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数学专业英语第二版-课文翻译-converted

2.4 整数、有理数与实数4－A Integers and rational numbersThere exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers.有一些R 的子集很著名，因为他们具有实数所不具备的特殊性质。

在本节我们将讨论这样的子集，整数集和有理数集。

To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers.我们从数字 1 开始介绍正整数，公理 4 保证了 1 的存在性。

1+1 用2 表示，2+1 用3 表示，以此类推，由 1 重复累加的方式得到的数字 1,2,3，…都是正的，它们被叫做正整数。

Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”.严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。

数学专业英语翻译2-1

Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. Formulas ,figures and charts are full of different symbols. Some of the 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 “=”. 数学家研究的是概念和命题，公理，公设，定义和定理都是命题。符号是数学中一个特殊而有用的工具，常用于表达概念和命题。公式，图形和图表都是不同的符号……..
第二章精读课文——入门必修
数学与计: useful terms and definitions of Mathematics,
equation
Difficult points:
Some mathematical terms
Requirements: 1. 掌握所讲课文的生词和词组 2. 理解并掌握课外作业里面的汉译英 3. 理解所讲段落的翻译技巧与方法
1－A What is mathematics

数学专业英语翻译2-5

6cfunction本小节重点掌握这种方法适用于翻译连环式后置定语或定语从句也就是当原文中的一个中心词组被若干个定语从句一环扣一环地修饰时可先将中心词组译出或把中心词组与靠近它的一个或两个后置定语从句译成汉语偏正词组然后顺着修饰环依次翻译其余成分
Mathematical English
Chapter 2 精读课文---入门必读
一个几何图形是满足一个或多个特殊条件的点集，比如平面上的曲线。通过把这些条件转化成含有坐标x和y的表达式，我们就得到了一个或多个能刻画该图形特征的方程。例如，如图2-5-2所示的中心在原点，半径为r的圆，令P是原上任意一点，假设P的坐标为(x, y).
本小节重点掌握
1 计算图形的面积是积分的一个重要应用。 2 微积分与解析几何在他们的发展史上已经互相交融在一起了。
5-A the coordinate system of Cartesian geometry
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 objects (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.

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数学专业英语（Doc版）.12数学专业英语－Linear ProgrammingLinear Programming is a relatively new branch of mathematics.The cornerstone of this exciting field was laid independently bu Leonid V. Kantorovich,a Russ ian mathematician,and by Tjalling C,Koopmans, a Yale economist,and George D. Dantzig,a Stanford mathematician. Kantorovich’s pioneering work was moti vated by a production-scheduling problem suggested by the Central Laboratory of the Len ingrad Plywood Trust in the late 1930’s. The development in the U nited States was influenced by the scientific need in World War II to solve lo gistic military problems, such as deploying aircraft and submarines at strategic positions and airlifting supplies and personnel.The following is a typical linear programming problem:A manufacturing company makes two types of television sets: one is black and white and the other is color. The company has resources to make at most 300 sets a week. It takes $180 to make a black and white set and $270 to make a color set. The company does not want to spend more than $64,800 a wee k to make television sets. If they make a profit of $170 per black and white set and $225 per color set, how many sets of each type should the company make to have a maximum profit?This problem is discussed in detail in Supplementary Reading Material Lesson 14.Since mathematical models in linear programming problems consist of linear in equalities, the next section is devoted to suchinequalities.Recall that the linear equation lx+my+n=0represents a straight line in a plane. Every solution (x,y) of the equation lx+my+n=0is a point on this line, and vice versa.An inequality that is obtained from the linear equation lx+my+n=0by replacin g the equality sign “=”by an inequality sign < (less than), ≤(less than or equal to), > (greater than), or ≥(greater than or equal to) is called a linear i nequality in two variables x and y. Thus lx+my+n≤0, lx+my+n≥0are all lin ear liequalities. A solution of a linear inequality is an ordered pair (x,y) of nu mbers x and y for which the inequality is true.EXAMPLE 1 Graph the solution set of the pair of inequalities SOLUTION Let A be the solution set of the inequality x+y-7≤0 and B be th at of the inequalit y x-3y +6 ≥0 .Then A∩B is the solution set of the given pair of inequalities. Set A is represented by the region shaded with horizontal lines and set B by the region shaded with vertical lines in Fig.1. Therefore thecrossed-hatched region represents the solution set of the given pair of inequali ties. Observe that the point of intersection (3.4) of the two lines is in the solu tion set.Generally speaking, linear programming problems consist of finding the maxim um value or minimum value of a linear function, called the objective function, subject to some linear conditions, called constraints. For example, we may wa nt to maximize the production or profit of a company or to maximize the num ber of airplanes that can land at or take off from an airport during peak hours; or we may want to minimize the cost of production or of transportation or to minimize grocery expenses while still meeting the recommended nutritional re quirements, all subject to certain restrictions. Linearprogramming is a very use ful tool that can effectively be applied to solve problems of this kind, as illust rated by the following example.EXAMPLE 2 Maximize the function f(x,y)=5x+7y subject to the constraintsx≥0 y≥0x+y-7≤02x-3y+6≥0SOLUTION First we find the set of all possible pairs(x,y) of numbers that s atisfy all four inequalities. Such a solution is called a feasible sulution of the problem. For example, (0,0) is a feasible solution since (0,0) satisf ies the giv en conditions; so are (1,2) and (4,3).Secondly, we want to pick the feasible solution for which the giv en function f (x,y) is a maximum or minimum (maximum in this case). S uch a feasible solution is called an optimal solution.Since the constraints x ≥0 and y ≥0 restrict us to the first quadrant, it follows from example 1 that the given constraints define the polygonal regi on bounded by the lines x=0, y=0,x+y-7=0, and 2x-3y+6=0, as shown in Fig.2.Fig.2.Observe that if there are no conditions on the values of x and y, then the f unction f can take on any desired value. But recall that our goal is to determi ne the largest value of f (x,y)=5x+7y where the values of x and y are restrict ed by the given constraints: that is, we must locate that point (x,y) in the pol ygonal region OABC at which the expression 5x+7y has the maximum possibl e value.With this in mind, let us consider the equation 5x+7y=C, where C is any n umber. This equation represents a family ofparallel lines. Several members of this family, corresponding to different values of C, are exhibited in Fig.3. Noti ce that as the line 5x+7y=C moves up through the polygonal region OABC, th e value of C increases steadily. It follows from the figure that the line 5x+7y =43 has a singular position in the family of lines 5x+7y=C. It is the line farth est from the origin that still passes through the set of feasible solutions. It yiel ds the largest value of C: 43.(Remember, we are not interested in what happen s outside the region OABC) Thus the largest value of the function f(x,y)=5x+7 y subject to the condition that the point (x,y) must belong to the region OAB C is 43; clearly this maximum value occurs at the point B(3,4).Fig.3.Consider the polygonal region OABC in Fig.3. This shaded region has the p roperty that the line segment PQ joining any two points P and Q in the regio n lies entirely within the region. Such a set of points in a plane is called a c onvex set. An interesting observation about example 2 is that the maximum va lue of the objective function f occurs at a corner point of the polygonal conve x set OABC, the point B(3,4).The following celebrated theorem indicates that it was not accidental.THEOREM (Fundamental theorem of linear programming) A linear objective function f defined over a polygonal convex set attains a maximum (or minim um) value at a corner point of the set.We now summarize the procedure for solving a linear programming problem:1.Graph the polygonal region determined by the constraints.2.Find the coordinates of the corner points of the polygon.3.Evaluate the objective function at the corner points.4.Identify the corner point at which the function has an optimal value.Vocabularylinear programming 线形规划 quadrant 象限objective function 目标函数 convex 凸的constraints 限制条件，约束条件 convex set 凸集feaseble solution 容许解，可行解corner point 偶角点optimal solution 最优解simplex method 单纯形法Notes1. A Yale economist, a Stanford mathematician 这里Yale Stanford 是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)2. subject to some lincar conditions 解作“在某些线形条件的限制下”。

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数学专业英语－MathematicansLeonhard Euler was born on April 15,1707,in Basel, Switzerland, the son of a mathematician and Caivinist pastor who wanted his son to become a pastor a s well. Although Euler had different ideas, he entered the University of Basel to study Hebrew and theology, thus obeying his father. His hard work at the u niversity and remarkable ability brought him to the attention of the well-known mathematician Johann Bernoulli (1667—1748). Bernoulli, realizing Euler’s tal ents, persuaded Euler’s father to change his mind, and Euler pursued his studi es in mathematics.At the age of nineteen, Euler’s first original work appeared. His paper failed to win the Paris Academy Prize in 1727; however this loss was compensated f or later as he won the prize twelve times.At the age of 28, Euler competed for the Pairs prize for a problem in astrono my which several leading mathematicians had thought would take several mont hs to solve.To their great surprise, he solved it in three days! Unfortunately, th e considerable strain that he underwent in his relentless effort caused an illness that resulted in the loss of the sight of his right eye.At the age of 62, Euler lost the sight of his left eye and thus became totally blind. However this did not end his interest and work in mathematics; instead, his mathematical productivity increased considerably.On September 18, 1783, while playing with his grandson and drinking tea, Eul er suffered a fatal stroke.Euler was the most prolific mathematician the world has ever seen. He made s ignificant contributions to every branch of mathematics. He had phenomenal m emory: He could remember every important formula of his time. A genius, he could work anywhere and under any condition.George cantor (March 3, 1845—June 1,1918),the founder of set theory, was bo rn in St. Petersburg into a Jewish merchant family that settled in Germany in 1856.He studied mathematics, physics and philosophy in Zurich and at the University of Berlin. After receiving his degree in 1867 in Berlin, he became a lecturer at the university of Halle from 1879 to 1905. In 1884,under the stra in of opposition to his ideas and his efforts to prove the continuum hypothesis, he suffered the first of many attacks of depression which continued to hospita lize him from time to time until his death.The thesis he wrote for his degree concerned the theory of numbers; however, he arrived at set theory from his research concerning the uniqueness of trigon ometric series. In 1874, he introduced for the first time the concept of cardinalnumbers, with which he proved that there were “more”transcendental numb ers than algebraic numbers. This result caused a sensation in the mathematical world and became the subject of a great deal of controversy. Cantor was troub led by the opposition of L. Kronecker, but he was supported by J.W.R. Dedek ind and G. Mittagleffer. In his note on the history of the theory of probability, he recalled the period in which the theory was not generally accepted and cri ed out “the essence of mathematics lies in its freedom!”In addition to his work on the concept of cardinal numbers, he laid the basis for the concepts of order types, transfinite ordinals, and the theory of real numbers by means of fundamental sequences. He also studied general point sets in Euclidean space a nd defined the concepts of accumulation point, closed set and open set. He wa s a pioneer in dimension theory, which led to the development of topology.Kantorovich was born on January 19, 1912, in St. Petersburg, now called Leni ngrad. He graduated from the University of Leningrad in 1930 and became a f ull professor at the early age of 22.At the age of 27, his pioneering contributi ons in linear programming appeared in a paper entitled Mathematical Methods for the Organization and planning of production. In 1949, he was awarded a S talin Prize for his contributions in a branch of mathematics called functional a nalysis and in 1958, he became a member of the Russian Academy of Science s. Interestingly enough, in 1965,kantorovich won a Lenin Prize fo r the same o utstanding work in linear programming for which he was awarded the Nobel P rize. Since 1971, he has been the director of the Institute of Economics of Ma nagement in Moscow.Paul R. Halmos is a distinguished professor of Mathematics at Indiana Univers ity, and Editor-Elect of the American Mathematical Monthly. He received his P h.D. from the University of Illinois, and has held positions at Illinois, Syracuse, Chicago, Michigan, Hawaii, and Santa Barbara. He has published numerous b ooks and nearly 100 articles, and has been the editor of many journals and se veral book series. The Mathematical Association of America has given him the Chauvenet Prize and (twice) the Lester Ford award for mathematical expositio n. His main mathematical interests are in measure and ergodic theory, algebraic, and operators on Hilbert space.Vito Volterra, born in the year 1860 in Ancona, showed in his boyhood his e xceptional gifts for mathematical and physical thinking. At the age of thirteen, after reading Verne’s novel on the voyage from earth to moon, he devised hi s own method to compute the trajectory under the gravitational field of the ear th and the moon; the method was worth later development into a general proc edure for solving differential equations. He became a pupil of Dini at the Scu ola Normale Superiore in Pisa and published many important papers while still a student. He received his degree in Physics at the age of 22 and was made full professor of Rational Mechanics at the same University only one year lat er, as a successor of Betti.Volterra had many interests outside pure mathematics, ranging from history to poetry, to music. When he was called to join in 1900 the University of Rome from Turin, he was invited to give the opening speech of the academic year. Volterra was President of the Accademia dei Lincei in the years 1923-1926. H e was also the founder of the Italian Society for the Advancement of Science and of the National Council of Research. For many years he was one of the most productive scientists and a very influential personality in public life. Whe n Fascism took power in Italy, Volterra did not accept any compromise and pr eferred to leave his public and academic activities.Vocabularypastor 牧师 hospitalize 住进医院theology 神学 thesis 论文strain 紧张、疲惫transcendental number 超越数relentless 无情的sensation 感觉，引起兴趣的事prolific 多产的controversy 争论，辩论depression 抑郁；萧条，不景气essence 本质，要素transfinite 超限的Note0. 本课文由几篇介绍数学家生平的短文组成，属传记式体裁。

数学专业英语课后答案

2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。

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数学专业英语－The Theory of GraphsIn this chapter, we shall introduce the concept of a graph and show that graph s can be defined by square matrices and versa.1.IntroductionGraph theory is a rapidly growing branch of mathematics. The graphs discusse d in this chapter are not the same as the graphs of functions that we studied previously, but a totally different kind.Like many of the important discoveries and new areas of learning, graph the ory also grew out of an interesting physical problem, the so-called Konigsberg bridge problem. (this problem is discussed in Section 2) The outstanding Swis s mathematician, Leonhard Euler (1707-1783) solved the problem in 1736, thus laying the foundation for this branch of mathematics. Accordingly, Euler is ca lled the father of graph theory.Gustay Robert Kirchoff (1824-1887), a German physicist, applied graph theor y in his study of electrical networks. In1847, he used graphs to solve systems of linear equations arising from electrical networks, thus developing an importa nt class of graphs called trees.In 1857, Arthur Caylcy discovered trees while working on differential equati ons. Later, he used graphs in his study of isomers of saturated hydrocarbons.Camille Jordan (1838-1922), a French mathematician, William Rowan Hamilt on, and Oystein Ore and Frank Harary, two American mathematicians, are also known for their outstanding contributions to graph theory.Graph theory has important applications in chemistry, genetics, management s cience, Markov chains, physics, psychology, and sociology.Throughout this chapter, you will find a very close relationship between gra phs and matrices.2.The Konigsberg Bridge ProblemThe Russian city of Konigsberg (now Kaliningrad, Russia) lies on the Pregel River.(See Fig.1) It consists of banks A and D of the river and the two island s B and C. There are seven bridges linking the four parts of the city. Residents of the city used to take evening walks from one section of the cit y to another and go over some of these bridges. This, naturally, suggested the following interesting problem: can one walk through the city crossing each bridge exactly once? The problem sounds simple, doesn’t it?You might want to try a few paths before going any further. After all, by the fundamental countin g principle, the number of possible paths cannot exceed 7!=5040. Nonetheless, it would be time consuming to look at each of them to find one that works.Fig .1 The city of KonigsbergIn 1736, Euler proved that no such walk is possible. In fact, he proved a far more general theorem, of which the Konigsberg bridge problem is a special ca se.Fig .2 A mathematical model for the Konigsberg bridge problemLet us construct a mathematical model for this problem.rcplace each area of the city by a point in a plane. The points A, B, C,and D denote the areas th ey represent and are called vertices. The arcs or lines joining these points repr esent the represent the respective bridges. (See图２)They are called edges. The Konigsberg bridge problem can now be stated as follows: Is it possible to tra ce this figure without lifting your pencil from paper or going over the same e dge twice? Euler proved that a figure like this can be traced without lifting pe ncil and without traversing the same edge twice if and only if it has no more than weo vertices with an odd number of edges joining them. Observe that m ore than two vertices in the figure have an odd number of edges connecting t hem-----in fact,they all do.1. GraphsLet us return to the example Friendly Airlines flies to the five cities, Boston (B), Chicago (C), Detroit (D), Eden (E), and Fairyland (F) as follows: it has direct daily flights from city B to cities C, D, and F, from C to B, D, and E; from D to B, C, and F, from E to C, and from F to B and D. This informat ion, though it sounds complicated, can be conveniently represented geometrically, as in 图3. Each city is represented by a heavy dot in the figure; an arc or a line segment between two dots indicates that ther e is a direct flight between these cities.What does this figure have in common with 图２? Both consist of points (denoted by thick dots ) co nnected by arcs or line segments. Such a figure is called a graph. The points are the vertices of the gra ph; the arcs and line segments are its edges. More generally, we make the following definition:A graph consists of a finite set of points, together with arcs or line segments connecting some of them. These points are called the vertices of the graph; the arcs and line segments are called the edges og thegraph. The vertices of graph are usually denoted by the letters A, B, C, and so on. An edge joining th e vertices A and B is denoted by AB or A-B.Fig .3图２and 图3 are graphs. Other graphs are shown in 图4. The graph in图２has four vertices A, B, C, and D, and seven edges AB, AB, AC, BC, BD, CD, and BD. For the graph in图4b, there are four vertices, A, B, C, and D, but only two edges AD and CD. Consider the graph in图4c, it contains an ed ge emanating from and terminating at the same vertex A. Such an edge is called a loop. The graph in 图4d contains two edges between the vertices A and C and a loop at the vertex C.The number of edges meeting at a vertex A is called the valence or degree of the vertex, denoted by v (A). For the graph in图4b, we have v(A)=1, v(B)=0, v(C)=1, and v(D)=2. In图4b, we have v(A)=3, v(B) =2, and v(C)=4.A graph can conveniently be described by using a square matrix in which the entry that belong to the row headed by X and the column by Y gives the number of edges from vertex X to vertex Y. This m atrix is called the matrix representation of the graph; it is usually denoted by the letter M.The matrix representation of the graph for the Konigsberg problem isClearly the sum of the entries in each row gives the valence of the corresponding vertex. We have v(A) =3, v(B)=5, v(C)=3, as we would expect.Conversely, every symmetric square matrix with nonnegative integral entries can be considered the ma trix representation of some graph. For example, consider the matrixA B C DClearly, this is the matrix representation of the graph in 图5.VocabularyNetwork 网络Electrical network 电网络Isomer 异构体emanate 出发，引出Saturated hydrocarbon 饱和炭氢化合物terminate 终止，终结Genetics 遗传学valence 度Management sciences 管理科学node 结点Markov chain 马尔可夫链interconnection 相互连接Psychology 心理学 Konigsberg bridge problem 康尼格斯堡桥问题Sociology 社会学Line-segment 线段Notes1. Camille Jordan, a French mathematician, William Rowan Hamilton and . . .注意：a French mathematician 是Camille Jordan 的同位语不要误为W.R.Hamilton 是a French mathe matician 同位语这里关于W.R.Hamilton 因在本文前几节已作介绍，所以这里没加说明。

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数学专业英语（Doc版）.20数学专业英语－Sequences and SeriesSeries are a natural continuation of our study of functions. In the previous cha pter we found howto approximate our elementary functions by polynomials, with a certain error te rm. Conversely, one can define arbitrary functions by giving a series for them. We shall see how in the sections below.In practice, very few tests are used to determine convergence of series. Esse ntially, the comparision test is the most frequent. Furthermore, the most import ant series are those which converge absolutely. Thus we shall put greater emp hasis on these.Convergent SeriesSuppose that we are given a sequcnce of numbersa1,a2,a3…i.e. we are given a number a n, for each integer ｎ＞1.We form the sumsS n=a1+a2+…+a nIt would be meaningless to form an infinite suma1+a2+a3+…because we do not know how to add infinitely many numbers. However, if ou r sums S n approach a limit as n becomes large, then we say that the sum of our sequence converges, and we now define its sum to be that limit.The symbols∑a=1 ∞a nwill be called a series. We shall say that the series converges if the sums app roach a limit as n becomes large. Otherwise, wesay that it does not converge, or diverges. If the seriers converges, we say that the value of the series is∑a=1∞=lim a→∞S n=lim a→∞(a1+a2+…+a n)In view of the fact that the limit of a sum is the sum of the limits, and other standard properties of limits, we get: THEOREM 1. Let{ a n}and { b n}(n=1,2,…)be two sequences and assume that the series∑a=1∞a n∑a=1∞b nconverge. Then ∑a=1∞(a n + b n ) also converges, and is equal to the sum of the two series. If c is a number, then ∑a=1∞c a n=c∑a=1∞a nFinally, if s n=a1+a2+…+a n and t n=b1+b2+…+b n then∑a=1∞a n ∑a=1∞b n=lim a→∞s n t nIn particular, series can be added term by term. Of course , they cannot be multiplied term by term.We also observe that a similar theorem holds for the difference of two serie s.If a series ∑a n co nverges, then the numbers a n must approach 0 as n beco mes large. However, there are examples of sequences {an} for which the serie s does not converge, and yet lim a→∞a n=0Series with Positive TermsThroughout this section, we shall assume that our numbers a n are ＞0. Then t he partial sumsS n=a1+a2+…+a nare increasing, i.e.s1＜s2 ＜s3＜…＜s n＜s n+1＜…If they are approach a limit at all, they cannot become arbitrarily large. Thus i n that case there is a number B such that S n< Bfor all n. The collection of numbers {s n} has therefore a least upper bound ,i.e. there is a smallest number S such that s n<s< p="">for all n. In that case , the partial sums s n approach S as a limit. In other wo rds, given any positive number ε>0, we have S –ε< s n < Sfor all n .sufficiently large. This simply expresses the fact that S is the least o f all upper bounds for our collection of numbers s n. We express this as a theo rem.THEOREM 2. Let{a n}(n=1,2,…)be a sequence of numbers>0 and letS n=a1+a2+…+a nIf the sequence of numbers {s n} is bounded, then it approaches a limit S , wh ich is its least upper bound.Theorem 3 gives us a very useful criterion to determine when a series with po sitive terms converges:THEOREM 3. Let∑a=1∞a n and∑a=1∞b n be two series , with a n>0 for all n an d b n>0 for all n. Assume that there is a number c such thata n< cb nfor all n, and that∑a=1∞b n converges. Then ∑a=1∞a n converges, and∑a=1∞a n ≤c∑a=1∞b nPROOF. We havea1+…+a n≤cb1+…+cb n=c(b1+…+b n)≤c∑a=1∞b nThis means that c∑a=1∞b n is a bound for the partial sums a1+…+a n.The least u pper bound of these sums is therefore ≤c∑a=1∞b n, thereby proving our theore m.Differentiation and Intergration of Power Series.If we have a polynomiala0+a1x+…+a n x nwith numbers a0,a1,…,a n as coefficients, then we know how to find its derivati ve. It is a1+2a2x+…+na n x n–1. We would like to say that the derivative of a ser ies can be taken in the same way, and that the derivative converges whenever the series does.THEOREM 4. Let r be a number >0 and let ∑a n x n be a series which conv erges absolutely for ∣x∣<r.<="" n="" n-1also="" p="" series="" the="" then="" x="" y="" ∑na="">A similar result holds for integration, but trivially. Indeed, if we have a series ∑a=1∞a n x n which converge s absolutely for ∣x∣<="" the="" then="">∑a=1∞a n/n+1 x n+1=x∑a=1∞a n x n∕n+1has terms whose absolute value is smaller than in the original series.The preceding result can be expressed by saying that an absolutely converge nt series can be integrated and differentiated term by term and and still yields an absolutely convergent power series.It is natural to expect that iff (x)=∑a=1∞a n x n,then f is differentiable and its derivative is given by differentiating the series t erm by term. The next theorem proves this.THEOREM 5. Letf (x)=∑a=1∞a n x nbe a power series, which converges absolutely for∣x∣<="" ble="" differentia="" f="" for="" is="" p="" then="" ∣x∣f′(x)=∑a=1∞na n x n-1.THEOREM 6. Let f (x)=∑a=1∞a n x n be a power series, which converges abs o lutely for ∣x∣<="" the="" then="">∫f (x)d x=∑a=1∞a n x n+1∕n+1is valid in the interval ∣x∣<r.< p="">We omit the proofs of theorems 4,5 and 6.Vocabularysequence 序列positive term 正项series 级数alternate term 交错项approximate 逼近,近似 partial sum 部分和elementary functions 初等函数 criterion 判别准则(单数)section 章节 criteria 判别准则(多数)convergence 收敛(名词) power series 幂级数convergent 收敛(形容词) coefficient 系数absolute convergence 绝对收敛 Cauchy sequence 哥西序列diverge 发散radius of convergence 收敛半径term by term 逐项M-test M—判别法Notes1. series一词的单数和复数形式都是同一个字.例如:One can define arbitrary functions by giving a series for them(单数)The most important series are those which converge absolutely(复数)2. In view of the fact that the limit of a sum of the limits, and other standard properties of limits, we get:Theorem 1…这是叙述定理的一种方式: 即先将事实说明在前面,再引出定理. 此句用in view of the fact that 说明事实,再用we get 引出定理.3. We express this as a theorem.这是当需要证明的事实已再前面作了说明或加以证明后,欲吧已证明的事实总结成定理时,常用倒的一个句子,类似的句子还有(参看附录Ⅲ):We summarize this as the following theorem; Thus we come to the following theorem等等.4. The least upper bound of these sums is therefore ≤c∑a=1∞b n, thereby proving our theorem.最一般的定理证明格式是”给出定理…定理证明…定理证毕”,即thereby proving our theorem;或we have thus proves the theorem 或This completes the proof等等作结尾(参看附录Ⅲ).5. 本课文使用较多插入语.数学上常见的插入语有:conversely; in practice; essentially; in particular; ind eed; in other words; in short; generally speaking 等等.插入语通常与句中其它成份没有语法上的关系,一般用逗号与句子隔开,用来表示说话者对句子所表达的意思的态度.插入语可以是一个词,一个短语或者一个句子.ExerciseⅠ. Translate the following exercises into Chinese:1. In exercise 1 through 4,a sequence f (n) is defined by the formula given. In each case, (ⅰ)Determine whether the sequence (the formulae are omitted).2. Assume f is a non–negative function defined for all x>1. Use the methodsuggested by the proof of the integral test to show that∑k=1n-1f(k)≤∫1n f(x)d x ≤∑k=2n f(k)Take f(x)=log x and deduce the inequalitiesc?n n?c-n< n！<="" p="">Ⅱ. The proof of theorem 4 is given in English as follows(Read the proof through and try to learn how a theorem is proved, then translate this proof into Chinese ):Proof of theorem 4 Since we are interested in the absolute convergence. We may assume that a n>0 for all n. Let 0<x<cWe may write n a n x n =a n(n1/n x)n. Then for all n sufficiently large, we conclude that n1/n xx n with∑a n c n to conclude that∑na n x n converges. Since∑na n x n-1=1n/x∑na n x n, we have proved theorem 4.Ⅲ. Recall from what you have learned in Calculus about (ⅰ) Cauchy sequence and (ⅱ) the radius of c onvergence of a po wer series.Now give the definitions of these two terms respectively.Ⅳ. Translate the following sentences into Chinese:1. 一旦我们能证明,幂级数∑a n z n在点z=z1收敛,则容易证明,对每一z1∣z∣<∣z1∣,级数绝对收敛;2. 因为∑a n z n在z=z1收敛,于是,由weierstrass的M—判别法可立即得到∑a n z n在点z,∣z∣<z1的绝对收敛性;< p="">3. 我们知道有限项和中各项可以重新安排而不影响和的值,但对于无穷级数,上述结论却不总是真的</z1的绝对收敛性;<></x</r.<></s<>。

数学专业英语【第二版】（吴炯圻）

数学专业英语【第二版】1- A 什么是数学数学来自于人的社会实践，例如，工业和农业生产、商业活动、军事行动和科研工作。

与数学反过来，为实践服务和所有字段中的伟大作用。

没有现代的科学和技术分支机构可以定期制定中的数学，应用无早有需要的人来了数字和形式的概念。

然后，开发出的几何土地和三角测量的问题来自测量的问题。

若要对付一些更复杂的实际问题，男子成立，然后解决方程未知号码，因此代数发生。

17 世纪前, 男子向自己限于小学数学，即几何、三角和代数，只有常量被认为在其中。

17 世纪产业的快速发展促进了经济和技术的进展和所需变量的数量、处理从常量到带来两个分支的数学-解析几何和微积分，属于高等数学，现在有很多分支机构，其中有数学分析、高等代数、微分方程的高等数学中的可变数量的飞跃函数理论等。

数学家研究理念和主张。

所有命题公理、假设、定义和定理都。

符号是一种特殊和功能强大的数学工具，用于表示很多时候的理念和主张。

公式、数字和图表是阿拉伯数字1,2,3,4,5,6,7,8,9,0 与另外的符号"+"、减法"-"，乘"*"，除"\"和平等"="。

数学中的结论得到主要由逻辑推理和计算。

长期的数学史上，以中心地点的数学方法被占领逻辑扣除。

现在，由于电子计算机是迅速发展和广泛应用，计算的作用变得越来越多重要。

在我们这个时代计算不只用于处理大量的信息和数据，而且还进行一些只是可以做的工作较早前的逻辑推理，例如，大部分的几何定理的证明。

1--B 方程方程是平等的语句的两个相等的数字或数字符号之间。

因此(a-5）= 一5a 和x 3 = 5 是方程。

方程的两种——身份和方程的条件。

方程的算术或代数的身份。

这种方程中两名成员是相似的或成为相似的指示操作的性能。

因此12-2=2+8,(m+n)(m-n) = m n 是身份。

1—c 比与测量今天的思想沟通往往根据编号和数量的比较。

数学专业英语第二版吴炯圻

数学专业英语第二版1-A 什么是数学数学来自于人(de)社会实践,例如,工业和农业生产、商业活动、军事行动和科研工作.与数学反过来,为实践服务和所有字段中(de)伟大作用.没有现代(de)科学和技术分支机构可以定期制定中(de)数学,应用无早有需要(de)人来了数字和形式(de)概念.然后,开发出(de)几何土地和三角测量(de)问题来自测量(de)问题.若要对付一些更复杂(de)实际问题,男子成立,然后解决方程未知号码,因此代数发生.17 世纪前, 男子向自己限于小学数学,即几何、三角和代数,只有常量被认为在其中.17 世纪产业(de)快速发展促进了经济和技术(de)进展和所需变量(de)数量、处理从常量到带来两个分支(de)数学-解析几何和微积分,属于高等数学,现在有很多分支机构,其中有数学分析、高等代数、微分方程(de)高等数学中(de)可变数量(de)飞跃函数理论等.数学家研究理念和主张.所有命题公理、假设、定义和定理都.符号是一种特殊和功能强大(de)数学工具,用于表示很多时候(de)理念和主张.公式、数字和图表是阿拉伯数字 1,2,3,4,5,6,7,8,9,0 与另外(de)符号"+"、减法"-",乘"",除"\"和平等"=".数学中(de)结论得到主要由逻辑推理和计算.长期(de)数学史上,以中心地点(de)数学方法被占领逻辑扣除.现在,由于电子计算机是迅速发展和广泛应用,计算(de)作用变得越来越多重要.在我们这个时代计算不只用于处理大量(de)信息和数据,而且还进行一些只是可以做(de)工作较早前(de)逻辑推理,例如,大部分(de)几何定理(de)证明.1--B 方程方程是平等(de)语句(de)两个相等(de)数字或数字符号之间.因此(a-5） = 一 5a 和 x 3 = 5 是方程.方程(de)两种——身份和方程(de)条件.方程(de)算术或代数(de)身份.这种方程中两名成员是相似(de)或成为相似(de)指示操作(de)性能.因此 12-2=2+8,(m+n)(m-n) = m n 是身份.1—c 比与测量今天(de)思想沟通往往根据编号和数量(de)比较.当你描述为 6 英尺高(de)人时,你比较他更小(de)单位,称为脚(de)高度.当一个人描述为昂贵(de)商品时,他指相对于其他相似或不同(de)商品这种商品(de)海岸.如果你说你(de)起居室(de)尺寸由 24 18 英尺,一个人可以判断房间(de)一般形式比较尺寸.当纳税人说他(de)城市政府支出 42%(de)税一美元作教学用途(de)时他知道 42 美分(de)每 100 美分用于此目(de).化学家和物理学家不断比较测量(de)数量在实验室里.家庭主妇比较时测量数量(de)烤(de)成分.与他规模图纸建筑师和他工作绘图(de)机器草拟比较成品中相应实际长度在绘图中(de)行(de)长度.定义.一个数量为另一种像数量是第一次(de)商(de)比例除以第二个.比率是一个分数,而关于一小部分(de)所有规则都适用于比.我们写(de)比率与分数线、斜线号、司标志,或符号"："（这读"是"）.因此,3 到 4 is3/ 3 和 4,被称为比例(de)条款(de)比率.很重要(de)了解比像数量(de)商学生.以某个角度向一条线段(de)比率已没有意义；他们不是同一种(de)数量.我们找到一条线段,第二个线段(de)比例或一个角,第二个角度(de)比例.这是我们做(de)测量它们并寻找他们(de)测量(de)商.测量必须表达相同(de)单位.S 比始终是一个抽象(de)数字；即,它有没有单位.它是一个数字,认为除了它所来自(de)测量单位.除非有相反是一个重要(de)原因,应最简单(de)形式表示(de)比率.在前面(de)示例(de)客厅尺寸在哪里由 24 18 英尺,最终和长度(de)比率是宽度(de) 3:4.但不是 18:24.第二章：几何与三角2.2 A 为什么要研究几何我们为什么研究几何开始此文本研究(de)学生也许会问,"什么是几何.什么可以预料从这项研究获得"许多高校领导已经认识到正面(de)好处可以获得所有人学习数学(de)这个分支.这种明显(de)从这一事实它们需要(de)几何研究作为这些院校预科(de)先决条件.几何巴比伦和埃及(de)尼罗河流域洪水淹没土地测量中很久以前了它(de)起源.希腊单词几何被从土力工程处、地球"意义"和美唐、意义"度量值".早作为公元前 2000 年,我们发现(de)这些土地测量师人家重建消失(de)地标和边界(de)利用几何(de)真理.几何是一门科学,由线条(de)形式处理.几何(de)研究是培养成功(de)工程师、科学家建筑师和草拟(de)重要组成部分.木匠、钳工、石匠、艺术家和设计师所有应用几何在他们(de)行业中(de)事实.在本课程学生将学到很多关于几何数字如线条、角度、三角形、圆和设计和多种模式.所得(de)几何研究(de)最重要(de)目标之一使学生在他(de)听力、阅读和思维更重要.学习几何他远离盲目接受语句和思想(de)实践领导和教想清楚与批判前形成(de)结论.有几何(de)学生可以获得许多其他不太直接(de)利益.这些人当中必须包括训练在英语语言(de)精确使用和分析(de)能力一种新形势或成基本部件,以及利用毅力、创意和解决问题(de)逻辑论证(de)问题.欣赏大自然(de)创作将几何研究(de)副产品.学生还应制定数学和数学家,我们(de)文化和文明(de)贡献(de)认识.一些几何术语1.固体和飞机.固是三维图.固体(de)常见示例是多维数据集、球、圆柱、圆锥和金字塔.多维数据集有六个面光滑、平整(de).这些面孔被称为平面曲面,或只是飞机、平面有两个方面.长度和宽度.表面(de)黑板或桌面是平面曲面(de)一个例子.2、线条和线段.我们都很熟悉,但很难词(de)定义.一线可由在一张纸上移动(de)钢笔或铅笔标记(de)代表.一条线,可被视为有只有一维,长度.虽然当我们绘制一条直线,我们给它(de)宽度和厚度,我们认为只(de)跟踪(de)长度,考虑行时.点有没有长度、没有宽度和没有厚度,但标记(de)位置.我们都熟悉用这种表达式作为铅笔点和针点.我们点表示一个小点,并将其命名(de)旁边,打印为大写字母 A 点 ' 图 2-2-1.在行(de)标记上它(de)两个点,用大写字母或附近(de)一个小字母命名.图 2-2-2 (de)直线是读"AB 线"行 l".直线延伸到无穷远(de)两个方向并没有结束.线上(de)两个点之间(de)部分称为行被称为一条线段.一条线段两个端点(de)命名.因此,图 2-2-2,我们称为 AB 线 l (de)一条线段.当不会混淆可能导致,表达"线段 AB 通常由 AB 段或更换,简单地说,行 AB.有三种线路：直线、断(de)线和曲线.弯曲(de)线条或,简单地说,曲线不是其中(de)任何部分是直行.断(de)线联接、直线线段组成为能得到一个唯一(de)图 2-2-3.3.部分(de)一个圆.圆是封闭(de)躺在一个平面,其中(de)所有点都都距离称为中心一个固定(de)点.一个圆(de)符号.图 2-2-4,O 是 ABC 中心,或简单(de) 圆点从圆(de)中心绘制(de)线段是圆(de)半径（复数,半径）.OA、转播,业主立案法团是中华民国(de)半径圆(de)直径是通过中心圆圈圈上(de)终结点(de)一条线段.一个直径等于两个半径.弦是任何加入圆上(de)两个点(de)线段.教育署是图 2-2-4 圈(de)弦.从这个定义很明显直径是弦.一条弧线,如弧 AE,其中由 AE 表示圆(de)任何部分.A、 E 点圆分为轻微弧 AE 和主要弧安倍.直径分为两个弧形称为半圆(de)一个圆.如 AB 和 BCD.周围是一圈(de)长度.涉及字母标识是数值(de)真正(de)任何一组字母它.因此身份 (+ 2) x = ax + 2 x 成为 3 (7++ 2) = 21++ 6 或 27 = 27,时,例如,x = 3,和= 7.这是事实,只有某些值(de)一封信中,或某些集(de)方程相关(de)值(de)两个或更多(de)信,是方程(de)条件或简单(de)方程.Thus3x-5 = 7 是真正为 x = 4 只；和 2 x-y = 10 是真正为 x = 6和 y = 2 和许多其他值对 x 和 y.任何数字或数字符号,满足这个方程(de)方程根.要获取(de)根或方程根(de)称为方程(de)求解.有很多种(de)方程.他们是线性方程组,二次 .解方程就发现未知词(de)价值.要做到这一点,我们必须,当然,改变条款有关直到未知(de)词单独一侧(de)方程,从而使它等于东西(de)另一边.然后,我们获得(de)价值与未知(de)问题(de)答案.若要求解方程,因此,移动和更改有关(de)条款,不做公式不真实,直到未知(de)数量只是意味着提起一侧,无论哪一方.方程(de)很大(de)使用.我们可以使用许多数学问题中(de)方程.我们可能会发现几乎每一个问题给我们一个或多个语句什么是等于什么等于某事；这给了我们方程,与我们可能工作如果我们需要.2.2 C 三角函数和直角三角形(de)解决方案相互依存(de)边和角(de)三角形.我们知道这从几何.三角开始通过显示这种依赖性之间(de)边和角(de)三角形(de)确切性质.为此目(de)三角雇佣双方(de)比率.这些比率称为三角函数.6 三角函数(de)直角三角形,为 A,任何急性角表示,如下所示；这些函数（比率) 是极为重要(de)三角学(de)研究.他们必须载入史册.最重要(de)应用程序之一是三角(de)三角形(de)解决方案.让我们现在占用(de)直角三角形(de)解决方案.一个三角形组成(de)六个部分、三边和三个角度.若要解决一个三角形是找到不给予(de)部分.直角三角形有一个角度,以正确(de)角度,总是给予.因此当双方或一方和急性角度来看,有一个直角三角形可以得到解决.解直角三角形(de)一般指示提供如下.第三章3—A 符号指示集一组(de)概念如此广泛利用整个现代数学(de)认识是所需(de)所有大学生.集是通过集合中一种抽象方式(de)东西(de)数学家谈(de)一种手段.集,通常用大写字母： A、 B、 C、进程运行·、 X、 Y、 Z ；由小写字母指定元素： a、 b (de) c、进程运行·,若 x、 y z.我们用特殊符号 x∈S 意味着 x 是 S (de)一个元素或属于美国(de) x如果 x 不属于 S,我们写xS.≠当方便时,我们应指定集(de)元素显示在括号内；例如,由符号表示(de)积极甚至整数小于 10 集 {2,468} {2,,进程运行·} 作为显示(de)所有积极甚至整数集,而三个点等(de)发生.点(de)和等等(de)意思是清楚时,才使用.上市(de)大括号内(de)一组成员方法有时称为名册符号.涉及到另一组(de)第一次基本概念是平等(de)集.DEFINITIONOFSETEQUALITY.两组 A 和 B,据说是平等(de) （或相同(de)）如果它们包含完全相同(de)元素,在这种情况下,我们写 A = B.如果其中一套包含在另一个元素,我们说这些集是不平等,我们写 A = B.EXAMPLE1.根据对这一定义,由于他们都是由构成(de)这四个整数 2, 和 8 两套 {2,468} 和 {2,864} 一律平等.因此,当我们用来描述一组(de)名册符号,元素(de)显示(de)顺序无关.动作.集 {2,468} 和 {2,2,4,4,6,8} 是平等(de)即使在第二组,每个元素 2 和 4 两次列出.这两组包含(de)四个要素 2,468 和无他人；因此,定义要求我们称之为这些集平等. 此示例显示了我们也不坚持名册符号中列出(de)对象是不同.类似(de)例子是一组在密西西比州,其值等于{M、我、 s、 p} 一组单词中(de)字母,组成四个不同字母 M、我、 s 和体育3—B 子集S.从给定(de)集 S,我们可能会形成新集,称为.(de)子集例如,组成(d e)那些正整数小于 10 整除 4 （集合{8 毫米}） (de)一组一般是(de)所有甚至小于 10.整数集(de)一个子集,我们有以下(de)定义. 子集(de)定义.A一组据说是B,集(de)一个子集,我们写A B每当A(de)每个元素也属于B.我们还说包含B A或B包含. 关系称为集.A和B(de)声明并不排除可能性,B.事实上,我们可能B A和B A,但只有当A和B都具有相同(de)元素发生这种情况.换句话说,A = B当且仅当B和B A.这一命题是上述定义(d e)平等和包容(de)直接后果.如果A和B,但A≠B,然后我们说(de)就是你(de)真子集我们表明这通过编写B.在所有(de)应用程序集理论,我们有一套固定事先,S,我们只关心这给定组(de)子集.底层(de)设置(de)不同而有所不同从一个应用程序,到另一台；它将转交作为每个特定(de)话语(de)通用组.符号{X∣X∈S和X满足P},将指定(de)所有元素X在S中满足该属性集体育当通用设置为我们所指(de)id(de)理解,我们省略参照以S,我们只需写{X∣X满足P}.这读取 '"集(de)所有这种x满足p.' "在此方法中指定(de)设置说笔下定义(de)属性,例如,所有正实数(de)一组可以被指定为{X∣X大于 0} ；通用集S,在这种情况下理解为所有实数集.当然,这封信x是个笨蛋,并可由任何其他方便(de)符号替换.因此,我们可以写{x∣x大于0} = {y∣y大于0} = {t∣t大于 0} 等等.它有可能设置为不包含任何元素.这套被称为空集或无效设置,并将由symbolφ表示.我们会考虑φto是每一集(de)一个子集.有些人觉得很有用(de)一套类似于一个容器（例如,一个袋子或框）包含某些对象,其元素.空集则类似于一个空(de)容器.为了避免逻辑(de)困难,我们必须区分元素x和集{x}(de)唯一元素是x,(A box with a hat in it is conceptually distinct from t he hat itself.)尤其是,空(de)setφis集合{φ}不相同.事实上,空设置φcontains没有元素而集{φ}有一个元素φ（一个框,其中包含一个空框不是空(de)）.组成一个元素(de)集合,有时也称为一个元素集.3— C在讨论任何分支(de)数学,它分析、代数、几何,最好使用符号和集理论(de)术语.这一问题,在世纪后期开发(de)布尔和康,已对 19 20世纪数学发展(de)深远(de)影响.它具有统一许多看似已断开连接(de)想法,并有助减少许多数学概念(de)逻辑基础,优雅和有系统(de)方式.集理论彻底治疗需要长时间(de)讨论,我们认为这本书(de)范围.幸运(de)是,基本(de) noticns 是几号中,并有可能发展(de)非正式讨论通过集理论思想工作知识(de)方法.实际上,我们将讨论不是一个新(de)理论作为协议有关(de)精确(de)术语,我们要将应用到更多或更少(de)熟悉(de)想法.在数学中,'"集'"一词用于表示作为单个实体(de)集合称为想查看(de)对象(de)集合,作为'"群'",这类名词(de)'"'"部落,'人群'"'",团队',是所有示例(de)集合,集合中(de)各个对象称为元素或一组(de)成员他们都说属于或载于一组.反过来,集包含或由其元素(de)表示.我们须主要兴趣(de)数学对象集：集数字、集(de)曲线、集(de)几何图形,等等.在许多应用程序,它是方便快捷(de)处理中,没有什么特别(de)集假定在集合中(de)各个对象(de)性质有关.这些称为抽象集.抽象集理论已经发展到处理(de)任意对象,这种集合,并从这种普遍性理论派生其权力.第四章4-A存在某些子集(de) R (de)区分,因为他们不共享(de)所有实数(de)特殊属性.在本节中,我们将讨论两个这样(de)子集、整数、有理数.若要引进(de)正整数,我们开始数字 1,在其存在(de)公理 4 保证.2、 2++ 3 1 (de)数由表示数字 1++ 1,依此类推.数字 1 2、 3,… …,以这种方式获得(de)重复加 1 (de)是积极(de)而他们则称为正整数.严格地说,这说明(de)正整数还没有完全完成因为我们已经没有详细解释我们"等等",什么意思表达或者"重复加 1".虽然直观(de)表达意义上去很清晰,实数系统精心治疗有必要给(de)正整数更精确(de)定义.有许多方法来执行此操作.一种方便(de)方法是先介绍一个感应集(de)概念.感应集(de)定义.实数集称为感应集,如果它具有以下两个属性：(a) 1 号是集.(b) 在一组,每个 x 数字 x + 1 是还在一组.例如,R 是归纳(de)组.所以是 R + 一组.现在我们将定义为那些属于每个感应集(de)实数(de)正整数.Definition (de)正整数.实数称为一个正整数,如果它属于每一个感应集.让 P 表示这个集合(de)所有(de)正整数.然后,P 是本身感应集因为 (a) 它包含 1,并且 (b) 它包含 x + 1,每当它包含 x.由于 P 成员属于每个感应集,我们称之为 P (de)最小(de)感应集.此属性(de)设置 P 形成一种类型(de)推理逻辑基础诱导,详细(de)讨论,其中有 4 部这个介绍数学家调用证明(de).消极(de)正整数称为负(de)整数.正整数,连同负整数和0(zero),从我们称之为简单(de)整数集设置 Z.实数系统彻底治疗,它会有必要在此阶段,证明某些定理(de)整数.例如,sum、差异,或产品(de)两个整数,但两个整数(de)商不需要是一个整数.然而,我们不应进入该等债权证明表(de)详细信息.整数(de)商 / b （其中 b 0) 被称为有理数.有理数,由 Q,表示(de)一组包含 Z 为子集.读者应意识到由 Q 满足所有字段公理和顺序公理.为此,我们说(de)有理数集是一个有序(de)字段.不在 Q (de)实数称为非理性.4-B读者是无疑熟悉实数(de)一条直线上(de)点(de)几何表示形式.代表0,有权代表 1,在图 2-4-1 所示(de) 0 及另一人,选择一个点.此选项确定规模.如果一个采用一套合适(de)欧几里德几何公理,然后每个真实(de)数字对应于这条线上(de)一个点,相反,在行上(de)每个点对应于一个且仅一个实数.为此线通常称为真正(de)直线或实轴,而且很习惯使用单词实际数量和互换点.因此我们经常讲点(de) x,而不是点对应(de)实数.实数(de)订购关系有一个简单(de)几何解释.如果 x < y、点 x 位于左侧(de)点(de) y,如图 2-4-1 所示.正数躺到左侧(de) 0 0,负数(de)权利.如果 < b、点 x 满足不等式 < x < b 当且仅当 x 是之间和 b.此设备(de)几何表示实数是非常有价值(de)工具,有助我们去发现和更好地了解实数(de)某些属性.然而,读者应意识到必须将所有属性都被视为定理(de)实数(de)推断出从不涉及任何几何公理.这并不意味着人不应该让几何研究(de)实数属性中(de)使用.相反,几何往往表明特定(de)定理证明(de)方法和有时几何参数是比纯粹(de)解析证明（一个完全取决于公理(de)实数）更加出色.在这本书中,几何参数用于很大程度上有助于激励或澄清特定(de)讨论.不过,所有(de)重要定理(de)证明以解析(de)窗体.4 C首先,让我们把注意力转移到分数.你肯定见过表达式"一半"和"季度",分母等于 2 或 4 时使用它们.1/3 是读"三分之一".其他分数是以相同(de)方式读取(de).因此,我们读 1/5,1/6,1/7,1/10,1/25,1/100 为五分之一,六分之一,六分之一、七分之一,十分之一,1:25 上午和百分之一.这些表达式被视为名词,因此可能产生(de)复数形式.因此,我们作为三分之二；读了 2/3同样 5 6,9 105/100 是读取六分之五、九十分之五百分之一.但是,如果分母(de)最后一位是 1 或 2,然后我们不要读分数以上述方式.例如,我们宣布 5/21 作为"五个超过 21".此方法还在其它情况下使用.如果分数不是一项共同 .,1/1089年或 501/1205年）,然后我们说"一对千和 89 个"或"五一百以上十二百和五".下一步,让我们检查小数.他们是很简单(de)发音.只读取数(de)整数部分普通(de)方式,然后说"点"（代表"小数点"）,然后后另读取一个小数位.因此读取十二-点-六-五；Π正确,6 位小数,等于three-point-one-four-one-five-nine-two,正确(de)五个重要(de)数字,等于 three-point-one-four-one-six.小于一个小数部分时,一般不写了,例如,,但仅限于英格兰.56）.56 读"点-5-6",.0007 读"点-团团转-团团转-团团转-七"或通常更-三-0 (de)七.现在代数表达式,分数再次读取"指针经过".（2a-1)/(ax+b) 对 ax + b 读取 2a-1 和括号表示"到"一词如（a+b)(a-b) 读"加进减号 b b".权力指数或指数由表示."平方"和 3"桂鱼",或"到第三个"指数(de)读取索引 2.其它指数",第四,第五,减号第二部分,第 n,以"读取.身份 ^3++ b ^3 = (+ b)(a^2-ab+b^2) 读取"多次元(de)罗宋汤(de)脓 b 到平方(de) ab 加 b 平方等于 a + b".或与方程 x^(-2/3) + √^2 = 0 读取"x,减号(de)三分之二,加上第五根(de)平方等于零".。

数学专业英语(Doc版).18

数学专业英语－Polya’s Craft of DiscoveryGeorge Polya has a scientific career extending more than seven decades. Abrill iant mathematician who has made fundamental contributions in many fields. Po lya has also been a brilliant teacher, a teacher’s teacher and an expositor. Pol ya believes that there is a craft of discovery. He believes that the ability to di scover and the ability to invent can be enchanced by skillful teaching which al erts the student to the principles of discovery and which gives him an opportu nity to practise these principles.In a series of remarkable books of great richness, the first of which was publi shed in 1945. Polya has crystallized these principles of discovery and invention out of his vast experience, and has shared them with us both in precept and in example.These books are a treasure-trove of strategy, know-how, rules of th umb, good advice, anecdote, mathematical history, together with problem after problem at all levels and all of unusual mathematical interest. Polya places a g lobal plan for “How to Solve It”in the endpapers of his book of that name:HOW TO SOLVE ITFirst: You have to understand the problem.Second: Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be fo und. You should obtain eventually a plan of the solution.Third: Carry out your plan.Fourth: Examine the solution obtained.These precepts are then broken down to “molecular”level on the opposite e ndpaper. There, individual strategies are suggested which might be called into play at appropriate momentsm, such as:If you cannot solve the proposed problem, look around for an appropriate relat ed problem.Work backwardsWork forwardsNarrow the conditionWiden the conditionSeek a counter exampleGuess and testDivide and conquerChange the conceptual modeEach of these heuristic principles is amplified by numerous appropriate exampl es.Subsequent investigators have carried Polya’s ideas forward in a number of w ays. A.H.Schoenfeld has made an interesting tabulation of the most frequently used heuristic principles in college-level mathematics. We have appended it her e.Frequently Used HeuristicsAnalysis1)Draw a diagram if at all possible2)Examine special cases:a)Choose special values to exemplify the problem and get a “feel”for it.b)Examine limiting cases to explore the range of possibilitiesc)Set any integer parameters equal to 1,2,3,…,in sequence, and look for an i nductive pattern.3)Try to simplify the problem bya)exploiting symmetry, orb)“Without Loss of Generality”arguments (including scaling)Exploration1)Consider essentially equivalent problems:a)Replacing conditions by equivalent ones.b)Re-combining the elements of the problem in different ways.c)Introduce auxiliary elements.d)Re-formulate the problem byI) change of perspective or notationII) considering argument by contradiction or contrapositiveIII) assuming you have a solution , and determining its properties2)Consider slightly modified problems:a)Choose subgoals (obtain partial fulfillment of the conditions)b)Relax a condition and then try to re-impose it .c)Decompose the domain of the problem and work on it case by case .3)Consider broadly modified problems:a)Construct an analogous problem with fewer variables .b)Hold all but one variable fixed to determine that variable’s impact .c)Try to exploit any related problems which have similarI) formII) “givens”III) conclusionsRemember: when dealing with easier related problems , you should try to expl oit both the RESULT and the METHOD OF SOLUTION on the given proble m .Verifying your solution1)Does your solution pass these specific tests:a)Does it use all the pertinent data?b)Does it conform to reasonable estimates or predictions?c)Does it withstand tests of symmetry, dimension analysis , or scaling?2)Does it pass these general tests?a)Can it be obtained differently?b)Can it be sudstantiated by special cases?c)Can it be reduced to known results?d)Can it be used to generate something you know?Vocabularycraft 技巧enchance 增强alert 警觉，机警precept 箴言，格言treasure trove 宝藏anecdote 轶事，趣闻auxiliary 辅助的appropriate 适当的heuristic 启发式的amplified 扩大，详述append 附加，追加exploration 探查，细查perspective 透视contrapositive 对换的relax 放松decompose 分解pertinent 适当的substantiate 证实，证明Notes1．A brilliant mathematician who has made fundamentral contributions in many fields,Polya has also been a brilliant teacher, a teacher’s teacher, and an expositor.意思是：Polya，一个在许多领域中都作出重要贡献的数学家，也是一位出色的教师，教师的教师和评注家。

数学专业英语第二版课文翻译

数学专业英语第二版课文翻译部分翻译2.4 整数、有理数与实数4－A Integers and rational numbersThere exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers.有一些R的子集很著名，因为他们具有实数所不具备的特殊性质。

在本节我们将讨论这样的子集，整数集和有理数集。

To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers.我们从数字1开始介绍正整数，公理4保证了1的存在性。

1+1用2表示，2+1用3表示，以此类推，由1重复累加的方式得到的数字1,2,3，…都是正的，它们被叫做正整数。

数学专业英语词汇对照表

数学专业英语词汇英汉对照Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数Mmain effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariate analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way ANOV A 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Qqadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ssample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数2. 最优化方法词汇英汉对照表Aactive constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子Bbandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界Ccoefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomial interpolation method 三次多项式插值法curve-fitting 曲线拟合Ddata-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search method 直接搜索法direction of search 搜索方向discontinuous 不连续Eeigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的Ffeasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶GGauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradient method 梯度法Hhandle 句柄Hessian matrix 海色矩阵Iindependent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代JJacobian 雅可比矩阵LLagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardt method列文伯格-马夸尔特法line search 一维搜索linear 线性的linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解Mmedium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的norm 范数Oobjective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统Pparameter 参数partial derivatives 偏导数polynomial interpolation method多项式插值法Qquadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划Rreal-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性Sscalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method序列二次规划法simplex search method 单纯形法solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点step length 步长subspace trust region method 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵Ttermination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepest descent 最速下降法transpose 转置Uunconstrained 无约束的under-determined system 负定系统Vvariable 变量vector 矢量Wweighting matrix 加权矩阵3 样条词汇英汉对照表Aapproximation 逼近array 数组a spline in b-form/b-spline b样条a spline of polynomial piece /ppform spline 分段多项式样条Bbivariate spline function 二元样条函数break/breaks 断点Ccoefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation三次样条插值/三次样条内插curve 曲线Ddegree of freedom 自由度dimension 维数Eend conditions 约束条件Iinput argument 输入参数interpolation 插值/内插interval 取值区间Kknot/knots 节点Lleast-squares approximation 最小二乘拟合Mmultiplicity 重次multivariate function 多元函数Ooptional argument 可选参数order 阶次output argument 输出参数Ppoint/points 数据点Rrational spline 有理样条rounding error 舍入误差（相对误差）Sscalar 标量sequence 数列（数组）spline 样条spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量Wweight/weights 权重4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图converge 收敛coordinate 坐标系Ddecomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditionsDirichlet边界条件Eeigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解Ggeneralized Neumann boundary condition推广的Neumann边界条件geometry 几何形状geometry description matrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface（GUI）图形用户界面Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环Mmachine precision 机器精度mixed boundary condition 混合边界条件NNeuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量PParabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变plane stress 平面应力Poisson's equation 泊松方程polygon 多边形positive definite 正定Qquality 质量Rrefined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数Ssingular 奇异的。

数学专业英语(2)

Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@§2.2 Geometry and TrigonometryTEXT A Why study geometry?Why do we study geometry? The student beginning the study of this text may well ask, “What is geometry? What can I expect to gain fro m this study?”Many leading institutions of higher learning have 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. Geometry had its origin long ago in the measurements by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River. The Greek word geometry is derived from geo, meaning “earth”, and metron, meaning “measure”. As early as2000 B.C. we find the land surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry.Geometry is a science that deals with forms made by lines. A study of geometry is an essential part of the training of the successful engineer, scientist, architect, and draftsman. The carpenter, machinist, stonecutter, artist, and designer all apply the facts of geometry in their trades. In this course the student will learn a great deal about geometric figures such as lines, angles, triangles, circles, and designs and patterns of many kinds. 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 forming conclusions.There are many other less direct benefits the student of geometry may gain. Among these one must include training in the exact use of the English language and in the ability to analyze a new situation or problem into its basic parts, and utilizing perseverance, originality, and logical reasoning in solving the problem. An appreciation for the orderliness and beauty of geometric forms that abound in man’s works and of the creations of nature will be a byproduct of the study of geometry. The student should also develop an awareness of the contributions of mathematics and mathematicians to our culture and civilization.TEXT B Some geometrical terms1. Solids and planes.A solid is a three-dimensional figure. Common examples of solid are cube, sphere, cylinder, cone and pyramid.A cube has six faces which are smooth and flat. These faces are called plane surfaces or simply planes. A plane surface has two dimensions, length and width .The surface of a blackboard or a tabletop is an example of a plane surface.2. Lines and line segments.We are all familiar with lines, but it is difficult to define the term. A line may be represented by the mark made by moving a pencil or pen across a piece of paper. A line may be considered as having only one dimension, length. Although when we draw a line we give it breadth and thickness, we think only of the length of the trace when considering the line. A point has no length, no width, and no thickness, but marks a position. We arefamiliar with such expressions as pencil point and needle point. We represent a point by a small dot and name it by a capital letter printed beside it, as “point A” in Fig. 2-2-1.The line is named by labeling two points on it with capital letters or one small letter near it. The straight line in Fig. 2-2-2 is read “line AB” or “line l”. A straight line extends infinitely far in two directions and has no ends. The part of the line between two points on the line is termed a line segment. A line segment is named by the two end points. Thus, in Fig. 2-2-2, we refer to AB as line segment of line l. When no confusion may result, the expression “line segment AB” is often replace d by segment AB or, simply, line A B.There are three kinds of lines: the straight line, the broken line, and the curved line. A curved line or, simply, curve is line no part of which is straight. A broken line is composed of joined, straightline segments, as ABCDE of Fig. 2-2-3.3. Parts of a circle.A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point called the center (Fig. 2-2-4). The symbol for a circle is ⊙. In Fig. 2-2-4, O is the center of ⊙ABC, or simply of ⊙O. A line segment drawn from the center of the circle to a point on the circle is a radius (plural, radii) of the circle. OA, OB, and OC are radii of ⊙O, A diameter of a circle is a line segment through the center of the circle with endpoints on the circle. A diameter is equal to two radii.A chord is any line segment joining two points on the circle. ED is a chord of the circle in Fig. 2-2-4.From this definition is should be apparent that a diameter is a chord. Any part of a circle is an arc, such as arc AE. Points A and E divide the circle into minor arc AE and major arc ABE. A diameter divides a circle into two arcs termed semicircles, such as AB. Thecircumference is the length of a circle.SUPPLEMENT A Ruler-and-compass constructionsA number of ancient problems in geometry involve the construction of lengths or angles using only an idealised ruler and compass. The ruler is indeed a straightedge, and may not be marked; the compass may only be set to already constructed distances, and used to describe circular arcs.Some famous ruler-and-compass problems have been proved impossible, in several cases by the results of Galois theory. In spite of these impossibility proofs, some mathematical amateurs persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially soluble provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compasses alone. Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books.Squaring the circle The most famous of these problems, “squaring the circle”, involves constructing a square with the same area as a given circle using only ruler and compasses. Squaring the circle has been proved impossible, as it involves generating a transcendental ratio, namely 1:√π.Only algebraic ratios can be constructed with ruler and compasses alone. The phrase “squaring the circle” is often used to mean “doing the impossible”for this reason.Without the constraint of requiring solution by ruler and compasses alone, the problem is easily soluble by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.Doubling the cube Using only ruler and compasses, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.Angle trisection Using only ruler and compasses, construct an angle that is one-third of a given arbitrary angle. This requires taking the cube root of an arbitrary complex number with absolute value 1 and is likewise impossible.Constructing with only ruler or only compassIt is possible, as shown by Georg Mohr, to construct anything with just a compass that can be constructed with ruler and compass. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but given a circle and its center, they can be constructed.Problem How can you determine the midpoint of any given line segment with only compass?SUPPLEMENT B Archimedes and On the Sphere and the CylinderArchimedes(287 BC-212 BC) was an Ancient mathematician, astronomer, philosopher, physicist and engineer born in the Greek seaport colony of Syracuse. He is considered by some math historians to be one of history's greatest mathematicians, along with possibly Newton, Gauss and Euler.He was an aristocrat, the son of an astronomer, but little is known of his early life except that he studied under followers of Euclid in Alexandria, Egypt before returning to his native Syracuse, then an independent Greek city-state. Several of his books were preserved by the Greeks and Arabs into the Middle Ages, and, fortunately, the Roman historian Plutarch described a few episodes from his life. In many areas of mathematics as well as in hydrostatics and statics, his work and results were not surpassed for over 1500 years!He approximated the area of circles (and the value of ¼) by summing the areas of inscribed and circumscribed rectang les, and generalized this "method of exhaustion," by taking smaller and smaller rectangular areas and summing them, to find the areas and even volumes of several other shapes. This anticipated the results of the calculus of Newton and Leibniz by almost 2000 years!He found the area and tangents to the curve traced by a point moving with uniform speed along a straight line which is revolving with uniform angular speed about a fixed point. This curve, described by r = a in polar coordinates, is now called the "spiral of Archimedes." With calculus it is an easy problem; without calculus it is very difficult.The king of Syracuse once asked Archimedes to find a way of determining if one of his crowns was pure gold without destroying the crown in the process. The crown weighed the correct amount but that was not a guarantee that it was pure gold. Thestory is told that as Archimedes lowered himself into a bath he noticed that some of the water was displaced by his body and flowed over the edge of the tub. This was just the insight he needed to realize that the crown should not only weigh the right amount but should displace the same volume as an equal weight of pure gold. He was so excited by this idea that he reportedly ran naked through the streets shouting "Eureka" ("I have found it")."Give me a place to stand and I will move the earth" was his boast when he discovered the laws of levers and pulleys. Since it was impossible to challenge that statement directly, he was asked to move a ship which had required a large group of laborers to put into position. Archimedes did so easily by using a compound pulley system.During the war between Rome and Carthage, the Roman fleet decided to attack Syracuse, but Archimedes had been at work devising a few surprises. There were catapults with adjustable ranges which could throw objects which weighted over 500 pounds. The ships which survived the catapults were met with poles which reached over the city walls and dropped heavy stones onto the ships. Large grappling hooks attached to levers lifted the ships out of the water and then dropped them. During another failed assault, it is said that Archimedes had the soldiers of Syracuse use specially shaped and shined shields to focus the sunlight onto the sails to set them afire. This was more than the terrified sailors could stand, and the fleet withdrew. Unfortunately, the city began celebrating a bit early, and Marcellus captured Syracuse by attackingfrom the landward side during the celebration. "Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming upon him, commanded him to follow to Marcellus, which he declined to do before he had worked out his problem to a demonstration; the soldier, enraged, drew his sword and ran him through." (Plutarch)Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with the ratio of the cylinder's volume to that of the sphere. Archimedes considered the discovery of this ratio the greatest of all his accomplishments.Archimedes Discovers the Volume of a SphereArchimedes balanced a cylinder, a sphere, and a cone. All of the following dimensions shown in blue are equal.Archimedes imagined taking a circular slice out of all three solids.He then imagined hanging the cylinder and the sphere from point A and suspending the solids at point F (the fulcrum).By the law of the lever Archimedes showed that2r ⨯ (cone volume + sphere volume) = 4r ⨯ (cylinder volumes)Since cylinder volume = base ⨯ height = πr2⨯2r = 2πr3and cone volume = 1/3base ⨯ height = 1/3[π⨯ (2r)2] ⨯ (2r) = 8/3πr3so sphere volume = 2 cylinder volumes - cone volume = 4/3πr3Problem 1 Can you obtain the volume of a cone by the same argument above?Problem 2 (about π) Among the earliest Chinese circle-squarers mention must be made of Chang Hung in the first place. Chang’s calculation of the circle, however, has been lost, although his value of π is given in commentary on Arithmetic in Nine Sections in the form that the ratio of the square of the circular circumference to that of the perimeter of the circumscribed squareis 5 to 8. This is equivalent to taking π at ____.。

数学专业英语第二版的课文翻译

线性方程，二次方程等。

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 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 without making the equation untrue,until only the unknown quantity is left on one side ,no matter which side.解方程意味着求未知项的值，为了求未知项的值，当然必须移项，直到未知项单独在方程的一边，令其等于方程的另一边，从而求得未知项的值，解决了问题。

因此解方程意味着进行一系列的移项和同解变形，直到未知量被单独留在方程的一边，无论那一边。

Equation are of very great use. We can use equation in many mathematical problems. 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，可以用方程解决很多数学问题。

《数学专业英语》课件

The course provides learners with feedback on their performance and guidance on areas where improvement is needed to help them achieve their learning goals effectively
Basic mathematical terms
This includes words like "equation," "function," "integral," etc., which are essential for understanding mathematical concepts and e combinations of symbols that represent relationships between mathematical entities They are used to express mathematical ideas conceptually and precisely
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THANKS
Application of Mathematics Professional English
Use English to write research articles for international journals or conferences, enabling more people to learn about and cite your article.
Summary and Outlook
01
Deepen the understanding of mathematical English vocabulary and grammar

1-数学专业英语的特点

以单词形式出现的术语最基本。如 integer（整数）、diameter（直径） differential（微分）triangle（三角形）parallelogram（平行四边形） fractal（分形）sheaf（层、簇） continuum（连续统）probability（概率）homeomorphic（同胚的）homologous （同调的） holomorphic（全纯的）
3.主动语态句型也多数用于强调事实，而不是强调行为发出者及其情感
Given 0 ，there exists a number N>0 such that an a for all n N .
Since h(x) is harmonic on a neighborhood of
B(a,r), we have h(x)d (x) h(a). B
integers a, b, and c can satisfy the an bn cn
equation for any integer value of n greater than two.
2、非限定动词使用频率高
非限定动词包括不定式、分词（现在分词、过去分词）和动名词，使用频率高，经常用来代替从句，达到简练的要求
3、半专业性术语穿插频繁、词义多变
又如，power常表示“能力、体力”，电学中是“动力、电力、功率、倍率、度”，政治中是“权力”，数学中作“幂、乘方”解。再如，set在数学中是“集或集合”(n)、“令、假定”(v)，日常用语中可做动词、名词形容词，词意繁多。
3、半专业性术语穿插频繁、Байду номын сангаас义多变
又如，power常表示“能力、体力”，电学中是“动力、电力、功率、倍率、度”，政治中是“权力”，数学中作“幂、乘方”解。再如，set在数学中是“集或集合”(n)、“令、假定”(v)，日常用语中可做动词、名词形容词，词意繁多。