数学专业英语只是部分不是很完整课件.ppt

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1、数学专业英语只是部数学专业英语只是部分不是很完整分不是很完整第1页，此课件共60页哦From the early need of man came the concepts of numbers and forms.Then,geometry developed out of problems of measuring land,and trigonometry came from problems of surveying.To deal with some more complex practical problems,man established and then solved equ

2、ation with unknown numbers,thus algebra occurred.很很早早的的时候候，人人类的的需需要要产生生了了数数和和形形的的概概念念。接接着着，测量量土土地地问题形形成成了了几几何何学学，测量量问题产生生了了三三角角学学。为了了处理理更更复复杂的的实际问题，人人类建建立和解决了立和解决了带未知数的方程，从而未知数的方程，从而产生了代数学。生了代数学。Before 17th century,man confined himself to the elementary mathematics,i.e.,geometry,trigonometry and alg

3、ebra,in which only the constants are considered.17世世纪前前，人人类局局限限于于只只考考虑常常数数的的初初等等数数学学，即即几几何何学学，三三角角学学和和代代数数学。学。第2页，此课件共60页哦The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities.The leap from constants to varia

4、ble quantities brought about two new branches of mathematics-analytic geometry and calculus,which belong to the higher mathematics.17世世纪工工业的的快快速速发展展推推动了了经济技技术的的进步步，从从而而遇遇到到需需要要处理理变量量的的问题。从从常常量量到到变量量的的跳跳跃产生生了了两两个个新新的的数数学学分分支支-解析几何和微解析几何和微积分，他分，他们都属于高等数学。都属于高等数学。Now there are many branches in higher m

5、athematics,among which are mathematical analysis,higher algebra,differential equations,function theory and so on.现在在高高等等数数学学里里面面有有很很多多分分支支，其其中中有有数数学学分分析析，高高等等代代数数，微微分方程，函数分方程，函数论等。等。第3页，此课件共60页哦Mathematicians study conceptions and propositions,Axioms,postulates,definitions and theorems are all propo

6、sitions.Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.数数学学家家研研究究的的是是概概念念和和命命题，公公理理，公公设，定定义和和定定理理都都是是命命题。符符号号是数学中一个特殊而有用的工具，常用于表达概念和命是数学中一个特殊而有用的工具，常用于表达概念和命题。Formulas,figures and charts are full of different symbols.Some of t

7、he best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition“+”,subtraction“-”,multiplication“”,division“”and equality“=”.公式，公式，图形和形和图表都是不同的符号表都是不同的符号.第4页，此课件共60页哦The conclusions in mathematics are obtained mainly by logical deductions and computation.Fo

8、r a long period of the history of mathematics,the centric place of mathematics methods was occupied by the logical deductions.数数学学结论主主要要由由逻辑推推理理和和计算算得得到到。在在数数学学发展展历史史的的很很长时间内，内，逻辑推理一直占据着数学方法的中心地位。推理一直占据着数学方法的中心地位。Now,since electronic computers are developed promptly and used widely,the role of compu

9、tation 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 work that merely could be done earlier by logical deductions,for example,the proof of most of geometrical theorems.现在在，由由于于电子子计算算机机的的迅迅速速发展展和和广广泛泛使使用

10、用，计算算机机的的地地位位越越来来越越重重要要。现在在计算算机机不不仅用用于于处理理大大量量的的信信息息和和数数据据，还可可以以完完成成一一些些之之前前只只能能由由逻辑推推理理来来做做的的工工作作，例例如如，证明明大大多数的几何定理。多数的几何定理。第5页，此课件共60页哦An equation is a statement of the equality between two equal numbers or number symbols.1B Equation等式是关于两个数或者数的符号相等的一种描述。等式是关于两个数或者数的符号相等的一种描述。Equations are of two

11、kinds-identities and equations of condition.An arithmetic or an algebraic identity is an equation.In such an equation either the two members are alike,or become alike on the performance of the indicated operation.等等式式有有两两种种恒恒等等式式和和条条件件等等式式。算算术或或者者代代数数恒恒等等式式都都是是等等式式。这种等式的两端要么一种等式的两端要么一样，要么，要么经过执行指定的运

12、算后行指定的运算后变成一成一样。第6页，此课件共60页哦An identity involving letters is true for any set of numerical values of the letters in it.含有字母的恒等式含有字母的恒等式对其中字母的任一其中字母的任一组数数值都成立。都成立。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

13、 an equation of condition,or simply an equation.Thus 3x-5=7 is true for x=4 only;and 2x-y=10 is true for x=6 and y=2 and for many other pairs of values for x and y.一一个个等等式式若若仅仅对其其中中一一个个字字母母的的某某些些值成成立立，或或对其其中中两两个个或或者者多多个个字字母母的的若若干干组相相关关的的值成成立立，则它它是是一一个个条条件件等等式式，简称称方方程程。因因此此3x-5=7仅当当x=4 时成成立立，而而2x-y=0

14、，当当x=6,y=2时成成立立，且且对x,y的的其其他他许多多对值也成立。也成立。第7页，此课件共60页哦A root of an equation is any number or number symbol which satisfies the equation.To obtain the root or roots of an equation is called solving an equation.方程的根是方程的根是满足方程的任意数或者数的符号。求方程根的足方程的任意数或者数的符号。求方程根的过程被称程被称为解方解方程。程。There are various kinds of

15、equations.They are linear equation,quadratic equation,etc.方程有很多种，例如：方程有很多种，例如：线性方程，二次方程等。性方程，二次方程等。第8页，此课件共60页哦To solve an equation means to find the value of the unknown term.To do this,we must,of course,change the terms about until the unknown term stands alone on one side of the equation,thus mak

16、ing it equal to something on the other side.We then obtain the value of the unknown and the answer to the question.解解方方程程意意味味着着求求未未知知项的的值，为了了求求未未知知项的的值，当当然然必必须移移项，直直到到未未知知项单独独在在方方程程的的一一边，令令其其等等于于方方程程的的另另一一边，从从而而求求得未知得未知项的的值，解决了，解决了问题。To solve the equation,therefore,means to move and change the terms

17、 about without making the equation untrue,until only the unknown quantity is left on one side,no matter which side.因因此此解解方方程程意意味味着着进行行一一系系列列的的移移项和和同同解解变形形，直直到到未未知知量量被被单独留在方程的一独留在方程的一边，无，无论那一那一边。第9页，此课件共60页哦Equations are of very great use.We can use equations in many mathematical problems.We may noti

18、ce 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 to.方方程程作作用用很很大大，可可以以用用方方程程解解决决很很多多数数学学问题。注注意意到到几几乎乎每每一一个个问题都都给出出一一个个或或多多个个关关于于一一个个事事情情与与另另一一个个事事情情相相等等的的陈述述，这就就给出出了了方方程程，利利用用该方方程程，如如果我果我们需要的需要的

19、话，可以解方程。，可以解方程。第10页，此课件共60页哦Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics.2A Why study geometry?许多居于领导地位的学术机构承认，所有学习这个数学分支的人都将得到确实的受益。This is evident from the fact that they require study of geometry as a

20、 prerequisite to matriculation in those schools.许多学校把几何的学习作为入学考试的先决条件，从这一点上可以证明。第11页，此课件共60页哦Geometry had its origin long ago in the measurement by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River.几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河洪水淹没的土地。The greek word geometry is deri

21、ved from geo,meaning“earth”and metron,meaning“measure”.希腊语几何来源于geo，意思是”土地“，和metron 意思是”测量“。第12页，此课件共60页哦As early as 2000 B.C.we find the land surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry.公元前2000年之前，我们发现这些民族的土地测量者利用几何知识重新确定消失了的土地标志和边

22、界。One of the most important objectives derived from a study of geometry is making the student be more critical in his listening,reading and thinking.In studying geometry he is led away from the practice of blind acceptance of statements and ideas and is taught to think clearly and critically before

23、forming conclusions.几何的学习使学生在思考问题时更周密、审慎，他们将不会盲目接受任何结论.第13页，此课件共60页哦A solid is a three-dimensional figure.Common examples of solids are cube,sphere,cylinder,cone and pyramid.2B Some geometrical terms立体是一个三维图形，立体常见的例子是立方体，球体，柱体，圆锥和棱锥。A cube has six faces which are smooth and flat.These faces are call

24、ed plane surfaces or simply planes.立方体有6个面，都是光滑的和平的，这些面被称为平面曲面或者简称为平面。第14页，此课件共60页哦A plane surface has two dimensions,length and width.The surface of a blackboard or of a tabletop is an example of a plane surface.平面曲面是二维的，有长度和宽度，黑板和桌子上面的面都是平面曲面的例子。A circle is a closed curve lying in one plane,all po

25、ints of which are equidistant from a fixed point called the center.平面上的闭曲线当其中每点到一个固定点的距离均相当时叫做圆。固定点称为圆心。第15页，此课件共60页哦A line segment drawn from the center of the circle to a point on the circle is a radius of the circle.The circumference is the length of a circle.经过圆心且其两个端点在圆周上的线段称为这个园的直径，这条曲线的长度叫做周长

26、。One of the most important applications of trigonometry is the solution of triangles.Let us now take up the solution to right triangles.三角形最重要的应用之一是解三角形，现在我们来解直角三角形。第16页，此课件共60页哦A triangle is composed of six parts three sides and three angles.To solve a triangle is to find the parts not given.一个三角形由

27、6个部分组成，三条边和三只角。解一个三角形就是要求出未知的部分。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 side and an acute angle,are given.如果三角形的三个部分（其中至少有一个为边）为已知，则此三角形就可以解

28、出。直角三角形的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一锐角为已知，则此直角三角形可解。第17页，此课件共60页哦The concept of a set has been utilized so extensively throughout modern mathematics that an understanding of it is necessary for all college students.Sets are a means by which mathematicians talk of collections of things in an abstrac

29、t way.3A Notations for denoting sets集合论的概念已经被广泛使用，遍及现代数学，因此对大学生来说，理解它的概念是必要的。集合是数学家们用抽象的方式来表述一些事物的集体的工具。Sets usually are denoted by capital letters;elements are designated by lower-case letters.集合通常用大写字母表示，元素用小写字母表示。第18页，此课件共60页哦We use the special notation to mean that“x is an element of S”or“x belo

30、ngs to S”.If x does not belong to S,we write .我们用专用记号来表示x是S的元素或者x属于S。如果x不属于S，我们记为。When convenient,we shall designate sets by displaying the elements in braces;for example,the set of positive even integers less than 10 is displayed as 2,4,6,8 whereas the set of all positive even integers is displayed

31、 as 2,4,6,the three dots taking the place of“and so on.”如果方便，我们可以用在大括号中列出元素的方式来表示集合。例如，小于10的正偶数的集合表示为2,4,6,8，而所有正偶数的集合表示为2,4,6,三个圆点表示“等等”。第19页，此课件共60页哦The dots are used only when the meaning of“and so on”is clear.The method of listing the members of a set within braces is sometimes referred to as th

32、e roster notation.只有当省略的内容清楚时才能使用圆点。在大括号中列出集合元素的方法有时被归结为枚举法。The first basic concept that relates one set to another is equality of sets:联系一个集合与另一个集合的第一个基本概念是集合相等。第20页，此课件共60页哦DEFINITION OF SET EQUALITY Two sets A and B are said to be equal(or identical)if they consist of exactly the same elements,in

33、 which case we write A=B.If one of the sets contains an element not in the other,we say the sets unequal and we write AB.集合相等的定义 如果两个集合A和和B确切包含同样的元素,则称二者相等，此时记为A=B。如果一个集合包含了另一个集合以外的元素，则称二者不等，记为AB。第21页，此课件共60页哦EXAMPLE 1.According to this definition,the two sets 2,4,6,8 and 2,8,6,4 are equal since the

34、y both consist of the four integers 2,4,6 and 8.Thus,when we use the roster notation to describe a set,the order in which the elements appear is irrelevant.根据这个定义，两个集合2,4,6,8和2,8,6,4是相等的，因为他们都包含了四个整数2,4,6,8。因此，当我们用枚举法来描述集合的时候，元素出现的次序是无关紧要的。第22页，此课件共60页哦EXAMPLE 2.The sets 2,4,6,8 and 2,2,4,4,6,8 are

35、equal even though,in the second set,each of the elements 2 and 4 is listed twice.Both sets contain the four elements 2,4,6,8 and no others;therefore,the definition requires that we call these sets equal.例2.集合2,4,6,8 和2,2,4,4,6,8也是相等的，虽然在第二个集合中，2和4都出现两次。两个集合都包含了四个元素2,4,6,8，没有其他元素，因此，依据定义这两个集合相等。This

36、example shows that we do not insist that the objects listed in the roster notation be distinct.A similar example is the set of letters in the word Mississippi,which is equal to the set M,i,s,p,consisting of the four distinct letters M,i,s,and p.这个例子表明我们没有强调在枚举法中所列出的元素要互不相同。一个相似的例子是，在单词Mississippi中字母

37、的集合等价于集合M,i,s,p,其中包含了四个互不相同的字母M,i,s,和p.第23页，此课件共60页哦From a given set S we may form new sets,called subsets of S.For example,the set consisting of those positive integers less than 10 which are divisible by 4(the set 4,8)is a subset of the set of all even integers less than 10.In general,we have the

38、following definition.3B Subsets一个给定的集合S可以产生新的集合，这些集合叫做S的子集。例如，由可被4除尽的并且小于10的正整数所组成的集合是小于10的所有偶数所组成集合的子集。一般来说，我们有如下定义。第24页，此课件共60页哦In all our applications of set theory,we have a fixed set S given in advance,and we are concerned only with subsets of this given set.The underlying set S may vary from o

39、ne application to another;it will be referred to as the universal set of each particular discourse.（35页第二段）当我们应用集合论时，总是事先给定一个固定的集合S，而我们只关心这个给定集合的子集。基础集可以随意改变，可以在每一段特定的论述中表示全集。第25页，此课件共60页哦It is possible for a set to contain no elements whatever.This set is called the empty set or the void set,and wi

40、ll be denoted by the symbol .We will consider to be a subset of every set.（35页第三段）一个集合中不包含任何元素，这种情况是有可能的。这个集合被叫做空集，用符号表示。空集是任何集合的子集。Some people find it helpful to think of a set as analogous to a container(such as a bag or a box)containing certain objects,its elements.The empty set is then analogous

41、 to an empty container.一些人认为这样的比喻是有益的，集合类似于容器（如背包和盒子）装有某些东西那样，包含它的元素。第26页，此课件共60页哦To avoid logical difficulties,we must distinguish between the elements x and the set x whose only element is x.In particular,the empty set is not the same as the set .（35页第四段）为了避免遇到逻辑困难，我们必须区分元素x和集合x，集合 x中的元素是x。特别要注意的是

42、空集和集合是不同的。In fact,the empty set contains no elements,whereas the set has one element.Sets consisting of exactly one element are sometimes called one-element sets.事实上，空集不含有任何元素，而有一个元素。由一个元素构成的集合有时被称为单元素集。第27页，此课件共60页哦Diagrams often help us visualize relations between sets.For example,we may think of

43、a set S as a region in the plane and each of its elements as a point.Subsets of S may then be thought of the collections of points within S.For example,in Figure 2-3-1 the shaded portion is a subset of A and also a subset of B.（35页第五段）图解有助于我们将集合之间的关系形象化。例如，可以把集合S看作平面内的一个区域，其中的每一个元素即是一个点。那么S的子集就是S内某些

44、点的全体。例如，在图2-3-1中阴影部分是A的子集，同时也是B的子集。第28页，此课件共60页哦Visual aids of this type,called Venn diagrams,are useful for testing the validity of theorems in set theory or for suggesting methods to prove them.Of course,the proofs themselves must rely only on the definitions of the concepts and not on the diagram

45、s.这种图解方法，叫做文氏图，在集合论中常用于检验定理的有效性或者为证明定理提供一些潜在的方法。当然证明本身必须依赖于概念的定义而不是图解。第29页，此课件共60页哦As mentioned earlier,one of the applications of the integral is the calculation of area.Ordinarily,we do not talk about area by itself,instead,we talk about the area of something.This means that we have certain object

46、s(polygonal regions,circular regions,parabolic segments etc.)whose areas we wish to measure.If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects,we must first find an effective way to describe these objects.就像前面提到的，积分的一个应用就是面积的计算，通常我们不讨论面积本身，相反

47、，是讨论某事物的面积。这意味着我们有些想测量的面积的对象（多边形区域，圆域，抛物线弓形等），如果我们希望获得面积的计算方法以便能够用它来处理各种不同类型的图形，我们就必须首先找出表述这些对象的有效方法。5-A the coordinate system of Cartesian geometry第30页，此课件共60页哦The most primitive way of doing this is by drawing figures,as was done by the ancient Greeks.A much better way was suggested by Rene Descar

48、tes,who introduced the subject of analytic geometry(also known as Cartesian geometry).Descartes idea was to represent geometric points by numbers.The procedure for points in a plane is this ：描述对象最基本的方法是画图，就像古希腊人做的那样。R 笛卡儿提出了一种比较好的方法，并建立了解析几何（也称为笛卡儿几何）这门学科。笛卡儿的思想就是用数来表示几何点，在平面上找点的过程如下：5-A the coordin

49、ate system of Cartesian geometry第31页，此课件共60页哦Two perpendicular reference lines(called coordinate axes)are chosen,one horizontal(called the“x-axis”),the other vertical(the“y-axis”).Their point of intersection denoted by O,is called the origin.On the x-axis a convenient point is chosen to the right of

50、 O and its distance from O is called the unit distance.Vertical distances along the Y-axis are usually measured with the same unit distance,although sometimes it is convenient to use a different scale on the y-axis.Now each point in the plane(sometimes called the xy-plane)is assigned a pair of numbe