New Words & Expressions: differentiable 可微的 polynomial 多项式exponential 指数的 rational function 有理函数 integrate 求积分 trigonometric function 三角函数 logarithm 对数 inverse trigonometric function 反三角函数,2.9 微分方程简介 Introduction to Differential Equations,Key points: Introduction to Differential Equations Difficult points: Applications of matrices,Requirements:,1. 理解微分方程的分类。 2. 理解矩阵学习的重要性。,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.,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。,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 .,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。,These equations are called differential equations, 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 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。,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)=ex .,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)=ex 满足这个等式。,We shall see presently that every solution of (9.1) must be of the form f(x)=Cex , 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)=Cex这种形式，这里C可以是任何常数。,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。,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。,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.,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。,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。,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。,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(P90 生词与词组二): consistent 相容的 matrix 矩阵 column 列 reducible 可简化的 determinate 行列式 row 行 inverse 逆 simultaneous linear equations 联立线性方程组,2.10 线性空间中的相关与无关集 Dependent and Independent Sets in a Linear Space,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,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 .,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。,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