數(shù)學專業(yè)英語吳炯圻第學習教案

1、數(shù)學數(shù)學(shxu)專業(yè)英語吳炯圻第專業(yè)英語吳炯圻第第一頁，共21頁。New Words & Expressions(二二)P59:measurable 可測的可測的 mapping 映射映射(yngsh)第1頁/共21頁第二頁，共21頁。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. 幾乎每個人都熟悉圖形，圖表幾乎每個人都熟悉圖形，圖表(tbio)，曲線，公式和蓋，曲線，公式和蓋洛普民意測驗。洛普民意測驗。第2頁/共21頁第三頁，共21頁。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，我們在本節(jié)中，我們(w

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

5、把一個鋼制的彈簧拉伸到超過其自然長度的距離(jl)為為x時所需要的力時所需要的力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.這個公式是在這個公式是在17世紀中葉被胡克世紀中葉被胡克(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，它用來表示力關(guān)于位移的函數(shù)。定律，它用來表示力關(guān)于位移的函數(shù)。第4頁/共21頁第五頁，共21頁。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)是它棱長的函數(shù)。如果棱長為是它棱長的函數(shù)。如果棱長為x，那么體積那么體積(tj)的公式為：的公式為： V=x3。第5頁/共21頁第六頁，共21頁。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. 素數(shù)是大于素數(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)。開始的幾個素數(shù)是的正整數(shù)。開始的幾個素數(shù)是2,3,5,7,11,13,17,19.第6頁/共21頁第七頁，共21頁。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.對于一個給定的實數(shù)對于一個給定的實數(shù)x0，數(shù)出小于或者等于，數(shù)出小于或者等于x的素數(shù)的素數(shù)的個數(shù)是有可能的。這個的個數(shù)是有可能的。這個

9、(zh ge)數(shù)稱為數(shù)稱為x的函數(shù)，盡管的函數(shù)，盡管還沒有一個簡單代數(shù)式可以由已知的還沒有一個簡單代數(shù)式可以由已知的x計算計算(不通過計數(shù)不通過計數(shù)求求)出它的值。出它的值。第7頁/共21頁第八頁，共21頁。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)”這個詞是由萊布尼茨引入到數(shù)學這個詞是由萊布尼茨引入到數(shù)學(shxu)中的，他主中的，他主要使

10、用這個術(shù)語來指代某種數(shù)學要使用這個術(shù)語來指代某種數(shù)學(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.后來人們才認識到，萊布尼茨的函數(shù)思想適用后來人們才認識到，萊布尼茨的函數(shù)思想適用(shyng)的范圍太過局限了，這個術(shù)語的含義從那時起已經(jīng)過的范圍太過局限了，這個術(shù)語的含義從那時起已經(jīng)過了多次

11、推廣。了多次推廣。第8頁/共21頁第九頁，共21頁。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：給定兩個如今，從本質(zhì)上講，函數(shù)的定義如下：給定兩個(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應(yīng)。一一對

12、應(yīng)。第9頁/共21頁第十頁，共21頁。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中的元素相對應(yīng)的中的元素相對應(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。(值域可能是整個值域可能是整個(zhn

13、gg)集集合合Y,也可能不是。也可能不是。)第10頁/共21頁第十一頁，共21頁。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.英語字母和希臘字母表通常用于表示函數(shù)。為此，英語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/共21頁第十二頁，共21頁。

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是一個給定是一個給定(i dn)的函數(shù)，的函數(shù)，x

15、是它定義域中的一個是它定義域中的一個點，符號點，符號f(x)表示值域中按照表示值域中按照f對應(yīng)于對應(yīng)于x的點，它叫做的點，它叫做f在在x點點的值或者的值或者x在在f下的像。符號下的像。符號f(x)讀作讀作“f of x.”第12頁/共21頁第十三頁，共21頁。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)學中，很少有個概念象函數(shù)的概念那樣，起那在數(shù)學中，很少有個概念象函數(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)，需要知道這個概念是，需要知道這個概念是如何發(fā)展起來的。如何發(fā)展起來的。This concept, like many others, originates in physics. 這個概念這個概念(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。理學。第13頁/共21頁第十四頁，共21頁。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)學物理量是數(shù)學(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世紀后期稱為函數(shù)關(guān)系。世紀后期稱為函數(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/共21頁第十五頁，共21頁。The study of such relations led people in the 18th century to think of a function relation as nothing but a formula. 對

19、這種關(guān)系的研究導致了對這種關(guān)系的研究導致了18世紀的人們認為函數(shù)關(guān)系只世紀的人們認為函數(shù)關(guān)系只不過是一個不過是一個(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應(yīng)關(guān)系，這個至于如何建立這種對應(yīng)關(guān)系，這個(zh ge)定義并未詳定義并未詳細規(guī)定。細規(guī)定。第15頁/共21頁第十六頁，共21頁。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世紀世紀(shj)的數(shù)學所假定的那樣，用公式來的數(shù)學所假定的那樣，用公式來建立，但同樣也可以用統(tǒng)計表那樣的表格或用某種建立，但同樣也可以用統(tǒng)計表那樣的表格或用某種其他的描述方式來建立。其他的描述方式來建立。第16頁/共21頁第十七頁，共21頁。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，這顯然是一個時間的函數(shù)。但典型的例子是室溫，這顯然是一個時間的函數(shù)。但是這個函數(shù)不能用公式來表示，不過可以用表格的是這個函數(shù)不能用公式來表示，不過可以用表格的形式來記錄形式來記錄(jl)或者用一種自動裝置以圖標形式來或者用一種自動裝置以圖標形式來追蹤。追蹤。第17頁/共21頁第十八頁，共21頁。The modern definition of a function y of x is simply a mapping from a space X to another