數(shù)學專業(yè)英語第二版的課文翻譯

1、fromman1AWhatismathematicsMathematicscomesssocialpractice,forexample,industrialandagriculturalproduction,commercialactivitmilitaroperationies,ysandmathematicresearches.Andinturn,sscientificserveandtechnologicalthepracticeandplays數(shù)軍事行動和科學技術(shù)研究。反過沒有應(yīng)用數(shù)學，任何一個現(xiàn)early need of man came thegreatroleinallfiel

2、ds.Nomodernscientificandtechnologicalbranchescouldberegularlydevelopedwithouttheapplicationofmathematics.學來源于人類的社會實踐，比如工農(nóng)業(yè)生產(chǎn)，商業(yè)活動，來，數(shù)學服務(wù)于實踐，并在各個領(lǐng)域中起著非常重要的作用。在的科技的分支都不能正常發(fā)展。Fromtheconceptsofnumbersandforms.Then,geometrydevelopedoutofproblemsofmeasuringland,andtrigonometrycamefromproblemsofsurveying.e

3、stablishedTodealwithsomesolveandthendmorecomplexequatiowitnhpracticalunknownproblems,mannumbers,thusalgebraoccurred.Before17thcentury,elementarymathematics,i.e.,geometry,manconfinedtrigonometryhimselftotheandalgebra,inwhichonlytheconstantsareconsidered.形式的概念，接著，測量土地的需要形成了幾何，很早的時候，人類的需要產(chǎn)生了數(shù)和由于測量的需要產(chǎn)生

4、了三角幾何，為了處理更復雜的實際問題，人類建立和解決了帶未知參數(shù)的方程，從而產(chǎn)生了代數(shù)學，17世紀前，人類局限于只考慮常數(shù)的初等數(shù)學，即幾何，三角幾何和代數(shù)。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世紀工業(yè)的快andsoon.17速發(fā)展推動了經(jīng)濟技術(shù)的進從而遇到需要處理變量的問題，從常數(shù)帶變量的跳躍產(chǎn)步，生了兩個新的數(shù)學分支-一解析幾何和微積分，他們都屬于高等數(shù)學，現(xiàn)在高等數(shù)學里面有很Mathemat

6、icians study多分支，其中有數(shù)學分析，高等代數(shù)，微分方程，函數(shù)論等。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.定義和定理都是命題。且full are ofmathematics ofand the signs addition,differe ntaresymbols.the Arabicsubtraction數(shù)學家研究的是概念和命題，公理，公 設(shè)，符號是數(shù)學中一個特殊而有用的工常用于表達概念和命題。式，圖表都是不同的符號? ? .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.在數(shù)學發(fā)展歷史的很長時間 內(nèi)，計算機的迅速發(fā)展和廣泛使邏輯推理一直占據(jù)著數(shù)學方法的中心地數(shù)學結(jié)論主要由邏輯推理和計算得到,現(xiàn)在，由于電子 現(xiàn)在計算機不僅用于處理大量用，信息和數(shù)據(jù), 作，例如，大多數(shù)幾何定理的證計算機的地位越來越重要,還可以完成一些之前只能由邏輯推理來做的工明。1BEquationAnequationisastatementoftheequal

10、itybetweentwoequalnumbersornumbersymbols.Equationareoftwokinds-一identitiesandequationsofcondition.Anarithmeticoranalgebraicidentityisanequation.In such aneither theequation twomembersare alike.Orbecome alike on the performance of the indicatedoperation.個數(shù)或者數(shù)的符號相等的一種描述。等式有兩種-恒等式和條件等式等式是關(guān)于兩 算術(shù)或者代數(shù)恒等式是

11、等式。這種等式的兩端要么一樣，要么經(jīng)過執(zhí)行指定的運算后變成一 樣。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.含有字母的恒等式對其中字母的任一組數(shù)值都成立。一個等式若僅僅對其中一個字母的某些值成立，或?qū)ζ渲袃蓚€或著多個字母的若干組相關(guān)的值成立，則它是一個條件等式，簡稱方程。因僅當x=4時成立，2x-y=0,當時成立，且對 x,此3x-5=7 而x=6,y=2y的其他許多對值也成立。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.方程的根是滿足方程的任意數(shù)或者數(shù)的符號。方程有很多種，例如：線性方程，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.為了求未知項的值，當然必須移項,邊，解決了問題。從而求得未知項的值，形，解方程意味著求未知項的值，直到未知項單獨在方程的一令其等于方程的另一邊，因此解方程意味著進行一系列的移項和同解變直到Equationareofverygreat未知量被單獨留在方程的一邊，無論那一邊。use.Wecanuseequationinmanymathematicalproblems.Wemaynoticethatalmosteveryproblemgivesusoneormorestatementsthatsometh

16、ingisequaltosomething,thisgivesusequations,withwhichwemayworkif方程作用很大，可以用方程解決很多數(shù)學問題。注意到幾乎每一個問題weneedit.都給由一個或多個關(guān)于一個事情與另一個事情相等的陳述，這就給由了方程，利用該方程，如果我們需要的話，可以解方程。2AWhystudygeometry?Manyleadinginstitutionsofhigherlearninghaverecognizedthatpositivebenefitscanbegainedbyallwhostudythisbranchofmathematics.Th

17、isisevidentfromthefactthattheyrequirestudyofgeometryasaprerequisitetomatriculationinthoseschools.多居于領(lǐng)導地位的學術(shù)機構(gòu)承認，所有學習這個數(shù)學分支的人都將得到確實的受 皿校把幾何的學習作為入學考試的先決條件，從這一點上可以證明。origin long ago in the measurement by the Babylonians andGeometry許多學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幾何學起源于很久以前巴比倫人和埃及人測量他們被尼羅河geo ,意思是” 土地 和metron 意思是”測量thetruthsofgeometry.洪水淹沒的土地，希臘語幾何來源于公元前2

19、000年之前，我們發(fā)現(xiàn)這些民族的土地測量者利用幾何知識重新確定消失了的土地標志和邊界。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.柱體，圓錐和棱錐。立方體有 簡稱為平面。平面曲面是二維的, 例子。立體是一個三維圖形，立體常見的例子是立方體,6個面，都是光滑的和平的，這些面被稱為平面曲面或者 有長度和寬度，黑板和桌子上面的面都是平面曲面的球體,2 - C三角函數(shù)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個部

22、分組成，三條邊和三只角。解一個三角形就是要求由未知的部分。如果三角angle, are given.三角形最重要的應(yīng)用之一是解三角形,現(xiàn)在我們來解直角三角 形。一個三角 形形的三個部分（其中至少有一個為邊）為已知，則此三角形就可以解由。直角三角形的一只角，即直角，總是已知的。因此，如果它的兩邊，或一邊和一銳角為已知，則此直角三角形可解9-AIntroductionAlargevarietyofscientificproblemsariseinwhichonetriestodeterminesomethingfromitsrateofchange.Forexample,wecouldtrytoc

23、omputethepositionofamovingparticlefromaknowledgeofitsvelocityoracceleration.Orasubstancmayberadioactiveedisintegratingataknownrateandwemayberequiredtodeterminetheamountofmaterialpresentafteragiventime.大量的科學問題需要人們根據(jù)事物的變化率來確定該事物，例如，我們可以由已知速度或者加速度來計算移動粒子的位又如，某種放射性物質(zhì)可能正在以已知的速度進行衰變，物質(zhì)的總量unknownfunctio需要我

24、們確定在給定的時間后遺留Inexampleslikethese,wearetryingtodetermineannequationfrominvolvinprescribedatinformationexpressedthinformofanfunction.Theseleastthequationsoneofcallearedthederivativesdifferentialofunknownequations,andtheirstudyformsoneofthemostchallengingbranchesofmathematics.似的例子中，我們力求由方程的形式表示的信息來確定未知函

25、數(shù)，在類知函數(shù)的一個導數(shù)。門分這些方程稱為微分方程,而這種方程至少包含了未對其研究形成了數(shù)學中最具有挑戰(zhàn)性的一支。Thestudyofdifferentialequationsisonepartofmathematicsthat,perhapsmorethananyother,hasbeendirectlyinspiredbymechanics,astronomy,andmathematicalphysics.其他分支更多的直接受到力學，天文學和數(shù)學物理的推動。微分方程的研究是數(shù)學的一部分，也許比Itshistorybeganintheth17thsimplecenturywhenNewton

26、,differentialequationsLeibniz,arisingandfrommechanics.Theseearlydiscoveries,beginningabout1690,graduallyledtothedevelopmentofalotofkindsofdifferentialequation.“specialtricks微分方程起源于努力家族解決了一些來自幾何和力學的簡單的微分方程。開始于漸引起了解某些特殊類型的微分方程的大量特殊技巧的發(fā)展。tricksareapplicableinrelativelyfewcases,theydoenableustosolvediff

27、erentmanyialequationsthatariseinisofimportance.Sometheirstudypracticalofandsomeoftheproblemswhichtheyhelpussolvearediscussedneartheendofthischapter.Bernoullisproblems”forsolv17insolvedsomegeometryingcertainspecial世紀，當時牛頓，萊布尼茨,波and1690年的早期發(fā)現(xiàn)，逐Althoughthesespecialmechanicsthesegeometryand,specialmetho

28、ds盡管這些特殊的技巧只是用于相對較少的幾種情況，但他們能夠解決力學和幾何中由現(xiàn)的許多微分方程，因此，他們的研究具有重要的實際應(yīng)用。特殊的技巧和有助于我們解決的一些問題將在本章最后討論。Experiencehasshownso這些i difficul that t is tto obtain mathematicalotheoriesfmuchgeneralityaboutsolutionofdifferentialequations,exceptfor經(jīng)驗表明afewtypes.Among these are the除了幾個典型方程外，很難得到微分方程解的一般性數(shù)學理論。so-calledl

29、ineardifferentialequationswhichoccurinagreatvarietyofscientific學問題中。applications ofmatricesproblems.在這些典型方程中，有一個稱為線性微分方程，由現(xiàn)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.同的領(lǐng)域中，矩陣的應(yīng)用一直以驚人的速度不斷增加。近年來

31、，在數(shù)學和許多各種不物理學上起著主要的作用。解決應(yīng)用微分方程,在研究量子力學時，特別是在空氣動力學，矩陣理論在現(xiàn)代應(yīng)力和結(jié)構(gòu)分析中的問題，要用矩陣方法。心理學研究上一種最強有力的數(shù)學方法是因子分析，這也廣泛的使用in矩陣（方）法.Recentdevelopmentsmathematicaleconomicsandinproblemsofbusinessadministrationhaveledtoextensiveuseofmatrixmethods.Thebiologicalsciences,andinparticulargenetics,usematrixadvantage，ftechni

32、questogood.Nomatterwhatthestudentsieldofmajointerethmatricerstis,knowledgeoferudimentsofsislikelytobroadentherangeofliteraturethathecanreadwithunderstanding.近年來，在數(shù)學經(jīng)濟學和商業(yè)管理問題方面的發(fā)展已經(jīng)導致廣泛的使用矩陣生物科學，法。特別在遺傳學方面，用矩陣的技術(shù)很有成效?？赡軘U大他能讀懂的文獻的范圍Thei不管學生主要興趣是什solutio nof ntheequations n n unknowns is one ofimporta

33、nt矩陣基本原理的知識 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個未知數(shù)的 n個聯(lián)立一次（線性） 方程是應(yīng)用數(shù)學的一個belief is now

34、 known to beo in the solution f theseequations.重要問題。解析幾何的發(fā)明者和現(xiàn)代代數(shù)計數(shù)法的創(chuàng)始人之一笛卡兒相信，所有的問題最后都能約簡為解一組聯(lián)立一次方程。Althoughthisuntenable,weknowthatalargegroupofsignificantappliedproblemsfrommanydifferentdisciplinesarereducibletosuchequations.Manyoftheapplications,requirethesolutionofalargenumberofsimultaneouslin

35、earequations,sometimesinthehundreds.Theadventofcomputershasmadethematrixmethodseffectiveformidableproblems.雖然這種信念現(xiàn)在認為是站不住腳的，但是，我們知道，從許多不同的學科里的一大群重要的應(yīng)用問題都可以約化為這類的方程。許多應(yīng)用要求解大量的，往往數(shù)以百計的聯(lián)立一次方程，計算機的發(fā)明已經(jīng)使得矩陣方法在解這些難以解決的問題方面非?；钴S。Example1.solvethesimultaneousequationsforx1x2,andx3.解聯(lián)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.例題iniweseethatthenunknownsis從上面的討論，我們看到解有n個

37、未知數(shù)的n個聯(lián)立一次方程問題化成求系數(shù)的矩陣的逆矩陣的問題。因此，在矩陣論的書中，用大量的篇幅來講求逆矩陣的技巧就不奇怪了。Ofcourse,wewillnotinourlimitedtreatmentdiscusssuchtechniques.Notonlyarematrixmethodsusefulinsolvingsimultaneousequations,buttheyarealsousefulindiscoveringwhetherornotthesetofequationsareconsistent,inthesensethattheyleadtosolutions,andindi

38、scoveringwhetherornotthesetofequationaredeterminate,inthesensethattheyleadtouniquesolution.當然，我們在這有限的敘述中不會討論這類的技巧。矩陣方法不僅在解聯(lián)立方程中有用，而且在發(fā)現(xiàn)方程組是否相容，印方程組是否有解的問題，以及方程組是否是確定的，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包含變量的語句，比 如propositionsx>3can be"x+y=3",produced" x+y=z ”斷中fromsuch常出現(xiàn)在數(shù)學論和計算機程序中, 值，若未給語句中的所有變量賦則不能判定該語句是真是由這種語句生成命題的方法OThe

40、statement than 3“假，“x is greater本節(jié)要討論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.是語句的主語 質(zhì)。語句“x大于3”分成兩部分，第一部分，變量,第二部分，謂語，“大于3”，指的是語句主語具有的性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.是變量。語句P(x)把語句 “x大于3 ”記為P(x),其中而x也稱為命題函數(shù) P在x點處的值。一旦賦予為一個命題，有了真值。11-BQuantifiersWhenallthe個值，語句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當命題函數(shù)所有變量都賦值時,可從命題函數(shù)中得到命題。這里討mathematica l結(jié)果語句有真值，但是還有另外一種方式，稱為量詞化，論兩種量詞化方法，也就是全稱量詞化和存在量詞化。Manystatementsassertthatapropertyistrueforallvaluesofavariableinaparticulardomain,calledtheuniverseofdiscourse.Suchastatementisexpressedusingauniversalquantification.Theuniversalquantificationofapropositionalfunct