentialequations,exceptfor经验表明afewtypes.Among these are the除了几个典型方程外，很难得到微分方程解的一般性数学理论。so-calledl

29、ineardifferentialequationswhichoccurinagreatvarietyofscientific学问题中。applications ofmatricesproblems.在这些典型方程中，有一个称为线性微分方程，由现在大量的科10-C Applications of matrices In recent years theinmathematics anddiverswith remarkablespeed. Matrixin many theoryplaysfiel ds centra lhave increasedrole in modernphysicsin

30、thestudyofquantummechanics.Matrixmethodsareusedtosolveproblemsinapplieddifferentialequations,specifically,intheareastudie sanalysis factorofaerodynamics,stressandstructureanalysis.Oneofthemostpowerfulmathematicamethodsforpsychologicalsubjectthatmakeswideuseofmatrixmethods.同的领域中，矩阵的应用一直以惊人的速度不断增加。近年来

31、，在数学和许多各种不物理学上起着主要的作用。解决应用微分方程,在研究量子力学时，特别是在空气动力学，矩阵理论在现代应力和结构分析中的问题，要用矩阵方法。心理学研究上一种最强有力的数学方法是因子分析，这也广泛的使用in矩阵（方）法.Recentdevelopmentsmathematicaleconomicsandinproblemsofbusinessadministrationhaveledtoextensiveuseofmatrixmethods.Thebiologicalsciences,andinparticulargenetics,usematrixadvantage，ftechni

32、questogood.Nomatterwhatthestudentsieldofmajointerethmatricerstis,knowledgeoferudimentsofsislikelytobroadentherangeofliteraturethathecanreadwithunderstanding.近年来，在数学经济学和商业管理问题方面的发展已经导致广泛的使用矩阵生物科学，法。特别在遗传学方面，用矩阵的技术很有成效。可能扩大他能读懂的文献的范围Thei不管学生主要兴趣是什solutio nof ntheequations n n unknowns is one ofimporta

33、nt矩阵基本原理的知识 simultaneouslinearproblemsof appliedmathematics. Descartes, the inventor of analytic geometry and one offounder oalgebraithe s f modern cnotation,coulsolutiod ultimately be reduced to the nbelievealdthat l problemsof a set of simultaneouslinear解一有n个未知数的 n个联立一次（线性） 方程是应用数学的一个belief is now

34、 known to beo in the solution f theseequations.重要问题。解析几何的发明者和现代代数计数法的创始人之一笛卡儿相信，所有的问题最后都能约简为解一组联立一次方程。Althoughthisuntenable,weknowthatalargegroupofsignificantappliedproblemsfrommanydifferentdisciplinesarereducibletosuchequations.Manyoftheapplications,requirethesolutionofalargenumberofsimultaneouslin

35、earequations,sometimesinthehundreds.Theadventofcomputershasmadethematrixmethodseffectiveformidableproblems.虽然这种信念现在认为是站不住脚的，但是，我们知道，从许多不同的学科里的一大群重要的应用问题都可以约化为这类的方程。许多应用要求解大量的，往往数以百计的联立一次方程，计算机的发明已经使得矩阵方法在解这些难以解决的问题方面非常活跃。Example1.solvethesimultaneousequationsforx1x2,andx3.解联立方程求x1x2和x3。Fromtheaboved

36、iscussion,I erefore nott surprising that inbooks on the theoryi of matrices thes ,techniques offinding inverset matrices occupy hh considerable blemofsolvingnsimultaneouslinearequationreducedtotheproblemoffindingtheinverseofthematrixofcoefficients.例题iniweseethatthenunknownsis从上面的讨论，我们看到解有n个

37、未知数的n个联立一次方程问题化成求系数的矩阵的逆矩阵的问题。因此，在矩阵论的书中，用大量的篇幅来讲求逆矩阵的技巧就不奇怪了。Ofcourse,wewillnotinourlimitedtreatmentdiscusssuchtechniques.Notonlyarematrixmethodsusefulinsolvingsimultaneousequations,buttheyarealsousefulindiscoveringwhetherornotthesetofequationsareconsistent,inthesensethattheyleadtosolutions,andindi

38、scoveringwhetherornotthesetofequationaredeterminate,inthesensethattheyleadtouniquesolution.当然，我们在这有限的叙述中不会讨论这类的技巧。矩阵方法不仅在解联立方程中有用，而且在发现方程组是否相容，印方程组是否有解的问题，以及方程组是否是确定的，即是否只有一解等方面，都是有用的。11-A“x>3”x+y=3”predicatesStatementsinvolvingvariables,suchas"x+y=z"areoftenfoundinmathematicalassertion

39、andincomputerprograms.Thesestatementsareneithertruenorfalse when valuethe sspecifiedthof the variablesare notIn this sectionwewilldiscuss statementsways that包含变量的语句，比 如propositionsx>3can be"x+y=3",produced" x+y=z ”断中fromsuch常出现在数学论和计算机程序中, 值，若未给语句中的所有变量赋则不能判定该语句是真是由这种语句生成命题的方法OThe

40、statement than 3“假，“x is greater本节要讨论has twoparts.The first part, the variables, is the subject of the statement. The second part- the predicate, “is greater subject of the statement canthan3" -refersato propertythat thehave.是语句的主语 质。语句“x大于3”分成两部分，第一部分，变量,第二部分，谓语，“大于3”，指的是语句主语具有的性We can denotet

41、he statement“ x is greater than 3by P(x), where P denote the predicate“is greater than 3and has a truth value.是变量。语句P(x)把语句 “x大于3 ”记为P(x),其中而x也称为命题函数 P在x点处的值。一旦赋予为一个命题，有了真值。11-BQuantifiersWhenallthe个值，语句variablesP(x)就成inaandxisthevariable.ThestatementP(x)isalsosaidtobethevalueofthepropositionalfunct

42、ionPatx.onceavaluehasbeenassignedtothevariablex,thestatementsP(x)becomesapropositionpropositionalfunctionareassignedvalues,theresultingstatementhasatruth value. However,thereis anotherimportantway,calledquantification,tocreateapropositionfromapropositionalfunction.twoo quantificat types f ion quanti

43、fication and existential quantification .wi llbediscussedhere, namely, universal当命题函数所有变量都赋值时,可从命题函数中得到命题。这里讨mathematica l结果语句有真值，但是还有另外一种方式，称为量词化，论两种量词化方法，也就是全称量词化和存在量词化。Manystatementsassertthatapropertyistrueforallvaluesofavariableinaparticulardomain,calledtheuniverseofdiscourse.Suchastatementisexpressedusingauniversalquantification.Theuniversalquantificationofapropositionalfunct