數(shù)學(xué)專業(yè)英語(yǔ)吳炯圻第學(xué)習(xí)教案

1、數(shù)學(xué)數(shù)學(xué)(shxu)專業(yè)英語(yǔ)吳炯圻第專業(yè)英語(yǔ)吳炯圻第第一頁(yè)，共21頁(yè)。New Words & Expressions(二二)P59:measurable 可測(cè)的可測(cè)的 mapping 映射映射(yngsh)第1頁(yè)/共21頁(yè)第二頁(yè)，共21頁(yè)。Various fields of human have to do with relationships that exist between one collection of objects and another. 6-A Informal description of functions各行各業(yè)各行各業(yè)( xn y)的人們都必須處理一類事物與另一類

2、的人們都必須處理一類事物與另一類事物之間存在的關(guān)系。事物之間存在的關(guān)系。Graphs, charts, curves, tables, formulas, and Gallup polls are familiar to everyone who reads the newspapers. 幾乎每個(gè)人都熟悉圖形，圖表幾乎每個(gè)人都熟悉圖形，圖表(tbio)，曲線，公式和蓋，曲線，公式和蓋洛普民意測(cè)驗(yàn)。洛普民意測(cè)驗(yàn)。第2頁(yè)/共21頁(yè)第三頁(yè)，共21頁(yè)。These are merely devices for describing special relations in a quantitative

3、 fashion. Mathematicians refer to certain types of these relations as functions. 這些只是以定量的方式描述特定關(guān)系的方法。數(shù)學(xué)家將這些只是以定量的方式描述特定關(guān)系的方法。數(shù)學(xué)家將這些關(guān)系中的某些類型這些關(guān)系中的某些類型(lixng)視作函數(shù)。視作函數(shù)。In this section, we give an informal description of the function concept. A formal definition is given in Section 3. 在本節(jié)中，我們?cè)诒竟?jié)中，我們(w

4、men)給出一個(gè)非正式的描述函數(shù)的概念。給出一個(gè)非正式的描述函數(shù)的概念。在第在第3節(jié)給出一個(gè)正式的定義節(jié)給出一個(gè)正式的定義 。第3頁(yè)/共21頁(yè)第四頁(yè)，共21頁(yè)。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.把一個(gè)鋼制的彈簧拉伸到超過(guò)其自然長(zhǎng)度的距離

5、把一個(gè)鋼制的彈簧拉伸到超過(guò)其自然長(zhǎng)度的距離(jl)為為x時(shí)所需要的力時(shí)所需要的力F與與x成正比。即，成正比。即，F(xiàn)=cx,這里這里c是不是不依賴與依賴與x的數(shù)，叫做彈性系數(shù)。的數(shù)，叫做彈性系數(shù)。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.這個(gè)公式是在這個(gè)公式是在17世紀(jì)中葉被胡克世紀(jì)中葉被胡克(h k)發(fā)現(xiàn)的，叫做胡克發(fā)現(xiàn)的，

6、叫做胡克(h k)定律，它用來(lái)表示力關(guān)于位移的函數(shù)。定律，它用來(lái)表示力關(guān)于位移的函數(shù)。第4頁(yè)/共21頁(yè)第五頁(yè)，共21頁(yè)。EXAMPLE 2. The volume of a cube is a function of its edge-length. If the edges have length x, the volume V is given by the formula V=x3.立方體的體積立方體的體積(tj)是它棱長(zhǎng)的函數(shù)。如果棱長(zhǎng)為是它棱長(zhǎng)的函數(shù)。如果棱長(zhǎng)為x，那么體積那么體積(tj)的公式為：的公式為： V=x3。第5頁(yè)/共21頁(yè)第六頁(yè)，共21頁(yè)。EXAMPLE 3. A pr

7、ime 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?cái)?shù)是大于素?cái)?shù)是大于1且不能表示且不能表示(biosh)成成n=ab形式的整數(shù)，這形式的整數(shù)，這里里a和和b都是小于都是小于n的正整數(shù)。開(kāi)始的幾個(gè)素?cái)?shù)是的正整數(shù)。開(kāi)始的幾個(gè)素?cái)?shù)是2,3,5,7,11,13,17,19.第6頁(yè)/共21頁(yè)第七頁(yè)，共21頁(yè)。For a gi

8、ven 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.對(duì)于一個(gè)給定的實(shí)數(shù)對(duì)于一個(gè)給定的實(shí)數(shù)x0，數(shù)出小于或者等于，數(shù)出小于或者等于x的素?cái)?shù)的素?cái)?shù)的個(gè)數(shù)是有可能的。這個(gè)的個(gè)數(shù)是有可能的。這個(gè)

9、(zh ge)數(shù)稱為數(shù)稱為x的函數(shù)，盡管的函數(shù)，盡管還沒(méi)有一個(gè)簡(jiǎn)單代數(shù)式可以由已知的還沒(méi)有一個(gè)簡(jiǎn)單代數(shù)式可以由已知的x計(jì)算計(jì)算(不通過(guò)計(jì)數(shù)不通過(guò)計(jì)數(shù)求求)出它的值。出它的值。第7頁(yè)/共21頁(yè)第八頁(yè)，共21頁(yè)。The word “function” was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas. “函數(shù)函數(shù)”這個(gè)詞是由萊布尼茨引入到數(shù)學(xué)這個(gè)詞是由萊布尼茨引入到數(shù)學(xué)(shxu)中的，他主中的，他主要使

10、用這個(gè)術(shù)語(yǔ)來(lái)指代某種數(shù)學(xué)要使用這個(gè)術(shù)語(yǔ)來(lái)指代某種數(shù)學(xué)(shxu)公式。公式。It was later realized that Leibnizs idea of function was much too limited in its scope, and the meaning of the word has since undergone many stages of generalization.后來(lái)人們才認(rèn)識(shí)到，萊布尼茨的函數(shù)思想適用后來(lái)人們才認(rèn)識(shí)到，萊布尼茨的函數(shù)思想適用(shyng)的范圍太過(guò)局限了，這個(gè)術(shù)語(yǔ)的含義從那時(shí)起已經(jīng)過(guò)的范圍太過(guò)局限了，這個(gè)術(shù)語(yǔ)的含義從那時(shí)起已經(jīng)過(guò)了多次

11、推廣。了多次推廣。第8頁(yè)/共21頁(yè)第九頁(yè)，共21頁(yè)。Today, the meaning of function is essentially this: Given two sets, say X and Y, a function is a correspondence which associates with each element of X one and one only element of Y. 如今，從本質(zhì)上講，函數(shù)的定義如下：給定兩個(gè)如今，從本質(zhì)上講，函數(shù)的定義如下：給定兩個(gè)(lin )集合集合X 和和Y，函數(shù)是，函數(shù)是X中元素與中元素與Y中元素的中元素的一一對(duì)應(yīng)。一一對(duì)

12、應(yīng)。第9頁(yè)/共21頁(yè)第十頁(yè)，共21頁(yè)。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)集合集合X叫做函數(shù)的定義域，與叫做函數(shù)的定義域，與X中的元素相對(duì)應(yīng)的中的元素相對(duì)應(yīng)的Y中的元中的元素的集合叫做函數(shù)的值域。素的集合叫做函數(shù)的值域。(值域可能是整個(gè)值域可能是整個(gè)(zhn

13、gg)集集合合Y,也可能不是。也可能不是。)第10頁(yè)/共21頁(yè)第十一頁(yè)，共21頁(yè)。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.英語(yǔ)字母和希臘字母表通常用于表示函數(shù)。為此，英語(yǔ)字母和希臘字母表通常用于表示函數(shù)。為此，一些特定一些特定(tdng)的字母如：的字母如：f,g,h,頻繁使用。頻繁使用。第11頁(yè)/共21頁(yè)第十二頁(yè)，共21頁(yè)。

14、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.”如果如果f是一個(gè)給定是一個(gè)給定(i dn)的函數(shù)，的函數(shù)，x

15、是它定義域中的一個(gè)是它定義域中的一個(gè)點(diǎn)，符號(hào)點(diǎn)，符號(hào)f(x)表示值域中按照表示值域中按照f(shuō)對(duì)應(yīng)于對(duì)應(yīng)于x的點(diǎn)，它叫做的點(diǎn)，它叫做f在在x點(diǎn)點(diǎn)的值或者的值或者x在在f下的像。符號(hào)下的像。符號(hào)f(x)讀作讀作“f of x.”第12頁(yè)/共21頁(yè)第十三頁(yè)，共21頁(yè)。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 conce

16、pt of function在數(shù)學(xué)中，很少有個(gè)概念象函數(shù)的概念那樣，起那在數(shù)學(xué)中，很少有個(gè)概念象函數(shù)的概念那樣，起那么重要的作用的。因此么重要的作用的。因此(ync)，需要知道這個(gè)概念是，需要知道這個(gè)概念是如何發(fā)展起來(lái)的。如何發(fā)展起來(lái)的。This concept, like many others, originates in physics. 這個(gè)概念這個(gè)概念(ginin)像許多其他概念像許多其他概念(ginin)一樣，起源于物一樣，起源于物理學(xué)。理學(xué)。第13頁(yè)/共21頁(yè)第十四頁(yè)，共21頁(yè)。The physical quantities were the forerunners of mat

17、hematical variables, and relation among them was called a function relation in the later 16th century. 物理量是數(shù)學(xué)物理量是數(shù)學(xué)(shxu)變量的先驅(qū)，他們之間的關(guān)系在變量的先驅(qū)，他們之間的關(guān)系在16世紀(jì)后期稱為函數(shù)關(guān)系。世紀(jì)后期稱為函數(shù)關(guān)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

18、 s and t, it describes the way s varies with t. 例如例如, 代表一物體在若干秒代表一物體在若干秒t中下落若干英尺中下落若干英尺s的公式的公式(gngsh)s=16t2 就是就是s和和t之間的函數(shù)關(guān)系之間的函數(shù)關(guān)系, 它描述了它描述了s隨隨t 變變化的公式化的公式(gngsh)。第14頁(yè)/共21頁(yè)第十五頁(yè)，共21頁(yè)。The study of such relations led people in the 18th century to think of a function relation as nothing but a formula. 對(duì)

19、這種關(guān)系的研究導(dǎo)致了對(duì)這種關(guān)系的研究導(dǎo)致了18世紀(jì)的人們認(rèn)為函數(shù)關(guān)系只世紀(jì)的人們認(rèn)為函數(shù)關(guān)系只不過(guò)是一個(gè)不過(guò)是一個(gè)(y )公式罷了。公式罷了。Not specified by this definition is the manner of setting up the correspondence. 至于如何建立這種對(duì)應(yīng)關(guān)系，這個(gè)至于如何建立這種對(duì)應(yīng)關(guān)系，這個(gè)(zh ge)定義并未詳定義并未詳細(xì)規(guī)定。細(xì)規(guī)定。第15頁(yè)/共21頁(yè)第十六頁(yè)，共21頁(yè)。It may be done by a formula as the 18th century mathematics presumed but i

20、t can equally well be done by a tabulation such as a statistical chart, or by some other form of description.可以如可以如18世紀(jì)世紀(jì)(shj)的數(shù)學(xué)所假定的那樣，用公式來(lái)的數(shù)學(xué)所假定的那樣，用公式來(lái)建立，但同樣也可以用統(tǒng)計(jì)表那樣的表格或用某種建立，但同樣也可以用統(tǒng)計(jì)表那樣的表格或用某種其他的描述方式來(lái)建立。其他的描述方式來(lái)建立。第16頁(yè)/共21頁(yè)第十七頁(yè)，共21頁(yè)。A typical example is the room temperature, which obviously is

21、 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.典型的例子是室溫，這顯然是一個(gè)時(shí)間的函數(shù)。但典型的例子是室溫，這顯然是一個(gè)時(shí)間的函數(shù)。但是這個(gè)函數(shù)不能用公式來(lái)表示，不過(guò)可以用表格的是這個(gè)函數(shù)不能用公式來(lái)表示，不過(guò)可以用表格的形式來(lái)記錄形式來(lái)記錄(jl)或者用一種自動(dòng)裝置以圖標(biāo)形式來(lái)或者用一種自動(dòng)裝置以圖標(biāo)形式來(lái)追蹤。追蹤。第17頁(yè)/共21頁(yè)第十八頁(yè)，共21頁(yè)。The modern definition of a function y of x is simply a mapping from a space X to another