第二级,第三级,第四级,第五级,Mathematical English,New Words & Expressions:,approximate evaluation 近似估计 initial 初始的,disintegrate 解体，衰变 integrate 求积分,differentiable 可微的 polynomial 多项式,exponential 指数的 rational function 有理函数,2.9 微分方程简介,Introduction to Differential Equations,New Words & Expressions:2.9 微分,New Words & Expressions:,differential equation 微分方程,partial differential equation 偏微分方程,inverse trigonometric function 反三角函数,approximate evaluation 近似估计,initial value problem 初值问题,initial condition 初始条件,mathematical physics 数学物理,boundary condition 边界条件,2.9 Introduction to Differential Equations,New Words & Expressions:2.9 In,Key points:,Introduction to Differential Equations,Difficult points:,Applications of matrices,Key points:,Requirements:,1. 理解微分方程的分类。,2. 理解矩阵学习的重要性。,Requirements:,A large variety of scientific problems arise in which one tries to determine something from its,rate of change,.,9-A,Introduction,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。,For example , we could try to compute the position of a moving,particle,from a knowledge of its,velocity,or,acceleration,.,例如，我们可以由已知速度或者加速度来计算移动质点的位置.,A large variety of scientific,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.,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。,Or a radioactive substance may,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,derivative,s of the unknown function,.,在类似的例子中，我们力求由方程的形式表述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。,In examples like these, we ar,These equations are called,differential equation,s, and their study forms one of the most challenging branches of mathematics.,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个分支。,Differential equations are classified,under,two,main headings,: ordinary and partial, depending on whether the unknown is a function of just one variable or of two or more variables.,微分方程根据未知量是单变量函数还是多变量函数分成两个主题：常微分方程和偏微分方程。,These equations are called dif,A,simple,example of an,ordinary differential equation,is the relation,f,(,x,)=,f,(,x,) (9.1),which is satisfied, in particular by the,exponential function,f,(,x,)=e,x .,常微分方程的一个简单例子是,f,(,x,)=,f,(,x,) ，特别地，指数函数,f,(,x,)=e,x,满足这个等式。,We shall see presently that every solution of (9.1) must be,of the form,f,(,x,)=,C,e,x, where,C,may be any constant.,我们马上就会发现(9.1)的每一个解都一定是,f,(,x,)=,C,e,x,这种形式，这里,C,可以是任何常数。,A simple example of an ordinar,On the other hand, an equation like,is an example of a,partial differential equation,.,另一方面，如下方程是偏微分方程的一个例子。,This particular one, is called,Laplaces equation, appears in the theory of,electricity and magnetism,fluid mechanics, and elsewhere.,这个特殊的方程叫做拉普拉斯方程，出现于电磁学理论、流体力学理论以及其他理论中。,On the other hand, an equation,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.,微分方程起源于17世纪，当时牛顿，莱布尼茨，伯努利家族解决了一些来自几何和力学的简单的微分方程。,The study of differential equa,These early discoveries, beginning about 1690, gradually,led to,the development of a lot of “special,trick,s” for solving certain special kinds of differential equation.,开始于1690年的早期发现，逐渐导致了解某些特殊类型的微分方程的大量特殊技巧的发展。,These early discoveries, begin,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,.,尽管这些特殊的技巧只是适用于相对较少的几种情况，但他们能够解决许多出现于力学和几何中的微分方程，因此，他们的研究具有重要的实际应用。,Although these special tricks,Some of these special methods and some of the problems,which,they help us solve are discussed near the end of this chapter.,这些特殊的技巧和利用这些技巧可以解决的一些问题将在本章最后讨论。,Some of these special methods,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。,Experience has shown that it i,The simplest types of,linear differential equations,and some of their applications are also discussed in this,introductory,chapter. A more thorough study of linear equations is,carried out,in Volume .,线性微分方程的最简单类型及其应用也会在介绍性的本章中讨论。线性方程的深入研究将在第二卷中进行。,The simplest types of linear d,When we work with a differential equation such as (9.1), it is,customary,to write y,in place of,f,(,x,),and y,in place of,f,(,x,),the higher derivatives being denoted by y, y, etc.,9-B,When we work with a differenti,Of course, other letters such as,u,v,z, etc. are also used instead of,y,.,当然，其他字母，比如,u,，,v,，,z,等，也用来代替,y,。,By,the order of an equation,is meant,the order of the,highest,derivative which appears.,方程的阶指的是出现在其中的最高阶导数的阶数。,By,A,is meant,B.,A的含义为B; A即B; A指的是B 。,Of course, other letters such,For example, (9.1) is a,first-order equation,which may be written as,y,=y,. The differential equation,y,=,x,3,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,=,x,3,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 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。,For example, (9.1) is a first-,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, Y,x, 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, Y,x,。,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)。,where the expression f (x, y),say,where Q,is assumed to 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。,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。,say, where Q is assumed to be,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。,where C is any constant (usual,If it is not possible to evaluate the integral in (9.4),in terms of,familiar functions, such as,polynomial,s, rational functions,trigonometric,and,inverse trigonometric functions,logarithm,s, and,exponential,s , 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 information 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。,If it is not possible to evalu,EXAMPLE.,Linear motion determined from the velocity. Suppose a,particle,moves along a straight line in such a way that its velocity 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 that 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.,EXAMPLE. Linear motion determi,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 the 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.,This is all we can deduce abou,In some respects the example just solved is,typical of,what happens,in general,.,Some-where in the process of solving 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。,In some respects the example j,The way in which the arbitrary constant C,enters into,the solution will depend on the,nature,of the given differential 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=Ce,x,.,The way in which the arbitrary,In many problems it is necessary to select from the collection 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 value represents the,displacement,at some initial time.,In many problems it is necessa,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.,本小节重点掌握A differential equation,New Words & Expressions,component 分量，成分 matrix 矩阵,dependent 相关的 independent 无关的,finite 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 & Expressions2.10 线性,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 reducible 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,New Words & Expressions(P90 生词,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,.,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,In recent years the applicatio,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.,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,.,Matrix methods are used to sol,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,.,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。,Recent developments in mathema,No matter what the students field of,major,interest is , knowledge of 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。,No matter what the students f,Descartes, the inventor of,analytic geometry,and one of the,founder,s 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。,Descartes, the inventor of ana,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.,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。,Although this belief is now kn,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.,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。,Many of the applications, requ,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,.,(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。,From the above discussion, we,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。,It is therefore not surprising,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 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。,Not only are matrix methods u,作业：,P78: 1, 2(1),P87: 1, 2(1),作业： P78:,New Words & Expressions:,assert 断言，主张 predicate 谓词,conjunction 合取 quantifier 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,New Words & Expressions:2.11 数,Key points:,introduction to predicates and quantifiers,Difficult points:,special terminology peculiar to probability theory,Key points:,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。,Requirements:,Statement,s involving variables, such as,“,x,3”, “,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“,x,3”, “,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 will discuss the ways that propositions can be,produce,d 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。,Statements involving variables,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, “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。,The statement “x is greater th,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,) 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。,We can denote the statement “x,When 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.,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,When all the variables in a pr,Two types of quantification will be discussed here, namely,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。,Two types of quantification wi,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 using a universal quantification.,这样的语句可用全称量词化表示。,Many mathematical statements,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,)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.,The universal quantification o,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.,本小节重点掌握The ways that propositi,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 & Expressions2.12 概率,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 正态分布,enumeration,枚举、计数,New Words & Expressions(P107,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。,In discussions involving proba,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.,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。,Expressions of this sort have,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。,Because of the way probabilit,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.,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。,The universal set S can then b,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,event,s. 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。,Each element of S is called an,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,outcome,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,occurre,d.,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。,Assume we have a probability s,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验的任何结果都不是,A,中的元素。,Otherwise