英文翻译原文数学系
数学专业英语(Doc版).13
数学专业英语－Mathematical DiscoveryTo give the flavor of Polya’s thinking and writing in a very beautiful but sub tle case , a case that involve a change in the conceptual mode , I shall quote at length from his Mathematical Discovery (vol.II , pp.54 ff):EXAMPLE I take the liberty a little experiment with the reader , I shall sta te a simple but not too commonplace theorem of geometry , and then I shall t ry to reconstruct the sequence of idoas that led to its proof . I shall proceed s lowly , very slowly , revealing one clue after the other , and revealing each g radually . I think that before I have finished the whole story , the reader will seize the main idea (unless there is some special hampering circumstance ) . B ut this main idea is rather unexpected , and so the reader may experience the pleasure of a little discovery .A.If three circles having the same radius pass through a point , the circle th rough their other three points of intersection also has the same radius .Fig.1 Three circles through one point.This is the theorem that we have to prove . The statement is short and clea r , but does not show the details distinctly enough . If we draw a figure (Fig .1) and introduce suitable notation , we arrive at the following more explicit restatement :B . Three circles k , l , m have the same radius r and pass through the sa me point O . Moreover , l and m intersect in the point A , m and k in B , k and l inC . Then the circle e through A , B , C has also the radiusFig .2 too crowded .Fig .1 exhibits the four circles k , l , m , and e and their four points of in tersection A, B , C , and O . The figure apt to be unsatisfactory , however , for it is not simple , and it is still incomplete ; something seems to be missin g ; we failed to take into account something essential , it seems .We are dialing with circles . What is a circle ? A circle is determined by c enter and radius ; all its points have the same distance , measured by the leng th of the radius , from the center . We failed to introduce the common radius r , and so we failed to take into account an essential part of the hypothesis . Let us , therefore , introduce the centers , K of k , L of l , and M of m . Where should we exhibit the radius r ? there seems to be no reason to treat a ny one of the three given circles k ; l , and m or any one of the three points of intersection A , B , and C better than the others . We are prompted to connect all three centers with all the points of intersection of the respective circl e ; K with B , C , and O , and so forth .The resulting figure (Fig . 2) is disconcertingly crowded . There are so many lines , straight and circular , that we have much trouble old－fashioned maga zines . The drawing is ambiguous on purpose ; it presents a certain figure if you look t it in the usual way , but if you turn it to a certain position and lo ok at it in a certain peculiar way , suddenly another figure flashes on you , s uggesting some more or less witty comment on the first . Can you recognize i n our puzzling figure , overladen with straight and circles , a second figure th at makes sense ?We may hit in a flash on the right figure hidden in our overladen drawing , or we may recognize it gradually . We may be led to it by the effort to sol ve the proposed problem , or by some secondary , unessential circumstance . For instance , when we are about to redraw our unsatisfactory figure , we ma y observe that the whole figure is determined by its rectilinear part (Fig . 3) .This observation seems to be significant . It certainly simplifies the geometri c picture , and it possibly improves the logical situation . It leads us to restate our theorem in the following form .C . If the nine segmentsKO , KC , KB ,LC , LO , LA ,MB , MA , MO ,are all equal to r , there exists a point E such that the three segmentsEA , EB , EC ,are also equal to r .Fig . 3 It reminds you －of what ?This statement directs our attention to Fig . 3 . This figure is attractive ; it reminds us of something familiar . (Of what ?)Of course , certain quadrilaterals in Fig .3 . such as OLAM have , by hypo thesis , four equal sided , they are rhombi , A rhombus I a familiar object ; having recognized it , we can “see “the figure better . (Of what does the whole figure remind us ?)Oppositc sides of a rhombus are parallel . Insisting on this remark , we reali ze that the 9 segments of Fig . 3 . are of three kinds ; segments of the same kind , such as AL , MO , and BK , are parallel to each other . (Of what d oes the figure remind us now ?)We should not forget the conclusion that we are required to attain . Let us a ssume that the conclusion is true . Introducing into the figure the center E or the circle e , and its three radii ending in A , B , and C , we obtain (suppos edly ) still more rhombi , still more parallel segments ; see Fig . 4 . (Of wha t does the whole figure remind us now ?)Of course , Fig . 4 . is the projection of the 12 edges of a parallele piped h aving the particularity that the projection of all edges are of equal length .Fig . 4 of course !Fig . 3 . is the projection of a “nontransparent “parallelepiped ; we see o nly 3 faces , 7 vertices , and 9 edges ; 3 faces , 1 vertex , and 3 edges are invisible in this figure . Fig . 3 is just a part of Fig . 4 . but this part define s the whole figure . If the parallelepiped and the direction of projection are so chosen that the projections of the 9 edges represented in Fig . 3 are all equa l to r (as they should be , by hypothesis ) , the projections of the 3 remainin g edges must be equal to r . These 3 lines of length r are issued from the pr ojection of the 8th, the invisible vertex , and this projection E is the center o f a circle passing through the points A , B , and C , the radius of which is r .Our theorem is proved , and proved by a surprising , artistic conception of a plane figure as the projection of a solid . (The proof uses notions of solid g eometry . I hope that this is not a treat wrong , but if so it is easily redresse d . Now that we can characterize the situation of the center E so simply , it i s easy to examine the lengths EA , EB , and EC independently of any solid geometry . Yet we shall not insist on this point here .)This is very beautiful , but one wonders . Is this the “light that breaks fo rth like the morning . “the flash in which desire is fulfilled ? Or is it merel y the wisdom of the Monday morning quarterback ? Do these ideas work out in the classroom ? Followups of attempts to reduce Polya’s program to practi cal pedagogics are difficult to interpret . There is more to teaching , apparentl y , than a good idea from a master .——From Mathematical ExperienceVocabularysubtle 巧妙的，精细的clue 线索，端倪hamper 束缚，妨碍disconcert 使混乱，使狼狈ambiguous 含糊的，双关的witty 多智的，有启发的rhombi 菱形（复数）rhombus 菱形parallelepiped 平行六面体projection 射影solid geometry 立体几何pedagogics 教育学，教授法commonplace 老生常谈；平凡的。
数学专业英语第三版课文翻译章
数学专业英语第三版课文翻译章本文将根据数学专业英语第三版课文《Step by Step Thinking》进行翻译。
"Step by Step Thinking"is an article that introduces the concept of step-by-step thinking in mathematics.It highlights the importance of breaking down complex problems into smaller,more manageable steps in order to solve them effectively.The article begins by stating that step-by-step thinking is a fundamental skill in mathematics.It emphasizes the need to approach problems by breaking them downinto smaller components,as this helps to clarify the problem and identify potential solutions.The author argues that this approach is not only applicable tomathematics but also to various other fields,as it promotes clearer thinking and problem-solving abilities.The article then discusses the step-by-step thinking process in more detail.It suggests that the first step is tocarefully read and understand the problem, ensuring that all relevant information is identified.This is followed by breaking the problem down into smaller sub-problems or steps,each of which can be solved individually.The author emphasizes the need to be systematic and organized during this process,as it helps to prevent mistakes and confusion.Furthermore,the article highlights the importance of logical reasoning in step-by-step thinking.It states that each step should be justified with logical reasoning,ensuring that the solution is based on sound mathematical principles.The author advises against skipping steps or making assumptions without proper justification,as this can lead to erroneous results.The article also provides examples to illustrate the step-by-step thinking approach.It presents a complex problem and demonstrates how breaking it down into smaller steps can simplify the solution process.By solving each step individually and logically connecting them,the problem can be solved effectively.In conclusion,"Step by Step Thinking" emphasizes the significance of step-by-step thinking in mathematics and problem-solving. It encourages readers to approach problems systematically,breaking them down into smaller components,and justifying eachstep with logical reasoning.This approach promotes clearer thinking and enhances problem-solving abilities,not only in mathematics but also in other disciplines.。
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(Doc版).14
数学专业英语－MathematicansLeonhard Euler was born on April 15,1707,in Basel, Switzerland, the son of a mathematician and Caivinist pastor who wanted his son to become a pastor a s well. Although Euler had different ideas, he entered the University of Basel to study Hebrew and theology, thus obeying his father. His hard work at the u niversity and remarkable ability brought him to the attention of the well-known mathematician Johann Bernoulli (1667—1748). Bernoulli, realizing Euler’s tal ents, persuaded Euler’s father to change his mind, and Euler pursued his studi es in mathematics.At the age of nineteen, Euler’s first original work appeared. His paper failed to win the Paris Academy Prize in 1727; however this loss was compensated f or later as he won the prize twelve times.At the age of 28, Euler competed for the Pairs prize for a problem in astrono my which several leading mathematicians had thought would take several mont hs to solve.To their great surprise, he solved it in three days! Unfortunately, th e considerable strain that he underwent in his relentless effort caused an illness that resulted in the loss of the sight of his right eye.At the age of 62, Euler lost the sight of his left eye and thus became totally blind. However this did not end his interest and work in mathematics; instead, his mathematical productivity increased considerably.On September 18, 1783, while playing with his grandson and drinking tea, Eul er suffered a fatal stroke.Euler was the most prolific mathematician the world has ever seen. He made s ignificant contributions to every branch of mathematics. He had phenomenal m emory: He could remember every important formula of his time. A genius, he could work anywhere and under any condition.George cantor (March 3, 1845—June 1,1918),the founder of set theory, was bo rn in St. Petersburg into a Jewish merchant family that settled in Germany in 1856.He studied mathematics, physics and philosophy in Zurich and at the University of Berlin. After receiving his degree in 1867 in Berlin, he became a lecturer at the university of Halle from 1879 to 1905. In 1884,under the stra in of opposition to his ideas and his efforts to prove the continuum hypothesis, he suffered the first of many attacks of depression which continued to hospita lize him from time to time until his death.The thesis he wrote for his degree concerned the theory of numbers; however, he arrived at set theory from his research concerning the uniqueness of trigon ometric series. In 1874, he introduced for the first time the concept of cardinalnumbers, with which he proved that there were “more”transcendental numb ers than algebraic numbers. This result caused a sensation in the mathematical world and became the subject of a great deal of controversy. Cantor was troub led by the opposition of L. Kronecker, but he was supported by J.W.R. Dedek ind and G. Mittagleffer. In his note on the history of the theory of probability, he recalled the period in which the theory was not generally accepted and cri ed out “the essence of mathematics lies in its freedom!”In addition to his work on the concept of cardinal numbers, he laid the basis for the concepts of order types, transfinite ordinals, and the theory of real numbers by means of fundamental sequences. He also studied general point sets in Euclidean space a nd defined the concepts of accumulation point, closed set and open set. He wa s a pioneer in dimension theory, which led to the development of topology.Kantorovich was born on January 19, 1912, in St. Petersburg, now called Leni ngrad. He graduated from the University of Leningrad in 1930 and became a f ull professor at the early age of 22.At the age of 27, his pioneering contributi ons in linear programming appeared in a paper entitled Mathematical Methods for the Organization and planning of production. In 1949, he was awarded a S talin Prize for his contributions in a branch of mathematics called functional a nalysis and in 1958, he became a member of the Russian Academy of Science s. Interestingly enough, in 1965,kantorovich won a Lenin Prize fo r the same o utstanding work in linear programming for which he was awarded the Nobel P rize. Since 1971, he has been the director of the Institute of Economics of Ma nagement in Moscow.Paul R. Halmos is a distinguished professor of Mathematics at Indiana Univers ity, and Editor-Elect of the American Mathematical Monthly. He received his P h.D. from the University of Illinois, and has held positions at Illinois, Syracuse, Chicago, Michigan, Hawaii, and Santa Barbara. He has published numerous b ooks and nearly 100 articles, and has been the editor of many journals and se veral book series. The Mathematical Association of America has given him the Chauvenet Prize and (twice) the Lester Ford award for mathematical expositio n. His main mathematical interests are in measure and ergodic theory, algebraic, and operators on Hilbert space.Vito Volterra, born in the year 1860 in Ancona, showed in his boyhood his e xceptional gifts for mathematical and physical thinking. At the age of thirteen, after reading Verne’s novel on the voyage from earth to moon, he devised hi s own method to compute the trajectory under the gravitational field of the ear th and the moon; the method was worth later development into a general proc edure for solving differential equations. He became a pupil of Dini at the Scu ola Normale Superiore in Pisa and published many important papers while still a student. He received his degree in Physics at the age of 22 and was made full professor of Rational Mechanics at the same University only one year lat er, as a successor of Betti.Volterra had many interests outside pure mathematics, ranging from history to poetry, to music. When he was called to join in 1900 the University of Rome from Turin, he was invited to give the opening speech of the academic year. Volterra was President of the Accademia dei Lincei in the years 1923-1926. H e was also the founder of the Italian Society for the Advancement of Science and of the National Council of Research. For many years he was one of the most productive scientists and a very influential personality in public life. Whe n Fascism took power in Italy, Volterra did not accept any compromise and pr eferred to leave his public and academic activities.Vocabularypastor 牧师 hospitalize 住进医院theology 神学 thesis 论文strain 紧张、疲惫transcendental number 超越数relentless 无情的sensation 感觉，引起兴趣的事prolific 多产的controversy 争论，辩论depression 抑郁；萧条，不景气essence 本质，要素transfinite 超限的Note0. 本课文由几篇介绍数学家生平的短文组成，属传记式体裁。
数学专业英语(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.。
2023新译林版高中英语选择性必修二unit1课文原文及翻译
2023新译林版高中英语选择性必修二unit1课文原文及翻译Unit 1 Lesson TextI. The Beauty of MathematicsMathematics is the language of science and the foundation of modern civilization. It is a subject that can be both challenging and fascinating, and it plays a crucial role in our everyday lives. In this lesson, we will explore the beauty of mathematics and discover how it enriches our understanding of the world.Text A: The Language of ScienceMathematics is often referred to as the language of science. It provides scientists with a universal system of symbols and rules that allows them to describe and analyze natural phenomena. Whether it’s calculating the trajectory of a rocket or predicting the behavior of tiny particles, mathematics provides the necessary tools for scientists to make sense of the world around us.Text B: The Foundation of CivilizationMathematics has been the foundation of modern civilization for centuries. From the ancient Egyptians’ use of geometry to build pyramids to the development of calculus during the Scientific Revolution, mathematics has played a vital role in shaping our world. It enables us to solve complex problems, make accurate measurements, and design intricate structures.Without mathematics, many of the technological advancements we enjoy today would not be possible.Text C: Challenging and FascinatingWhile mathematics can be a challenging subject, it is also incredibly fascinating. It requires logical thinking, problem-solving skills, and a deep understanding of concepts. As students delve deeper into mathematics, they discover its elegance, patterns, and connections to other disciplines. The thrill of solving a difficult equation or discovering a new mathematical concept can be incredibly rewarding.II. 翻译1.文本A：科学的语言数学通常被称为科学的语言。
数学专业英语第二版 课文翻译
2.4 整数、有理数与实数4－A Integers and rational numbersThere exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers.有一些R的子集很著名，因为他们具有实数所不具备的特殊性质。
在本节我们将讨论这样的子集，整数集和有理数集。
To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers.我们从数字1开始介绍正整数，公理4保证了1的存在性。
1+1用2表示，2+1用3表示，以此类推，由1重复累加的方式得到的数字1,2,3，…都是正的，它们被叫做正整数。
Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”.严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。
数学英语词汇和数学学科中英文翻译
偏微分方程 partial differential equation
数理方程 equation of mathematical physics
代数拓扑学 algebraic topology
集合论 set theory
十六进制 hexadecimal system
权 weight, significance
进位 carry
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
体 solid
线段 segment
射线 radial
平行 parallel
相交 intersect
角 angle
角度 degree
弧度 radian
锐角 acute angle
直角 right angle
钝角 obtuse angle
平角 straight angle
周角 perigon
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
大于 is greater than
数学专业英语翻译
第一段翻译(2)：what is the exact value of the number pai?a mathematician made an experiment in order to find his own estimation of the number pai.in his experiment,he used an old bicycle wheel of diameter 63.7cm.he marked the point on the tire where the wheel was touching the ground and he rolled the wheel straight ahead by turning it 20 times.next,he measured the distance traveled by the wheel,which was 39.69 meters.he divided the number 3969 by 20*63.7 and obtained 3.115384615 as an approximation of the number pai.of course,this was just his estimate of the number pai and he was aware that it was not very accurate.数π的精确值是什么？一位数学家做了实验以便找到他自己对数π的估计。
在试验中，他用了一直径63.1厘米的旧自行车轮。
他在车轮接触地面的轮胎上做了标记，而且将车轮向前转动20次。
接下来，他测量了车轮经过的距离，是39.69米。
他用3969除20*63.7得到了数π的近似值3.115384615。
当然，这只是对数π的估计值，并且他也意识到不是很准确。
第二段翻译(5)：one of the first articles which we included in the "History Topics" section archive was on the history of pai.it is a very popular article and has prompted many to ask for a similar article about the number e.there is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one of pai.the number e is,compared to pai,a relative newcomer on the mathematical scene.我们包括在“历史专题”部分档案中的第一篇文章就是历史上的π，这是一篇很流行的文章，也促使许多人想了解下一些有关数e的类似文章。
数学专业英语课文翻译2-9,2-10
No matter what the students’ field of major interest is , knowledge of the rudiments of matrices is likely to broaden the range of literature that he can read with understanding .
2.9 微分方程简介 Introduction to Differential Equations
New Words & Expressions:
approximate evaluation 近似估计 initial 初始的
disintegrate 解体，衰变
differentiable 可微的 exponential 指数的 数
Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material present after a given time.
又如，某种放射性物质可能正在以已知的速度进行衰 变，需要我们确定在给定的时间后遗留物质的总量。
※In examples like these, we are trying to determine an unknown function from prescribed information expressed in the form of an equation involving at least one of the derivatives of the unknown function . 在类似的例子中，我们力求由方程的形式表述的信息 来确定未知函数，而这种方程至少包含了未知函数的 一个导数。
数学学科的英文翻译及相关术语解释和应用场景介绍
数学学科的英文翻译及相关术语解释和应用场景介绍Mathematics is a subject that is concerned with the study of numbers, quantities, and shapes. It is a fundamental discipline that has wide applications in science, engineering, economics, and other fields. In this article, we will explore the English translation of mathematical terms and their meanings, as well as the practical applications of mathematics.1. AlgebraAlgebra is a branch of mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols. It involves the use of variables, equations, and functions to solve problems. Algebra is used in many fields, including physics, engineering, and economics.代数是数学的一个分支，它涉及到数学符号的研究以及操作这些符号的规则。
它涉及到使用变量、方程和函数来解决问题。
代数在许多领域中得到了应用，包括物理学、工程学和经济学。
2. CalculusCalculus is a branch of mathematics that deals with the study of rates of change and slopes of curves. It involves the use of limits, derivatives, and integrals to solve problems related to motion, optimization, and other applications. Calculus is used in physics, engineering, and economics.微积分是数学的一个分支，它涉及到速率变化和曲线斜率的研究。
数学专业英语中英文对照翻译2.5[1]
![数学专业英语中英文对照翻译2.5[1]](https://img.taocdn.com/s3/m/1977e8d37c1cfad6195fa7c1.png)
2.5笛卡尔几何学的基本概念（basic concepts of Cartesian geometry）课文5-A the coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily , we do not talk about area by itself ,instead, we talk about the area of something . This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was to represent geometric points by numbers. The procedure for points in a plane is this ：Two perpendicular reference lines (called coordinate axes) are chosen, one horizonta l (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of Oand its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance ,although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the points.Figure 2-5-1 illustrates some examples.The point with coordinates (3,2) lies three units to the right of the y-axis and two units above the x-axis.The number 3 is called the x-coordinate of the point,2 its y-coordinate. Points to the left of the y-axis have a negative x-coordinate; those below the x-axis have a negtive y-coordinate. The x-coordinateof a point is sometimes called its abscissa and the y-coordinateis called its ordinate.When we write a pair of numberssuch as (a,b) to represent a point, we agree that the abscissa or x-coordinate,a is written first. For this reason, the pair(a,b) is often referred to as an ordered pair. It is clear that two ordered pairs (a,b) and (c,d) represent the same point if and only if we have a=c and b=d. Points (a,b) with both a and b positiveare said to lie in the first quadrant ,those with a<0 and b>0 are in the second quadrant ; and those with a<0 and b<0 are in the third quadrant ; and those with a>0 and b<0 are in the fourth quadrant. Figure 2-5-1 shows one point in each quadrant.The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin) . These lines determine three mutually perpendicular planes ,and each point in space can be completely described by specifying , with appropriate regard for signs ,its distances from these planes. We shall discuee three-dimensional Cartesian geometry in more detail later on ; for the present we confine our attention to plane analytic geometry.课文5-A：笛卡尔几何坐标系正如前面所提到的，积分应用的一种是计算面积。
高等数学教材翻译英文版
高等数学教材翻译英文版IntroductionIn recent years, the demand for high-quality translated textbooks has been on the rise, especially in the field of higher education. As English continues to be the dominant language in academic discourse, there is a growing need for translated versions of textbooks from different languages. This article aims to discuss the importance and challenges of translating a high-level mathematics textbook into English, and provide insights into how to effectively accomplish this task.Importance of Translating High-Level Mathematics Textbooks1. Global Accessibility: By translating high-level mathematics textbooks into English, we can ensure that these valuable educational resources are accessible to a wider audience worldwide. This is particularly beneficial for non-English speaking countries where English-language proficiency may still be a barrier to accessing advanced mathematical knowledge.2. Academic Exchange: Translated textbooks facilitate academic exchange between scholars and students from different countries. By sharing mathematical concepts and theories in a common language, researchers can collaborate more effectively and contribute to the global advancement of mathematics.3. Educational Equality: Translated textbooks promote educational equality by bridging the gap between English-speaking and non-English speaking students. Access to high-quality mathematics education should notbe limited by language barriers, and translating textbooks plays a crucial role in creating a level playing field for students worldwide.Challenges in Translating High-Level Mathematics Textbooks1. Technical Terminologies: High-level mathematics textbooks contain a multitude of technical terminologies that may not have direct translations in English. Translators must possess a strong mathematical background to accurately interpret and convey these complex concepts without losing their meaning.2. Linguistic Adaptation: The process of translating a mathematics textbook requires not only preserving the technical accuracy but also adapting the language style and structure to suit the English-speaking audience. The translator must find a balance between maintaining the rigor of the original text and ensuring clarity and comprehensibility for the target readers.3. Cultural Context: Mathematical concepts and theories may have cultural nuances that are difficult to translate directly. Translators need to possess a deep understanding of both the source and target cultures to accurately convey the intended meaning and context behind mathematical expressions.Approaches for Effective Translation1. Collaboration: Translating a high-level mathematics textbook should be a collaborative effort involving mathematicians, educators, and proficient translators. This interdisciplinary collaboration ensures the accuracy of the technical content and the effectiveness of the language adaptation.2. Review and Feedback: After the initial translation, it is essential to have the translated text reviewed by mathematics experts and educators. Their feedback and suggestions can further enhance the quality and accuracy of the translated textbook.3. Language Proficiency: The translator must possess an exceptional command of both the source language and English. A strong mathematical background combined with advanced language skills is crucial for accurately conveying complex mathematical concepts.ConclusionTranslating high-level mathematics textbooks into English is of great importance in today's globalized world. It promotes accessibility, academic exchange, and educational equality. However, it should be approached with caution due to the challenges it presents, such as technical terminology and cultural nuances. By adopting effective translation approaches, we can overcome these challenges and ensure the successful translation of high-level mathematics textbooks, contributing to the advancement of mathematics education worldwide.。
数学专业英语课文翻译
1-A：什么是数学数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
反过来，数学服务于实践并在所有领域扮演一个重要的角色。
没有数学的应用，现代化科学和技术的分支都不能有规律的发展。
从早期人类的需求引出了数和形状。
然后，几何学因测量陆续的发展出来，三角学来自于勘探问题。
为了处理一些更复杂的实践问题，人们建立了方程，通过求解方程的未知数，从而代数学出现了。
17世纪之前，人们局限于初等数学，例如几何、三角和代数，那些只考虑常数。
17世纪工业的迅速发展促进了经济学和科技的发展，并且我们需要处理变量。
从常数到变量的跳跃带来了两个属于高等数学的新的数学分支，解析几何和微积分学。
现在，高等数学中有了许多分支，数学分析、高等代数、微分方程、函数论等。
数学家们研究概念和命题。
公理、公社、定义和定理都是命题。
符号是一种特别并且很重要的数学工具，它常用于表示概念和命题。
公式、图形和表格充满着不同的符号。
阿拉伯数字1,2,3,4,5,6,7,8,9,0和加”+”减”-”乘”*”除”/”等号”=”使我们最熟悉的数学符号。
主要通过逻辑推导和计算来获得数学结论。
在数学史的很长的时期内，逻辑推论一直占据数学方法的中心地位。
现在，自从电子计算机迅速发展和广泛应用，计算的角色越来越重要。
现在，计算不仅用来处理信息与数据，而且用来完成一些在以前只能靠逻辑推理来做的工作，例如证明大多数的几何定理。
1-B：等式等式是关于两个数或数的符号相等的一种陈述。
因此a(a-5)=a^2-5a和x-3=5是等式。
等式有两种，恒等式和条件等式。
算术和代数恒等式是等式。
这种等式的两端要么一样，要么经过执行指定的运算后变成一样。
因此12-2=2+8，(m-n)(m+n)=m^2-n^2是恒等式。
含有字母的恒等式对其中字母的任何一组数值都成立。
因此恒等式x(a+2)=ax+2x变成3(7+2)=21+6或27=27，比如当x=3和a=7。
数学的英语文
数学的英语文English: Mathematics, often referred to as the language of the universe, is a fundamental discipline that explores patterns, structures, and relationships using symbols and logical reasoning. It encompasses a vast array of subfields, including algebra, geometry, calculus, statistics, and more, each with its own set of principles and techniques. Algebra delves into the manipulation of symbols and equations to solve unknowns, while geometry examines shapes, sizes, and spatial relationships. Calculus, on the other hand, deals with change and motion, providing tools to understand rates of change and accumulation. Statistics plays a crucial role in analyzing data and making inferences about populations based on samples. These diverse branches of mathematics find applications in virtually every field, from physics and engineering to economics and social sciences, underpinning technological advancements and scientific discoveries. Moreover, mathematics serves as a tool for problem-solving, fostering critical thinking skills and logical reasoning abilities essential for navigating the complexities of the modern world.中文翻译: 数学常被称为宇宙的语言，是一门探索模式、结构和关系的基础学科，利用符号和逻辑推理进行研究。
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英文翻译专业：数学与应用数学学号：081106117姓名：毛赛武Chapter2Functions and RelationsThe idea of associating an element from one setwith an element (or elements) from another is a fundamental one in mathematics.There are many different kinds of associations and a wide variety of notations used for expressing them.we will examine two kinds of associations:functions and relations,both of which have significant applications in computer science.We begin with functions,since you are more familiar with them, before considering the more general idea of relations.Recall that a function from A to B is a mapping from one set (the domain of the function) to another (the range of the function) where each element of the domain is mapped to one element of the range. For example,we could map each element in the set of integers to twice its value.In this case,if we call the function ƒ,one way of expressing the mapping is to write x)(=.It is importantf2xto specify the domain and range of a function.For example,if we wanted to restrict the preceding exampleto the integers,we could write Z f→:such thatxxf2)(=.Here,Z stands for the set of integers. On theother hand, we could have a function RRg→:defined by xxg2)(=. (R is the set of real numbers.) The functions g and ƒ are different but they agree on the set of integers.The critical idea is that a function maps each element in its domain to a unique element in its range.What is not allowed is for an element in the domain to be mapped to more than one element in the range. An example of a mapping that is not a function is not a function is one that maps each non-negative real number ɑto the numbers x such x=2. This mapping is not a function since, for example, 4 would get mapped to 2 and to -2. (There are many examples where this mapping fails to be a function. If it fails for just one value in the domain, we cannot call the mapping a function.)We illustrate a mapping that is a function and one that in not in the following diagrams:A B A B。
A function from A toB Not a function from A to BThere are so many functions that are important to us in computer science that we cannot possibly list them all. We will give a few examples of the most important ones used not only for carrying out computations, but also for analyzing the performance of programs.Exponential and Log FunctionsAn important class of functions for us are the exponential functions:b xx f )(. We call b the base of theexponential. Two bases are of particular importance for us:e(Euler s constant ≈2.7182818...) and 2. When you graphxor 2x(or any exponential with base largerthan 1), you should notice that the function grows very rapidly after a short while. The graphs of all exponentialGive three representations for the idea of associating any given real number with that number squared plus one.Is this Association a function?functions, with base greater than 1, have similar shapes.Recall that a polynomial over the reals,p ,in the variable x can be though of as a function of the formxa x a a a nnx x p ++++=...)(2210,where the coefficients a an,...,0are real numbers. The degree of )(x p is the largest nsuch that 0≠an. (If)(=x p then we say )(x p has degree-1.) As mentioned,exponential functions grow very rapidly. Indeed, if )(x p is any polynomial and1>b ,thenthere is a number Nsuch that if Nx >, then )(x p bx>.That is,eventually b xis greater than)(x p . This is trueregardlessofthedegreeof)(x p oritscoefficients,although for polynomials of particularly large degree, the first time b xexceeds)(x p (that is, thevalue of the smallestNmentioned previously)might beUse a symbolic computation software packageor a graphing calculator to compare the graphs of the identity function, the function that takes a real number and squares it, and an exponential function that takes any real number and raises 2 to that power.a rather larger number. For example,xxex31031000+> for all45>x .(You can check that 45 is the smallest such valuewhere this is true.)Another important class of functions is the collection of logarithm (or log) functions. The log functions "undo" the mappings of the exponential functions. We need to make this idea of "undoing, " or inverse,precise.The inverse of a function ƒ is another function, denoted by f 1-, such that xx f f=-))((1for all x in thedomain of ƒ. Since f 1- must itself be a function, ƒ has an inverse only if ƒ is one-to-one; that is, only if)()(y f x f = impliesyx =. The following are two functions;the first one is not one-to-one and but the second one is. A B A B。