数学专业英语第五讲精读课文.ppt

1、New Words A即B; A指的是B 。,For example, (9.1) is a first-order equation which may be written as y =y. The differential equation y =x3 y + sin (x y) is one of second order. 例如，（9.1）是一个一阶微分方程, 可写为y =y, y =x3 y + sin (x y) 是一个二阶微分方程。 In this chapter we shall begin our study with first-order equations which

2、 can be solved for y and written as follows. 本章将开始研究通过y 可以求解的并可以写成如下形式的一阶方程。,where the expression f (x, y) on the right has various special forms. A differentiable function y=Y(x) will be called a solution of (9.2) on an interval I if the function Y and its derivative Y satisfy the relation Y(x)=f x

3、, Yx for every x in I. 右边的函数 f (x, y) 具有不同的表达方式。称可微函数y=Y(x) 为一阶微分方程（9.2）在区间I上的一个解，如果对于区间I中的每一个x, 函数及其导数都满足关系Y(x)=f x, Yx。,The simplest case occurs when f (x, y) is independent of y. In this case, (9.2) becomes (9.3) y=Q (x). 最简单的情况是f (x, y)与y无关，在这种情况下，（9.2）成为(9.3) y=Q(x)。,say, where Q is assumed to

4、be a given function defined on some interval I. To solve the differential equation (9.3) means to find a primitive of Q. The Second fundamental theorem of calculus tells us how to do it when Q is continuous on an open interval I. 其中Q为定义在某区间I上的给定函数。解微分方程(9.3)即寻找Q的一个原函数。当Q在开区间I上连续时，由微积分第二基本定理可知如何求解此方程

5、。,We simply integrate Q and add any constant. 我们直接对Q积分并加上任意常数。 Integrate Q with respect to x. 对Q关于变量x进行积分。,where C is any constant (usually called an arbitrary constant of integration) . The differential equation (9.3) has infinitely many solutions , one for each value of C . 其中C是任意常数（通常被称为任意积分常数）。微

6、分方程（9.3）有无穷多解，每一个解对应一个常数C。,Thus , every solution of (9.3) is included in the formula (9.4) 因此, (9.3)的每一个解都包含在公式(9.4) 中。,If it is not possible to evaluate the integral in (9.4) in terms of familiar functions, such as polynomials, rational functions, trigonometric and inverse trigonometric functions,

7、logarithms, and exponentials , still we consider the differential equation as having been solved if the solution can be expressed in terms of integrals of known functions. In actual practice, there are various methods for obtaining approximate evaluations of integrals which lead to useful informatio

8、n about the solution . 多项式，有理函数，三角函数和反三角函数，对数和指数函数,Automatic high-speed computing machines are often designed with this kind of problem in mind. 自动高速计算机在设计时就考虑到这类问题的处理。,EXAMPLE. Linear motion determined from the velocity. Suppose a particle moves along a straight line in such a way that its velocity

9、 at time t is 2sint. Determine its position at time t. 例 直线运动的速度的确定。设质点沿直线运动的速度函数为2sint。试确定其在时刻t的位置。 Solution. If Y(t) denotes the position at time t measured from some starting point , then the derivative Y(t) represents the velocity at time t . We are given that Y(t)=2sint. Integrating , we find t

10、hat Y(t)=-2cos t+C. 解 令Y(t)表示从某起点开始至时刻t的位置，则导数Y(t)表示在时刻t的速度。由已知，Y(t)=2sint. 积分即可得 Y(t)=-2cos t+C.,This is all we can deduce about Y(t) from a knowledge of the velocity alone; 这是我们仅由速度可以推断出的所有Y(t); some other piece of information is needed to fix the position function. We can determine C if we know t

11、he value of Y at some particular instant. For example, if Y(0)=0; then C=2, and the position function is Y(t) =2-2cost. But if Y(0)=2, then C=4 and the position function is Y(t)=4-2cost.,In some respects the example just solved is typical of what happens in general.,Some-where in the process of solv

12、ing a first-order differential equation, an integration is required to remove the derivative y and in this step an arbitrary constant C appears. 在解一阶微分方程的过程中，为了消去导数y，需要在某一步进行积分，这时候就出现了一个任意常数。,The way in which the arbitrary constant C enters into the solution will depend on the nature of the given di

13、fferential equation. It may appear as an additive constant, as in Equation(9.4), but it is more likely to appear in some other way. For example, when we solve the equation y=y in Section 9.3 , we shall find that every solution has the form y=Cex.,In many problems it is necessary to select from the c

14、ollection of all solutions one having a prescribed value at some point. The prescribed value is called an initial condition, and the problem of determining such a solution is called an initial-value problem. This terminology originated in mechanics where, as in the above example , the prescribed val

15、ue represents the displacement at some initial time.,本小节重点掌握,如果一个微分方程的未知函数是多元函数，则称为偏微分方程。,A differential equation is called partial differential equation if the unknown of it is a function of two or more variables.,New Words & Expressions component 分量，成分 matrix 矩阵 dependent 相关的 independent 无关的 finit

16、e dimensional 有限维的 scalar 标量、数量 infinite dimensional 无限维的 n-tuple n元组 hold trivially 显然成立 span 张成、支撑 The Pythagorean identity 勾股定理/毕达哥拉斯等式,2.10 线性空间中的相关与无关集 Dependent and Independent Sets in a Linear Space,New Words & Expressions(P90 生词与词组二): consistent 相容的 matrix 矩阵 column 列 row 行 determinate 行列式 r

17、educible 可简化的 inverse 逆 matrix of coefficients 系数矩阵 entry of matrix 矩阵的元 square matrix 方阵 polynomial in x x的多项式 untenable 不可到达的 simultaneous linear equations 联立方程,In recent years the applications of matrices in mathematics and in many diverse fields have increased with remarkable speed. Matrix theor

18、y plays a central role in modern physics in the study of quantum mechanics.,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。,10-C Applications of matrices,Matrix methods are used to solve problems in applied differential equations, specifically, in the area of aerodynamics, stress a

19、nd structure analysis. One of the most powerful mathematical methods for psychological studies is factor analysis, a subject that makes wide use of matrix methods.,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 proble

20、ms of business administration have led to extensive use of matrix methods. The biological sciences, and in particular genetics, use matrix techniques to good advantage. 近年来，在数量经济学和企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。,No matter what the students field of major interest is , knowledge of

21、the rudiments of matrices is likely to broaden the range of literature that he can read with understanding . 不管学生主要兴趣是什么，矩阵基本原理的知识都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. 解一有n个未知数的n个联立（线性）方程组是应用数学的一个重要问题。,

22、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简化为解一组联立方程。,Although this belief is now known to b

23、e untenable , we know that a large group of significant applied problems from many different disciplines are reducible to such equations. 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简为这类的方程。,Many of the applications, require the solution of a large number of simultaneous linear equations, somet

24、imes in the hundreds . The advent of computers has made the matrix methods effective in the solution of these formidable problems. 许多应用要求解大量的，往往数以百计的联立方程，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。,From the above discussion, we see that the problem of solving n simultaneous linear equation in n unknowns is red

25、uced to the problem of finding the inverse of the matrix of coefficients. (P89 下数第9行) 从上面的讨论，我们看到解有n个未知数的n个联立方程问题简化成求系数矩阵的逆矩阵的问题。,It is therefore not surprising that in books on the theory of matrices the techniques of finding inverse matrices occupy considerable space. 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

26、怪了。,Of course , we will not in our limited treatment discuss such techniques. 当然，我们在这有限的叙述中不会讨论这类的技巧。,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 solu

27、tions, and in discovering whether or not the set of equations are determinate, in the sense that they lead to unique solution. 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，即是否有惟一解等方面，都是有用的。,作业：,P78: 1, 2(1) P87: 1, 2(1),New Words & Expressions: assert 断言，主张 predicate 谓词 conjunction 合取 quanti

28、fier 量词 connective 连词 quantification 量词化 disjunction 析取 statement 语句,2.11 数理逻辑入门 Elementary Mathematical Logic,Key points: introduction to predicates and quantifiers Difficult points: special terminology peculiar to probability theory,Requirements:,1. 了解谓词和量词的基本表示方法。 2 . 掌握概率论基本的表示方法。,Statements inv

29、olving variables, such as “x3”, “x+y=3”, “x+y=z” are often found in mathematical assertions and in computer programs.,11-A Predicates,包含变量的语句，比如“x3”, “x+y=3”, “x+y=z” 常出现在数学论断和计算机程序中。,These statements are neither true nor false when the values of the variables are not specified. In this section we w

30、ill discuss the ways that propositions can be produced from such statements.,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.,语句“x大于3”分成两部分，第一部分，变量，是语句的主语。,The second part-the predicate, “

31、is greater than 3”-refers to a property that the subject of the statement can have.,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 denotes the predicate “is greater than 3” and x is the variable.,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。,The statement P(x)

32、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.,语句P(x)也称为命题函数P在x点处的值。一旦赋予变量x一个值，语句P(x)就成为一个命题,有了真假值。,When all the variables in a propositional function are assigned v

33、alues, the resulting statement has a truth value. However, there is another important way, called quantification, to create a proposition from a propositional function.,当命题函数所有变量都赋值时，结果语句就有了真假值。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。,11-B Quantifiers,Two types of quantification will be discussed here, namely,

34、 universal quantification and existential quantification .,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.,许多数学语句认为，性质对论域这个特殊领域内的变量的所有值都成立。,Such a statement is expressed usi

35、ng a universal quantification.,这样的语句可用全称量词化表示。,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.,命题函数的全称量词化是一个命题，认为P(x)

36、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.,本小节重点掌握,本节要讨论由这种语句生成命题的方法。,The ways that propositions can be produced from such statements will be discussed in this section.,New Words & Expressions event 事件 certain event 必然事件 sample 样本 discernible 可识别的 population 总体 mathematical statistics 数理统计 probability theory概率

37、论 highlight 强调、凸显 sampling unit 样本单位 trial 实验、试用,2.12 概率论与数理统计 Probability Theory and Mathematical Statistics,New Words & Expressions(P107, 二) binomial 二项式 polynomial 多项式 combinatorial analysis 组合分析 commensurability 均匀 deviation 偏差 mean/standard deviation 平均偏差/标准差distribution 分布 normal distribution

38、正态分布 enumeration 枚举、计数,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.”,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。,Expressions of this sort have intui

39、tive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions.,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

40、. 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. 根据概率论实际应用的方式，把每一个概率空间(S, B, P)想象成对应于一个实际的或者概念上的试验是很方便的。,The universal set S can then be th

41、ought of as the collection of all conceivable outcomes of the experiment, as in the example of coin tossing discussed in the foregoing section. 全集S是试验中所有可能结果的集体，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 eve

42、nts. 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。,Assume we have a probability space (S, B, P ) associated with an experiment. Let A be an event, and suppose the experiment is performed and that its o

43、utcome is x. (In other words, let x be a point of S.),假设有一个对应于某一个试验的概率空间(S,B,P) 。A是一个事件，假设试验已经完成，结果是x(换句话说，x是S中的一个点)。,This outcome x may or may not belong to the set A. If it does, we say that the event A has occurred.,结果x可能属于集合A，也可能不属于A。如果属于，则称事件A发生。,否则，称事件A不发生，那么余事件发生。,如果A等于空集，事件A称为不可能事件，因为在这种情况下试

44、验的任何结果都不是A中的元素。,Otherwise, we say that the event A has not occurred, in which case , so the complementary event has occurred.,An event A is called impossible if , because in this case no outcome of the experiment can be an element of A.,The event A is said to be certain if A=S, because then every ou

45、tcome is automatically an element of A.,如果A=S，则称事件A是必然事件，因为每一个结果必然是A中的元素。,Each event A has a probability P(A) assigned to it by the probability function P.,每一个事件A都通过概率函数P被赋予一个概率P(A)。,The number P(A) is also called the probability that an outcome of the experiment is one of the elements of A. 数P(A)又称

46、为试验的结果，是A的一个元素的概率。,We also say that P(A) is the probability that the event A occurs when the experiments is performed. 也称P(A)是试验完成时事件A出现的概率。,The impossible event must be assigned probability zero because P is a finitely additive measure. However, there may be events with probability zero that are no

47、t impossible. 因为P是有限可加测度，所以不可能事件被赋予零概率。然而，也存在具有零概率的事件，但它并不是不可能事件。,In other words, some of the nonempty subsets of S may be assigned probability zero. The certain event S must be assigned probability 1 by the very definition of probability, but there may be other subsets as well that are assigned pro

48、bability 1. 换句话说，S的某个非空子集也可能被赋予零概率。仅根据概率的定义就得把必然事件S的概率指定为1，但是可能有别的子集其概率也是1.,Two events A and B are said to be equally likely if P(A)=P(B). The event A is called more likely than B if P(A)P(B), and at least as likely as B if P(A) P(B). 如果事件P(A)=P(B)，则A与B被称为同等可能。如果P(A)P(B), 则称事件A比事件B有更大的可能性，如果P(A) P(B

49、), 则称事件A至少和事件B的可能性一样大。 P101, Table 2-12-1. Glossary of Probability Terms,本小节重点掌握,在试验中事件A出现的概率记做P(A).,The probability that the event A occurs in an experiment is denoted by P(A).,作业：,P95 2: (2) P105 2: (4),美国语言文化,What do you call someone who speaks three languages? Trilingual. What do you call someon

50、e who speaks two languages? Bilingual. What do you call someone who speaks one language? An American. 你如何称呼能说三种语言的人？ Trilingual/说三国语言的人。能讲两种语言的人又如何称呼？ Bilingual/说两国语言的人。那么，只会一种语言的人呢？美国人。,To people in many countries, being bilingual or even trilingual is a way of life. But since the mother tongue of

51、most Americans is English-a language widely spoken around the world-they often dont feel the need to learn a foreign language. Moreover, people who live in the heartland of America have little contact with other linguistic groups, making foreign language skills irrelevant. 对许多国家的人而言，说两种或三种语言，是家常便饭。但

52、在美国，由于大多数人的母语为英语-一个世界上广泛通用的语言-因此美国人都不觉得有必要学习外语。此外，居住于美国大陆内地的人，较少与其它语言族群接触，因此外语能力更显得无关紧要。,Actually, though, this land of immigrants has always had people of many different nationalities-and languages. The 1990 census indicates that almost 14% of Americans speak a non-English language at home. Yet onl

53、y 3% reported that they spoke English not well or not at all. That means that slightly more than one out of 10 Americans could be considered bilingual. Besides that, many high school and college students-and even some elementary school students-are required to take a foreign language as a part of th

54、eir curriculum. In addition to old standbys like Spanish, German and French, more and more students are opting for Eastern European and Asian languages. Of course, not all students keep up their foreign language abilities. As the old saying goes, If you dont use it, you lose it. But still, a growing

55、 number of Americans are coming to appreciate the benefits of being multilingual.,事实上，在美国这块移民之地，一直住着来自不同国籍、说不同语言的人口。一九九年的人口普查显示，几乎百分之十四的美国人，在家里不讲英语。只有百分之三的人宣称他们英语说得不好或根本不会说。这表示十分之一强的美国人可视为说两种语言的。此外，许多中学生、大学生，甚至小学生在校的学科都必须加修一门外语。除了西班牙文、德文与法文等常选的第二语言外，愈来愈多的学生选修东欧和亚洲语言。当然，并非所有学生都能够学以致用。套句老话：不用就会丢掉。尽管如此

56、，愈来愈多的美国人已经开始意识到会说多种语言的好处。,Ethnic enclaves, found particularly in major metropolitan centers, have preserved the language and culture of American immigrants. Some local residents can function quite well in their native language, without having to bother learning English. Regions such as southern Flo

57、rida and the Southwest have numerous Spanish-speaking neighborhoods. In fact, Spanish speakers-numbering over 17 million-compose the largest non-English linguistic group in America. But Chinese, Vietnamese, Italian, Polish and many other ethnic groups add to the linguistic flavor of America. Foreign

58、 languages are so commonly used in some ethnic neighborhoods that visitors might think they are in another country!,在主要大都市里，有许多族群居住的社区，都还保存着移民的文化传统和语言。有些当地人在生活上只讲母语就够了，根本不需要学英语。佛罗里达州南部以及美国西南部等区，有许多说西班牙语的社区。事实上，说西班牙语的人口超过一千七百万，是美国最大的非英语系族群。不过，还有中国、越南、意大利、波兰与其它很多种族的族群，更使美国语言增色不少。在许多外来民族的后裔社区里，由于外语的广泛使

59、用，甚至会使访客有身置异国的感觉。,Although some Americans welcome this linguistic and cultural diversity, others have begun to fear that the English language is being threatened. Since the 1980s, the English Only movement has sought to promote legislation which would establish English as the official language and restrict the use of non-English languages. However, some groups, including TESOL, the organization for Teachers of English to Speakers of Other Languages, object to such language restrictionism. Th