数学专业英语27528
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
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.
数学专业英语（Doc版）.12
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”。
数学专业英语(Doc版).14
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。
数学专业英语(Doc版).25
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。
数学专业英语（Doc版）.20
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<>。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
第一章 数学专业英语的阅读和翻译初级阶段1.1 数学专业英语的基本特点一 注意对客观事实与真理的描述： 1. 语句事态的使用上常用一般现在时例 An equation is a statement of the equality between two equal numbers or number symbols.2. 被动语态出现频率高“It is …”句型也使用得多例 It is cleat that any function defined for all positive real x may be used to construct a sequence by restricting x to take only integer values.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例 1：Given ε> 0，there exits a number N>0. such that ε<-a a n for all n ≥N 例 2：Since h(x) is harmonic on a neighborhood of B(a,r), we have)()()(a h x d x h B=⎰∂σ“we have ” =one has 可省略 译为 “可以得出”什么结论 二 科学内容的完整性与表达形式的精炼性要求 三 数学的专业性十分典型1、有的概念可能有几个同义词 如“计算”有：count calculate calculation calculus compute computation 2、同一词根的次和词组很多，如：Integrability 可积性 integrable 可积的 integral 积分，积分的，整数的 integral calculus 积分学 integralization 整化 integrate 积分 3、半专业词汇多是出现频率高 如: function 函数functional 泛函 power 幂 set 假定，令4、数学词汇具有自己的特点（不易记忆）5、表示条件，推理根据的句型相对固定如(1)用if，when, as为连接的条件从句例：The function f(x) approaches infinity as x tends to zero.(2)用with 短语表示条件或补充条件例：Suppose D is an open set with its closure in G.(3)用such that 为连接词的从句表示条件或补充条件例：Suppose f(x) is a function on a domain D such that ︳f(x) ︳< M for all x ∈D，when M is a constant.Such that =satisfying(4)虚拟语气表示条件.(5)用特殊动词，如：suppose, let, set, assume等表示假设.(6)表示推理的根据常用“by”短语，也用“according to”例：By lemma 2we have x≥y6、形成了一批数学特殊记号和表达式(1)充分必要条件The sufficient and necessary condition when and only when =if and only =iff(2)表示事先任意取定的量例：对任意的ε﹥0 即 For any number ε﹥0,或Given ε﹥01.2 数学专业英语的阅读和翻译一、翻译技巧1、词义的选择及引申例 1.A take its value 3 take value “取值”2.Two and three make fiveMake “制造”引申为“等于”3.Year after year and century after century the moon goes through itscycles of changes.月亮的盈亏变化，一年又一年，一世纪又一世纪，周而复始。
2、词语的增减：例1.An arithmetic or an algebraic identity is an equation 代数和算术的恒等式都是等式。
2. After checking .we see -3.does not, but 3 does.省略了“Satisfy the original equation”3、词序的变动：例 1.such is the case （情况就是这样）2.The square root of a negative number is a pure imaginary（负数的平方根是纯虚数。
）3.something new 某种新东西4.the element know.5.the conclusion required .所需要的结论4、词类的转变例Applied mathematics aims at achieving the optimization.（应用数学的目的是达到最优化）“aims” v→n5、句子成分的转换例1.The same signs and symbols of mathematics are used throughout the world.（全世界都使用同样的数学记号和符号。
）2.The statement of the Gauss Theorem is as follows.（高斯定理叙述如下）6、语态与人称的改变：例 We call a triangle an obtuse triangle when one angle is an obtuse angle.(有一个角为钝角的三角形被称为钝角三角形)7、数词翻译的变动例1: “increase（be, go up）+ by +数字和倍数+…”的结构中，by后表示的是净增加数和倍数“increase（be, go up）+数字和倍数+…”的结构中，应将增加的倍数减一例2：“decreased(fall, drop, lower, decrease …)+ by +数字”的结构中，by后表示的是纯减少的量“decreased(drop, decrease …)+ to +数字”的结构中，“to”表示“到”的意思，所以常译为“降到…”，“减少到…”。
二、数学专业英语翻译的要求及步骤(一) 要求准确、通顺且符合专业规范（二）步骤：⎧⎪⎨⎪⎩一、正确地理解原文二、恰当的表达成中文三、认真地进行校对三、长句翻译的实例分析例： The use of logarithms has decreased the labor of computing to such an extent that many calculations ,which would require hours without the use of logarithms. can be performed with their aid in a small fraction of that time.译为“对数的使用把计算的劳动强度减少到这样的程度，即许多计算当未采用对数时需耗费数个小时，而现在由于借助对数只要用原来的很少的一部分时间就能完成”.长句翻译步骤：（1）抓句子主干及语法特征;（2）大意群划分;（3）小意群划分;（4）综合得出准确翻译.小学二（2）班班规一、安全方面1、每天课间不能追逐打闹。
2、中午和下午放学要结伴回家。
3、公路上走路要沿右边走，过马路要注意交通安全。
4、不能在上学路上玩耍、逗留。
二、学习方面1、每天到校后，不允许在走廊玩耍打闹，要进教室读书。
2、每节课铃声一响，要快速坐好，安静地等老师来上课。
3、课堂上不做小动作，不与同桌说悄悄话，认真思考，积极回答问题。
4、养成学前预习、学后复习的好习惯。
每天按时完成作业，保证字迹工整，卷面整洁。
5、考试时做到认真审题，不交头接耳，不抄袭，独立完成答卷。
三、升旗排队和两操方面1、升旗时，要快速出教室排好队，做到快、静、齐，安静整齐地排队走出课室门，班长负责监督。
2、上午第二节后，快速坐好，按要求做好眼保健操。
3、下午预备铃声一响，在座位上做眼保健操。
四、卫生方面1、每组值日生早晨7：35到校做值日。
2、要求各负其责，打扫要迅速彻底，打扫完毕劳动工具要摆放整齐。
3、卫生监督员（剑锋，锶妍，炜薪）要按时到岗，除负责自己的值日工作外，还要做好记录。
五、一日常规1、每天学生到齐后，班长要检查红领巾。
2、劳动委员组织检查卫生。
3、每天负责领读的学生要督促学生学习。
4、上课前需唱一首歌，由文娱委员负责。
5、做好两操。
6、放学后，先做作业，然后帮助家长至少做一件家务事。
7、如果有人违反班规，要到老师处说明原因。
班训：坐如钟站如松快如风静无声班规：课堂听讲坐如钟，精神集中认真听；排队升旗站如松，做操到位展雄风；做事迅速快如风，样样事情记得清；自习课上静无声，踏实学习不放松；个人努力进步快，团结向上集体荣；我为领巾添光彩，标兵集体记我功。
加分标准扣分标准注：每人基本分60分起，学期末核算总分，作为学期评先依据。