專業(yè)英語翻譯(第二章)

1、1A What is mathematicsMathematics comes from mans 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 fields. No modern

2、 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 man came the concepts of numbers and forms. Then, geometry developed out o

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

4、eometry, 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 technol

5、ogy and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics-analytic geometry and calculus, which belong to the higher mathematics. Now there are many branches in higher mathematics, among which are mathematical anal

6、ysis, higher algebra, differential equations, function theory 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 propositio

7、ns. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of additio

8、n, 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，常用于表達概念和命題。公式，圖表都是不同的符號.The conclusions in mathematics are obtained mainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathematics met

9、hods was occupied by the 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 also to carry out some work that m

10、erely could be done earlier by logical deductions, for example, the proof of most of geometrical 1theorems.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。1B Equation1An equation is a statement of the equality

11、 between two equal numbers or number symbols. Equation are of two kinds- identities and equations of condition.An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike. Or become alike on the performance of the indicated operation.等式是關(guān)于兩個數(shù)或者數(shù)的符號相等的一

12、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 of numerical values of the letters in it.An equation which is true only for certain values of a letter in it, or for certain sets of related values of two or more of its letters, is an equat

13、ion of condition, or simply an equation. Thus 3x-5=7 is true for x=4 only; and 2x-y=0 is true for x=6 and y=2 and for many other pairs of values 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的其他許多對值也成立。A root

14、 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 the unknown term. To do this , we must, of cour

15、se, 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 the question. To solve the equation, therefore, means to move and change the terms about without

16、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 many mathematical problems. W

17、e may 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。2A Why study geometry?Many leading institut

18、ions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics. This is 2evident from the fact that they require study of geometry as a prerequisite tomatriculation in those schools.許多居于領(lǐng)導(dǎo)地位的學(xué)術(shù)機構(gòu)承認(rèn)，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，許多學(xué)校把幾何的學(xué)習(xí)作為入學(xué)考試的先決條件，

19、從這一點上可以證明。Geometry had its origin long ago in themeasurement by the Babylonians and Egyptians of their lands inundated by thefloods of the Nile River. The greek word geometry is derived from geo, meaning“earth” and metron, meaning “measure” . As early as 2000 B.C. we find the landsurveyors of these

20、people re-establishing vanishing landmarks and boundariesby utilizing the truths of geometry .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。2B Some geometrical termsA solid is a three-dimensional figure. Common examples of solid

21、s are cube,sphere, cylinder, cone and pyramid.A cube has six faces which are smooth andflat. These faces are called plane surfaces or simply planes. A plane surface hastwo dimensions, length and width. The surface of a blackboard or of a tabletop isan 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

22、，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。3A Notations for denoting setsThe concept of a set has been utilized so extensively throughout modernmathematics that an understanding of it is necessary for all college students. Setsare a means by which mathematicia

23、ns talk of collections of things in an abstractway.Sets usually are denoted by capital letters; elements are designated by lower-caseletters.集合論的概念已經(jīng)被廣泛使用，遍及現(xiàn)代數(shù)學(xué)，因此對大學(xué)生來說，理解它的概念是必要的。集合是數(shù)學(xué)家們用抽象的方式來表述一些事物的集體的工具。集合通常用大寫字母表示，元素用小寫字母表示。We use the special notation to mean that “x is an element of S” or “x

24、 belongs to S”. If x does not belong to S,we write .When convenient, we shall designate sets by displaying the elements in braces;for example, the set of positive even integers less than 10 is displayed as 2,4,6,8whereas the set of all positive even integers is displayed as 2,4,6, the threedots taki

25、ng the place of “and so on.”我們用專用記號來表示x是S的元素或者x屬于S。如果x不屬于S，我們記為。如果方便，我們可以用在大括號中列出元素的方式來表示集合。例如，小于10的正偶數(shù)的集合表示為2,4,6,8，而所有正偶數(shù)的集合表示為2,4,6, 三個圓點表示 “等等”。The dots are used only when the meaning of “and so on” is clear. The method oflisting the members of a set within braces is sometimes referred to as the

26、 rosternotation.The first basic concept that relates one set to another is equality of sets:只有當(dāng)省略的內(nèi)容清楚時才能使用圓點。在大括號中列出集合元素的方法有時被歸結(jié)為枚舉法。聯(lián)系一個集合與另一個集合的第一個基本概念是集合相等DEFINITION OF SET EQUALITY Two sets A and B are said to be equal (or identical) if they consist of exactly the same elements, in which case w

27、e write A=B. If one of the sets contains an element not in the other, we say the sets unequal and we write AB.集合相等的定義 如果兩個集合A和B確切包含同樣的元素,則稱二者相等，此時記為A=B。如果一個集合包含了另一個集合以外的元素，則稱二者不等，記為AB。EXAMPLE 1. According to this definition, the two sets 2,4,6,8 and 2,8,6,4 are equal since they both consist of the f

28、our integers 2,4,6 and 8. Thus, when we use the roster notation to describe a set, the order in which the elements appear is irrelevant.根據(jù)這個定義，兩個集合2,4,6,8和2,8,6,4是相等的，因為他們都包含了四個整數(shù)2,4,6,8。因此，當(dāng)我們用枚舉法來描述集合的時候，元素出現(xiàn)的次序是無關(guān)緊要的EXAMPLE 2. The sets 2,4,6,8 and 2,2,4,4,6,8 are equal even though, in the second

29、set, each of the elements 2 and 4 is listed twice. Both sets contain the four elements 2,4,6,8 and no others; therefore, the definition requires that we call these sets equal.This example shows that we do not insist that the objects listed in the roster notation be distinct. A similar example is the

30、 set of letters in the word Mississippi, which is equal to the set M,i,s,p, consisting of the four distinct letters M,i,s, and p例2. 集合2,4,6,8 和2,2,4,4,6,8也是相等的，雖然在第二個集合中，2和4都出現(xiàn)兩次。兩個集合都包含了四個元素2,4,6,8，沒有其他元素，因此，依據(jù)定義這兩個集合相等.這個例子表明我們沒有強調(diào)在枚舉法中所列出的元素要互不相同。一個相似的例子是，在單詞Mississippi中字母的集合等價于集合M,i,s,p, 其中包含了四個

31、互不相同的字母M,i,s,和p.3B SubsetsFrom a given set S we may form new sets, called subsets of S. For example, the set consisting of those positive integers less than 10 which are divisible by 4 (the set 4,8) is a subset of the set of all even integers less than 10. In general, we have the following definitio

32、n.一個給定的集合S可以產(chǎn)生新的集合，這些集合叫做S的子集。例如，由可被4除盡的并且小于10的正整數(shù)所組成的集合是小于10的所有偶數(shù)所組成集合的子集。一般來說，我們有如下定義。In all our applications of set theory, we have a fixed set S given in advance, and we are concerned only with subsets of this given set. The underlying set S may vary from one application to another; it will be r

33、eferred to as the universal set of each particular discourse. （35頁第二段）當(dāng)我們應(yīng)用集合論時，總是事先給定一個固定的集合S，而我們只關(guān)心這個給定集合的子集?；A(chǔ)集可以隨意改變，可以在每一段特定的論述中表示全集。 To avoid logical difficulties, we must distinguish between the elements x and the set x whose only element is x. In particular, the empty set is not the same as

34、the set .In fact, the empty set contains no elements, whereas the set has one element. Sets consisting of exactly one element are sometimes called one-element sets.為了避免遇到邏輯困難，我們必須區(qū)分元素x和集合x，集合 x中的元素是x。特別要注意的是空集和集合是不同的。事實上，空集不含有任何元素，而有一個元素。由一個元素構(gòu)成的集合有時被稱為單元素集。Diagrams often help us visualize relations

35、 between sets. For example, we may think of a set S as a region in the plane and each of its elements as a point. Subsets of S may then be thought of the collections of points within S. For example, in Figure 2-3-1 the shaded portion is a subset of A and also a subset of B. （35頁第五段）圖解有助于我們將集合之間的關(guān)系形象

36、化。例如，可以把集合S看作平面內(nèi)的一個區(qū)域，其中的每一個元素即是一個點。 那么S的子集就是S內(nèi)某些點的全體。例如，在圖2-3-1中陰影部分是A的子集，同時也是B的子集。Visual aids of this type, called Venn diagrams, are useful for testing the validity of theorems in set theory or for suggesting methods to prove them. Of course, the proofs themselves must rely only on the definition

37、s of the concepts and not on the diagrams.這種圖解方法，叫做文氏圖，在集合論中常用于檢驗定理的有效性或者為證明定理提供一些潛在的方法。當(dāng)然證明本身必須依賴于概念的定義而不是圖解。 4A Integers and rational numbersThere exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discus

38、s such subsets, the integers and the rational numbers.有一些R的子集很著名，因為他們具有實數(shù)所不具備的特殊性質(zhì)。在本節(jié)我們將討論這樣的子集，整數(shù)集和有理數(shù)集。To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3, obtain

39、ed in this way by repeated addition of 1 are all positive, and they are called the positive integers.我們從數(shù)字1開始介紹正整數(shù)，公理4保證了1的存在性。1+1用2表示，2+1用3表示，以此類推，由1重復(fù)累加的方式得到的數(shù)字1,2,3，都是正的，它們被叫做正整數(shù)。Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in

40、 detail what we mean by the expressions “and so on”, or “repeated addition of 1”.嚴(yán)格地說，這種關(guān)于正整數(shù)的描述是不完整的，因為我們沒有詳細(xì)解釋“等等”或者“1的重復(fù)累加”的含義。Although the intuitive meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the posi

41、tive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set.雖然這些說法的直觀意思似乎是清楚的，但是在認(rèn)真處理實數(shù)系統(tǒng)時必須給出一個更準(zhǔn)確的關(guān)于正整數(shù)的定義。 有很多種方式來給出這個定義，一個簡便的方法是先引進歸納集的概念。DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive set if it has the f

42、ollowing two properties:The number 1 is in the set.For every x in the set, the number x+1 is also in the set.For example, R is an inductive set. So is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set. 現(xiàn)在我們來定義正整數(shù)，就是屬于每一個歸納集的實數(shù)。Let P deno

43、te the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P belong to every inductive set, we refer to P as the smallest inductive set.用P表示所有正整數(shù)的集合。那么P本身是一個歸納集，因為其中含1，滿足(a)；只要包含x就包含x+1, 滿足(b)

44、。由于P中的元素屬于每一個歸納集，因此P是最小的歸納集。This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this introduction.P的這種性質(zhì)形成了一種推理的邏輯基礎(chǔ)，數(shù)學(xué)家稱之為歸納證明，在介紹的第四部分將給出這種方法的詳細(xì)論述。The negatives of the positive integers

45、 are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply the set of integers.正整數(shù)的相反數(shù)被叫做負(fù)整數(shù)。正整數(shù)，負(fù)整數(shù)和零構(gòu)成了一個集合Z，簡稱為整數(shù)集。n a thorough treatment of the real-number system, it would be necessary at this stage to prove cert

46、ain theorems about integers. For example, the sum, difference, or product of two integers is an integer, but the quotient of two integers need not to ne an integer. However, we shall not enter into the details of such proofs.在實數(shù)系統(tǒng)中，為了周密性，此時有必要證明一些整數(shù)的定理。例如，兩個整數(shù)的和、差和積仍是整數(shù)，但是商不一定是整數(shù)。然而還不能給出證明的細(xì)節(jié)。 Quoti

47、ents of integers a/b (where b0) are called rational numbers. The set of rational numbers, denoted by Q, contains Z as a subset. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. R

48、eal numbers that are not in Q are called irrational整數(shù)a與b的商被叫做有理數(shù)，有理數(shù)集用Q表示，Z是Q的子集。讀者應(yīng)該認(rèn)識到Q滿足所有的域公理和序公理。因此說有理數(shù)集是一個有序的域。不是 6有理數(shù)的實數(shù)被稱為無理數(shù)。4B Geometric interpretation of real numbers as points on a lineThe reader is undoubtedly familiar with the geometric interpretation of real numbers by means of points

49、 on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 2-4-1. This choice determines the scale.毫無疑問，讀者都熟悉通過在直線上描點的方式表示實數(shù)的幾何意義。如圖2-4-1所示，選擇一個點表示0，在0右邊的另一個點表示1。這種做法決定了刻度。 If one adopts an appropriate set of axioms for Euclidean

50、geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number.如果采用歐式幾何公理中一個恰當(dāng)?shù)募?，那么每一個實數(shù)剛好對應(yīng)直線上的一個點，反之，直線上的每一個點也對應(yīng)且只對應(yīng)一個實數(shù)。For this reason the line is often called the real line or the real axis, and

51、it is customary to use the words real number and point interchangeably. Thus we often speak of the point x rather than the point corresponding to the real number.為此直線通常被叫做實直線或者實軸，習(xí)慣上使用“實數(shù)”這個單詞，而不是“點”。因此我們經(jīng)常說點x不是指與實數(shù)對應(yīng)的那個點。This does not mean that one should not make use of geometry in studying proper

52、ties of real numbers. On the contrary, the geometry often suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof (one depending entirely on the axioms for the real numbers).這并不意味著研究實數(shù)的性質(zhì)時不會應(yīng)用到幾何。相反，幾何經(jīng)常會為證明一些定理提供思路，有

53、時幾何討論比純分析式的證明更清楚。In this book, geometric arguments are used to a large extent to help motivate or clarity a particular discuss. Nevertheless, the proofs of all the important theorems are presented in analytic form.在本書中，幾何在很大程度上被用于激發(fā)或者闡明一些特殊的討論。不過，所有重要定理的證明必須以分析的形式給出。5-A The coordinate system of Cart

54、esian geometry As mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily we do not talk about area by itself, instead, we talk about the area of something. 就像前面提到的 積分的一個應(yīng)用就是計算面積. 通常我們不討論面積本身, 相反, 是討論某物的面積. This means that we have certain objects (polygonal

55、regions, circular regions, parabolic segments etc.) whose areas we wish to measure. 這意味著我們想測量一些物體的面積 多邊形區(qū)域 圓域 拋物弓形等。 If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects. 如果我們希望獲得面積

56、的計算方法以便能夠用它來處理各種不同類型的圖形 我們就必須首先找出表述這些圖形的有效方法。 The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. 描述圖形最原始的方法是畫圖, 就像古希臘人做的那樣 A much better way was 7suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry

57、). R.笛卡兒提出了一種好得多的辦法 并建立了解析幾何 也稱為笛卡兒幾何 這個學(xué)科。 Descartes idea was to represent geometric points by numbers. The procedure for points in a plane is this : 笛卡兒的思想就是用數(shù)來表示幾何點 在平面上找點的過程如下 Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the “ x-axis ”), the other

58、 vertical (the “ y-axis ”). Their point of intersection, denoted by O, is called the origin. 選兩條互相垂直的參考線(稱為坐標(biāo)軸), 其中一條是水平的(稱為x軸), 另一條是豎直的(稱為y軸). 它們的交點記為O, 稱為原點. On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance. 在x軸上 原點的右側(cè)選擇一個合適的點 該點與原點之間

59、的距離稱為單位長度。 Vertical distances along the y-axis are usually measured with the same unit distance, although sometimes it is convenient to use a different scale on the y-axis. 沿著y軸的豎直距離通常用同樣的單位長度來測量 不過有時采用不同的尺度(單位長度)較為方便。 Now each point in the plane (sometimes called the xy-plane) is assigned apair of numbers, called its coordinates. These number