数学专业英语第二版的课文翻译

1、1 A What is mathematics Mathematics comes from man'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 fields. No

2、 modern scientific and technological branches could be regularly developed without the application of mathematics.数学来源于人类的社会实践，比方工农业生产，商业活动， 军事行动和科学技术研究。反过 来，数学效劳于实践，并在各个领域中起着非常重要的作用。 没有应用数学，任何一个现 在的科技的 分支都 不能正 常开展。 From the early need of man came theconceptsof numbersand forms.Then,geometrydevelope

3、dout ofproblems of measuring land , and trigonometry came from problems ofsurveying .To deal with somemore complexpracticalproblems,manestablishedand then solvedequationwithunknownnumbers,thusalgebraoccurred.Before 17thcentury,manconfinedhimselfto theelementarymathematics,i.e., geometry,trigonometry

4、and algebra,inwhich only the constants are considered.很早的时候，人类的需要产生了数和形式的概念，接着， 测量土地的需要形成了几何， 出于测量的需要产生了三角几何，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。 The rapiddevelopment of industry in 17th century promoted the progress of economics and technology and requ

5、ired dealing with variable quantities.The leap from constantsto variable quantities brought about two newbranches of mathematicsanalytic geometry and calculus, whichbelong to the higher mathematics. Now there are many branches in higher mathematics, among which are mathematical analysis, higher alge

6、bra, differential equations, function theory and so on. 17世纪工业的快速开展推动了经济技术的进步， 从而遇到需要处理变量的问题，从常数带变量的跳跃产生了两个新的数学分支 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definitions and多分支，其中有数学分析，高等代数，微分方程，函数论等。 Mathematicians studyconceptions and propositions, Axioms, postulates, theorems are all propositions. Notations are

7、 a special and powerful tool of mathematics and are used to express conceptions and propositionsvery often. Formulas ,figuresand chartsSome of the best known symbolsofare full of different symbols.mathematics are the Arabicnumerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition, subtraction multiplic

8、ation, division and equality. 数学家研究的是概念和命题，公理，公设， 定义和定理都是命题。 符号是数学中一个特殊而有用的工具， 常用于表达概念和命题。 公式，图表都是不同的 符号.Thecon clusi onsin mathematicsare obtainedmainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathematics methods was occupied by t

9、he 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 workthat merely could be done ear

10、lier by logical deductions, for example, theproof of most of geometrical theorems.数学结论主要由逻辑推理和计算得到，在数学开展历史的很长时间内， 逻辑推理一直占据着数学方法的中心地位， 现在， 由于电子电脑的迅速开展和广泛使用，电脑的地位越来越重要，现在电脑不仅用于处理大量的信息和数据， 还可以完成一些之前只能由逻辑推理来做的工作， B Equation An equation is a statement例如， 大多数几何定理的证明。 1of the equality betweentwoequal numbe

11、rs or number symbols. Equation are of two kinds identities and equations of condition. An arithmetic or an algebraic identity is anare alike. Or等式是关于两 算术或者代数恒等式equation. In such an equation either the two members become alike on the performance of the indicated operation. 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

12、件等式。是等式。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 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 equation of condition, or

13、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. 含有字母的恒等式对其中字母的任一组数值都成立。一个等式假设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假设干组相关的值成立， 那么它是一个条件等式， 简称方程。因此 3x-5=7 仅当 x=4 时成立，而 2x-y=0 ，当 x=6,y=2 时成立，且对 x, y 的其他许多对值也成立。 A root of

14、 an equation is any number or number symbol which satisfies the equation. There are various kinds of equation. Theyare 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 equation meansto find the value of the unknown term. To do this , we must, of co

15、urse, 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 withou

16、t making the equation untrue, until only the unknown quantity is left on one side ,no matter which side.解方程意味着求未知项的值，为了求未知项的值， 当然必须移项， 直到未知项单独在方程的一边， 令其等于方程的另一边， 从而求得未知项的值， 解决了问题。 因此解方程意味着进行一系列的移项和同解变形， 直到 未知量被单独留在方程的一边，无论那一边。Equation are of very great use. Wecan use equation in many mathematical pr

17、oblems. We 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，可以用方程解决很多数学问题。注意到几乎每一个问题都给出一个或多个关于一个事情与另一个事情相等的陈述， 这就给出了方程， 利用该方程， 如果 我们需要的话，可以解方程。2A Why study geometry? Many le

18、ading institutions of higher learninghave recognized that positive benefits can be gained by all who study this branch of mathematics. This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.许多居于领导地位的学术机构成认， 所有学习这个数学分支的人都将得到确实的受益， 许多学 校把几

19、何的学习作为入学考试的先决条件，从这一点上可以证明。 Geometry had its origin long ago in the measurement by the Babylonians and Egyptians of their lands inundatedby the floods of the Nile River. The greek wordgeometry is derived from geo, meaning “ earth and metron, meaning “ measure . As early as 2000 B.C. we find the land

20、surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry .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洪水淹没的土地，希腊语几何来源于 geo ，意思是土地“，和 metron 意思是测量“。 公元前 2000 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 geometrical terms A solid is a three-dimensional figure

21、. Common examples of solids are cube, sphere, cylinder, cone and pyramid.A cube has six faces which are smooth and flat. These faces are calledplane surfaces or simply planes. A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop is an example of a plane

22、surface. 立体是一个三维图形， 立体常见的例子是立方体， 球体， 柱体，圆锥和棱锥。立方体有 6 个面，都是光滑的和平的，这些面被称为平面曲面或者简 称为平面。平面曲面是二维的，有长度和宽度，黑板和桌子上面的面都是平面曲面的例子。2C 三角函数于直角三角形的解 One of the most important applications of trigonometry is the solution of triangles. Let us now take up the solution to right triangles. A triangle is composed of si

23、x parts 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 sides, or one si

24、de and an acute 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 由6 个局部组成，三条边和三只角。解一个三角形就是要求出未知的局部。如果三角形的 三个局部其中至少有一个为边为，那么此三角形就可以解出。直角三角形的一只角， 即直角，总是的。因此，如果它的两边，或一边和一锐角为， 那么此直角三角形可解。9-A Introduction A large variety of scientific problems arise in which one tries to determine something from it

25、s 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 bedisintegrating at a known rate and we may be required to determine theamount of material present after a given time.大量的科学问题需要人们

26、根据事物的变化率来确定该事物，例如，我们可以由速度或者加速度来计算移动粒子的位置 又如，某种放射性物质可能正在以的速度进行衰变， 需要我们确定在给定的时间后遗留物质的总量。 In examples like these, we are trying to determine an unknownfunctionfrom prescribedinformationexpressedequation involving at least one of the derivativesin the form of an of the unknownfunction .These equations a

27、re called differentialequations, and their在类而这种方程至少包含了未study forms one of the most challenging branches of mathematics. 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数， 知函数的一个导数。 这些方程称为微分方程， 对其研究形成了数学中最具有挑战性的一门分 支。 The study of differential equations is one part of mathematics that,perhaps more than any other, has be

28、en directly inspired by mechanics, astronomy, and mathematical physics.微分方程的研究是数学的一局部，也许比其他分支更多的直接受到力学，天文学和数学物理的推动。 Its history began in the 17th century when Newton,Leibniz, and the Bernoullis solved somesimple differentialequationsarisingfrom problems in geometry andmechanics. These early discover

29、ies, beginning about 1690, gradually led to the development of a lot of“ special tricks for solving certain specialkinds of differential equation. 微分方程起源于 17 世纪，当时牛顿，莱布尼茨，波 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 1690 年的早期发现，逐 渐引起了解某些特殊类型的微分方程的大量特殊技巧的开展。Although these special tricks are applicable in relativel

30、y few cases, they do enable us to solvemany 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 the end of this chapter. 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

31、够解决力学和几何中出现的许多微分方程，因此， 他们的研究具有重要的实际应用。 这些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。Experience has shownthat it is difficult to obtain mathematical theories of much generality about solution of differential equations, except for a few types.经验说明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。Among these are theso-called linear differe

32、ntial equations which occur in a great variety of scientific problems. 在这些典型方程中，有一个称为线性微分方程，出现在大量的科 学问题中。 10-C Applications of matrices In recent years the applications of matrices in mathematics and in many diverse fields have increased with remarkable speed. Matrix theory plays a central role in m

33、odern 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 studies is factor analysis, a subje

34、ct that makes wide use of matrix methods.近年来，在数学和许多各种不同的领域中， 矩阵的应用一直以惊人的速度不断增加。 在研究量子力学时， 矩阵理论在现代 物理学上起着主要的作用。 解决应用微分方程， 特别是在空气动力学， 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 矩阵 方 法 . Recent developments in mathematical economics and in problems of business administration have led to extensi

35、ve 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 matricesis likely tobroaden the range of literature that he can read with understanding

36、 . 近 年来，在数学经济学和商业管理问题方面的开展已经导致广泛的使用矩阵法。 生物科学，特 别在遗传学方面，用矩阵的技术很有成效。 不管学生主要兴趣是什么， 矩阵根本原理的知识 可能扩大他能读懂的文献的范围。 The solution of n simultaneous linear equations in n unknowns is one of the important problems of applied mathematics. Descartes, the inventor of analytic geometry and one of the founders of mod

37、ern algebraic notation, believed that all problems could ultimately be reduced to the solution of a set of simultaneous linear equations.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重要问题。解析几何的创造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。 Although this belief is now known to be untenable , we know that a large g

38、roup 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 the so

39、lution of these formidable problems. 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， 电脑的创造已经使得矩阵方法在解这些难以解决的问题 方面非常活泼。Example 1. solve the simultaneous equations for x1 x2, and x3 .例题 1，解联立方程求 x1 x2 和 x3 。 From the above discussion,we see that theproblem of solvin

40、g n simultaneouslinear equationin n unknowns isreduced 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.从上面的讨论，我们看到解有 n 个未知数的 n 个联立一次方程问题化成求系

41、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。 Of course , we will not in our limited treatment discuss such techniques. Not only are matrix methods useful in solving simultaneous equations , but they are also useful in discovering whether or not the set of equations are consistent, in the sense that

42、 they lead to solutions, and in discovering whether or not the set of equation are determinate, in the sense that they lead to unique solution.当然，我们在这有限的表达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， 即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 predicatesStatements involving variables, such as “x>

43、;3, x+y=3 , x+y=z are often found in mathematical assertion and in computer programs. These statements are neither true nor false whenthe values of the variables are not discuss the ways that propositions statements. 包含变量的语句，比方 “x>3 和电脑程序中，假设未给语句中的所有变量赋值， 由这种语句生成命题的方法。 The statement “specified. I

44、n this section we will can be produced from such , x+y=3 , x+y=z 常出现在数学论断中 那么不能判定该语句是真是假， 本节要讨论 x is greater than 3 has two parts.The first part, the variables, is the subject of the statement. The second part- the predicate,“is greater than3-refersto a property that thesubject of the statement can

45、have.语句"x大于3分成两局部，第一局部，变量，是语句的主语。第二局部，谓语， “大于 3，指的是语句主语具有的性质。We can denotethe sta tement “x is greater than 3 by P(x), where P denote the predicate“is greater than 3 and x is the variable. The statement P(x) is also said tobe the value of the propositional function P at x. once a value has been

46、 assigned to the variable x, the statements P(x) becomes a proposition and has a truth value.把语句"x大于3记为P(x),其中P表示谓词 大于3，而x是变量。语句 P(x) 也称为命题函数 P 在 x 点处的值。一旦赋予 x 一个值，语句 P(x) 就成 为一个命题,有了真值。 11-B QuantifiersWhenall the variables in apropositional function are assigned values, the resulting statemen

47、t has atruth value. However, there is another important way, called quantification, to create a proposition from a propositional function. twotypes of quantificationwill be discussed here, namely, universalquantification and existential quantification .当命题函数所有变量都赋值时，结果语句有真值，但是还有另外一种方式，称为量词化，可从命题函数中得

48、到命题。这里讨论两种量词化方法，也就是全称量词化和存在量词化。 Many mathematical 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 for all value