数学专业英语(第2版)26课件

1、New Words & Expressions:alphabet 字母表 prime 素数，质数displacement 位移 proportional 成比例的domain 定义域 the real-valued function实值函数edge 棱，边 spring constant 弹性系数 graph 图，图形 limit 极限 stretch 拉伸 volume 体积，容积，卷 2.6 函数的概念与函数思想Function concept and function ideaNew Words & Expressions(二)P59:measurable 可测的 mapping 映射V

2、arious fields of human have to do with relationships that exist between one collection of objects and another. 6-A Informal description of functions各行各业的人们都必须处理一类事物与另一类事物之间存在的关系。Graphs, charts, curves, tables, formulas, and Gallup polls are familiar to everyone who reads the newspapers. 几乎每个人都熟悉图形，图

3、表，曲线，公式和盖洛普民意测验。These are merely devices for describing special relations in a quantitative fashion. Mathematicians refer to certain types of these relations as functions. 这些只是以定量的方式描述特定关系的方法。数学家将这些关系中的某些类型视作函数。In this section, we give an informal description of the function concept. A formal defini

4、tion is given in Section 3. 在本节中，我们给出一个非正式的描述函数的概念。在第3节给出一个正式的定义 。EXAMPLE 1. The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x. That is, F=cx, where c is a number independent of x called the spring constant.把一个钢制的弹簧拉伸到超过其自然长度的距离为x时所需要的力F与x成正比

5、。即，F=cx,这里c是不依赖与x的数，叫做弹性系数。This formula, discovered by Robert Hooke in the mid-17th century, is called Hookes law, and it is said to express the force as a function of the displacement.这个公式是在17世纪中叶被胡克发现的，叫做胡克定律，它用来表示力关于位移的函数。EXAMPLE 2. The volume of a cube is a function of its edge-length. If the ed

6、ges have length x, the volume V is given by the formula V=x3.立方体的体积是它棱长的函数。如果棱长为x，那么体积的公式为： V=x3。EXAMPLE 3. A prime is any integer n1 that cannot be expressed in the form n=ab, where a and b are positive integers, both less than n. The first few primes are 2,3,5,7,11,13,17,19. 素数是大于1且不能表示成n=ab形式的整数，

7、这里a和b都是小于n的正整数。开始的几个素数是2,3,5,7,11,13,17,19.For a given real number x0, it is possible to count the number of primes less than or equal to x. This number is said to be a function of x even though no simple algebraic formula is known for computing it (without counting) when x is known.对于一个给定的实数x0，数出小于

8、或者等于x的素数的个数是有可能的。这个数称为x的函数，尽管还没有一个简单代数式可以由已知的x计算(不通过计数求)出它的值。The word “function” was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas. “函数”这个词是由莱布尼茨引入到数学中的，他主要使用这个术语来指代某种数学公式。It was later realized that Leibnizs idea of function wa

9、s much too limited in its scope, and the meaning of the word has since undergone many stages of generalization.后来人们才认识到，莱布尼茨的函数思想适用的范围太过局限了，这个术语的含义从那时起已经过了多次推广。Today, the meaning of function is essentially this: Given two sets, say X and Y, a function is a correspondence which associates with each e

10、lement of X one and one only element of Y. 如今，从本质上讲，函数的定义如下：给定两个集合X 和Y，函数是X中元素与Y中元素的一一对应。The set X is called the domain of the function. Those elements of Y associated with the elements in X form a set called the range of the function. (This may be all of Y, but it need not be)集合X叫做函数的定义域，与X中的元素相对应的

11、Y中的元素的集合叫做函数的值域。(值域可能是整个集合Y,也可能不是。)Letters of the English and Greek alphabets are often used to denote functions. The particular letters f,g,h,F,G,H, and are frequently used for this purpose.英语字母和希腊字母表通常用于表示函数。为此，一些特定的字母如：f,g,h,频繁使用。If f is a given function and if x is an object of its domain, the n

12、otation f(x) is used to designate that object in the range which is associated to x by the function f; and it is called the value of f at x or the image of x under f. The symbol f(x) is read as “f of x.”如果f是一个给定的函数，x是它定义域中的一个点，符号f(x)表示值域中按照f对应于x的点，它叫做f在x点的值或者x在f下的像。符号f(x)读作“f of x.”Seldom has a sing

13、le concept played so important a role in mathematics as has the concept of function. It is desirable to know how the concept has developed. 6-C The concept of function在数学中，很少有个概念象函数的概念那样，起那么重要的作用的。因此，需要知道这个概念是如何发展起来的。This concept, like many others, originates in physics. 这个概念像许多其他概念一样，起源于物理学。The phy

14、sical quantities were the forerunners of mathematical variables, and relation among them was called a function relation in the later 16th century. 物理量是数学变量的先驱，他们之间的关系在16世纪后期称为函数关系。For example, the formula s=16t2 for the number of feet s a body falls in any number of seconds t is a function relation

15、between s and t, it describes the way s varies with t. 例如, 代表一物体在若干秒t中下落若干英尺s的公式s=16t2 就是s和t之间的函数关系, 它描述了s随t 变化的公式。The study of such relations led people in the 18th century to think of a function relation as nothing but a formula. 对这种关系的研究导致了18世纪的人们认为函数关系只不过是一个公式罢了。Not specified by this definition

16、is the manner of setting up the correspondence. 至于如何建立这种对应关系，这个定义并未详细规定。It may be done by a formula as the 18th century mathematics presumed but it can equally well be done by a tabulation such as a statistical chart, or by some other form of description.可以如18世纪的数学所假定的那样，用公式来建立，但同样也可以用统计表那样的表格或用某种其他

17、的描述方式来建立。A typical example is the room temperature, which obviously is a function of time. But this function admits of no formula representation, although it can be recorded in a tabular form or traced out graphically by an automatic device.典型的例子是室温，这显然是一个时间的函数。但是这个函数不能用公式来表示，不过可以用表格的形式来记录或者用一种自动装置以

18、图标形式来追踪。The modern definition of a function y of x is simply a mapping from a space X to another space Y. A mapping is defined when every point x of X has a definite image y, a point of Y. 现代给x的一个函数y所下的定义只是从一个空间X到另一个空间Y的映射。当X空间的每一个点x有一个确定的像点y，即Y空间的一点，那么映射就确定了。The mapping concept is close to intuition, and therefor