数学专业英语
数学专业英语
举一个简单的例子，当常微分方程满足f‟(x) = f(x) 的关系时，最特殊的情况是指数函数 f(x) = ex
We shall see presently that every solution of (9.1) f‟(x) = f(x) must be of the form f(x) = Cex , where C may be any constant.
The study of differential equations is one part of mathematics that, perhaps more than any other , has been directly inspired by mechanics, astronomy, and mathematical physics .Its history began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equations arising from problems in geometry and mechanics .
displacement Bernoulli
n.位移
n. (人名) 伯努利
The Bernoullis 伯努利(家族) mathematical physics 数学物理
A large variety of scientific problems arise in which one tries to determine something from its rate of change.
这些方程叫做微分方程，他们的研究方式 是最难的数学分ations are classified under two main headings: ordinary and partial, depending on whether the unknown is a function of just one variable or of two or more variables. classified 分类 variable 变量
数学专业英语2-2
学生通过几何的学习而达到的最主要目标是：在听， 读，和思考时变得更加审慎。在学习几何的过程中， 他们不再盲目地接受一些陈述和思想，而是在得出结 论之前学会了清楚和审慎的思考。
2－B Some geometrical terms
1. Solids and planes. A solid is a three-dimensional figure. Common examples of solids are cube, sphere, cylinder, cone and pyramid.
点没有长度，宽度和厚度，但是标记了一个位置。我们熟悉铅 笔尖，针尖这样的表达。我们可以用一个小圆点来表示一个点， 在它旁边用打印体大写字母来命名，如图2-2-1中的点A。
The line is named by labeling two points on it with capital letters or one small letter near it. The 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.
有三种线：直线，折线和曲线。曲线是指其中没有任何部分是 直的。折线是由连起来的直线段构成，如图2-2-3中的ABCDE.
数学专业英语(Doc版).Word4
数学专业英语－Continuous Functions of One Real VariableThis lesson deals with the concept of continuity, one of the most important an d also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concep t in an informal and intuitive way to give the reader a feeling for its meanin g.Roughly speaking the situation is this: Suppose a function f has the value f ( p )at a certain point p. Then f is said to be continuous at p if at every ne arby point x the function value f ( x )is close to f ( p ). Another way of pu tting it is as follows: If we let x move toward p, we want the corresponding f unction value f ( x )to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x. At each integer we have what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x a pproaches 2 from the left, f ( x )approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x )does appro ach f ( 2 )if we let x approach 2 from the right, but this by itself is not en ough to establish continuity at 2. In case like this, the function is called conti nuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating loo k into the exact meaning of continuity. It was not until late in the 18th centur y that discontinuous functions began appearing in connection with various kind s of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19th century to examine m ore carefully the exact meaning of the word “continuity”.A satisfactory mathematical definition of continuity, expressed entirely in term s of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789-1857). His definition, whi ch is still used today, is most easily explained in terms of the limit concept to which we turn now.The definition of the limit of a function.Let f be a function defined in some open interval containing a point p, altho ugh we do not insist that f be defined at the point p itself. Let A be a real n umber.The equationf ( x ) = Ais read “The limit of f ( x ), as x approached p, is equal to A”, or “f ( x )approached A as x approached p.”It is also written without the limit symb ol, as follows:f ( x )→A as x →pThis symbolism is intended to convey the idea that we can make f ( x )as close to A as we please, provided we choose x sufficiently close to p.Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.Definition of neighborhood of a point.Any open interval containing a point p as its midpoint is called a neighborho od of p.NOTATION. We denote neighborhoods by N ( p ), N1( p ), N2( p )etc. S ince a neighborhood N( p )is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p )by N ( p; r )if we wish to specify its radius. The inequalities p-r < x < p+r are equiv alent to –r<x-p<r,and to ∣x-p∣< r. Thus N ( p; r )consists of all points x whose distance from p is less than r.In the next definition, we assume that A is a real number and that f is a fun ction defined on some neighborhood of a point p(except possibly at p) . Th e function may also be defined at p but this is irrelevant in the definition.Definition of limit of a function.The symbolismf ( x ) = A or [ f ( x )→A as x→p ]means that for every neighborhood N1( A )there is some neighborhood N2( p)such thatf ( x )∈N1( A ) whenever x ∈N2( p ) and x ≠p (*)The first thing to note about this definition is that it involves two neighborho ods,N1( A) andN2( p). The neighborhood N1( A)is specified first; it tells us how close we wish f ( x )to be to the limit A. The second neighborhood, N2( p ),tells u s how close x should be to p so that f ( x ) will be within the first neighbor hood N1( A). The essential part of the definition is that, for every N1( A),n o matter how small, there is some neighborhood N2(p)to satisfy (*). In gener al, the neighborhood N2( p)will depend on the choice of N1( A). A neighbo rhood N2( p )that works for one particular N1( A) will also work, of course, for every larger N1( A), but it may not be suitable for any smaller N1( A).The definition of limit can also be formulated in terms of the radii of the n eighborhoodsN1( A)and N2( p ). It is customary to denote the radius of N1( A) byεan d the radius of N2( p)by δ.The statement f ( x )∈N1( A ) is equivalent to the inequality ∣f ( x ) –A∣<ε,and the statement x ∈N1( A) ,x ≠p ,is equivalent to the inequalities 0<∣x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such th at∣f ( x ) –A∣<εwhenever 0 <∣x –p∣<δ“One-sided”limits may be defined in a similar way. For example, if f ( x )→A as x→p through values greater than p, we say that A is right-hand limi t of f at p, and we indicate this by writingf ( x ) = AIn neighborhood terminology this means that for every neighborhood N1( A) ,t here is some neighborhood N2( p) such thatf ( x )∈N1( A) wheneve r x ∈N1( A) and x > pLeft-hand limits, denoted by writing x→p-, are similarly defined by restricti ng x to values less than p.If f has a limit A at p, then it also has a right-hand limit and a left-hand li mit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.The definition of continuity of a function.In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not b e equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.Definition of continuity of a function at a point.A function f is said to be continuous at a point p if( a ) f is defined at p, and ( b ) f ( x ) = f ( p )This definition can also be formulated in term of neighborhoods. A function f is continuous at p if for every neighborhood N1( f(p))there is a neighborhood N2(p)such thatf ( x ) ∈N1( f (p)) whenever x ∈N2( p).In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that ∣f( x ) –f ( p )∣< εwhenever ∣x –p∣< δIn the rest of this lesson we shall list certain special properties of continuou s functions that are used quite frequently. Most of these properties appear obvi ous when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these state ments are no more self-evident than the definition of continuity itself, and ther efore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axio m for the real number system.THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a cl osed interval [a, b] and assume that f ( a )an f ( b )have opposite signs. T hen there is at least one c in the open interval (a ,b) such that f ( c )= 0.THEOREM 2. Sign-preserving property of continuous functions. Let f be conti nuious at c and suppose that f ( c )≠0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ).THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x1<x2in [a, b] such that f ( x1 ) ≠f ( x2 ). T hen f takes every value between f ( x1) and f(x2)somewhere in the interval ( x1,x2 ).THEOREM 4. Boundedness theorem for continuous functions. Let f be contin uous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤M for all x in [a, b].THEOREM 5. (extreme value theorem) Assume f is continuous on a closed i nterval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = su p f and f ( d ) = inf f .Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum.Vocabularycontinuity 连续性 assume 假定,取continuous 连续的 specify 指定, 详细说明continuous function 连续函数statement 陈述,语句intuitive 直观的 right-hand limit 右极限corresponding 对应的left-hand limit 左极限correspondence 对应 restrict 限制于graph 图形 assertion 断定approach 趋近,探索,入门consequently 因而,所以tend to 趋向 prove 证明regardless 不管,不顾 proof 证明discontinuous 不连续的 bound 限界jump discontinuity 限跳跃不连续least upper bound 上确界mathematician 科学家 greatest lower bound 下确界formulate 用公式表示,阐述boundedness 有界性limit 极限maximum 最大值Interval 区间 minimum 最小值open interval 开区间 extreme value 极值equation 方程extremum 极值neighborhood 邻域increasing function 增函数midpoint 中点decreasing function 减函数symmetric 对称的strict 严格的radius 半径(单数) uniformly continuous 一致连续radii 半径(复数) monotonic 单调的inequality 不等式monotonic function 单调函数equivalent 等价的Notes1. It wad not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems.意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.这里It was not until …that译为“直到……才”2. The symbol f ( x ) = A means that for every ε> 0 ,there is a δ> 0, such that|f( x ) - A|<εwhenever 0 <|x –p |<δ注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.有时句中there is可换为there exists; such that可换为satisfying; whenever换成if或for.3. Let…and assume (suppose)…Then…这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课Let假设条件的大前提,assume (suppo se)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.4. Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:f ( x ) approaches A as x approaches p. Approach有时可代以tend to. 如f ( x )tends to A as x tends t o p.值得留意的是approach后不加to而tend之后应加to.5. as close to A as we please = arbitrarily close to A..ExerciseI. Fill in each blank with a suitable word to be chosen from the words given below:independent domain correspondenceassociates variable range(a) Let y = f ( x )be a function defined on [a, b]. Then(i) x is called the ____________variable.(ii) y is called the dependent ___________.(iii) The interval [a, b] is called the ___________ of the function.(b) In set terminology, the definition of a function may be given as follows:Given two sets X and Y, a function f : X →Y is a __________which ___________with each element of X one and only one element of Y.II. a) Which function, the exponential function or the logarithmic function, has the property that it satisfi es the functional equationf ( xy ) = f ( x ) + f ( v )b) Give the functional equation which will be satisfied by the function which you do not choose in (a).III. Let f be a real-valued function defined on a set S of real numbers. Then we have the following two definitions:i) f is said to be increasing on the set S if f ( x ) < f ( y )for every pair of points x and y with x < y.ii) f is said to have and absolute maximum on the set S if there is a point c in S such that f ( x ) < f ( c )for all x∈S.Now definea) a strictly increasing function;b) a monotonic function;c) the relative (or local ) minimum of f .IV. Translate theorems 1-3 into Chinese.V. Translate the following definition into English:定义:设E 是定义在实数集E上的函数,那么, 当且仅当对应于每一ε>0(ε不依赖于E上的点)存在一个正数δ使得当p和q属于E且|p –q| <δ时有|f ( p ) – f ( q )|<ε,则称f在E上一致连续.。
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, we say 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 converges, 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 thatS 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 thats n<Sfor 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 haveS –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. Then the series ∑na n x n-1also converges absolutel y for∣x∣<r.A similar result holds for integration, but trivially. Indeed, if we have a series ∑a=1∞a n x n which converges absolutely for ∣x∣<r, then the series∑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∣<r. Then f is differentia ble for ∣x∣<r, andf′(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 abso lutely for ∣x∣<r. Then the relation∫f (x)d x=∑a=1∞a n x n+1∕n+1is valid in the interval ∣x∣<r.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！<c•n n+1•c-nⅡ. 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<r, and let c be a number such that x<c<r. Recall that lim a→∞n1/n=1.We may write n a n x n =a n(n1/n x)n. Then for all n sufficiently large, we conclude that n1/n x<c. This is because n1/n comes arbitrarily close to x and x<c. Hence for all n sufficiently large, we have na n x n<a n c n. We can then compare the series ∑nax 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 power 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;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。
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).Word5
数学专业英语－Differential CalculusHistorical IntroductionNewton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insur mountable problems could be solved by more or less routine methods.The succ essful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch o f calculus,differential calculus.The central idea of differential calculus is the notion of derivative.Like the inte gral,the derivative originated from a problem in geometry—the problem of find ing the tangent line at a point of a curve.Unlile the integral,however,the deriva tive evolved very late in the history of mathematics.The concept was not form ulated until early in the 17th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special func tions.Fermat’s idea,basically very simple,can be understood if we refer to a curve a nd assume that at each of its points this curve has a definite direction that ca n be described by a tangent line.Fermat noticed that at certain points where th e curve has a maximum or minimum,the tangent line must be horizontal.Thus t he problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.This raises the more general question of determining the direction of the tange nt line at an arbitrary point of the curve.It was the attempt to solve this gener al problem that led Fermat to discover some of the rudimentary ideas underlyi ng the notion of derivative.At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of findin g the tangent line at a point of a curve.The first person to realize that these t wo seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they explo ited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.Although the derivative was originally formulated to study the problem of tang ents,it was soon found that it also provides a way to calculate velocity and,mo re generally,the rate of change of a function.In the next section we shall consi der a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.A Problem Involving VelocitySuppose a projectile is fired straight up from the ground with initial velocity o f 144 feet persecond.Neglect friction,and assume the projectile is influenced onl y by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at ti me t we woule have f(t)=144 t.In actual practice,gravity causes the projectile t o slow down until its velocity decreases to zero and then it drops back to eart h.Physical experiments suggest that as the projectile is aloft,its height f(t) is gi ven by the formula(1)f(t)=144t –16 t2The term –16t2is due to the influence of gravity.Note that f(t)=0 when t=0 a nd when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.The problem we wish to consider is this:To determine the velocity of the proj ectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we int roduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient.Change in distance during time interval =f(t+h)-f(t)/hThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positiv e or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute v alue.The limit process by which v(t) is obtained from the difference quotient is wri tten symbolically as follows:V(t)=lim(h→0)[f(t+h)-f(t)]/hThe equation is used to define velocity not only for this particular example bu t,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.The example describe in the foregoing section points the way to the introducti on of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this in terval and introduce the difference quotient[f(x+h)-f(x)]/hwhere the number h,which may be positive or negative(but not zero),is such th at x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to a s the average rate of change of f in the interval joining x to x+h.Now we let h approach zero and see what happens to this quotient.If the quot ient.If the quotient approaches some definite values as a limit(which implies th at the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as “f prime of x”).Thus the formal defi nition of f’(x) may be stated as follows:Definition of derivative.The derivative f’(x)is defined by the equationf’(x)=lim(h→o)[f(x+h)-f(x)]/hprovided the limit exists.The number f’(x) is also called the rate of change of f at x.In general,the limit process which produces f’(x) from f(x) gives a way of ob taining a new function f’from a given function f.This process is called differ entiation,and f’is called the first derivative of f.If f’,in turn,is defined on an interval,we can try to compute its first derivative,denoted by f’’,and is calle d the second derivative of f.Similarly,the nth derivative of f denoted by f^(n),is defined to be the first derivative of f^(n-1).We make the convention that f^(0) =f,that is,the zeroth derivative is the function itself.Vocabularydifferential calculus微积分differentiable可微的intergral calculus 积分学differentiate 求微分hither to 迄今 integration 积分法insurmountable 不能超越 integral 积分routine 惯常的integrable 可积的fuse 融合integrate 求积分originate 起源于sign-preserving保号evolve 发展，引出 axis 轴（单数）tangent line 切线 axes 轴（复数）direction 方向 contradict 矛盾horizontal 水平的contradiction 矛盾vertical 垂直的 contrary 相反的rudimentary 初步的，未成熟的composite function 合成函数，复合函数area 面积composition 复合函数intimately 紧密地interior 内部exploit 开拓，开发 interior point 内点inaugurate 开始 imply 推出，蕴含projectile 弹丸 aloft 高入云霄friction摩擦initial 初始的gravity 引力 instant 瞬时rate of change 变化率integration by parts分部积分attain 达到definite integral 定积分defferential 微分indefinite integral 不定积分differentiation 微分法 average 平均Notes1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.意思是：在很大程度上是牛顿和莱伯尼，他们相互独立地把积分学的思想发展到这样一种程度，使得迄今一些难于超越的问题可以或多或少地用通常的方法加以解决。
数学专业英语
数学专业英语Lesson 1Mathematics as a Language of Scienceassert vt. 断言；坚持主张；维护表明qualitative adj. 性质的；定性的quantitative adj. 量的；数量的；定量的；与数量有关的astronomy n. 天文学postulate n. 假定, 基本条件, 基本原理 vt. 要求, 假定 vi. 要求hypothetical adj. 假设的, 假定的,爱猜想的Lesson 2deduction n. 减除, 扣除, 减除额, 推论, 演绎induction n. 归纳；归纳法；归纳所得之结论verification n. 验证；证实correlate vt. 使相互关联 vi. 和...相关discard vt. 丢弃, 抛弃 v. 放弃discredit n. 不信任；失信consistent adj. 一致的, 调和的, 坚固的, [数、统]相容的inadequacy n. 不充分 ,不适当,不适合,不足额conic, conical adj 圆锥的；圆锥形的ellipse n. 椭圆, 椭圆形 ellipt (n.)hyperbolic adj. 双曲线的 hyperbola (n.)parabolic adj. 用寓言表达的: 抛物线的，像抛物线的 parabola (n.) algebraic adj. 代数的, 关于代数学的mineralogy n. 矿物学refraction n. 折光, 折射stimulus n. 刺激物, 促进因素, 刺激, 刺激impetus n. 冲力推动力；刺激Lesson 3Axioms, definitions and Theoremsaxiom n. [数]公理definition n. 阐明；确定定义；界说extravagant adj. 奢侈的, 浪费的, 过分的, 放纵的collinear adj. 在同一直线上的, 同线的convex adj. 凸出的；凸面的segment n. 部分；片段；节, 弓形；圆缺；弧形, 线段conswquently adv. 从而, 因此in terms of adv. 根据, 按照, 用...的话, 在...方面pretense n. 主张, 要求, 伪称, 借口, 自称Lesson 4Geometry and Geometrical termsterm n. 学期, 期限, 期间, 条款, 条件, 术语triangle n. [数]三角形, 三人一组, 三角关系parallelogram n. 平行四边形straight angle n. [数]平角right angle n. 直角acute angle n. 锐角obtuse angle n. 钝角reflex angle n. 优角rectilinear adj 直线的；由直线组成的；循直线进行的isosceles triangle n. 等腰三角形equilateral triangle n. 等边三角形right triangle n. 直角三角形obtuse triangle n. 钝角三角形acute triangle n. 锐角三角形equiangular triangle n. 正三角形,等角三角形hypotenuse n. （直角三角形的）斜边circle 圆center 中心；中央；圆心diameter n. 直径radius n. 半径, 范围, 辐射光线, 有效航程, 范围, 界限circumference n. 圆周, 周围Lesson 5The Method of Limitslimit n. 限度,极限,极点infinite adj. 无限的；无穷的infinitesimal adj. 无穷小的, 极小的, 无限小的calculus n. 微积分学, 结石exemplify vt. 例证, 例示, 作为...例子inscribe v. 记下polygon n. [数]多角形, 多边形diminish v. (使)减少, (使)变小curvilinear adj 曲线的, 由曲线组成的intuition n. 直觉, 直觉的知识integral n. [数学] 积分, 完整, 部分defective adj. 有缺陷的, (智商或行为有)欠缺的differential coefficient 微分系数arithmetical adj. 算术的, 算术上的convergence n. 集中, 收敛criterion n. (批评判断的)标准, 准据, 规范sequence n. 次序, 顺序, 序列irrational numbers n. [数]无理数domain ，定义域contradiction 矛盾reversal n. 颠倒, 反转, 反向, 逆转, 撤销Lesson 6Functioncontinuous variable 连续变量;［连续变数］variation 变分, 变化interval 区间independent variable 自变量dependent variable 应变量rectangular coordinate 直角坐标abscissa n. 〈数〉横坐标ordinate n. [数]纵线, 纵座标gradient adj. 倾斜的n. 梯度, 倾斜度, 坡度slope n. 斜坡, 斜面, 倾斜 v. (使)顺斜Lesson 7Differential and Integral calculusdifferential adj. 微分的n. 微分 (differentiation)Integral n. [数学] 积分, 完整, 部分 (integration)calculus n. 微积分学, 结石interrelation n. 相互关系trigonometry n. 三角法exponential adj. 指数的, 幂数的logarithm n. [数] 对数derivative n. 导数；微商tangent n. 切线, [数]正切counterclockwise adj. 反时针方向的adv. 反时针方向 (clockwise) definite integral 定积分approximation n. 接近, 走近, [数]近似值culminate v. 达到顶点mean n. 平均数, 中间, 中庸differential equation 微分方程extreme value n. 极值multiple integral 多重积分double integralline integralfunctional analysis 泛函分析Lesson 8 The Concept of Cardinal Number (I)cardinal number n. 基数（如： 1, 2, 3, ... 有别于序数）denumerable adj. 可数的aggregate n. 合计, 总计, 集合体adj. 合计的, 集合的, 聚合的v. 聚集, 集合, 合计purport n. 主旨 v. 声称fancier n. 空想家, 培育动物(或植物)的行家, 爱好者sniff v. 用力吸, 嗅, 闻到, 发觉, 轻视, 用力吸气n. 吸, 闻, 吸气声, 嗤之以鼻scheme n. 安排, 配置, 计划, 阴谋, 方案, 图解, 摘要v. 计划, 设计, 图谋, 策划, * n.（计算数学）方法，格式superior n. 长者, 高手, 上级adj. 较高的, 上级的, 上好的, 出众的, 高傲的cumbersome adj. 讨厌的, 麻烦的, 笨重的instruction n. 指示, 用法说明(书), 教育, 指导, 指令drastically adv. 激烈地, 彻底地conservation 守衡律quadrature n. 求积, 求积分interpolation n. 插值extrapolation n. [数]外推法, 推断internal point 内点identical adj. 同一的, 同样的generalized solution 广义解functional 泛函hydrodynamics 流体力学，水动力学divergence 发散（性），梯度，发散量play an important (fundamental ... ) role 起着重要的（...）作用integro-interpolation method 积分插值法Variational method 变分方法comparatively adv. 比较地, 相当地deficiency n. 缺乏, 不足fictive adj. 虚构的, 想象上的, 虚伪的self-adjoint (nonself-adjoint) 自治的，自伴的，自共轭的finite element method 有限元法spline approximation 样条逼近Particles-in-the-Cell 网格质点法herald n. 使者, 传令官, 通报者, 先驱, 预兆vt. 预报, 宣布, 传达, 欢呼advection n. 水平对流phenomenological adj. 现象学的, 现象的fluctuation n. 波动, 起伏optimism n. 乐观, 乐观主义pessimism n. 悲观, 悲观主义unjustified adj 未被证明其正确的mean-square 均方dispersion n. [数] 离差, 差量Polynomial n adj. [数]多项式的interpolation 插值arithmetic n. 算术, 算法rounding errors 舍入误差multiple n. 倍数, 若干subjective adj. 主观的, 个人的objective adj. 客观的,outcome n. 结果, 成果pattern n. 样品toss v. 投, 掷exhaust vt. 用尽, 耗尽, 抽完, 使精疲力尽divisible adj. 可分的dice, die n. 骰子assign vt. 分配, 指派attach vt. 缚上, 系上, 贴上v. 配属, 隶属于pitfall n. 缺陷chairperson 主席mechanics n. (用作单数)机械学、力学, (用作复数)技巧, 结构statics n. [物]静力学dynamics n. 动力学adequately adv. 充分地celestial adj. 天上的macroscopic adj. 肉眼可见的, 巨观的classical field theory 经典场理论rigit adj. 刚硬的, 刚性的, 严格的elastic adj. 弹性的plastic n. 可塑的,塑性的,塑料的quantum n. 量, 额, [物] 量子, 量子论inception n. 起初, 获得学位pertain v. 适合, 属于gravitation n. 地心吸力, 引力作用tide n. 潮, 潮汐, 潮流, 趋势monumental adj. 纪念碑的, 纪念物的, 不朽的, 非常的encompass v. 包围, 环绕, 包含或包括某事物ingredient n. 成分, 因素acquainted adj. 有知识的, 知晓的synonymous adj. 同义的configuration n. 构造, 结构, 配置, 外形reference n. 提及, 涉及, 参考, 参考书目inertia n. 惯性, 惯量attribute 特性momentum n. 动量proportional adj. 比例的, 成比例的, 相称的, 均衡的designate 指明negligible adj. 可以忽略的, 不予重视的projectile n. 射弹 adj. 发射的ballistics n. 弹道学, 发射学intractable adj. 难处理的{Mechanics of a Particlein consequence of adv. 由于的...缘故exert vt. 尽(力), 施加(压力等), 努力v. 发挥, 竭尽全力, 尽galaxy n. 星系, 银河, 一群显赫的人, 一系列光彩夺目的东furnish vt. 供应, 提供, 装备, 布置v. 供给torque n. 扭矩, 转矩moment 力矩的friction 摩擦dissipation n. 消散, 分散, 挥霍, 浪费, 消遣, 放荡, 狂饮infer v. 推断Hooke s Law and Its Consequenceselasticity n. 弹力, 弹性constitutive adj. 构成的, 制定的atomistic adj. 原子论的crack n. 裂缝, 噼啪声v. (使)破裂, 裂纹, (使)爆裂continuum mechanics n. 连续介质力学superposition n. 重叠, 重合, 叠合strain n. 过度的疲劳, 紧张, 张力, 应变vt. 扭伤, 损伤v. 拉紧, 扯紧, (使)紧张, 尽力thermodynamics n. [物] 热力学reckon vt. 计算, 总计, 估计, 猜想vi. 数, 计算, 估计, 依赖, 料想lesson 20strength 强度load 载荷empirical 以经验为依据的member 构件isolated 孤立的segment 部分、段、节stress 应力strain 应变tension 拉伸shear 剪切bend 弯曲torsion 扭转、扭力insofar 在……范围cohesive 内聚性的tensile 拉力、张力stiffness 硬度furnish 供给Lesson 23 Fluid Mechanicseruption 喷发、爆发turbulent 湍流laminar 层流isothermal 等温isotropic 各向同性prevalent 普遍的、流行的tornado 旋风、飓风eddy 旋涡viscosity 粘性、粘度nonviscous 无粘性的rotation 旋转adiabatic 绝热的reversible 可逆的isentropic 等熵的instant 瞬时的streamline 流线stream tube 流管tangential 切线的incompressible 不可压缩的resultant 合成的,组合的downstream 下游的,顺流的elbow 弯管,肘similitude 相似性hydraulic 水力的,水力学的predominante 占主导地位spillway (河或水坝的)放水道,泄洪道prototype 原型,样板Lesson 24 Mechanical Vibration repetitive 重复的,反复的periodic 周期的,定期的tidal 潮的,像潮的stationary 固定的,不动的vibratory 振动的,摆动的propagation 传播couple v .连接,连合acoustic 听觉的,声学的annoyance 烦恼,困惑adjacent 接近的,邻近的damp 阻尼,衰减restore 复职,归还neutral 平衡exciting force 激励力resonant adj. 共振的,谐振的stiffness 刚度,刚性proportionality 成比例地inclusion 包含,包括magnitude 数值,大小substantially adv. 实质上的perturb 干扰,扰乱resonance n. 共振vibratory adj. 振动的, 可知的perceptible 可见的,可知的adudible 听得见的,可闻的foregoing 前述的impulsive 冲击的shock 冲击Fourier series 傅里叶级数excitation 激发,激励discrete 分离,离散的contend with 向…作斗争compressor 压气机fatigue 疲劳perceptible 可见的,可知觉的shredder 切菜器disposal 处理urban 都市的metropolitan 大都市的at-grade 在同一水平面上elevated 高架的guideway 导轨Lesson 25 A prefect to the Continuum Mechanics preface 序言continuum连续 pl. continuua rigid body 刚体contemporary 当代的,同时期的widespread 分布广的, 普及的accommodate 容纳,使适应medium 介质plasticity 塑性residual 剩余的,残留的creep 蠕变,爬行,塑性变形aging 老化polymeric聚合(物)的sandy 沙的,沙质的aubterranean 地下的,隐藏的essence 精髓,本质thermodynamics 热力学self-similar 自相似expedient 方便的sonsolidate 把…联合为一体,统一justify 证明…有理radically 根本地,本质上deliberate 从容不迫的,深思熟虑Lesson 33 what is a computer Attribute v. 赋予medieval 中世纪的astronomer 天文学家Mars 火星resemble vt. 像,相似tedious adj. 冗长乏味的pulp 浆状物,果肉filter vt.过滤underlying adj. 潜在的, 基本的ore n. 矿沙,矿石perceive v. 察觉,看见intervention n. 干涉,插入intelligent adj. 有智力的,聪明的Lesson 34 A computer system manipulate vt. 操纵,使用chip n. 芯片etch vt. 蚀刻,蚀镂fingernail 指甲mount vt. 安装,安置assemble vt. 集合,聚集cabinet 橱柜execute vt. 执行,实现paycheck n.支付薪金的支票bar chart 直方图joystick 游戏杆encounter vt. 遇到,遇上Mathematical Modelingindustry n. 工业, 产业, 行业, 勤奋commerce n. 商业complexity n. 复杂(性), 复杂的事物, 复杂性career n. (原意:道路, 轨道)事业, 生涯, 速度outset n. 开端, 开始essence n. 基本, [哲]本质, 香精advocation n. (=advocacy)拥护支持provision n. 供应, (一批)供应品, 预备, 防备, 规定publicize v. 宣扬roundabout adj. 迂回的, 转弯抹角的n. 道路交叉处的环形路, 迂回路线, 兜圈子的话trial-error vt. n. 试制, 试生产maneuverability n. 可操作性, 机动性vehicle n. 交通工具, 车辆, 媒介物, 传达手段junction n. 连接, 接合, 交叉点, 汇合处ponder v. 沉思, 考虑contrive v. 发明, 设计, 图谋snooker n. (=snooker pool)彩色台球, 桌球context n. 上下文, 文章的前后关系deviation n. 背离数学专业英语－Groups and RingsDuring the present century modern abstract algebra has become more and more important as a tool for research not only in other branches of mathematics bu t even in other sciences .Many discoveries in abstract algebra itself have been made during the past years and the spirit of algebraic research has definitely t ended toward more abstraction and rigor so as to obtain a theory of greatest p ossible generality. In particular, the concepts of group ,ring,integral domain and field have been emphasized.The notion of an abstract group is fundamental in all sciences ,and it is certai nly proper to begin our subject with this concept. Commutative additive groupsare made into rings by assuming closure with respect to a second operation h aving some of the properties of ordinary multiplication. Integral domains and fi elds are rings restricted in special ways and may be fundamental concepts and their more elementary properties are the basis for modern algebra.GroupsDEFINITION A non-empty set G of elements a,b,…is said to form a group with respect to 0 if:I.G is closed with respect to 0II.The associative law holds in G, that isaо(bоc)=(aоb)оcfor every a, b, c of GⅢ. For every a and b of G there exist solutions χand Уin G of the equ ationsaοχ=b yοa=bA group is thus a system consisting of a set of elements and operation οwit h respect to which G forms a group. We shall generally designate the entire s ystem by the set G of its elements and shall call G a group. The notation use d for the operation is generally unimportant and may be taken in as convenien t a way as possible.DEFINITION A group G is called commutative or abelian ifaοb=bοaFor every a and b of G.An elementary physical example of an abelian group is a certain rotation grou p. We let G consist of the rotations of the spoke of a wheel through multiples of 90ºand aοb be the result of the rotation a followed by the rotation b. T he reader will easily verify that G forms a group with respect to οand that aοb=bοa. There is no loss of generality when restrict our attention to multipl icative groups, that is, write ab in stead of aοb.EQUIVALENCEIn any study of mathematical systems the concept of equivalence of systems of the same kind always arises. Equivalent systems are logically distinct but weusually can replace any one by any other in a mathematical discussion with no loss of generality. For groups this notion is given by the definition: let G an d G´be groups with respective operations o and o´,and let there be a1-1 corr espondenceS : a a´ (a in G and a´in G´)between G and G´such that(aοb)´=a´οb´for all a, b of G. then we call G and G´equivalent(or simply, isomorphic)grou ps.The relation of equivalence is an equivalence relation in the technical sense in the set of all groups. We again emphasize that while equivalent groups may be logically distinct they have identical properties.The groups G and G´of the above definition need not be distinct of course a nd o´may be o. when this is the case the self-equivalence S of G is called a n automorphism.I: a aOf G, but other automorphisms may also exist.RingsA ring is an additive abelian groupB such thatI.the set B is closed with respect to a second operation designated by multiplication; that is , every a and b of B define a unique element ab of B. II.multiplication is associative; that isa (bc) = (ab)cfor every a, b, c of B.Ⅲ. The distributive lawsa (b+c) = ab +ac (b+c) a=ba +cahold for every a, b, c of B.The concept of equivalence again arises. We shall writeB ≌B′to mean that B and B′are equivalent.VocabularyGroup 群rigor 严格ring 环 generalization 推广integral domain 整环Abelian group 阿贝尔群commutative additive group 可交换加法群 rotation 旋转automorphism 自同构数学专业英语－Historical introduction of CalculusThe Two Basic Concepts of CalculusThe remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in phys ics,engineering,chemistry,geology,biology, and other fields including,rather recentl y,some of the social sciences.To give the reader an idea of the many different types of problems that can b e treatedby the methods of calculus,we list here a few sample questions.With what speed should a rocket be fired upward so that it never returns to e arth? What is the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter Ｌ? What volume of material is removed fr om a solid sphere of radius 2 r if a hole of redius r is drilled through the ce nter? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour,by how much will it increase at th e end of two hours? If a ten-pound force stretches an elastic spring one inch,h ow much work is required to stretch the spring one foot?These examples,chosen from various fields,illustrate some of the technical quest ions that can be answered by more or less routine applications of calculus.Calculus is more than a technical tool－it is a collection of fascinating and ex eiting idea that have interested thinking men for centuries.These ideas have to do with speed,area,volume,rate of growth,continuity,tangent line,and otherconcept s from a varicty of fields.Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is it s unifying power.Most of these ideas can be formulated so that they revolve a round two rather specialized problems of a geometric nature.We turn now to a brief description of these problems.Consider a cruve C which lies above a horizontal base line such as that show n in Fig.1. We assume this curve has the property that every vertical line inter sects it once at most.The shaded portion of the figure consists of those pointe which lie below the curve C , above the horizontal base,and between two para llel vertical segments joining C to the base.The first fundamental problem of c alculus is this: To assign a number which measures the area of this shaded re gion.Consider next a line drawn tangent to the curve,as shown in Fig.1. The second fundamental problem may be stated as follows:To assign a number which me asures the steepness of this line.Basically,calculus has to do with the precise formulation and solution of these two special problems.It enables us to define the concepts of area and tangent l ine and to calculate the area of a given region or the steepness of a given an gent line. Integral calculus deals with the problem of area while differential cal culus deals with the problem of tangents.Historical BackgroundThe birth of integral calculus occurred more than 2000 years ago when the Gr eeks attempted to determine areas by a procees which they called the method of exhaustion.The essential ideas of this ,method are very simple and can be d escribed briefly as follows:Given a region whose area is to be determined,we inscribe in it a polygonal region which approximates the given region and whos e area we can easily compute.Then we choose another polygonal region which gives a better approximation,and we continue the process,taking polygons with more and more sides in an attempt to exhaust the given region.The method is illustrated for a scmicircular region in Fig.2. It was used successfully by Arch imedes(287－212 B.C.) to find exact formulas for the area of a circle and a fe w other special figures.The development of the method of exhaustion beyond the point to which Ar chimcdcs carried it had to wait nearly eighteen centuries until the use of algeb raic symbols and techniques became a standard part of mathematics. The eleme ntary algebra that is familiar to most high-school students today was completel y unknown in Archimedes’time,and it would have been next to impossible to extend his method to any general class of regions without some convenient w ay of expressing rather lengthy calculations in a compact and simpolified form.A slow but revolutionary change in the development of mathematical notations began in the 16th century A.D. The cumbersome system of Roman numerals was gradually displaced by the Hindu-Arabic characters used today,the symbol s “+”and “-”were introduced for the forst time,and the advantages of the decimal notation began to be recognized.During this same period,the brilliant su ccesse of the Italian mathematicians Tartaglia,Cardano and Ferrari in finding al gebraic solutions of cubic and quadratic equations stimulated a great deal of ac tivity in mathematics and encouraged the growth and acceptance of a new and superior algebraic language. With the wide spread introduction of well-chosen algebraic symbols,interest was revived in the ancient method of exhaustion an d a large number of fragmentary results were discovered in the 16 th century by such pioneers as Cavalieri, Toricelli, Roberval, Fermat, Pascal, and Wallis.Fig.2. The method of exhaustion applied to a semicircular region.Gradually the method of exhaustion was transformed into the subject now calle d integral calculus,a new and powerful discipline with a large variety of applic ations, not only to geometrical problems concerned with areas and volumes but also to jproblems in other sciences. This branch of mathematics, which retaine d some of the original features of the method of exhaustion,received its bigges t impetus in the 17 th century, largely due to the efforts of Isaac Newion (16 42—1727) and Gottfried Leibniz (1646—1716), and its development continued well into the 19 th century before the subject was put on a firm mathematical basis by such men as Augustin-Louis Cauchy (1789-1857) and Bernhard Riem ann (1826-1866).Further refinements and extensions of the theory are still being carried o ut in contemporary mathematicsVocabularygeology 地质学decimal 小数，十进小数biology 生物学discipline 学科social sciences 社会科学 contemporary 现代的disk (disc) 圆盘bacteria 细菌isosceles triangle 等腰三角形 elastic 弹性的perimeter 周长 impetus 动力volume 体积 proportional to 与…成比例center 中心 inscribe 内接steepness 斜度 solid sphere 实心球method of exhaustion 穷举法 refinement 精炼，提炼polygon 多边形，多角形 cumbersome 笨重的，麻烦的polygonal 多角形fragmentary 碎片的，不完全的approximation 近似，逼近 background 背景学专业英语－How to Organize a paper (For Beginers)?The usual journal article is aimed at experts and near-experts, who are the peo ple most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There ar e many good ways of organizing a paper which can be learned by studying pa pers of the better expositors. The following suggestions describe a standard acc eptable style.Choose a title which helps the reader place in the body of mathematics. A use less title: Concerning some applications of a theorem of J. Doe. A. good titlecontains several well-known key words, e. g. Algebraic solutions of linear parti al differential equations. Make the title as informative as possible; but avoid re dundancy, and eschew the medieval practice of letting the title serve as an infl ated advertisement. A title of more than ten or twelve words is likely to be m iscopied, misquoted, distorted, and cursed.The first paragraph of the introduction should be comprehensible to any mathe matician, and it should pinpoint the location of the subject matter. The main p urpose of the introduction is to present a rough statement of the principal resul ts; include this statement as soon as it is feasible to do so, although it is som etimes well to set the stage with a preliminary paragraph. The remainder of th e introduction can discuss the connections with other results.It is sometimes useful to follow the introduction with a brief section that estab lishes notation and refers to standard sources for basic concepts and results. N ormally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.The section following the introduction should contain the statement of one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a ma jor theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts.Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by J.E.Littlewood, in wh ich the contrast between barbaric and civilized proofs is beautifully and amusin gly portrayed. To achieve conceptual proofs, it is often helpful for an author t o adopt an initial attitude such as one would take when communicating mathe matics orally (as when walking with a friend). Decide how to state results wit h a minimum of symbols and how to express the ideas of the proof without c omputations. Then add to this framework the details needed to clinch the resul ts.Omit any computation which is routine (i.e. does not depend on unexpected tri cks). Merely indicate the starting point, describe the procedure, and state the o utcome.It is good research practice to analyze an argument by breaking it into a succe ssion of lemmas, each stated with maximum generality. It is usually bad practi ce to try to publish such an analysis, since it is likely to be long and unintere sting. The reader wants to see the path-not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice l ater on.The rudiments of grammar are important. The few lines written on the blackbo ard during an hour’s lecture are augmented by spoken commentary, and aat t he end of the day they are washed away by a merciful janitor. Since the publ ished paper will forever speak for its author without benefit of the cleansing s ponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless a ppropriate.The barbarism called the dangling participle has recently become more prevalen t, but not less loathsome. “Differentiating both sides with respect to x, the eq uation becomes---”is wrong, because “the equation”cannot be the subject th at does the differentiation. Write instead “differentiating both sides with respec t to x, we get the equation---,”or “Differentiation of both sides with respect to x leads to the equation---”Although the notion has gained some currency, it is absurd to claim that infor mal “we”has no proper place in mathematical exposition. Strict formality is appropriate in the statement of a theorem, and casual chatting should indeed b e banished from those parts of a paper which will be printed in italics. But fif teen consecutive pages of formality are altogether foreign to the spirit of the t wentieth century, and nearly all authors who try to sustain an impersonal digni fied text of such length succeed merely in erecting elaborate monuments to slu msiness.A sentence of the form “if P,Q”can be understood. However “if P,Q,R,S,T”is not so good, even if it can be deduced from the context that the third co mma is the one that serves the role of “then.”The reader is looking at the paper to learn something, not with a desire for mental calisthenics.Vocabularypreliminary 序,小引(名)开端的,最初的(形)eschew 避免medieval 中古的，中世纪的inflated 夸张的comprehensible 可领悟的，可了解的pinpoint 准确指出（位置）weave 插入，嵌入unobtrusivcly 无妨碍地triviality 平凡琐事barbarism 野蛮，未开化portray 写真，描写clinch 使终结rudiment 初步，基础commentary 注解，说明janitor 看守房屋者sponge 海绵dangling participle 不连结分词prevalent 流行的，盛行loathsome 可恶地absurd 荒谬的banish 排除sustain 维持，继续slumsiness 粗俗，笨拙monument 纪念碑calisthenics 柔软体操，健美体操notes1. 本课文选自美国数学会出版的小册子A mamual for authors of mathematical paper的一节，本文对准备投寄英文稿件的读者值得一读。
数学专业英语(Doc版).11
数学专业英语－Linear AlgebraFor the definition that follows we assume that we are given a particular field K. The scalars to be used are to be elements of K.DEFINITION. A vector space is a set V of elements called vectors satisfyi ng the following axioms.(A) To every pair, x and y ,of vectors in V corresponds a vector x+y,call ed the sum of x and y, in such a way that.(1) addition is commutative, x + y = y + x.(2) addition is associative, x + ( y + z ) = ( x + y ) + z.(3) there exists in V a unique vector 0 (called the origin ) such that x + 0 = x for every vector x , and(4) to every vector x in V there corresponds a unique vector - x such that x + ( - x ) = 0.(B) To every pair,αand x , where αis a scalar and x is a vector in V ,the re corresponds a vector αx in V , called the product of αand x , in such a way that(1) multiplication by scalars is associative,α(βx ) = (αβ) x(2) 1 x = x for every vector x.(C) (1) multiplication by scalars is distributive with respect to vector addition,α( x + y ) = αx+βy , and(2)multiplication by vectors is distributive with respect to scalar addition,(α+β) x = αx + βx .The relation between a vector space V and the underlying field K is usually d escribed by saying that V is a vector space over K . The associated field of s calars is usually either the real numbers R or the complex numbers C . If V i s linear space and M真包含于V , and if αu -v belong to M for every u an d v in M and every α∈ K , then M is linear subspace of V . If U = { u 1,u 2,…} is a collection of points in a linear space V , then the (linear) span of the set U is the set of all points o the form ∑c i u i, where c i∈ K ,and all but a finite number of the scalars c i are 0.The span of U is al ways a linear subspace of V.A key concept in linear algebra is independence. A finite set { u 1,u 2,…, u} is said to be linearly independent in V if the only way to write 0 = ∑kc i u i is by choosing all the c i= 0 . An infinite set is linearly independent if every finite set is independent . If a set is not independent, it is linearlyd ependent, and in this case, some point in the set can be written as a linear co mbination of other points in the set. A basis for a linear space M is an indep endent set that spans M . A space M is finite-dimensional if it can be spanne d by a finite set; it can then be shown that every spanning set contains a basi s, and every basis for M has the same number of points in it. This common number is called the dimension of M .Another key concept is that of linear transformation. If V and W are linear sp aces with the same scalar field K , a mapping L from V into W is called lin ear if L (u + v ) = L( u ) + L ( v ) and L ( αu ) = αL ( u ) for ever y u and v in V and αin K . With any I , are associated two special linear spaces:ker ( L ) = null space of L = L-1 (0)= { all x ∈V such that L ( X ) = 0 }Im ( L ) = image of L = L( V ) = { all L( x ) for x∈V }.Then r = dimension of Im ( L ) is called the rank of L. If W also has dime nsion n, then the following useful criterion results: L is 1-to-1 if and only if L is onto.In particular, if L is a linear map of V into itself, and the only solu tion of L( x ) = 0 is 0, then L IS onto and is therefore an isomorphism of V onto V , and has an inverse L -1. Such a transformation V is also said to b e nonsingular.Suppose now that L is a linear transformation from V into W where dim ( V ) = n and dim ( W ) = m . Choose a basis {υ 1 ,υ 2 ,…,υn} for V and a basis {w 1 ,w2 ,…,w m} for W . Then these define isomorphisms of V onto K n and W onto K m, respectively, and these in turn induce a linear transfor mation A between these. Any linear transformation ( such as A ) between K n and K m is described by means of a matrix ( a), according to the formula Aij( x ) = y , where x = { x1, x 2,…, x n} y = { y1, y 2,…, y m} and Y j =Σn j=i a ij x i I=1,2,…,m.The matrix A is said to represent the transformation L and to be the represent ation induced by the particular basis chosen for V and W .If S and T are linear transformations of V into itself, so is the compositic tra nsformation ST . If we choose a basis in V , and use this to obtain matrix re presentations for these, with A representing S and B representing T , then ST must have a matrix representation C . This is defined to be the product AB o f the matrixes A and B , and leads to the standard formula for matrix multipli cation.The least satisfactory aspect of linear algebra is still the theory of determinants even though this is the most ancient portion of the theory, dating back to Lei bniz if not to early China. One standard approach to determinants is to regard an n -by- n matrix as an ordered array of vectors( u 1 , u 2,…, u n) and t hen its determinant det ( A ) as a function F( u 1 , u 2 ,…, u n) of these n vectors which obeys certain rules.The determinant of such an array A turns out to be a convenient criterion for characterizing the nonsingularity of the associated linear transformation, since d et ( A ) = F ( u 1, u 2,…, u n) = 0 if and only if the set of vectors u i ar e linearly dependent. There are many other useful and elegant properties of det erminants, most of which will be found in any classic book on linear algebra. Thus, det ( AB ) = det ( A ) det ( B ), and det ( A ) = det ( A') ,where A' is the transpose of A , obtained by the formula A' =( a ji ), thereby rotating the array about the main diagonal. If a square matrix is triangular, meaning th at all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.Another useful concept is that of eigenvalue. A scalar is said to be an eigenva lue for a transformation T if there is a nonzero vector υwith T (υ) λυ. It is then clear that the eigenvalues will be those numbers λ∈ K such that T -λI is a singular transformation. Any vector in the null space of T -λI is called an eigenvector of T associated with eigenvalue λ, and their span the eigenspace, E λ. It is invariant under the action of T , meaning that T carrie s Eλinto itself. The eigenvalues of T are then exactly the set of roots of the polynomial p(λ) =det ( T -λI ).If A is a matrix representing T ,then one h as p (λ) det ( A -λI ), which permits one to find the eigenvalues of T easil y if the dimension of V is not too large, or if the matrix A is simple enough. The eigenvalues and eigenspaces of T provide a means by which the nature and structure of the linear transformation T can be examined in detail.Vocabularylinear algebra 线性代数non-singular 非奇异field 域isomorphism 同构vector 向量isomorphic 同构scalar 纯量,无向量 matrix 矩阵(单数)vector space 向量空间matrices 矩阵(多数)span 生成,长成determinant 行列式independence 无关(性),独立(性) array 阵列dependence 有关(性) diagonal 对角线linear combination 线性组合 triangular 三角形的basis 基(单数) entry 表值,元素basis 基(多数) eigenvalue 特征值,本征值dimension 维eigenvector 特征向量linear transformation 线性变换 invariant 不变,不变量null space 零空间 row 行rank 秩 column 列singular 奇异 system of equations 方程组homogeneous 齐次Notes1. If U = { u 1, u 2,…}is a collection of points in a linearspace V , then the (linear) span of the set U is the set of all points of the form ∑c i u i , w where c i ∈K ,and all but a finite number of scalars c I are 0.意思是:如果U = { u 1, u 2,…}是线性空间V 的点集,那么集 U 的(线性)生成是所有形如∑c i u i的点集,这里c i ∈ K ,且除了有限个c i外均为0.2. A finite set { u 1, u 2,…, u k}is said to be linearly independent if the only way to write 0 = ∑c i u I is by choosing all the c i= 0.这一句可以用更典型的句子表达如下: A finite set { u 1, u 2,…, u k} is said to be linearly independen t in V if ∑c i u i is by choosing all the c i= 0.这里independent 是形容词,故用linearly修饰它. 试比较F(x) is a continuous periodic function.这里periodi c 是形容词但它前面的词却用continuous 而不用continuously,这是因为continuous 这个词不是修饰periodi c而是修饰作为整体的名词periodic function.3. Then these define isomorphisms of V onto K n and W onto K M respectively, and these in turn inducea linear transformation A between these.这里第一个these代表前句的两个基(basis);第二个these代表isomorphisms;第三个these代表什么留给读者自己分析.4. The least satisfactory aspect of linear algebra is still the theory of determinants-意思是:线性代数最令人不满意的方面仍是有关行列式的理论.least satisfactory 意思是:最令人不满意.5. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.意思是:如果方阵是三角形的,即所有在主对角线上方的元素均为零,那末det( A ) 刚好就是对角线元素的乘积.这里meaning that 可用that is to say 代替,turns out to be解为”结果是”.ExerciseI. Answer the following questions:1. How can we define the linear independence of an infinite set?2. Let T be a linear transformation (T: V →W ) whose associated matrix is A.Give a criterion for the non-singularity of the transformation T.3. Where is the entry a45of a m -by- n matrix( m>4; n>5) located ?4. Let A , B be two rectangular matrices.Under what condition is the product matrix well-defined ?II.Translate the following two examples and their proofs into Chinese:1.Example1. Let u k= t k ,k=0,1,2,... and t real. Show that the set ｛u 0,u1,u2,…｝is independent.Proof: By the definition of independence of an infinite set, it suffices to show that for each n ,the n+ 1 polynomials u0,u1,...,u n are independent.A relation of the form ∑n k=0c k u k=0 means ∑n k=0c k t k=0 for all t.When t=0,this gives c0=0.Differentiating both sides of ∑n k=0c k t k=0 and setting t=0,we fi nd that c1=0.Repeating the process,we find that each cocfficient is zero2. Example 2. Let V be afinite dimensional linear space, Then every finite basis for V has the same nu mber of elements.Proof: Let S and T be two finite bases for V. Suppose S consists of k elemnts and T consists of m e lements.Since S is independent and spans V ,every set of k+1 elements in V is dependent.Therefore eve ry set of more than k elements in V is dependent. Since T is an independent set , we must have m<k. The same argument with S and T interchanged shows that k<m. Hence k=m.III.Translate the following sentences into English:1.设 A 是一矩阵。
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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。
Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.（2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。
No modern scientific and technological branches could be regularly developed without the application of mathematics.（3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。
Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.（4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。
Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。
Equation is different from arithmetic identity in that it contains unknown quantity which can join operations.（6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。
Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it.（7）方程很有用，可以用它来解决许多实际应用问题。
Equations are of very great use. We can use equations in many mathematical problems.（8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。
To solve the equation means to move and change the terms about without making the equation untrue, until the root of the equation is obtained, which is the value of unknown term.英译汉1.Algebra has evolved from the operations and rules of arithmetic.The study of arithmetic begins with addition,multiplication,subtraction,and division of numbers:4+7,37×682,49-22,40÷8.In algebra we introduce symbols or letters—such as a,b,c,d,x,y,z—to denote arbitrary numbers and，instead of special cases，we often consider general statements：a+b,cd,x-y,x÷a.代数是从算术的运算和规则当中逐渐发展起来的，算术的研究是从数的加减乘除开始的。
例如4+7，37×682，49-22，40÷8。
在代数学里,我们采用符号或字母。
例如a,b,c,d,x,y,z来表示任意的数字，而不考虑那些特殊情况。
我们经常考虑的是一般的表达式，例如a+b,cd,x-y,x÷a。
2.The language of algebra serves a twofold purpose.First,we may use it as a shorthand to abbreviate and simplify long or complicated statements.Second,it proves a convenient means of generalizing many specific statements.代数的语言有两个作用。
第一个是使用它作为一种速记法去缩减和减化那些又长又复杂的表达。
第二，它被证明是一种概括许多具体的表达方式的便捷途径。
3.Many expressions involve two or more operations.Grouping symbols tell us which operation is to be done first.The common grouping symbols are parentheses,(),brackets.[],and the fraction bar,—.For example,in the expression2(3+4),we do the addition first and then we do the multiplication:2(3+4)=2(7)=14.许多数学表达式包含两个或更多的运算。
分组符号告诉我们哪一个运算先做。
常见的分组符号是圆括号，方括号和分数线。
例如，在数学表达公式2（3+4）里。
我们先做加法再做乘法2（3+4）=2（7）=142.2 几何与三角词组翻译1.学会institution，建筑师architect, 机械师machinist, 制图员draftsman, 测量者surveyor, 木匠carpenter2.点point, 端点endpoint, 线line, 直线straight line, 线段line segment, 曲线curved line, 折线broken line, 射线ray , 平面plane,曲面curved surface3.立体solid, 柱体cylinder, 立方体cube,球sphere, 棱锥pyramid,圆锥cone ,4.圆circle,圆心center, 直径diameter, 半径radius, 半圆semicircle, 弦chord, 弧arc, 优弧major arc, 劣弧minor arc5.角angle, 边side, 三角形triangle, 直角三角形right triangle,斜边hypotenuse, 直角边right-angle side6.长度length,宽度breadth/width,厚度thickness, 位置position7.几何的geometrical,立体的three-dimensional , 弯曲的curved,等距离的equidistant ,无限的infinite8.培养创造力train originality,必须的毅力necessary perseverance ,提高鉴赏力raise/improve the appreciation ability9.消失了的边界vanishing boundaries/landmarks,有序性和优美感orderliness and sense of beauty, 几何图形大量存在geometric forms aboundin , 定理成立的先决条件a prerequisite to a theorem汉译英（1）许多专家都认为数学是学习其他科学技术的必备基础和先决条件。
Many experts recognize that mathematics is the necessary foundation and prerequisite of studying other science technology.（2）西方国家的专家认为几何起源于巴比伦和埃及人的土地测量技术，其实中国古代的数学家对几何做了许多出色的研究。