数学专业英语第二版的课文翻译

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 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，比如工农业生产，商业活动， 军事行动和科学技术研究。反过来，数学服务于实践，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发展。From the early need of man came the concepts of numbers and forms. Then, geometry developed out of 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. , geometry, trigonometry and algebra, in which 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 rapid development of industry in 17th century promoted the progress of economics and technology 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 analysis, higher algebra, differential equations, function theory and so on. 17世纪工业的快速发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多分支，其中有数学分析，高等代数，微分方程，函数论等。Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. 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 addition, subtraction , multiplication, 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……..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 methods 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 merely could be done earlier by logical deductions, for example, the proof of most of geometrical theorems. 数学结论主要由逻辑推理和计算得到，在数学发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，由于电子计算机的迅速发展和广泛使用，计算机的地位越来越重要，现在计算机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，例如，大多数几何定理的证明。 1－B Equation An equation is a statement of the equality 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. 等式是关于两个数或者数的符号相等的一种描述。等式有两种－恒等式和条件等式。算术或者代数恒等式是等式。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 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。一个等式若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若干组相关的值成立，则它是一个条件等式，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 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 means to find the value of the unknown term. To do this , we 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 the question. To solve the equation, therefore, means to move and change the terms about without 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. We can use equation in many mathematical problems. 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。

2－A Why study geometry? Many leading institutions of higher learning

have 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承认，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 inundated by the floods 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 land 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. Common examples of solids are cube, sphere, cylinder, cone and pyramid.

A cube has six faces which are 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 an example of a plane 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。 2－C 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 six 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 side 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个部分组成，三条边和三只角。解一个三角形就是要求出未知的部分。如果三角形的三个部分（其中至少有一个为边）为已知，则此三角形就可以解出。直角三角形的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一锐角为已知，则此直角三角形可解。

9-A Introduction A large variety of scientific problems arise in which one tries to determine something from 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 given time. 大量的科学问题需要人们根据事物的变化率来确定该事物，例如，我们可以由已知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已知的速度进行衰变，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 unknown function from prescribed information expressed in the form of an equation involving at least one of the derivatives of the unknown function . These equations are called differential equations, and their study forms one of the most challenging branches of mathematics. 在类似的例子中，我们力求由方程的形式表示的信息来确定未知函数，而这种方程至少包含了未知函数的一个导数。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 been directly inspired by mechanics, astronomy, and mathematical physics. 微分方程的研究是数学的一部分，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 some simple differential equations arising from problems in geometry and mechanics. These early discoveries, beginning about 1690, gradually led to the development of a lot of “special tricks” for solving certain special kinds 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发展。 Although these special 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 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 shown that it is difficult to obtain mathematical theories of much generality about solution of differential equations, except for a few types. 经验表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 the so-called linear differential 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 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 studies is factor analysis, a subject 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 extensive use of matrix methods. The biological sciences, and in particular genetics, use matrix techniques to good advantage. No matter what the students’ f ield of major interest is , knowledge of the rudiments of matrices is likely to broaden the range of literature that he can read with understanding . 近年来，在数学经济学和商业管理问题方面的发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，矩阵基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 modern 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。解析几何的发明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 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 the solution 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，计算机的发明已经使得矩阵方法在解这些难以解决的问题方面非常活跃。

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 the problem of solving 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. 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 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，我们在这有限的叙述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>3”, ”x+y=3”, ”x+y=z” are often found in mathematical assertion and in computer programs. These statements are neither true nor false when the values of the variables are not specified. In this section we will discuss the ways that propositions can be produced from such statements. 包含变量的语句，比如“x>3”, ”x+y=3”, ”x+y=z” 常出现在数学论断中和计算机程序中，若未给语句中的所有变量赋值，则不能判定该语句是真是假，本节要讨论由这种语句生成命题的方法。 The statement “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 than 3”-refers to a property that the subject of the statement can have. 语句“x大于3”分成两部分，第一部分，变量，是语句的主语。第二部分，谓语，“大于3”，指的是语句主语具有的性质。 We can denote the statement “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 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 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 QuantifiersWhen all the variables in a propositional function are assigned values, the resulting statement has a truth 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, universal quantification 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 values of x in the universe of discourse. The universe of discourse specifies the possible values of the variable x. 许多数学语句认为，在特殊领域内对变量所有值都成立的性质称为论域。这样的语句可用全称量词化表示。命题函数的全称量词化是一个命题，认为P(x)对论域中x 的所有值P(x)都是真的。论域指定变量x 的可能取值. 12-A Special terminology peculiar to probability theory In discussions involving probability, one often sees phrases from everyday language such as “two events are equally likely,” “an event is impossible,” or “an event is certain to occur.” expressions of this sort have intuitive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions. 在讨论概率论时，会常常从日常用语中看到这样的语句：两个事件是同等可能的，一个事件是不可能的，一个事件肯定发生，这种表达方式非常直观，在数学讨论中，乐于使用这样有色彩的语言，而且使用起来很有帮助。 Before we can do so, however, it is necessary to

explain the meaning of this language in terms of the fundamental concepts of our theory. 但是，在我们这么做之前，有必要根据我们理论的基本概念来解释这种语句的含义。 Because of the way probability is used in practice, it is convenient to imagine that each probability space (S,B,P) is associated with a real or conceptual experiment. The universal set S can then be thought of as the collection of all conceivable outcomes of the experiment, as in the example of coin tossing discussed in the foregoing section. 根据概率论实际应用的方式，很方便的把每一个概率空间（S,B,P ）对应于一个实际的或者概念上的试验。全集S 是试验中所有可信结果的集体，就像前面章节讨论的掷硬币的例子。 Each element of S is called an outcome or a sample and the subsets of S that occur in the Boolean algebra B are called events. The reasons for this terminology will become more apparent when we treat some examples. S的每一个元素称为结果或者样本，在布尔代数B 中出现的S 的子集称为事件，为什么使用这个术语在我们举例后就会很明显。