数学专业英语(吴炯圻-第2版)2-6课件

1、New Words 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 single concept played so important a role in mathematics as has the concept of function. It is

2、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 physical quantities were the forerunners of mathematical variables, and relation among them

3、 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 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

4、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 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 do

5、ne 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的描述方式来建立。,A typical example is the room temperature, which obviously is a function

6、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图标形式来追踪。,The modern definition of a function y of x is simply a mapping from a sp

7、ace 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 therefore desirable to serve as a basis of the function concept. Moreover, as the space concept is incorporated in this modern definition, its generality contributes much to the g