1、1 A What is mathematics Mathematics comes from man's social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. And in turn, mathematics serves the practice and plays a great role in all fields. No

2、 modern scientific and technological branches could be regularly developed without the application of mathematics.數(shù)學(xué)來源于人類的社會(huì)實(shí)踐，比方工農(nóng)業(yè)生產(chǎn)，商業(yè)活動(dòng)， 軍事行動(dòng)和科學(xué)技術(shù)研究。反過 來，數(shù)學(xué)效勞于實(shí)踐，并在各個(gè)領(lǐng)域中起著非常重要的作用。 沒有應(yīng)用數(shù)學(xué)，任何一個(gè)現(xiàn) 在的科技的 分支都 不能正 常開展。 From the early need of man came theconceptsof numbersand forms.Then,geometrydevelope

3、dout ofproblems of measuring land , and trigonometry came from problems ofsurveying .To deal with somemore complexpracticalproblems,manestablishedand then solvedequationwithunknownnumbers,thusalgebraoccurred.Before 17thcentury,manconfinedhimselfto theelementarymathematics,i.e., geometry,trigonometry

4、and algebra,inwhich only the constants are considered.很早的時(shí)候，人類的需要產(chǎn)生了數(shù)和形式的概念，接著， 測量土地的需要形成了幾何， 出于測量的需要產(chǎn)生了三角幾何，為了處 理更復(fù)雜的實(shí)際問題，人類建立和解決了帶未知參數(shù)的方程，從而產(chǎn)生了代數(shù)學(xué)， 17 世紀(jì) 前，人類局 限于只考慮常 數(shù)的初等 數(shù)學(xué)， 即幾何 ，三角幾 何和代數(shù)。 The rapiddevelopment of industry in 17th century promoted the progress of economics and technology and requ

5、ired dealing with variable quantities.The leap from constantsto variable quantities brought about two newbranches of mathematicsanalytic geometry and calculus, whichbelong to the higher mathematics. Now there are many branches in higher mathematics, among which are mathematical analysis, higher alge

6、bra, differential equations, function theory and so on. 17世紀(jì)工業(yè)的快速開展推動(dòng)了經(jīng)濟(jì)技術(shù)的進(jìn)步， 從而遇到需要處理變量的問題，從常數(shù)帶變量的跳躍產(chǎn)生了兩個(gè)新的數(shù)學(xué)分支 解析幾何和微積分，他們都屬于高等數(shù)學(xué)，現(xiàn)在高等數(shù)學(xué)里面有很definitions and多分支，其中有數(shù)學(xué)分析，高等代數(shù)，微分方程，函數(shù)論等。 Mathematicians studyconceptions and propositions, Axioms, postulates, theorems are all propositions. Notations are

7、 a special and powerful tool of mathematics and are used to express conceptions and propositionsvery often. Formulas ,figuresand chartsSome of the best known symbolsofare full of different symbols.mathematics are the Arabicnumerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition, subtraction multiplic

8、ation, division and equality. 數(shù)學(xué)家研究的是概念和命題，公理，公設(shè)， 定義和定理都是命題。 符號是數(shù)學(xué)中一個(gè)特殊而有用的工具， 常用于表達(dá)概念和命題。 公式，圖表都是不同的 符號.Thecon clusi onsin mathematicsare obtainedmainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathematics methods was occupied by t

9、he logical deductions. Now , since electronic computers are developed promptly and used widely, the role of computation becomes more and more important. In our times, computation is not only used to deal with a lot of information and data, but also to carry out some workthat merely could be done ear

10、lier by logical deductions, for example, theproof of most of geometrical theorems.數(shù)學(xué)結(jié)論主要由邏輯推理和計(jì)算得到，在數(shù)學(xué)開展歷史的很長時(shí)間內(nèi)， 邏輯推理一直占據(jù)著數(shù)學(xué)方法的中心地位， 現(xiàn)在， 由于電子電腦的迅速開展和廣泛使用，電腦的地位越來越重要，現(xiàn)在電腦不僅用于處理大量的信息和數(shù)據(jù)， 還可以完成一些之前只能由邏輯推理來做的工作， B Equation An equation is a statement例如， 大多數(shù)幾何定理的證明。 1of the equality betweentwoequal numbe

11、rs or number symbols. Equation are of two kinds identities and equations of condition. An arithmetic or an algebraic identity is anare alike. Or等式是關(guān)于兩 算術(shù)或者代數(shù)恒等式equation. In such an equation either the two members become alike on the performance of the indicated operation. 個(gè)數(shù)或者數(shù)的符號相等的一種描述。 等式有兩種恒等式和條

12、件等式。是等式。這種等式的兩端要么一樣，要么經(jīng)過執(zhí)行指定的運(yùn)算后變成一樣。 An identity involving letters is true for any set of numerical values of the letters in it. An equation which is true only for certain values of a letter in it, or for certain sets of related values of two or more of its letters, is an equation of condition, or

13、simply an equation. Thus 3x-5=7 is true for x=4 only; and 2x-y=0 is true for x=6 and y=2 and for many other pairs of values for x and y. 含有字母的恒等式對其中字母的任一組數(shù)值都成立。一個(gè)等式假設(shè)僅僅對其中一個(gè)字母 的某些值成立， 或?qū)ζ渲袃蓚€(gè)或著多個(gè)字母的假設(shè)干組相關(guān)的值成立， 那么它是一個(gè)條件等式， 簡稱方程。因此 3x-5=7 僅當(dāng) x=4 時(shí)成立，而 2x-y=0 ，當(dāng) x=6,y=2 時(shí)成立，且對 x, y 的其他許多對值也成立。 A root of

14、 an equation is any number or number symbol which satisfies the equation. There are various kinds of equation. Theyare linear equation, quadratic equation, etc. 方程的根是滿足方程的任意數(shù)或者 數(shù)的符號。 方程有很多種， 例如： 線性方程， 二次方程等。 To solve an equation meansto find the value of the unknown term. To do this , we must, of co

15、urse, change the terms about until the unknown term stands alone on one side of the equation, thus making it equal to something on the other side. We then obtain the value of the unknown and the answer to the question. To solve the equation, therefore, means to move and change the terms about withou

16、t making the equation untrue, until only the unknown quantity is left on one side ,no matter which side.解方程意味著求未知項(xiàng)的值，為了求未知項(xiàng)的值， 當(dāng)然必須移項(xiàng)， 直到未知項(xiàng)單獨(dú)在方程的一邊， 令其等于方程的另一邊， 從而求得未知項(xiàng)的值， 解決了問題。 因此解方程意味著進(jìn)行一系列的移項(xiàng)和同解變形， 直到 未知量被單獨(dú)留在方程的一邊，無論那一邊。Equation are of very great use. Wecan use equation in many mathematical pr

17、oblems. We may notice that almost every problem gives us one or more statements that something is equal to something, this gives us equations, with which we may work if we need it.方程作用很大，可以用方程解決很多數(shù)學(xué)問題。注意到幾乎每一個(gè)問題都給出一個(gè)或多個(gè)關(guān)于一個(gè)事情與另一個(gè)事情相等的陳述， 這就給出了方程， 利用該方程， 如果 我們需要的話，可以解方程。2A Why study geometry? Many le

18、ading institutions of higher learninghave recognized that positive benefits can be gained by all who study this branch of mathematics. This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.許多居于領(lǐng)導(dǎo)地位的學(xué)術(shù)機(jī)構(gòu)成認(rèn)， 所有學(xué)習(xí)這個(gè)數(shù)學(xué)分支的人都將得到確實(shí)的受益， 許多學(xué) 校把幾

19、何的學(xué)習(xí)作為入學(xué)考試的先決條件，從這一點(diǎn)上可以證明。 Geometry had its origin long ago in the measurement by the Babylonians and Egyptians of their lands inundatedby the floods of the Nile River. The greek wordgeometry is derived from geo, meaning “ earth and metron, meaning “ measure . As early as 2000 B.C. we find the land

20、surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry .幾何學(xué)起源于很久以前巴比倫人和埃及人測量他們被尼羅河洪水淹沒的土地，希臘語幾何來源于 geo ，意思是土地“，和 metron 意思是測量“。 公元前 2000 年之前，我們發(fā)現(xiàn)這些民族的土地測量者利用幾何知識重新確定消失了的土 地標(biāo)志和邊界。2 B Some geometrical terms A solid is a three-dimensional figure

21、. Common examples of solids are cube, sphere, cylinder, cone and pyramid.A cube has six faces which are smooth and flat. These faces are calledplane surfaces or simply planes. A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop is an example of a plane

22、surface. 立體是一個(gè)三維圖形， 立體常見的例子是立方體， 球體， 柱體，圓錐和棱錐。立方體有 6 個(gè)面，都是光滑的和平的，這些面被稱為平面曲面或者簡 稱為平面。平面曲面是二維的，有長度和寬度，黑板和桌子上面的面都是平面曲面的例子。2C 三角函數(shù)于直角三角形的解 One of the most important applications of trigonometry is the solution of triangles. Let us now take up the solution to right triangles. A triangle is composed of si

23、x parts three sides and three angles. To solve a triangle is to find the parts not given. A triangle may be solved if three parts (at least one of these is a side ) are given. A right triangle has one angle, the right angle, always given. Thus a right triangle can be solved when two sides, or one si

24、de and an acute angle, are given. 三角形最重要的應(yīng)用之一是解三角形， 現(xiàn)在我們來解直角三角形。 一個(gè)三角形 由6 個(gè)局部組成，三條邊和三只角。解一個(gè)三角形就是要求出未知的局部。如果三角形的 三個(gè)局部其中至少有一個(gè)為邊為，那么此三角形就可以解出。直角三角形的一只角， 即直角，總是的。因此，如果它的兩邊，或一邊和一銳角為， 那么此直角三角形可解。9-A Introduction A large variety of scientific problems arise in which one tries to determine something from it

25、s rate of change. For example , we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration. Or a radioactive substance may bedisintegrating at a known rate and we may be required to determine theamount of material present after a given time.大量的科學(xué)問題需要人們

26、根據(jù)事物的變化率來確定該事物，例如，我們可以由速度或者加速度來計(jì)算移動(dòng)粒子的位置 又如，某種放射性物質(zhì)可能正在以的速度進(jìn)行衰變， 需要我們確定在給定的時(shí)間后遺留物質(zhì)的總量。 In examples like these, we are trying to determine an unknownfunctionfrom prescribedinformationexpressedequation involving at least one of the derivativesin the form of an of the unknownfunction .These equations a

27、re called differentialequations, and their在類而這種方程至少包含了未study forms one of the most challenging branches of mathematics. 似的例子中， 我們力求由方程的形式表示的信息來確定未知函數(shù)， 知函數(shù)的一個(gè)導(dǎo)數(shù)。 這些方程稱為微分方程， 對其研究形成了數(shù)學(xué)中最具有挑戰(zhàn)性的一門分 支。 The study of differential equations is one part of mathematics that,perhaps more than any other, has be

28、en directly inspired by mechanics, astronomy, and mathematical physics.微分方程的研究是數(shù)學(xué)的一局部，也許比其他分支更多的直接受到力學(xué)，天文學(xué)和數(shù)學(xué)物理的推動(dòng)。 Its history began in the 17th century when Newton,Leibniz, and the Bernoullis solved somesimple differentialequationsarisingfrom problems in geometry andmechanics. These early discover

29、ies, beginning about 1690, gradually led to the development of a lot of“ special tricks for solving certain specialkinds of differential equation. 微分方程起源于 17 世紀(jì)，當(dāng)時(shí)牛頓，萊布尼茨，波 努力家族解決了一些來自幾何和力學(xué)的簡單的微分方程。開始于 1690 年的早期發(fā)現(xiàn)，逐 漸引起了解某些特殊類型的微分方程的大量特殊技巧的開展。Although these special tricks are applicable in relativel

30、y few cases, they do enable us to solvemany differential equations that arise in mechanics and geometry, so their study is of practical importance. Some of these special methods and some of the problems which they help us solve are discussed near the end of this chapter. 盡管這些特殊的技巧只是用于相對較少的幾種情況，但他們 能

31、夠解決力學(xué)和幾何中出現(xiàn)的許多微分方程，因此， 他們的研究具有重要的實(shí)際應(yīng)用。 這些特殊的技巧和有助于我們解決的一些問題將在本章最后討論。Experience has shownthat it is difficult to obtain mathematical theories of much generality about solution of differential equations, except for a few types.經(jīng)驗(yàn)說明除了幾個(gè)典型方程外，很難得到微分方程解的一般性數(shù)學(xué)理論。Among these are theso-called linear differe

32、ntial equations which occur in a great variety of scientific problems. 在這些典型方程中，有一個(gè)稱為線性微分方程，出現(xiàn)在大量的科 學(xué)問題中。 10-C Applications of matrices In recent years the applications of matrices in mathematics and in many diverse fields have increased with remarkable speed. Matrix theory plays a central role in m

33、odern physics in the study of quantum mechanics. Matrix methods are used to solve problems in applied differential equations , specifically, in the area of aerodynamics, stress and structure analysis. One of the most powerful mathematical methods for psychological studies is factor analysis, a subje

34、ct that makes wide use of matrix methods.近年來，在數(shù)學(xué)和許多各種不同的領(lǐng)域中， 矩陣的應(yīng)用一直以驚人的速度不斷增加。 在研究量子力學(xué)時(shí)， 矩陣?yán)碚撛诂F(xiàn)代 物理學(xué)上起著主要的作用。 解決應(yīng)用微分方程， 特別是在空氣動(dòng)力學(xué)， 應(yīng)力和結(jié)構(gòu)分析中的問題，要用矩陣方法。心理學(xué)研究上一種最強(qiáng)有力的數(shù)學(xué)方法是因子分析， 這也廣泛的使用 矩陣 方 法 . Recent developments in mathematical economics and in problems of business administration have led to extensi

35、ve use of matrix methods. The biological sciences, and in particular genetics, use matrix techniques to good advantage. No matter what the students ' field of major interest is , knowledge of the rudiments of matricesis likely tobroaden the range of literature that he can read with understanding

36、 . 近 年來，在數(shù)學(xué)經(jīng)濟(jì)學(xué)和商業(yè)管理問題方面的開展已經(jīng)導(dǎo)致廣泛的使用矩陣法。 生物科學(xué)，特 別在遺傳學(xué)方面，用矩陣的技術(shù)很有成效。 不管學(xué)生主要興趣是什么， 矩陣根本原理的知識 可能擴(kuò)大他能讀懂的文獻(xiàn)的范圍。 The solution of n simultaneous linear equations in n unknowns is one of the important problems of applied mathematics. Descartes, the inventor of analytic geometry and one of the founders of mod

37、ern algebraic notation, believed that all problems could ultimately be reduced to the solution of a set of simultaneous linear equations.解一有 n 個(gè)未知數(shù)的 n 個(gè)聯(lián)立一次線性方程是應(yīng)用數(shù)學(xué)的一個(gè)重要問題。解析幾何的創(chuàng)造者和現(xiàn)代代數(shù)計(jì)數(shù)法的創(chuàng)始人之一笛卡兒相信， 所有的問題最后都 能約簡為解一組聯(lián)立一次方程。 Although this belief is now known to be untenable , we know that a large g

38、roup of significant applied problems from many different disciplines are reducible to such equations. Many of the applications, require the solution of a large number of simultaneous linear equations ,sometimes in the hundreds . The advent of computers has made the matrix methods effective in the so

39、lution of these formidable problems. 雖然這種信念現(xiàn)在認(rèn)為是站不住腳的，但是，我們知道，從許 多不同的學(xué)科里的一大群重要的應(yīng)用問題都可以約化為這類的方程。許多應(yīng)用要求解大量 的，往往數(shù)以百計(jì)的聯(lián)立一次方程， 電腦的創(chuàng)造已經(jīng)使得矩陣方法在解這些難以解決的問題 方面非常活潑。Example 1. solve the simultaneous equations for x1 x2, and x3 .例題 1，解聯(lián)立方程求 x1 x2 和 x3 。 From the above discussion,we see that theproblem of solvin

40、g n simultaneouslinear equationin n unknowns isreduced to the problem of finding the inverse of the matrix of coefficients.It is therefore not surprising that in books on the theory of matrices the techniques of finding inverse matrices occupy considerable space.從上面的討論，我們看到解有 n 個(gè)未知數(shù)的 n 個(gè)聯(lián)立一次方程問題化成求系

41、數(shù)的矩陣的逆矩陣 的問題。因此，在矩陣論的書中，用大量的篇幅來講求逆矩陣的技巧就不奇怪了。 Of course , we will not in our limited treatment discuss such techniques. Not only are matrix methods useful in solving simultaneous equations , but they are also useful in discovering whether or not the set of equations are consistent, in the sense that

42、 they lead to solutions, and in discovering whether or not the set of equation are determinate, in the sense that they lead to unique solution.當(dāng)然，我們在這有限的表達(dá)中不會(huì)討論這類的技巧。矩陣方法不僅在解聯(lián)立方程中有用， 而且在發(fā)現(xiàn)方程組是否相容， 即方程組是否有解的問題， 以及 方程組是否是確定的，即是否只有一解等方面，都是有用的。11-A predicatesStatements involving variables, such as “x>

43、;3, x+y=3 , x+y=z are often found in mathematical assertion and in computer programs. These statements are neither true nor false whenthe values of the variables are not discuss the ways that propositions statements. 包含變量的語句，比方 “x>3 和電腦程序中，假設(shè)未給語句中的所有變量賦值， 由這種語句生成命題的方法。 The statement “specified. I

44、n this section we will can be produced from such , x+y=3 , x+y=z 常出現(xiàn)在數(shù)學(xué)論斷中 那么不能判定該語句是真是假， 本節(jié)要討論 x is greater than 3 has two parts.The first part, the variables, is the subject of the statement. The second part- the predicate,“is greater than3-refersto a property that thesubject of the statement can

45、have.語句"x大于3分成兩局部，第一局部，變量，是語句的主語。第二局部，謂語， “大于 3，指的是語句主語具有的性質(zhì)。We can denotethe sta tement “x is greater than 3 by P(x), where P denote the predicate“is greater than 3 and x is the variable. The statement P(x) is also said tobe the value of the propositional function P at x. once a value has been

46、 assigned to the variable x, the statements P(x) becomes a proposition and has a truth value.把語句"x大于3記為P(x),其中P表示謂詞 大于3，而x是變量。語句 P(x) 也稱為命題函數(shù) P 在 x 點(diǎn)處的值。一旦賦予 x 一個(gè)值，語句 P(x) 就成 為一個(gè)命題,有了真值。 11-B QuantifiersWhenall the variables in apropositional function are assigned values, the resulting statemen

47、t has atruth value. However, there is another important way, called quantification, to create a proposition from a propositional function. twotypes of quantificationwill be discussed here, namely, universalquantification and existential quantification .當(dāng)命題函數(shù)所有變量都賦值時(shí)，結(jié)果語句有真值，但是還有另外一種方式，稱為量詞化，可從命題函數(shù)中得

48、到命題。這里討論兩種量詞化方法，也就是全稱量詞化和存在量詞化。 Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the universe of discourse. Such a statement is expressed using a universal quantification. The universal quantification of a propositional function is the proposition that assert that P(x) is true for all value