数学专业英语(吴炯圻)(6)

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。

12

If f is a given function and if x is an object of its domain, the notation 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.”

13

Seldom has a single 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。

14

The physical 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 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。

15

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。

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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

用某种其他的描述方式来建立。

17

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图标形式来追踪。

第二版 课文翻译及课后习题

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。

19

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.

这个映射概念接近于直观，因此，值得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，它的普及性对函数概念的普及性有很大的贡献。

New Words & Expressions:

assume 假定 sequence 序列，数列

converge 收敛 series 级数，序列

diverge 发散 subscript 下标

imaginary part 虚部 succession 连贯性 imply 蕴含，推出 successor 后继

recursion formula 递推公式

2.7 序列及其极限

Sequences and Their Limits

1

Key points:

the definition of sequences

Difficult points:

some relevant terms

2

In everyday usage of the English language, the words “sequence” and “series” are synonyms, and

they are used to suggest a succession of things or events arranged in some order.

7-A The definition of sequences

在日常英语中，单词“sequence”和“series”是同义词，用以表示按某种次序排列的一串

东西或事件。

In mathematics these words have special technical meanings.

在数学中这些单词有特殊的专业含义。

3

The word “sequence” is employed as in the common use of the term to convey the idea of a set

of things arranged in order, but the word “series” is used in a somewhat different sense.

像通常用法一样，术语“sequence”用以表达按次序排列的一串东西的意思，但是“series”

一词则用于表示别的意思（级数）。

The concept of a sequence will be discussed in this section, and series will be defined in section

11.

序列的概念在本节讨论，级数的概念将在第11节定义。

第二版 课文翻译及课后习题

4

If for every positive integer n there is associated a real or complex number an, then the ordered set a1 , a2, …, an ,… is said to define an infinite sequence.

如果对每一个正整数 n都有一个实数或复数an与之对应, 则有序集a1 , a2, , an , 称为一个无穷序列.

The important thing here is that each member of the set has been labeled with an integer so that we may speak of the first term a1, the second term a2, and, in general, the nth term an. 这里重要的是集合中的每一个元素都由一个整数标记，因此我们可以说第一项 , 第二项, 一般的，第n项

5

Each term has a successor and hence there is no “last ” term.

每一项都有下面的一项，因此没有最后一项。

The most common examples of sequences can be constructed if we give some rule or formula for describing the nth term.

如果我们给出描述第n项的规则或者公式，就可以构造出序列的常见例子。

6

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