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(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。
数学专业英语
MATHS ENGLISHabsolute value 绝对值 acceptable region 接受域 additivity 可加性alternative hypothesis 对立假设 analysis of covariance 协方差分析analysis of variance 方差分析 arithmetic mean 算术平均值 association 相关性 assumption checking 假设检验 availability 有效度 band 带宽bar chart 条形图 beta-distribution 贝塔分布 between groups 组间的binomial distribution 二项分布binomial test 二项检验center of gravity 重心 central tendency 中心趋势 hi-square distribution 卡方分布 chi-square test 卡方检验 classify 分类 cluster analysis 聚类分析coefficient 系数 coefficient of correlation 相关系数 collinearity 共线性 components 构成，分量 compound 复合的 confidence interval 置信区间consistency 一致性continuous variable 连续变量control charts 控制图 correlation 相关 covariance 协方差 covariance matrix 协方差矩阵 critical point 临界点 critical value 临界值 cross tab 列联表 cubic term 三次项 cumulative distribution function 累加分布函数curve estimation 曲线估计 default 默认的 deleted residual 剔除残差density function 密度函数dependent variable 因变量design of experiment 试验设计 df.(degree of freedom) 自由度 diagnostic 诊断discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析 D-optimal design D-优化设计 effects of interaction 交互效应eigenvalue 特征值equal size 等含量estimation of parameters 参数估计 estimations 估计量 exact value 精确值 expected value 期望值 exponential指数的 exponential distribution 指数分布 extreme value 极值 factor analysis 因子分析 factor score 因子得分 factorial designs 析因设计 factorial experiment 析因试验fitted line 拟合线fitted value 拟合值fixed variable 固定变量fractional factorial design 部分析因设计 F-test F检验 full factorial design 完全析因设计 gamma distribution 伽玛分布 geometric mean 几何均值 harmonic mean 调和均值 heterogeneity 不齐性 histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性 hypothesis test 假设检验independence独立independent variable 自变量independent-samples 独立样本index of correlation 相关指数interclass correlation 组内相关 interval estimate 区间估计inverse 倒数的iterate 迭代kurtosis 峰度large sample problem 大样本问题least-significant difference 最小显著差数 least-square estimation 最小二乘估计 least-square method 最小二乘法 level of significance 显著性水平 leverage value 中心化杠杆值 life test 寿命试验likelihood function 似然函数 likelihood ratio test 似然比检验 linear estimator 线性估计linear model 线性模型 linear regression 线性回归 linear relation 线性关系 linear term 线性项 logarithmic 对数的 logarithms 对数 lost function 损失函数 main effect 主效应matrix 矩阵 maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差 mean sum of square 均方和 measure 衡量 media中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数 Monte Carle method 蒙特卡罗法 moving average移动平均值 multicollinearity 多元共线性 multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数 multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析 multiple regression equation 多元回归方程 multiple response 多响应 multivariate analysis 多元分析negative nonadditively 不可加性 nonlinear 非线性 nonlinear regression 非线性回归 noparametric tests 非参数检验 normal distribution 正态分布null hypothesis 零假设number of cases 个案数one-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类 optimal 优化的 optimum allocation 最优配制order statistics 次序统计量 origin 原点 orthogonal 正交的 outliers 异常值paired observations 成对观测数据paired-sample 成对样本parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数 partial regression coefficient 偏回归系 percentiles 百分位数 pie chart 饼图 point estimate 点估计poisson distribution 泊松分布 polynomial curve 多项式曲线 polynomial regression 多项式回归 polynomials 多项式 positive relationship 正相关 power 幂 P-P plot P-P概率图 predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率 probability density function 概率密度函数 quadratic 二次的 Q-Q plot Q-Q概率图 quadratic term 二次项 quality control 质量控制 quantitative 数量的，度量quartiles 四分位数 random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数 regression line 回归线rejection region 拒绝域residual 残差 residual sum of squares 剩余平方和 risk function 风险函数 robustness 稳健性 root mean square 标准差 row 行 run test 游程检验sample size 样本容量 sample space 样本空间 sampling 取样sampling inspection 抽样检验 scatter chart 散点图 S-curve S形曲线sets 集合sign test 符号检验significance level 显著性水平significance testing 显著性检验significant digits 有效数字skewed distribution 偏态分布 small sample problem 小样本问题 sort 排序sources of variation 方差来源 ion 标准离差 standard error of mean 均值的标准误差 statistical quality control 统计质量控制 std. residual 标准残差 stepwise regression analysis 逐步回归 strong assumption 强假设 stud. deleted residual 学生化剔除残差 stud. residual 学生化残差subsamples 次级样本 sufficient statistic 充分统计量 sum of squares 平方和t-distribution t分布test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验 test of independence 独立性检验 test rules 检验法则test statistics 检验统计量 testing function 检验函数 timeseries 时间序列 tolerance limits 容许限 trimmed mean 截尾均值 true value 真值 t-test t检验 two-tailed test 双侧检验unbiased estimation 无偏估计 unbiasedness 无偏性 uniform distribution 均匀分布 value of estimator 估计值variance 方差 variance components 方差分量 variance ratio 方差比weighted average 加权平均值 within groups 组内的 Z score Z分数 active constraint 活动约束 active set method 活动集法 analytic gradient 解析梯度 approximate 近似 arbitrary 强制性的 argument 变量attainment factor 达到因子 bandwidth 带宽 be equivalent to 等价于best-fit 最佳拟合 coefficient 系数 complex-value 复数值 component 分量constrained 有约束的 constraint function 约束函数 converge 收敛cubic polynomial interpolation method 三次多项式插值法 curve-fitting 曲线拟合 data-fitting 数据拟diagonal 对角的 direct search method 直接搜索法direction of search 搜索方向eigenvalue 特征值empty matrix 空矩阵exceeded 溢出的feasible solution 可行解finite-difference 有限差分 first-order 一阶 Gauss-Newton method 高斯-牛顿法 goal attainment problem 目标达到问题 gradient method 梯度法 handle 句柄 Hessian matrix 海色矩阵 independent variables 独立变量inequality 不等式infeasibility 不可行性initial feasible solution 初始可行解 initialize 初始化 invoke 激活 iteration 迭代Jacobian 雅可比矩阵 Lagrange multiplier 拉格朗日乘子 large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardt method 列文伯格-马夸尔特法 line search 一维搜索linear equality constraints 线性等式约束 linear programming problem 线性规划问题local solution 局部解 medium-scale 中型的 mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法 multi objective 多目标的 norm 范数 observed data 测量数据optimization routine 优化过程optimizer 求解器over-determined system 超定系统 partial derivatives 偏导数polynomial interpolation method 多项式插值法quadrati二次的quadratic interpolation method 二次内插法quadratic programming 二次规划real-value 实数值 residuals 残差robust 稳健的robustness 稳健性，鲁棒性scalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method 序列二次规划法 simplex search method 单纯形法sparse matrix 稀疏矩阵 sparsity pattern 稀疏模式 sparsity structure 稀疏结构 starting point 初始点 step length 步长 subspace trust region method 子空间置信域法symmetric matrix 对称矩阵termination message 终止信息 termination tolerance 终止容限 the exit condition 退出条件 the method of steepest descent 最速下降法 transpose 转置unconstrained 无约束的under-determined system 负定系统weighting matrix 加权矩阵approximation 逼近a spline in b-form/b-spline b样条 a spline of polynomial piece /ppform spline 分段多项式样条bivariate spline function 二元样条函数break/breaks 断点coefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条 cubic spline interpolation 三次样条插值/三次样条内插 curve 曲线 degree of freedom 自由度 end conditions 约束条件input argument 输入参数 interpolation 插值/内插 interval 取值区间knot/knots 节点least-squares approximation 最小二乘拟合 multiplicity 重次 multivariate function 多元函数 optional argument 可选参数 output argument 输出参数point/points 数据点rational spline 有理样条rounding error 舍入误差（相对误差）sequence 数列（数组spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线 spline interpolation 样条插值/样条内插 spline surface 样条曲面 smoothing spline 平滑样条 tolerance 允许精度univariate function 一元函数 absolute error 绝对误差 absolute tolerance 绝对容限adaptive mesh 适应性网格 boundary condition 边界条件 contour plot 等值线图coordinate 坐标系decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditions 边界条件eigenvalue 特征值 elliptic 椭圆形的 error estimate 误差估计exact solution 精确解 generalized Neumann boundary condition 推广的Neumann 边界条件geometry description matrix 几何描述矩阵 geometry matrix 几何矩阵 graphical user interface（GUI）图形用户界面 hyperbolic 双曲线的 initial mesh 初始网格 jiggle 微调Lagrange multipliers 拉格朗日乘子 Laplace equation 拉普拉斯方程 linear interpolation 线性插值machine precision 机器精度mixed boundary condition 混合边界条件Neuman boundary condition Neuman边界条件 node point 节点 nonlinear solver 非线性求解器normal vector 法向量Parabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变 plane stress 平面应力 Poisson's equation 泊松方程 polygon 多边形positive definite 正定refined triangular mesh 加密的三角形网格relative tolerance 相对容限 relative tolerance 相对容限 residual norm 残差范数 singular 奇异的postulate假定, 基本条件, 基本原理,要求, 假定,要求conic, conical圆锥的；圆锥形的ellipse椭圆, 椭圆形ellipt hyperbolic 双曲线的parabolic用寓言表达的: 抛物线的，像抛物线的algebraic代数的, 关于代数学的mineralogy 矿物学axiom公理collinear在同一直线上的同线的convex 凸出的；凸面的triangle三角形, 三人一组, 三角关系parallelogram平行四边形straight angle平角right angle 直角acute angle锐角obtuse angle钝角reflex angle优角rectilinear直线的；由直线组成的；循直线进行的isosceles triangle等腰三角形equilateral triangle等边三角形right triangle n. 直角三角形obtuse triangle钝角三角形acute triangle锐角三角形equiangular triangle正三角形,等角三角形hypotenuse（直角三角形的）斜边infinitesimal 无穷小的, 极小的, 无限小的calculus 微积分学, 结石inscribe 记下polygon多角形, 多边形curvilinear曲线的, 由曲线组成的intuition 直觉, 直觉的知识integral积分, 完整, 部分defective有缺陷的, (智商或行为有)欠缺的differential coefficient 微分系数irrational numbers无理数domain 定义域contradiction 矛盾continuous variable 连续变量;［连续变数］variation 变分, 变化independent variable 自变量dependent variable 应变量rectangular coordinate 直角坐标abscissa〈数〉横坐标ordinate纵线, 纵座标differential 微分的,微分(differentiation)Integral 积分, 完整, 部分(integration) trigonometry 三角法exponential 指数的, 幂数的logarithm 对数derivative导数；微商tangent 切线正切definite integral 定积分culminate 达到顶点differential equation 微分方程extreme value 极值multiple integral 多重积分functional analysis 泛函分析cardinal number 基数（如：1, 2, 3, ... 有别于序数）denumerable可数的aggregate 合计, 总计, 集合体，合计的, 集合的, 聚合的，聚集, 集合, 合计purport主旨，声称superior 长者, 高手, 上级，较高的, 上级的, 上好的, 出众的, 高傲的cumbersome 讨厌的, 麻烦的, 笨重的drastically 激烈地, 彻底地conservation 守衡律quadrature求积, 求积分interpolation插值extrapolation外推法, 推断internal point 内点generalized solution 广义解hydrodynamics 流体力学，水动力学divergence 发散（性），梯度，发散integro-interpolation method 积分插值法Variational method 变分方法comparatively 比较地, 相当地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 使者, 传令官, 通报者, 先驱, 预兆，预报, 宣布, 传达, 欢呼advection水平对流fluctuation波动, 起伏mean-square 均方dispersion离差, 差量nterpolation 插值divisible 可分的dice, die 骰子pitfall 缺陷celestial天上的macroscopic肉眼可见的, 巨观的classical field theory 经典场理论rigit 刚硬的, 刚性的, 严格的quantum量, 额, [物] 量子, 量子论inception 起初, 获得学位pertain 适合, 属于encompass 包围, 环绕, 包含或包括某事物ingredient 成分, 因素acquainted有知识的, 知晓的 synonymous同义的configuration 构造, 结构, 配置, 外形inertia 惯性, 惯量attribute 特性momentum动量designate 指明projectile 射弹，发射的ballistics 弹道学, 发射学intractable 难处理的furnish 供应, 提供, 装备, 布置torque n. 扭矩, 转矩moment 力矩的dissipation 消散, 分散, 挥霍, 浪费, 消遣, 放荡, 狂饮constitutive构成的, 制定的continuum mechanics 连续介质力学superposition重叠, 重合, 叠合reckon 计算, 总计, 估计, 猜想，数, 计算, 估计, 依赖, 料想strength 强度load 载荷empirical 以经验为依据的insofar 在……范围cohesive 内聚性的stiffness 硬度furnish 供给turbulent 湍流laminar 层流isothermal 等温isotropic 各向同性eddy 旋涡viscosity 粘性、粘度adiabatic 绝热的reversible 可逆的 isentropic 等熵的stream tube 流管 tangential 切线的incompressible 不可压缩的similitude 相似性hydraulic 水力的,水力学的spillway (河或水坝的)放水道,泄洪道prototype 原型,样板vibratory 振动的,摆动的propagation 传播acoustic 听觉的,声学的damp 阻尼,衰减restore 复职,归还neutral 平衡 exciting force 激励力resonant共振的,谐振的stiffness 刚度,刚性magnitude 数值,大小substantially实质上的perturb 干扰,扰乱Fourier series 傅里叶级数shredder 切菜器metropolitan 大都市的at-grade 在同一水平面上elevated 高架的guide way 导轨rigid body 刚体medium 介质aging 老化polymeric聚合(物)的consolidate 把…联合为一体,统一radically 根本地,本质上deliberate 从容不迫的,深思熟虑Attribute赋予medieval 中世纪的etch 蚀刻,蚀镂fingernail 指甲bar chart 直方图joystick 游戏杆trial-error 试制, 试生产junction n. 连接, 接合, 交叉点, 汇合处contrive v. 发明, 设计, 图谋snooker (=snooker pool)彩色台球, 桌球****公理 axiom 命题 proposition 被加数augend , summand 加数addend 被减数minuend 减数subtrahend 差remainder 被乘数multiplicand, faciend 乘数multiplicator 积 product 被除数 dividend 除数 divisor 商 quotient 大于等于 is equal or greater than 小于等于 is equal or lesser than 运算符operator 算术平均数geometric mean n个数之积的n次方根（reciprocal） x的倒数为1/x 有理数 rational number 无理数irrational number 整数 integer小数点 decimal point分数 fraction 分子 numerator 分母 denominator 比 ratio 十进制 decimal system 二进制binary system 十六进制 hexadecimal system 权 weight, significance 截尾 truncation 四舍五入 round 下舍入 round down 上舍入 round up 有效数字significant digit 无效数字 insignificant digit 代数 algebra 单项式monomial 多项式polynomial, multinomial 系数coefficient 未知数 unknown, x-factor, y-factor, z-factor 等式，方程式 equation 一次方程simple equation 二次方程quadratic equation 三次方程cubic equation 四次方程 quartic equation 阶乘 factorial 对数logarithm 指数，幂 exponent 乘方 power 二次方，平方 square 三次方，立方 cube 四次方 the power of four, the fourth power n次方 the power of n, the nth power 开方 evolution, extraction 二次方根，平方根 square root 三次方根，立方根 cube root 四次方根 the root of four, the fourth root n次方根 the root of n, the nth root 坐标系coordinates 坐标轴 x-axis, y-axis, z-axis 横坐标 x-coordinate 纵坐标y-coordinate 原点origin 象限quadrant 截距(有正负之分)intercede （方程的）解solution 线段 segment 射线 radial 平行parallel 相交intersect 角度degree 弧度radian 钝角obtuse angle 平角 straight angle 周角 perigon 底 base 锐角三角形 acute triangle 直角边 leg 斜边 hypotenuse 勾股定理 Pythagorean theorem 钝角三角形 obtuse triangle 不等边三角形 scalene triangle 等腰三角形isosceles triangle 等边三角形equilateral triangle 四边形quadrilateral 平行四边形parallelogram 周长perimeter 全等congruent 三角 trigonometry 正弦 sine 余弦 cosine 正切 tangent 余切 cotangent 正割 secant 余割 cosecant 反正弦 arc sine 反余弦 arc cosine 反正切 arc tangent 反余切 arc cotangent 反正割arc secant 反余割 arc cosecant 集合aggregate 空集 void 子集subset 交集intersection 并集union 补集complement 映射mapping 定义域 domain, field of definition 值域 range 单调性monotonicity 图象 image 数列，级数 series 导数 derivative 无穷小infinitesimal 复数complex number 矩阵matrix 行列式determinant 半圆 semicircle 扇形 sector 环 ring 椭圆 ellipse 圆周 circumference 轨迹 locus, loca(pl.) 平行六面体 parallelepiped 立方体 cube 七面体 heptahedron 八面体 octahedron 九面体 enneahedron 十面体 decahedron 十一面体 hendecahedron 十二面体 dodecahedron 二十面体 icosahedron 多面体 polyhedron 四面体 tetrahedron 五面体pentahedron 六面体hexahedron 菱形rhomb, rhombus, rhombi(pl.), diamond 正方形 square 梯形 trapezoid 直角梯形 right trapezoid 等腰梯形 isosceles trapezoid 五边形 pentagon 六边形 hexagon 七边形heptagon 八边形 octagon 九边形 enneagon 十边形 decagon 十一边形hendecagon 十二边形dodecagon 多边形polygon 正多边形equilateral polygon 相位 phase 振幅 amplitude 内心 incentre(BrE), incenter(AmE) 外心 excentre(BrE), excenter(AmE) 旁心 escentre(BrE), escenter(AmE) 垂心orthocentre(BrE), orthocenter(AmE) 重心barycentre(BrE), barycenter(AmE) 内切圆 inscribed circle 外切圆circumcircle 方差variance 标准差root-mean-square deviation, standard deviation 百分点 percentage 百分位数 percentile 排列permutation 分布 distribution 正态分布 normal distribution 非正态分布abnormal distribution 条形统计图bar graph 柱形统计图histogram 折线统计图 broken line graph 曲线统计图 curve diagram 扇形统计图pie diagram**** mutually disjoint events 互不相交事件mutually disjoint subsets 互不相交子集 mutually independent events 互相独立事件myria 万 myriad 无数的 multiplicity 重数 mid square method 平方取中法 midperpendicular 中垂线 minor 子式 minor arc 劣弧 mixed number 带分数 regular convergence 正则收敛 relative discriminant 相对判别式relative error 相对误差 relative extremum 局部极值 ricci equatoin 李奇恒等式ricci identity 李奇恒等式riemann function 黎曼函数riemann integral 黎曼积分 right direct product 右直积 right endpoint 右端点 right inner product 右内积 ring of integers 整数环 ring of matrices 矩阵环 root mean square error 均方根差 root of equation 方程式的根 rotation of axes 坐标轴的旋转 rotation of co ordinate system 坐标轴的旋转 round off error 舍入规则 round up error 舍入规则 runge kutta method 龙格库塔法 n disk n维圆盘 nth member 第n项 nth partial quotient 第n偏商 nth power operation n次幂运算 nth root n次根 nth term 第n项 n times continuously differentiable n次连续可微的natural injection 自然单射natural isomorphism 自然等necessary and sufficient conditions 必要充分的条件necessary and sufficient statistic 必要充分统计量 neutral element 零元素 neutral line 中线 nonhomogeneous linear boundary value problem 非齐次线性边值问题 nonhomogeneous linear differential equation 非齐次线性微分方程nonhomogeneous linear system of differential equations 非齐次线性微分方程组interval algebra 区间代数 interval analysis 区间分析 interval closed at the right 右闭区间 interval estimation 区域估计 intervalfunction 区间函数 interval graph 区间图 interval of convergence 收敛区间 interval of definition 定义区间 interval topology 区间拓扑irreducible set 不可约集 irreducible r module 不可约r模 periodical decimal fraction 循环十进小数 pentad 拼五小组 pentadecagon 十五边形pentagon 五角形 pentagonal number 五角数 pentagonal pyramid 五角锥pentagram 五角星 pentahedron 五面体 pentaspherical coordinates 五球坐标 penalty method 补偿法 pascal distribution 帕斯卡分布 partition function 分折函数 partial differential equation of elliptic type 椭圆型偏微分方程 partial differential equation of first order 一阶偏微分方程 partial differential equation of hyperbolic type 双曲型偏微分方程 partial differential equation of mixed type 混合型偏微分方程partial differential equation of parabolic type 抛物型偏微分方程partial differential operator 偏微分算子parametric test 参数检验particular solution 特解parallelogram axiom 平行四边形公理orthogonality relation 正交关系 ordinary differential equation 常微分方程optimal value function 最优值函数opposite angles 对角opposite category 对偶范畴 one to one mapping 一一映射 onto mapping 满射 open mapping theorem 开映射定理 one to many mapping 一对多映射one sided limit 单侧极限 numerical solution of linear equations 线性方程组的数值解法 null set 空集 null solution 零解 third boundary condition 第三边界条件two sided neighborhood 双侧邻域unbiased estimating equation 无偏估计方程unbounded function 无界函数unbounded quantifier 无界量词uncertainty principle 测不准原理uncorrelated random variables 不相关随机变量 undetermined coefficient 末定系数 velocity distribution 速度分布 velocity optimal 速度最优的weak approximation theorem 弱逼近定理 weak completeness 弱完备性weak continuity 弱连续性 weak convergence 弱收敛 wiener measure 维纳测度word group 自由群 sample correlation coefficient 样本相关系数sample covariance 样本协方差schwarz inequality 施瓦尔兹不等式second boundary condition 诺伊曼边界条件second comparison test 第二比较检验second limit theorem 第二极限定理 self adjoint differential equation 自伴微分方程 semimajor axis 半长轴semiminor axis 半短轴 sentential calculus 命题演算 set of measure zero 零测度集 set topology 集论拓扑simple connectedness 单连通性slope function 斜率函数 solution curve 积分曲线 solution domain 解域solution set of equation 方程的解集spatial co ordinate 空间坐标specific address 绝对地址spherical bessel function 球贝塞耳函数 spherical cap 球冠 spherical coordinates 球极坐标 spherical curvature 球面曲率 spherical shell 球壳 spherical zone 球带spline function 样条函数spline interpolation 样条内插stability conditions 稳定条件 statistical hypothesis testing 统计假设检验strict inequality 严格不等式strict isotonicity 严格保序性strict isotony 严格保序性strict increasing 严格递增system of partial differential equations 偏微分方程组system of ordinary differential equations 常微分方程组system of linear homogeneousequations 线性齐次方程组 system of linear inhomogeneous equations 线性非齐次方程组system of inequalities 联立不等式system of polarcoordinates 极坐标系system of variational equations 变分方程组system with concentrated parameters 集中参数系统system with distributedparameters 分布参数系统 t1topological space t1拓扑空间 t2topologicalspace t2拓扑空间 t3topological space 分离空间 t4topological space 正则拓扑空间 t5 topological space 正规空间 t6topological space 遗传正规空间 tangent cone 切线锥面 telegraph equation 电报方程 theorem for damping 阻尼定理****充分条件sufficient condition必要条件necessary condition 充要条件sufficient and necessary condition……的充要条件是……… if and only if …****abscissa 横坐标 alternatingseries 交错级数 angle of the sector 扇形角 arbitrary constant 任意常数 augmented matrix 增广矩阵 axis of parabola 拋物线的轴 axis of revolution 旋转轴 axis of rotation 旋转轴 binomial series 二项级数binomial theorem 二项式定理 binomial distribution 二项分布 bisectionmethod 分半法；分半方法 bounded above 有上界的；上有界的 boundedbelow 有下界的；下有界的bounded function 有界函数boundedsequence 有界序列brace 大括号bracket 括号Cartesian coordinates 笛卡儿坐标 certain event 必然事件 circumcentre 外心；外接圆心 circumcircle 外接圆 classical theory of probability 古典概率论 cofactor 余因子; 余因式 common denominator 同分母；公分母 commondifference 公差 common divisor 公约数；公约 common logarithm 常用对数 common multiple 公位数；公倍 common ratio 公比 commutative law 交换律 compasses 圆规 Cauchy-Schwarz inequality 柯西 - 许瓦尔兹不等式central limit theorem 中心极限定理 centripedal acceleration 向心加速度concave downward 凹向下的concurrent 共点concyclic 共圆concyclic points 共圆点Euclidean geometry 欧几里德几何Euler'sformula 尤拉公式；欧拉公式 even function 偶函数 even number 偶数（2）博奕 Gaussian distribution 高斯分布 greatest term 最game （1）对策；大项 greatest value 最大值 harmonic mean (1) 调和平均数; (2) 调和中项 harmonic progression 调和级数 higher order derivative 高阶导数improper fraction 假分数improper integral 广义积分; 非正常积分implicit function 隐函数 incircle 内切圆 inclined plane 斜 included angle 夹角 indefinite integral 不定积分 initial condition 原始条件；初值条件 initial-value problem 初值问题 interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角isosceles triangle 等腰三角形 iterate (1)迭代值; (2)迭代 Lagrange interpolating polynomial 拉格朗日插值多项代Laplace expansion 拉普拉斯展式 lemniscate 双纽线 left hand limit 左方极限 limiting case 极限情况limiting position 极限位置line of best-fit 最佳拟合line segment 线段 logarithmic equation 对数方程 mathematical analysis 数学分析mathematical induction 数学归纳法monotonic decreasing function 单调递减函数 monotonic convergence 单调收敛性 monotonic increasing function 单调递增函数multiple-angle formula 倍角公式multiple root 多重根 mutually disjoint 互不相交 mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直 numerical method 计算方法；数值法oblique cone 斜圆锥 orthogonal circles 正交圆 orthogonality 正交性oscillatory convergence 振动收敛性 ordinary differential equation 常微分方程pairwise mutually exclusive events 两两互斥事件place holder 补位数字 point of inflection (inflexion) 拐点; 转折点Pisson distribution 泊松分布point-slope form 点斜式polar coordinate plane 极坐标平面polynomial equation 多项式方程posterior probability 后验概率; 事后概率premultiply 前乘; 自左乘prime factor 质因子；质因素 prime number 素数；质数 principal angle 主角principal axis 主轴 principal value 主值 prior probability 先验概率; 事先概率 probability density function 概率密度函数 product and sum formula 和积互变公式 product sample space 积样本空间 product to sum formula 积化和差公式 proof by contradiction 反证法; 归谬法 proper fraction 真分数proper integral 正常积分proper subset 真子集propositional calculus 命题演算propositional inference 命题推演protractor 量角器Pythagoras' theorem 勾股定理Pythagorean triplet 毕氏三元数组 quadratic convergence 二阶收敛性 quadrature 求积法 quotient set 商集 radial component 沿径分量 radical axis 根轴range 值域；区域；范围；极差；分布域 rationalization 有理化 raw data 原始数据 rectifiable 可求长的 reciprocal 倒数 rectangular coordinate plane 直角坐标平面 recurrence formula 递推公式 reducibility 可约性; 可化简性 reflexive relation 自反关系 reference angle 参考 reference line 基准线 reflex angle 优角；反角 region of acceptance 接受区域region of convergency 收敛区域 region of rejection 否定区域 right circular cone 直立圆锥（体） resolution of vector 向量分解; 矢量分解right hand limit 右方极限 right prism 直立棱柱；直立角柱(体) right pyramid 直立棱锥；直立角锥(体) right-angled triangle 直角二角形scalene triangle 不等边三角形；不规则三角形 scatter diagram 散点图scientific notation 科学记数法semi-conjugate axis 半共轭轴semi-transverse axis 半贯轴semi-vertical angle 半顶角separable differential equation 可分微分方程septic equation 七次方程set square 三角尺；三角板 shaded portion 有阴影部分 significance level 显著性水平 significant figure 有效数字 similar triangles 相似三角形simple iteration method 简单迭代法simple pendulum 单摆Simpson's integral 森逊积分 standard deviation 标准差；标准偏离 standard normal distribution 标准正态分布; 标准常态分布 stationary point 平稳点; 逗留点; 驻点 strictly monotonic 严格单调 statistical chart 统计分析submultiple angle formula 半角公式subsidiary angle 辅助角substitution 代入; 代入法successive approximation 逐次逼近法successive derivative 逐次导数 successive differentiation 逐次微分法suffix 下标 sum to infinity 无限项之和 sum to product formula 和化积公式 superimposing 迭合 supplementary angle 补角 surjection 满射symmetric relation 对称关系 tautology 恒真命题；恒真式 Taylor’s expansion 泰勒展开式 Taylor’s series 泰勒级数 Taylor’s theorem 泰勒定理 test criterion 检验标准 test of significance 显著性检验 to the nearest 至最接近之torque 转矩torus 环面transcendental function 超越函数 transformation of variable 变数转换 transitive 可传递的 transpose of matrix 倒置矩阵；转置矩阵 transversal 截 ；横截的 triangle law of addition 三角形加法 travel graph 行程图 tree diagram 树形图trapezoidal integral 梯形积分truncated Taylor’s series 截断泰勒级数 two-tailed test 双尾检验；只端检验 type I error I 型误差type II error II型误差unbiased estimator 无偏估计量undetermined coefficient 待定系数 unique solution 唯一解 vertical asymptote 垂直渐近线 vertically opposite angles 对顶角 without loss of generality 不失一般性****分子Numerator 分母Denominator 阿拉伯数字Hindu-Arabic numeral假分数Improper fraction 最大公因子Highest Common Factor (H.C.F.) 最小公倍数 Lowest Common Multiple (L.C.M.) 行列式 determinant****interval closed at the right 右闭区间 interval of convergence 收敛区间interval of definition 定义区间invariance theorem 不变性定理 invariant of an equation 方程的不变量 inverse circular function 反三角函数 inverse hyperbolic function 反双曲函数inversion formula 反演公式 isotonic injective mapping 保序单射映射jacobi identity 雅可比恒等式 jump point 跳跃点 law of double negation 双重否定律 law of inertia 惯性律law of large numbers 大数定律 leading ideal 猪想 liouville theorem 刘维尔定理 lipschitz condition 李普希茨条件 markov transform 马尔可夫变换 mathematical approximation 数学近似法 mathematical model 数学模型 maximum condition 极大条件 maximum deviation 最大偏差 mean square deviation 方差 mean square of error 误差的均方 meromorphic function 亚纯函数柱形统计图 histogram 折线统计图 broken line graph 曲线统计图 curve diagram 扇形统计图 pie diagram排列 permutatio内切圆 inscribed circle 外切圆 circumcircle正多边形 equilateral polygon metric space 度量空间 metric subspace 度量子空间method of runge kutta type 朗格库塔型的方法 method of steepest ascent 最速上升法 method of steepest descent 最速下降法method of finite elements 有限元法method of fractional steps 分步法method of exhaustion 穷竭法 method of approximation 近似法 method of artificial variables 人工变量法method of balayage 扫除法method of characteristic curves 特者法 method of comparison 比较法 method of conjugate gradients 共轭梯度法lateral area 侧面积 last multiplier 最后乘子large sample test 大样本检验lattice constant 点阵常数lattice design 格子设计method of difference 差分法method of elimination 消元法method of estimation 估计法meromorphic differential 亚纯微分 median 中位数 measuring rule 量尺 mean term 内项mean term 内项mean term 内项irreducibility criterion 不可约性判别准则irreducible polynomial 不可约多项式 irreducible generating set 不可约生成集 irregular divisor class 非正则因子类 irregular point 非正则点irregular singular point 非正则奇点isometric circle 等距圆isometric embedding 等距嵌入isomorphic field 同构域isomorphic graph 同构图isomorphic group 同构群isomorphic image 同构象isothermal parameter 等温参数iterated function 叠函数iterated integral 累积分joint distribution 联合分布 jordan algebra 约当代数kernel of an integral equation 积分方程的核l'hospital rule 洛必达规则laboratory system of coordinates 实验室坐标系 labyrinth 迷宫lacation principle 介值定理lag correlation coefficient 滞后相关系数lag regression 落后回归laguerre differential equation 拉盖尔微分方程lame equation 拉梅方程language of formula 公式语言 laplace beltrami operator 拉普拉斯贝尔特拉米算子lateral area 侧面积 last multiplier 最后乘子large sample test 大样本检验lattice constant 点阵常数lattice design 格子设计 left adjoint 左伴随的 left derivative 左导数left differential 左微分 left direct product 左直积 left end point 左端点 left length 左长 left limit value 左极限值left multiplication ring 左乘环length of curve 曲线的长 length of normal 法线的长 levi decomposition 列维分解 limes inferior 下极限 limes superior 上极限limit circle 极限圆 limit circle type 极限圆型logarithm to the base 10 常用对数logarithmic normal distribution 对数正态分布logic of relations 关系逻辑magic circle 幻圆magic cube 幻立方manifold without boundary 无边廖many valued mapping 多值映射marginal distribution density function 边缘分布密度函数 marginal distribution function 边缘分布函数mathematical programming 数学规划 mathematical random sample 数学随机样本 mathematical statistics 数理统计 maximum likelihood estimating function 极大似然估计量 independent variable 自变量 dependent variable 应变量equiangular triangle 正三角形,等角三角形命题proposition 差remainder 积product 除数divisor 商quotient 截尾truncation 未知数unknown, x-factor, y-factor, z-factor 阶乘 factorial 集合 aggregate 空集 void 子集 subset 交集 intersection 并集 union 补集 complement 映射 mapping 勾股定理 Pythagorean theorem 菱形 rhomb, rhombus, rhombi(pl.), diamond 双曲线 hyperbola 抛物线 parabola topology of bounded convergence 有界收敛拓扑toroid 超环面 toroidal coordinates 圆环坐标trace of dyadic 并向量的迹transcendental integral function 超越整函数transformation formulas of the coordinates 坐标的变换公式 transformation to principal axes 轴变换 transversal lines 截线trapezoid method 梯形公式trefoil knot 三叶形纽结truth function 真值函项two sided test 双侧检定 two sided neighborhood 双侧邻域two sided surface 双侧曲面two termed expression 二项式ultrahyperbolic equation 超双曲型方程。
数学专业英语(Doc版).Word6
数学专业英语－First Order Differential EquationsA differential equation is an equation between specified derivatives of a functio n, itsvalves,and known quantities.Many laws of physics are most simply and naturall y formu-lated as differential equations (or DE’s, as we shall write for short).For this r eason,DE’shave been studies by the greatest mathematicians and mathematical physicists si nce thetime of Newton..Ordinary differential equations are DE’s whose unknowns are functions of a s ingle va-riable;they arise most commonly in the study of dynamic systems and electric networks.They are much easier to treat than partial differential equations,whose unknown functionsdepend on two or more independent variables.Ordinary DE’s are classified according to their order. The order of a DE is d efined asthe largest positive integer, n, for which an n-th derivative occurs in the equati on. Thischapter will be restricted to real first order DE’s of the formΦ(x, y, y′)=0 (1)Given the function Φof three real variables, the problem is to determine all re al functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the follo wing sense.DEFINITION A solution of (1)is a differentiable function f(x) such thatΦ(x. f(x),f′(x))=0 for all x in the interval where f(x) is defined.EXAMPLE 1. In the first-other DEthe function Φis a polynomial function Φ(x, y, z)=x+ yz of three variables i n-volved. The solutions of (2) can be found by considering the identityd(x²+y²)/d x=2(x+yyˊ).From this identity,one sees that x²+y²is a con-stant if y=f(x) is any solution of (2).The equation x²+y²=c defines y implicitly as a two-valued function of x,for any positive constant c.Solving for y,we get two solutions,the(single-valued) functions y=±(c-x²)0.5,for each positive constant c.The graphs of these so-lutions,the so-called solution curves,form two families of scmicircles,which fill t he upper half-plane y>0 and the lower half-plane y>0,respectively.On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solu tionswhich cross the x-axis,except possibly at the origin.This fact is easily overlook ed,because the solution curves appear to cross the x-axis;hence yˊdoes not exist, and the DE (2) is not satisfied there.The preceding difficulty also arises if one tries to solve the DE(2)for yˊ. Div iding through by y,one gets yˊ=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.DEFINITION. A normal first-order DE is one of the formyˊ=F(x,y) (3)In the normal form yˊ=-x/y of the DE (2),the function F(x,y) is continuous i n the upper half-plane y>0 and in the lower half-plane where y<0;it is undefin ed on the x-axis.Fundamental Theorem of the Calculus.The most familiar class of differential equations consists of the first-order DE’s of the formSuch DE’s are normal and their solutions are descried by the fundamental tho rem of the calculus,which reads as follows.FUNDAMENTAL THEOREM OF THE CALCULUS. Let the function g(x)i n DE(4) be continuous in the interval a<x<b.Given a number c,there is one an d only one solution f(x) of the DE(4) in the interval such that f(a)=c. This sol ution is given by the definite integralf(x)=c+∫a x g(t)dt , c=f(a) (5)This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a<x<b in the xy-plane.Second,it describes in precise terms the class of functions g(x) consid ered.And third, it asserts the existence and uniqueness of a solution,given the “initial condition”f(a)=c.We recall that the definite integral∫a x g(t)dt=lim(maxΔt k->0)Σg(t k)Δt k , Δt k=t k-t k-1 (5ˊ)is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral ∫g(x) dx to give meanin g to the definite integral ∫a x g(t)dt,provided only that g(t) is continuous.Such f unctions as the error function crf x =(2/(π)0.5)∫0x e-t²dt and the sine integral f unction SI(x)=∫x∞[(sin t )/t]dt are indeed commonly defined as definite int egrals.Solutions and IntegralsAccording to the definition given above a solution of a DE is always a functi on. For example, the solutions of the DE x+yyˊ=0 in Example I are the func tions y=±(c-x²)0.5,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily describ ed by the equation x²+y²=c,describing a family of circles centered at the origi n.In what sense can such a family of curves be considered as a solution of th e DE ?To answer this question,we require a new notion.DEFINITION. An integral of DE(1)is a function of two variables,u(x,y),whic h assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.In the above example, the function u(x,y)=x²+y²is an integral of the DE x +yyˊ=0,because,upon replacing the variable y by any function ±( c-x²)0.5,we obtain u(x,y)=c.The second-order DEd²x/dt²=-x (2ˊ)becomes a first-order DE equivalent to (2) after setting dx/dx=y:y ( dy/dx )=-x (2)As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE.When th e DE (2ˊ)is interpreted as equation of motion under Newton’s second law,the integrals c=x²+y²represent curves of constant energy c.This illustrates an important prin ciple:an integral of a DE representing some kind of motion is a quantity that r emains unchanged through the motion.Vocabularydifferential equation 微分方程 error function 误差函数ordinary differential equation 常微分方程 sine integral function 正弦积分函数order 阶,序 diameter 直径derivative 导数 curve 曲线known quantities 已知量replace 替代unknown 未知量substitute 代入single variable 单变量strip 带形dynamic system 动力系统 exact differential 恰当微分electric network 电子网络line integral 线积分partial differential equation 偏微分方程path of integral 积分路径classify 分类 endpoints 端点polynomial 多项式 general solution 通解several variables 多变量parameter 参数family 族rigorous 严格的semicircle 半圆 existence 存在性half-plane 半平面 initial condition 初始条件region 区域uniqueness 唯一性normal 正规,正常Riemann sum 犁曼加identity 恒等(式)Notes1. The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs i n the question.这是另一种定义句型,请参看附录IV.此外要注意nth derivative 之前用an 不用a .2. This chapter will be restricted to real first order differential equations of the formΦ(x,y,yˊ)=0意思是;文章限于讨论形如Φ(x,y,yˊ)=0的实一阶微分方程.有时可以用of the type代替of the form 的用法.The equation can be rewritten in the form yˊ=F(x,y).3. Dividing through by y,one gets yˊ=-x/y,…划线短语意思是:全式除以y4. As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE这里x²+y²=c 因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.5. Their solutions are described by the fundamental theorem of the calculus,which reads as follows.意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下.句中reads as follows 就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.ExerciseⅠ.Translate the following passages into Chinese:1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫c M(x,y) dx +N(x,y) dy is the same for all paths of int egration c in D, which have the same endpoints.Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.2. For any normal first order DE yˊ=F(x,y) and any initial x0 , the initial valve problem consists of finding the solution or solutions of the DE ,for x>x0 which assumes a given initial valve f(x0)=c.3. To show that the initial valve problem is well-set requires proving theorems of existence (there isa solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on t he initial value).Ⅱ. Translate the following sentences into English:1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有dч(x)/dx=f (x,ч(x))2) 两边从x0到x取定积分得ч(x)-ч(x0)=∫x0x f(x,ч(x)) dx x0<x<x0+h3) 把y0=ч(x0)代入上式, 即有ч(x)=y0+∫x0x f(x,ч(x)) dx x0<x<x0+h4) 因此y=ч(x) 是积分方程y=y0+∫x0x f (x,y) dx定义于x0<x<x0+h 的连续解.Ⅲ. Translate the following sentences into English:1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数f (x, dy/dx) 有连续的偏导数.2) 引入参数dy/dx=p, 则已给方程变为y=f (x,p).3) 在y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为y=f (x,ч(x,c)).6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解 x=ψ(p,c)y=f(ψ(p,c)p)其中p是参数,c是任意常数.。