數(shù)學(xué)專業(yè)英語第二版的課文翻譯

1、v1.0可編輯可修改1A What is mathematics Mathematics comes from maH s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. And in turn, mathematics serves the practice and plays a great role in all field

2、s. No modern scientific and technological branches could be regularly developed without the application of mathematics.數(shù)學(xué)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，軍事行動和科學(xué)技術(shù)研究。反過來，數(shù)學(xué)服務(wù)于實踐，并在各個領(lǐng)域中起著非常重要的作用。沒有應(yīng)用數(shù)學(xué)，任何一個現(xiàn)在的科技的分支都不能正常發(fā)展。From the early need of mancamethe concepts of numbersand forms. Then, geometry develope

3、d out of problems of measuring land , and trigonometry came from problems of surveying . To deal with some more complexpractical problems, man established and then solved equation with unknown numbers ,thus algebra occurred. Before 17th century, man confined himself to the elementary mathematics, .

4、, geometry, trigonometry and algebra, in which only the constants are considered.很早的時候，人類的需要產(chǎn)生了數(shù)和形式的概念，接著，測量土地的需要形成了幾何，出于測量的需要產(chǎn)生了三角幾何，為了處理更復(fù)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(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)。 The rapid development of industry in 17th century promoted the progress of economics and tec

5、hnology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematicsanalyticgeometry and calculus, which belong tothe higher mathematics. Now there are many branches in higher mathematics, among which are mathematical ana

6、lysis, higher algebra, differentialequations, functiontheory and so on. 17世紀(jì)工業(yè)的快速發(fā)展推動了經(jīng)濟技術(shù)的進(jìn)步，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)生了兩個新的數(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。Mathematicians study conceptions and propositions, Axioms, postulates,definitions and theorems are all proposition

7、s. Notations are a special and powerful 11v1.0可編輯可修改tool of mathematics and are used to express conceptions and propositions very often. Formulas figures and charts are full of different symbols.Someof the best knownsymbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of

8、addition, subtraction , multiplication, division and equality.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。符號是數(shù)學(xué)中一個特殊而有用的工具，常用于表達(dá)概念和命題。公式，圖表都是不同的符號 .The conclusions in mathematicsare obtained mainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathemat

9、ics methods was occupied bythe logical deductions. Now , since electronic computers are developed promptly and used widely, the role of computation becomes more and more important. In our times, computation is not only used to deal with a lot of information and data, but alsoto carry out some work t

10、hat merely could be done earlier by logical deductions, for example, the proof of most of geometrical theorems.數(shù)學(xué)結(jié)論主要由邏輯推理和計算得到，在數(shù)學(xué)發(fā)展歷史的很長時間內(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，由于電子計算機的迅速發(fā)展和廣泛使用，計算機的地位越來越重要，現(xiàn)在計算機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做的工作，例如，大多數(shù)幾何定理的證明。1 B Equation An equation is a statement of the equ

11、ality between two equalnumbers or number symbols. Equation are of two kinds- identities and equationsof condition. An arithmetic or an algebraic identity is an equation. In such anequation either the two members are alike. Or become alike on the performance ofthe indicated operation.等式是關(guān)于兩個數(shù)或者數(shù)的符號相等

12、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)過執(zhí)行指定的運算后變成一樣。An identity involving letters is true for any set ofnumerical values of the letters in it. An equation which is true only for certainvalues of a letter in it, or for certain sets of related values of two or more offor x=4 only; and 2x-y=

13、0 is trueits letters, is an equation of condition, or simply an equation. Thus 3x-5=7 is truefor x=6 and y=2 and for many other pairs of values23v1.0可編輯可修改for x and y.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。因此 3x-5=7僅當(dāng)x=4時成立，而2x-y=0 ,當(dāng)x=6,y=2時成立，且對 x, y 的其他許

14、多對值也成立。 A root of an equation is any number or number symbol which 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 equation means to find the value of theunknown term. To do this , we

15、 must, of course, change the terms about until the unknown term stands alone on one side of the equation, thus making it equal to something on the other side. We then obtain the value of the unknown and the answer to thequestion. To solve the equation, therefore, means to move and change the terms a

16、bout without making the equation untrue, until only the unknown quantity is left on one side ,no matter which side.解方程意味著求未知項的值，為了求未知項的值，當(dāng)然必須移項，直到未知項單獨在方程的一邊，令其等于方程的另一邊，從而求得未知項的值，解決了問題。因此解方程意味著進(jìn)行一系列的移項和同解變形，直到未知量被單獨留在方程的一邊，無論那一邊。Equation are of very great use. We can use equation in manymathematical

17、 problems. Wemay notice that almost every problem gives us one or more statements that something is equal to something, this gives us equations, with which we may work if we need it.方程作用很大，可以用方程解決很多數(shù)學(xué)問題。注意到幾乎每一個問題都給出一個或多個關(guān)于一個事情與另一個事情相等的陳述，這就給出了方程，利用該方程，如果我們需要的話，可以解方程。2 A Why study geometry Many lead

18、ing institutions of higher learning haverecognized that positive benefits can be gained by all who study this branch ofmathematics. This is evident from the fact that they require study of geometry asa prerequisite to matriculation in those schools.許多居于領(lǐng)導(dǎo)地位的學(xué)術(shù)機構(gòu)承認(rèn)，33v1.0可編輯可修改所有學(xué)習(xí)這個數(shù)學(xué)分支的人都將得到確實的受益，許

19、多學(xué)校把幾何的學(xué)習(xí)作為入學(xué)考試的先決條件，從這一點上可以證明。Geometry had its origin long ago in the measurement bythe Babylonians and Egyptians of their lands inundated by the floods of the NileRiver. The greek word geometry is derived from geo,meaning “earth " and metron,meaning “measure” . As early as 2000 . we find the

20、land surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths ofgeometry .幾何學(xué)起源于很久以前巴比倫人和埃及人測量他們被尼羅河洪水淹沒的土地， 希臘語幾何來源于 geo ,意思是"土地"，和 metron意思是"測量"。公元前 2000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(biāo)志和邊界。2 B Somegeometrical terms A solid is a t

21、hree-dimensional figure. Commonexamples of solids are cube, sphere, cylinder, cone and pyramid. A cube has six faces whichare smooth and flat. These faces are called plane surfaces or simply planes. A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop is

22、 an example of a plane surface.立體是一個三維圖形，立體常見的例子是立方體，球體，柱體，圓錐和棱錐。立方體有 6個面，都是光滑的和平的，這些面被稱為平 面曲面或者簡稱為平面。平面曲面是二維的，有長度和寬度，黑板和桌子上面的面都是平面 曲面的例子。2C三角函數(shù)于直角三角形的解One of the most important applicationsof trigonometry is the solution of triangles. Let us now take up the solution toright triangles. A triangle is

23、 composed of sixparts three sides and three angles.To solve a triangle is to find the parts not given. A triangle may be solved if three parts (at least one of these is a side ) are given. A right triangle has one angle, the right angle, always given. Thus a right triangle can be solved when two sid

24、es, or one side and an acute angle, are given.三角形最重要的應(yīng)用之是解三角形，現(xiàn)在我們來解直角三角形。一個三角形由6個部分組成，三條邊和三只角。解一個三角形就是要求出未知的部分。如果三角形的三個部分(其中至少有一個為邊)為已知，則此三角形就44v1.0可編輯可修改可以解出。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-A Introduction A large variety of scientific problems arise in which one triesto determ

25、ine something from its rate of change. For example , we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration. Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material present after a

26、 given time.大量的科學(xué)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間后遺留物質(zhì)的總量。In examples like these, we are trying to determinean unknown function from prescribed information expressed in the form of an equation involving at least one of the derivatives of the unkn

27、own function . These equations are called differential equations, and their study forms one of the most challenging branches of mathematics.在類似的例子中，我們力求由方程的形式表示的信息來確定未知函數(shù)，而這種方程至少包含了未知函數(shù)的一個導(dǎo)數(shù)。這些方程稱為微分方程，對其研究形成了數(shù)學(xué)中最具有挑戰(zhàn)性的一門分支。The study of differential equations is one partof mathematics that, perhaps

28、more than any other, has been directly inspired by mechanics, astronomy, and mathematical physics. 微分方程的研究是數(shù)學(xué)的部分， 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。Its history began in the17th centurywhen Newton, Leibniz, and the Bernoullis solved some simpledifferentialequations arising from problems in geometry and m

29、echanics. These earlydiscoveries, beginning about 1690, gradually led to the development of a lot ofaspecial tricks" for solving certain special kinds of differential equation.微分方程起源于17世紀(jì)，當(dāng)時牛頓，萊布尼茨，波努力家族解決了一些來自幾何和力學(xué)的簡55v1.0可編輯可修改單的微分方程。開始于 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。Although these s

30、pecial tricks are applicable in relatively few cases, they do enable us to solve many differential equations that arise in mechanics and geometry, so their study is of practical importance. Some of these special methods and some of the problems which they help us solve are discussed near theend of t

31、his chapter.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(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際應(yīng)用。這些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。Experience has shown that it isdifficult to obtain mathematical theories of much generality about solution of differential equations, except for a few types.經(jīng)驗表明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(xué)理論。Among

32、these are the so-called linear differentialequations which occur in a great variety of scientific problems.在這些典型方程中，有一個稱為線性微分方程，出現(xiàn)在大量的科學(xué)問題中。10-C Applications of matricesIn recent years the applications of matrices in mathematics and in many diversefields have increased with remarkable speed. Matrix

33、theory plays a central role in modern physics in the study of quantum mechanics. Matrix methods are used to solve problems in applied differential equations , specifically, in the area of aerodynamics, stress and structure analysis. One of the most powerful mathematical methods for psychological stu

34、dies is factor analysis, a subject that makes wide use of matrix methods.近年來，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，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，特別是在空氣動力學(xué)，應(yīng)力和結(jié)構(gòu)分析中的問題，要用矩陣方法。心理學(xué)研究上一種最強有力的數(shù)學(xué)方法是因子分析，這也廣泛的使用矩陣（方）法.Recent developments inmathematical economics and in problems of business administration

35、 have led to extensive use of matrix methods. The biological sciences, and in particular genetics, use matrix techniques to good advantage. No matter what the students ' field of major interest is , knowledge of the rudiments of matrices is likely to broaden the range 66of literature that he can

36、 read with understanding .近年來，在數(shù)學(xué)經(jīng)濟學(xué)和商業(yè)管理問題方面的發(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可能擴大他能讀懂的文獻(xiàn)的范 圍。The solution of n simultaneous linear equations in n unknowns is one of theimportant problems of applied mathematics. Descartes, the inventor of analyticgeometry and one of the

37、founders of modern algebraic notation, believed that allproblems could ultimately be reduced to the solution of a set of simultaneous linear equations. 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(xué)的一個重要問題。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組聯(lián)立次方程。Although this belief is now known to be untenable , we knowthat a

38、large group of significant applied problems from many different disciplines are reducible to such equations. Many of the applications, require the solution of a large number of simultaneous linear equations ,sometimes in the hundreds . The advent of computers has made the matrix methods effective in

39、 the solution of these formidable problems.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常活躍。Example 1. solve the simultaneous equations for x1 x2, and x3 .例題 1,解聯(lián)立方程求 x1 x2 和 x3 。 From the above discussion, we see that the problem of s

40、olving n simultaneous linear equation in n unknowns is reduced to the problem of finding the inverse of the matrix of coefficients. It is therefore not surprising that in books on the theory of matrices the techniques of finding inverse matrices occupy considerable space.從上面的討論，我們看到解有n個未知數(shù)的n個聯(lián)立一次方程問

41、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。Of course , we will not in our limited treatment discuss suchtechniques. Not only are matrix methods useful in solving simultaneous equations , 77v1.0可編輯可修改but they are also useful in discovering whether or not the set of equations areconsistent, in

42、the sense that they lead to solutions, and in discovering whetheror not the set of equation are determinate, in the sense that they lead to uniquesolution. 當(dāng)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，即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 predicatesStatementsinvolvingvariables,such as

43、"x>3" , " x+y=3" , " x+y=z" are often found in mathematical assertion and incomputer programs. These statements are neither true nor false when the values ofthe variables are not specified. In this section we will discuss the ways thatpropositions can be produced fro

44、m such statements. 包含 變量的 語句， 比如 “x>3"，" x+y=3"，" x+y=z”常出現(xiàn)在數(shù)學(xué)論斷中和計算機程序中，若未給語句中的所有變量賦值，則不能判定該語句是真是假，本節(jié)要討論由這種語句生成命題的方法。Thestatement “x is greater than 3" has two parts. The fir st part, the variables, isthe subject of the statement. The second part-the predicate, “is great

45、er than3" -refers to a property that the subject of the statement can have.語句 "x 大于3”分成兩部分，第一部分，變量，是語句的主語。第二部分，謂語，“大于3”，指的是語句主語具有的性質(zhì)。We can denote the statemen t “x is greater than 3" by P(x),where P denote the predicate “is greater than 3" and x is the variable. Thestatement P(x

46、) is also said to be the value of the propositional function P at x.once a value has been assigned to the variable x, the statements P(x) becomes aproposition and has a truth value. 把語句"x大于3"記為P(x),其中P表示謂詞“大于3”,而x是變量。語句P(x)也稱為命題函數(shù) P在x點處的值。一旦賦予x 一個值，語句P(x)就成為一個命題，有了真值。11-B QuantifiersWhen a

47、ll the variables in a propositional function are assigned values, the resulting statement has a truth88v1.0可編輯可修改value. However, there is another important way, called quantification, to create a proposition from a propositional function. two types of quantification will be discussed here, namely, u

48、niversal quantification and existential quantification . 當(dāng)命題函數(shù)所有變量都賦值時，結(jié)果語句有真值，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。Manymathematical statements assert that a property is true for all values of a variable in a particular domain, called the universe of discourse. Such a statement is expressed using a universal quantification. The universal quantification of a propositional function is the proposition that assert that P(x) is true