數(shù)學(xué)專業(yè)英語(yǔ)第二版的課文翻譯[教材]1,AWhatismathematicsMathematicscomesfromman’ssocialpractice,forexample,industrialandagriculturalproduction,commercialactivities,militaryoperationsandscientificandtechnologicalresearches.Andinturn,mathematicsservesthepracticeandplaysagreatroleinallfields.Nomodernscientificandtechnologicalbranchescouldberegularlydevelopedwithouttheapplicationofmathematics.數(shù)學(xué)來(lái)源于人類的社會(huì)實(shí)踐，比如工農(nóng)業(yè)生產(chǎn)，商業(yè)活動(dòng)，軍事行動(dòng)和科學(xué)技術(shù)研究。反過(guò)來(lái)，數(shù)學(xué)服務(wù)于實(shí)踐，并在各個(gè)領(lǐng)域中起著非常重要的作用。沒(méi)有應(yīng)用數(shù)學(xué)，任何一個(gè)現(xiàn)在的科技的分支都不能正常發(fā)展。Fromtheearlyneedofmancametheconceptsofnumbersandforms.Then,geometrydevelopedoutofproblemsofmeasuringland,andtrigonometrycamefromproblemsofsurveying.Todealwithsomemorecomplexpracticalproblems,manestablishedandthensolvedequationwithunknownnumbers,thusalgebraoccurred.Before17thcentury,manconfinedhimselftotheelementarymathematics,i.e.,geometry,trigonometryandalgebra,inwhichonlytheconstantsareconsidered.很早的時(shí)候，人類的需要產(chǎn)生了數(shù)和形式的概念，接著，測(cè)量土地的需要形成了幾何，出于測(cè)量的需要產(chǎn)生了三角幾何，為了處理更復(fù)雜的實(shí)際問(wèn)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(xué)，17世紀(jì)前，人類局限于只考慮常數(shù)的初等數(shù)學(xué)，即幾何，三角幾何和代數(shù)。Therapiddevelopmentofindustryin17thcenturypromotedtheprogressofeconomicsandtechnologyandrequireddealingwithvariablequantities.Theleapfromconstantstovariablequantitiesbroughtabouttwonewbranchesofmathematics----analyticgeometryandcalculus,whichbelongtothehighermathematics.Nowtherearemanybranchesinhighermathematics,amongwhicharemathematicalanalysis,higheralgebra,differentialequations,functiontheoryandsoon.17世紀(jì)工業(yè)的快速發(fā)展推動(dòng)了經(jīng)濟(jì)技術(shù)的進(jìn)步，從而遇到需要處理變量的問(wèn)題，從常數(shù)帶變量的跳躍產(chǎn)生了兩個(gè)新的數(shù)學(xué)分支-----解析幾何和微積分，他們都屬于高等數(shù)學(xué)，現(xiàn)在高等數(shù)學(xué)里面有很多分支，其中有數(shù)學(xué)分析，高等代數(shù)，微分方程，函數(shù)論等。Mathematiciansstudyconceptionsandpropositions,Axioms,postulates,definitionsandtheoremsareallpropositions.Notationsareaspecialandpowerfultoolofmathematicsandareusedtoexpressconceptionsandpropositionsveryoften.Formulas,figuresandchartsarefullofdifferentsymbols.SomeofthebestknownsymbolsofmathematicsaretheArabicnumerals1,2,3,4,5,6,7,8,9,0andthesignsofaddition,subtraction,multiplication,divisionandequality.數(shù)學(xué)家研究的是概念和命題，公理，公設(shè)，定義和定理都是命題。符號(hào)是數(shù)學(xué)中一個(gè)特殊而有用的工具，常用于表達(dá)概念和命題。公式，圖表都是不同的符號(hào)……..Theconclusionsinmathematicsareobtainedmainlybylogicaldeductionsandcomputation.Foralongperiodofthehistoryofmathematics,thecentricplaceofmathematicsmethodswasoccupiedbythelogicaldeductions.Now,sinceelectroniccomputersaredevelopedpromptlyandusedwidely,theroleofcomputationbecomesmoreandmoreimportant.Inourtimes,computationisnotonlyusedtodealwithalotofinformationanddata,butalsotocarryoutsomeworkthatmerelycouldbedoneearlierbylogicaldeductions,forexample,theproofofmostofgeometricaltheorems.數(shù)學(xué)結(jié)論主要由邏輯推理和計(jì)算得到，在數(shù)學(xué)發(fā)展歷史的很長(zhǎng)時(shí)間內(nèi)，邏輯推理一直占據(jù)著數(shù)學(xué)方法的中心地位，現(xiàn)在，由于電子計(jì)算機(jī)的迅速發(fā)展和廣泛使用，計(jì)算機(jī)的地位越來(lái)越重要，現(xiàn)在計(jì)算機(jī)不僅用于處理大量的信息和數(shù)據(jù)，還可以完成一些之前只能由邏輯推理來(lái)做的工作，例如，大多數(shù)幾何定理的證明。1,BEquationAnequationisastatementoftheequalitybetweentwoequalnumbersornumbersymbols.Equationareoftwokinds----identitiesandequationsofcondition.Anarithmeticoranalgebraicidentityisanequation.Insuchanequationeitherthetwomembersarealike.Orbecomealikeontheperformanceoftheindicatedoperation.等式是關(guān)于兩個(gè)數(shù)或者數(shù)的符號(hào)相等的一種描述。等式有兩種,恒等式和條件等式。算術(shù)或者代數(shù)恒等式是等式。這種等式的兩端要么一樣，要么經(jīng)過(guò)執(zhí)行指定的運(yùn)算后變成一樣。Anidentityinvolvinglettersistrueforanysetofnumericalvaluesofthelettersinit.Anequationwhichistrueonlyforcertainvaluesofaletterinit,orforcertainsetsofrelatedvaluesoftwoormoreofitsletters,isanequationofcondition,orsimplyanequation.Thus3x-5=7istrueforx=4only;and2x-y=0istrueforx=6andy=2andformanyotherpairsofvaluesforxandy.含有字母的恒等式對(duì)其中字母的任一組數(shù)值都成立。一個(gè)等式若僅僅對(duì)其中一個(gè)字母的某些值成立，或?qū)ζ渲袃蓚€(gè)或著多個(gè)字母的若干組相關(guān)的值成立，則它是一個(gè)條件等式，簡(jiǎn)稱方程。因此3x-5=7僅當(dāng)x=4時(shí)成立，而2x-y=0，當(dāng)x=6,y=2時(shí)成立，且對(duì)x,y的其他許多對(duì)值也成立。Arootofanequationisanynumberornumbersymbolwhichsatisfiestheequation.Therearevariouskindsofequation.Theyarelinearequation,quadraticequation,etc.方程的根是滿足方程的任意數(shù)或者數(shù)的符號(hào)。方程有很多種，例如:線性方程，二次方程等。Tosolveanequationmeanstofindthevalueoftheunknownterm.Todothis,wemust,ofcourse,changethetermsaboutuntiltheunknowntermstandsaloneononesideoftheequation,thusmakingitequaltosomethingontheotherside.Wethenobtainthevalueoftheunknownandtheanswertothequestion.Tosolvetheequation,therefore,meanstomoveandchangethetermsaboutwithoutmakingtheequationuntrue,untilonlytheunknownquantityisleftononeside,nomatterwhichside.解方程意味著求未知項(xiàng)的值，為了求未知項(xiàng)的值，當(dāng)然必須移項(xiàng)，直到未知項(xiàng)單獨(dú)在方程的一邊，令其等于方程的另一邊，從而求得未知項(xiàng)的值，解決了問(wèn)題。因此解方程意味著進(jìn)行一系列的移項(xiàng)和同解變形，直到未知量被單獨(dú)留在方程的一邊，無(wú)論那一邊。Equationareofverygreatuse.Wecanuseequationinmanymathematicalproblems.Wemaynoticethatalmosteveryproblemgivesusoneormorestatementsthatsomethingisequaltosomething,thisgivesusequations,withwhichwemayworkifweneedit.方程作用很大，可以用方程解決很多數(shù)學(xué)問(wèn)題。注意到幾乎每一個(gè)問(wèn)題都給出一個(gè)或多個(gè)關(guān)于一個(gè)事情與另一個(gè)事情相等的陳述，這就給出了方程，利用該方程，如果我們需要的話，可以解方程。2,AWhystudygeometry?Manyleadinginstitutionsofhigherlearninghaverecognizedthatpositivebenefitscanbegainedbyallwhostudythisbranchofmathematics.Thisisevidentfromthefactthattheyrequirestudyofgeometryasaprerequisitetomatriculationinthoseschools.許多居于領(lǐng)導(dǎo)地位的學(xué)術(shù)機(jī)構(gòu)承認(rèn)，所有學(xué)習(xí)這個(gè)數(shù)學(xué)分支的人都將得到確實(shí)的受益，許多學(xué)校把幾何的學(xué)習(xí)作為入學(xué)考試的先決條件，從這一點(diǎn)上可以證明。GeometryhaditsoriginlongagointhemeasurementbytheBabyloniansandEgyptiansoftheirlandsinundatedbythefloodsoftheNileRiver.Thegreekwordgeometryisderivedfromgeo,meaning“earth”andmetron,meaning“measure”.Asearlyas2000B.C.wefindthelandsurveyorsofthesepeoplere-establishingvanishinglandmarksandboundariesbyutilizingthetruthsofgeometry.幾何學(xué)起源于很久以前巴比倫人和埃及人測(cè)量他們被尼羅河洪水淹沒(méi)的土地，希臘語(yǔ)幾何來(lái)源于geo，意思是”土地“，和metron意思是”測(cè)量“。公元前2000年之前，我們發(fā)現(xiàn)這些民族的土地測(cè)量者利用幾何知識(shí)重新確定消失了的土地標(biāo)志和邊界。2,BSomegeometricaltermsAsolidisathree-dimensionalfigure.Commonexamplesofsolidsarecube,sphere,cylinder,coneandpyramid.Acubehassixfaceswhicharesmoothandflat.Thesefacesarecalledplanesurfacesorsimplyplanes.Aplanesurfacehastwodimensions,lengthandwidth.Thesurfaceofablackboardorofatabletopisanexampleofaplanesurface.立體是一個(gè)三維圖形，立體常見(jiàn)的例子是立方體，球體，柱體，圓錐和棱錐。立方體有6個(gè)面，都是光滑的和平的，這些面被稱為平面曲面或者簡(jiǎn)稱為平面。平面曲面是二維的，有長(zhǎng)度和寬度，黑板和桌子上面的面都是平面曲面的例子。2,C三角函數(shù)于直角三角形的解Oneofthemostimportantapplicationsoftrigonometryisthesolutionoftriangles.Letusnowtakeupthesolutiontorighttriangles.Atriangleiscomposedofsixpartsthreesidesandthreeangles.Tosolveatriangleistofindthepartsnotgiven.Atrianglemaybesolvedifthreeparts(atleastoneoftheseisaside)aregiven.Arighttrianglehasoneangle,therightangle,alwaysgiven.Thusarighttrianglecanbesolvedwhentwosides,oronesideandanacuteangle,aregiven.三角形最重要的應(yīng)用之一是解三角形，現(xiàn)在我們來(lái)解直角三角形。一個(gè)三角形由6個(gè)部分組成，三條邊和三只角。解一個(gè)三角形就是要求出未知的部分。如果三角形的三個(gè)部分(其中至少有一個(gè)為邊)為已知，則此三角形就可以解出。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,wecouldtrytocomputethepositionofamovingparticlefromaknowledgeofitsvelocityoracceleration.Oraradioactivesubstancemaybedisintegratingataknownrateandwemayberequiredtodeterminetheamountofmaterialpresentafteragiventime.大量的科學(xué)問(wèn)題需要人們根據(jù)事物的變化率來(lái)確定該事物，例如，我們可以由已知速度或者加速度來(lái)計(jì)算移動(dòng)粒子的位置.又如，某種放射性物質(zhì)可能正在以已知的速度進(jìn)行衰變，需要我們確定在給定的時(shí)間后遺留物質(zhì)的總量。Inexampleslikethese,wearetryingtodetermineanunknownfunctionfromprescribedinformationexpressedintheformofanequationinvolvingatleastoneofthederivativesoftheunknownfunction.Theseequationsarecalleddifferentialequations,andtheirstudyformsoneofthemostchallengingbranchesofmathematics.在類似的例子中，我們力求由方程的形式表示的信息來(lái)確定未知函數(shù)，而這種方程至少包含了未知函數(shù)的一個(gè)導(dǎo)數(shù)。這些方程稱為微分方程，對(duì)其研究形成了數(shù)學(xué)中最具有挑戰(zhàn)性的一門分支。Thestudyofdifferentialequationsisonepartofmathematicsthat,perhapsmorethananyother,hasbeendirectlyinspiredbymechanics,astronomy,andmathematicalphysics.微分方程的研究是數(shù)學(xué)的一部分，也許比其他分支更多的直接受到力學(xué)，天文學(xué)和數(shù)學(xué)物理的推動(dòng)。Itshistorybeganinthe17thcenturywhenNewton,Leibniz,andtheBernoullissolvedsomesimpledifferentialequationsarisingfromproblemsingeometryandmechanics.Theseearlydiscoveries,beginningabout1690,graduallyledtothedevelopmentofalotof“specialtricks”forsolvingcertainspecialkindsofdifferentialequation.微分方程起源于17世紀(jì)，當(dāng)時(shí)牛頓，萊布尼茨，波努力家族解決了一些來(lái)自幾何和力學(xué)的簡(jiǎn)單的微分方程。開(kāi)始于1690年的早期發(fā)現(xiàn)，逐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。Althoughthesespecialtricksareapplicableinrelativelyfewcases,theydoenableustosolvemanydifferentialequationsthatariseinmechanicsandgeometry,sotheirstudyisofpracticalimportance.Someofthesespecialmethodsandsomeoftheproblemswhichtheyhelpussolvearediscussedneartheendofthischapter.盡管這些特殊的技巧只是用于相對(duì)較少的幾種情況，但他們能夠解決力學(xué)和幾何中出現(xiàn)的許多微分方程，因此，他們的研究具有重要的實(shí)際應(yīng)用。這些特殊的技巧和有助于我們解決的一些問(wèn)題將在本章最后討論。Experiencehasshownthatitisdifficulttoobtainmathematicaltheoriesofmuchgeneralityaboutsolutionofdifferentialequations,exceptforafewtypes.經(jīng)驗(yàn)表明除了幾個(gè)典型方程外，很難得到微分方程解的一般性數(shù)學(xué)理論。Amongthesearetheso-calledlineardifferentialequationswhichoccurinagreatvarietyofscientificproblems.在這些典型方程中，有一個(gè)稱為線性微分方程，出現(xiàn)在大量的科學(xué)問(wèn)題中。10-CApplicationsofmatricesInrecentyearstheapplicationsofmatricesinmathematicsandinmanydiversefieldshaveincreasedwithremarkablespeed.Matrixtheoryplaysacentralroleinmodernphysicsinthestudyofquantummechanics.Matrixmethodsareusedtosolveproblemsinapplieddifferentialequations,specifically,intheareaofaerodynamics,stressandstructureanalysis.Oneofthemostpowerfulmathematicalmethodsforpsychologicalstudiesisfactoranalysis,asubjectthatmakeswideuseofmatrixmethods.近年來(lái)，在數(shù)學(xué)和許多各種不同的領(lǐng)域中，矩陣的應(yīng)用一直以驚人的速度不斷增加。在研究量子力學(xué)時(shí)，矩陣?yán)碚撛诂F(xiàn)代物理學(xué)上起著主要的作用。解決應(yīng)用微分方程，特別是在空氣動(dòng)力學(xué)，應(yīng)力和結(jié)構(gòu)分析中的問(wèn)題，要用矩陣方法。心理學(xué)研究上一種最強(qiáng)有力的數(shù)學(xué)方法是因子分析，這也廣泛的使用矩陣(方)法.Recentdevelopmentsinmathematicaleconomicsandinproblemsofbusinessadministrationhaveledtoextensiveuseofmatrixmethods.Thebiologicalsciences,andinparticulargenetics,usematrixtechniquestogoodadvantage.Nomatterwhatthestudents’fieldofmajorinterestis,knowledgeoftherudimentsofmatricesislikelytobroadentherangeofliteraturethathecanreadwithunderstanding.近年來(lái)，在數(shù)學(xué)經(jīng)濟(jì)學(xué)和商業(yè)管理問(wèn)題方面的發(fā)展已經(jīng)導(dǎo)致廣泛的使用矩陣法。生物科學(xué)，特別在遺傳學(xué)方面，用矩陣的技術(shù)很有成效。不管學(xué)生主要興趣是什么，矩陣基本原理的知識(shí)可能擴(kuò)大他能讀懂的文獻(xiàn)的范圍。Thesolutionofnsimultaneouslinearequationsinnunknownsisoneoftheimportantproblemsofappliedmathematics.Descartes,theinventorofanalyticgeometryandoneofthefoundersofmodernalgebraicnotation,believedthatallproblemscouldultimatelybereducedtothesolutionofasetofsimultaneouslinearequations.解一有n個(gè)未知數(shù)的n個(gè)聯(lián)立一次(線性)方程是應(yīng)用數(shù)學(xué)的一個(gè)重要問(wèn)題。解析幾何的發(fā)明者和現(xiàn)代代數(shù)計(jì)數(shù)法的創(chuàng)始人之一笛卡兒相信，所有的問(wèn)題最后都能約簡(jiǎn)為解一組聯(lián)立一次方程。Althoughthisbeliefisnowknowntobeuntenable,weknowthatalargegroupofsignificantappliedproblemsfrommanydifferentdisciplinesarereducibletosuchequations.Manyoftheapplications,requirethesolutionofalargenumberofsimultaneouslinearequations,sometimesinthehundreds.Theadventofcomputershasmadethematrixmethodseffectiveinthesolutionoftheseformidableproblems.雖然這種信念現(xiàn)在認(rèn)為是站不住腳的，但是，我們知道，從許多不同的學(xué)科里的一大群重要的應(yīng)用問(wèn)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(jì)的聯(lián)立一次方程，計(jì)算機(jī)的發(fā)明已經(jīng)使得矩陣方法在解這些難以解決的問(wèn)題方面非常活躍。Example1.solvethesimultaneousequationsforx1x2,andx3.例題1，解聯(lián)立方程求x1x2和x3。Fromtheabovediscussion,weseethattheproblemofsolvingnsimultaneouslinearequationinnunknownsisreducedtotheproblemoffindingtheinverseofthematrixofcoefficients.Itisthereforenotsurprisingthatinbooksonthetheoryofmatricesthetechniquesoffindinginversematricesoccupyconsiderablespace.從上面的討論，我們看到解有n個(gè)未知數(shù)的n個(gè)聯(lián)立一次方程問(wèn)題化成求系數(shù)的矩陣的逆矩陣的問(wèn)題。因此，在矩陣論的書中，用大量的篇幅來(lái)講求逆矩陣的技巧就不奇怪了。Ofcourse,wewillnotinourlimitedtreatmentdiscusssuchtechniques.Notonlyarematrixmethodsusefulinsolvingsimultaneousequations,buttheyarealsousefulindiscoveringwhetherornotthesetofequationsareconsistent,inthesensethattheyleadtosolutions,andindiscoveringwhetherornotthesetofequationaredeterminate,inthesensethattheyleadtouniquesolution..當(dāng)然，我們?cè)谶@有限的敘述中不會(huì)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，即方程組是否有解的問(wèn)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-ApredicatesStatementsinvolvingvariables,suchas“x>3”,”x+y=3”,”x+y=z”areoftenfoundinmathematicalassertionandincomputerprograms.Thesestatementsareneithertruenorfalsewhenthevaluesofthevariablesarenotspecified.Inthissectionwewilldiscussthewaysthatpropositionscanbeproducedfromsuchstatements.包含變量的語(yǔ)句，比如“x>3”,”x+y=3”,”x+y=z”常出現(xiàn)在數(shù)學(xué)論斷中和計(jì)算機(jī)程序中，若未給語(yǔ)句中的所有變量賦值，則不能判定該語(yǔ)句是真是假，本節(jié)要討論由這種語(yǔ)句生成命題的方法。Thestatement“xisgreaterthan3”hastwoparts.Thefirstpart,thevariables,isthesubjectofthestatement.Thesecondpart-thepredicate,“isgreaterthan3”-referstoapropertythatthesubjectofthestatementcanhave.語(yǔ)句“x大于3”分成兩部分，第一部分，變量，是語(yǔ)句的主語(yǔ)。第二部分，謂語(yǔ)，“大于3”，指的是語(yǔ)句主語(yǔ)具有的性質(zhì)。Wecandenotethestatement“xisgreaterthan3”byP(x),wherePdenotethepredicate“isgreaterthan3”andxisthevariable.ThestatementP(x)isalsosaidtobethevalueofthepropositionalfunctionPatx.onceavaluehasbeenassignedtothevariablex,thestatementsP(x)becomesapropositionandhasatruthvalue.把語(yǔ)句“x大于3”記為P(x),其中P表示謂詞“大于3”，而x是變量。語(yǔ)句P(x)也稱為命題函數(shù)P在x點(diǎn)處的值。一旦賦予x一個(gè)值，語(yǔ)句P(x)就成為一個(gè)命題,有了真值。11-BQuantifiersWhenallthevariablesinapropositionalfunctionareassignedvalues,theresultingstatementhasatruthvalue.However,thereisanotherimportantway,calledquantification,tocreateapropositionfromapropositionalfunction.twotypesofquantificationwillbediscussedhere,namely,universalquantificationandexistentialquantification.當(dāng)命題函數(shù)所有變量都賦值時(shí)，結(jié)果語(yǔ)句有真值，但是還有另外一種方式，稱為量詞化，可從命題函數(shù)中得到命題。這里討論兩種量詞化方法，也就是全稱量詞化和存在量詞化。Manymathematicalstatementsassertthatapropertyistrueforallvaluesofavariableinaparticulardomain,calledtheuniverseofdiscourse.Suchastatementisexpressedusingauniversalquantification.TheuniversalquantificationofapropositionalfunctionisthepropositionthatassertthatP(x)