数学专业英语(吴炯圻)(5)

function. It is desirable to know how the concept has developed.

第二版 课文翻译及课后习题

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。

6-C The concept of function

11

This concept, like many others ,originates in physics. The physical quantities were the forerunners

of mathematical variables. And relation among them was called a function relation in the later

16th century.

这个概念像许多其他概念一样，起源于物理学。物理的量是数学的变量的先驱，他们之间的

关系在16世纪后期称为函数关系。

6-C The concept of function

12

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. the study of such

relations led people in the 18th century to think of a function relation as nothing but a formula.

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，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。

6-C The concept of function

13

Only after the rise of modern analysis in the early 19th century could the concept of function be

extended. In the extended sense , a function may be defined as follows: if a variable y depends

on another variable x in such a way that to each value of x corresponds a definite value of y, then

y is a function of x. this definition serves many a practical purpose even today.

只有在19世纪初期现代分析出现以后，函数的概念才得以扩大。在扩大的意义上讲，函数

可定义如下：如果一变量y随着另一个变量x而变换，即x的每一个值都和y的一定值相对

应，那么，y就是x的函数。这个定义甚至在今天还适用于许多实际的用途。

6-C The concept of function

14

Not specified by this definition is the manner of setting up the correspondence. 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.

至于如何建立这种对应关系，这个定义并未详细规定。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。

6-C The concept of function

15

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 but 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代表，但可以用

第二版 课文翻译及课后习题

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。

6-C The concept of function

16

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 definition image y, a point of Y. 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 generality of the function concept.

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。这个映射概念接近

于直观，因此，很可能作为函数概念的一个基础。此外，由于这个现代的定义中体现了空间

的概念，所以，它的普遍性对函数概念的普遍性有很大的贡献。

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 idea

1

New Words & Expressions(二)P59:

admit 准许 mapping 映射

forerunner 先行者 presume 假定

incorporate 并入，结合 trace 追踪

2

Various 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，图表，曲线，公式和盖洛普民意测验。

3

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 definition is

given in Section 3.

第二版 课文翻译及课后习题

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。

4

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。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 Hooke’s 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。

5

EXAMPLE 2. The volume of a cube is a …… 此处隐藏：4278字，全部文档内容请下载后查看。喜欢就下载吧 ……