数学专业英语作业

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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2-2

A line segment is named by the two end points. When no confusion result, the expression “line segment AB” is often replaced by segment AB or, simply, line AB.
学生通过几何的学习而达到的最主要目标是：在听，读，和思考时变得更加审慎。在学习几何的过程中，他们不再盲目地接受一些陈述和思想，而是在得出结论之前学会了清楚和审慎的思考。
2－B Some geometrical terms
1. Solids and planes. A solid is a three-dimensional figure. Common examples of solids are cube, sphere, cylinder, cone and pyramid.
点没有长度，宽度和厚度，但是标记了一个位置。我们熟悉铅笔尖，针尖这样的表达。我们可以用一个小圆点来表示一个点，在它旁边用打印体大写字母来命名，如图2-2-1中的点A。
The line is named by labeling two points on it with capital letters or one small letter near it. The straight line extends infinitely far in two directions and has no ends. The part of the line between two points on the line is termed a line segment.
有三种线：直线，折线和曲线。曲线是指其中没有任何部分是直的。折线是由连起来的直线段构成，如图2-2-3中的ABCDE.

数学专业英语课后答案

数学专业英语课后答案1、He kept walking up and down, which was a sure（）that he was very worried. [单选题] *A. sign(正确答案)B. characterC. natureD. end2、20．Jerry is hard-working. It’s not ______ that he can pass the exam easily. [单选题] * A．surpriseB．surprising (正确答案)C．surprisedD．surprises3、--What’s the weather like today?--It’s _______. [单选题] *A. rainB. windy(正确答案)C. sunD. wind4、Many of my classmates are working _______volunteers. [单选题] *A. as(正确答案)B. toC. atD. like5、Mary is interested ______ hiking. [单选题] *A. onB. byC. in(正确答案)D. at6、We need a _______ when we travel around a new place. [单选题] *A. guide(正确答案)B. touristC. painterD. teacher7、While they were in discussion, their manager came in by chance. [单选题] *A. 抓住时机C. 碰巧(正确答案)D. 及时8、Our campus is _____ big that we need a bike to make it. [单选题] *A. veryB. so(正确答案)C. suchD. much9、57．Next week will be Lisa's birthday. I will send her a birthday present ________ post. [单选题] *A．withB．forC．by(正确答案)D．in10、The manager demanded that all employees _____ on time. [单选题] *A. be(正确答案)B. areC. to be11、He has two sisters but I have not _____. [单选题] *A. noneB. someC. onesD. any(正确答案)12、______! It’s not the end of the world. Let’s try it again.（）[单选题] *A. Put upB. Set upC. Cheer up(正确答案)D. Pick up13、My brother usually _______ his room after school. But now he _______ soccer. [单选题] *A. cleans; playsB. cleaning; playingC. cleans; is playing(正确答案)D. cleans; is playing the14、The museum is _______ in the northeast of Changsha. [单选题] *A. sitB. located(正确答案)C. liesD. stand15、______ my great joy, I met an old friend I haven' t seen for years ______ my way ______ town. [单选题] *A. To, in, forB. To, on, to(正确答案)C. With, in, toD. For, in, for16、In the closet（）a pair of trousers his parents bought for his birthday. [单选题] *A. lyingB. lies(正确答案)c. lieD. is lain17、44．—Hi, Lucy. You ________ very beautiful in the new dress today.—Thank you very much. [单选题] *A．look(正确答案)B．watchC．look atD．see18、They all choose me ______ our class monitor.（）[单选题] *A. as(正确答案)B. inC. withD. on19、My mother’s birthday is coming. I want to buy a new shirt ______ her.（）[单选题] *A. atB. for(正确答案)C. toD. with20、--Could you please tell me _______ to get to the nearest supermarket?--Sorry, I am a stranger here. [单选题] *A. whatB. how(正确答案)C. whenD. why21、?I am good at schoolwork. I often help my classmates _______ English. [单选题] *A. atB. toC. inD. with(正确答案)22、He was very excited to read the news _____ Mo Yan had won the Nobel Prize for literature [单选题] *A. whichB. whatC. howD. that(正确答案)23、Becky is having a great time ______ her aunt in Shanghai. （）[单选题] *A. to visitB. visitedC. visitsD. visiting(正确答案)24、In the future, people ______ a new kind of clothes that will be warm when they are cold, and cool when they’re hot.（）[单选题] *A. wearB. woreC. are wearingD. will wear(正确答案)25、This message is _______. We are all _______ at it. [单选题] *A. surprising; surprisingB. surprised; surprisedC. surprising; surprised(正确答案)D. surprised; surprising26、Your homework must_______ tomorrow. [单选题] *A. hand inB. is handed inC. hands inD. be handed in(正确答案)27、Since the war their country has taken many important steps to improve its economic situation. [单选题] *A. 制定B. 提出C. 讨论D. 采取(正确答案)28、How _______ Grace grows! She’s almost as tall as her mother now. [单选题] *A. cuteB. strongC. fast(正确答案)D. clever29、_____ is not known yet. [单选题] *A. Although he is serious about itB. No matter how we will do the taskC. Whether we will go outing or not(正确答案)D. Unless they come to see us30、You cannot see the doctor _____ you have made an appointment with him. [单选题] *A. exceptB.evenC. howeverD.unless(正确答案)。

数学专业英语(Doc版).22

数学专业英语－The Normal DistributionWe shall begin by considering some simple continuously variable quantity like stature.We know this varies greatly from one individual to another ,and may al so expect to find certain average differences between people drawn from differ ent social classes or living in different geographical areas,etc.Let us suppose th at a socio-medical survey of a particular community has provided us with a re presentative sample of 117 males whose heights are distributed as shown in th e first and third columns of Table 1.Table 1.Distribution of stature in 117 malesWe shall assume that the original measurements were made as accurately as possible,but that they are given here only to the mearest 0.02 m (i.e.2 cm).Thu s the group labeled “1.66”contains all those men whose true measurements were between 1065 and 1067 m.One si biable to run into trouble if the exact methods of recording the measurements and grouping them are not specified ex actly.In the example just given the mid-point of the interval labeled”1.66”m. But suppose that the original readings were made only to the nearest 0.01 m (i.e. 1 cm )and then “rounded up “to the nearest multiple of 0.02 m.We sho uld then have “1.65”, which covers the range 1.645 to 1.655,included with “1.66”.The interval “1.66”would then contain all measurements lying betwe en 1.645 m and 1.665 m .for which the mid-point is 1.655 m. The difference of 5 mm from the supposed value of 1.66 m could lead to serious inaccuracy in certain types of investigation.A convenient visual way of presenting such data is shown in fig. 1, in which the area of e ach rectangle is ,on the scale used, equal to the observed proportion or percentage of indiv iduals whose height falls in the corresponding group.The total area covered by all the recta ngles therefore adds up to unity or 100per cent .This diagram is called a histogram.It is ea sily constructed when ,sa here ,all the groups are of the same width.It is also easily adapt ed to the case when the intervals are uneqal, provided we remember that the areas of the rec tangles must be proportional to the numbers of units concerned.If, for example, we wished to group togcther the entries for the three groups 1.80,1.82 and 1.84 m,totaling 7 individuals or 6 per cent of the total,then we should need a rectangle whose base covered 3working grou ps on the horizontal scale but whose height was only 2 units on the vertical scale shown in the diagram.In this way we can make allowance for unequal grouping intervals ,but it is usua lly less troublesome if we can manageto keep them all the same width.In some books histogram s are drawn so that the area of each rectangle is equal to the actual number (instend of the proportion) of individuals in the corresponding group.It is better, however, to use proport ions, sa different histograms can then be compared directly.----------------------------------------------The general appearance of the rectangles in Fig.1 is quite striking ,especially the tall hump in the centre and the rapidly falling tails on each side.There are certain minor irregularities in the pattern, and these would, in general ,be more ronounced if the size of the sample were smaller. Conversely, weth larger sa mples we usually find that the set of rectangles presents a more regular appearance. This suggests that if we had a very large number of measurements ,the ultimate shape of the picture for a suitably small wi dth of rectangle would be something very like a smooth curve,Such a curve could be regarded as represe nting the true ,theoretical or ideal distribution of heights in a very(or ,better,infinitely)large population of individuals.What sort of ideal curve can we expect ? There are seveala theoretical reasons for expecting the so-call ed Gaussiao or “normal “curve to turn up in practice;and it is an empirical fact that such a curve lften describes with sufficient accuracy the shape of histograms based on large numbers of obscrvations. More over,the normal curve is one of the easiest to handle theoretically,and it leads to types of statistical analy sis that can be carried out with a minimum amount of computation. Hence the central importance of this distribution in statistical work .The actual mathematical equation of the normal curve is where u is the mean or average value and is t he standard deviation, which is a measure of the concentration of frequency about the mean. More will b e said about and later .The ideal variable x may take any value from to .However ,some real measureme nts,like stature, may be essentially positive. But if small values are very rare ,the ideal normal curve ma y be a sufficiently close approximation. Those readers who are anxious to avoid as much algebraic mani pulation as possible can be reassured by the promise that no derect use will be made in this book of th e equation shown. Most of the practical numerical calculations to which it leads are fairly simple.Fig. 1 shows a normal curve, with its typocal symmetrical bell shape , fitted by suitable methods to the data embodied in the rectangles. This is not to say that the fitted curve is actually the t rue, ideal one t o which the histogram approxime.tes; it is merely the best approximation we can find.The mormal curve used above is the curve we have chosen to represent the frequency distribution of st ature for thr ideal or infinitely large population. This ideal poplation should be contrasted with the limite d sample of obsrever. Values that turns up on any occasion when we make actual measurements in the r eal world. In the survey mentioned above we had a sample of 117 men .If the community were sufficie ntly large for us to collect several samples of this size, we should find that few if any of the correspon ding histograms were exactly the same ,although they might all be taken as illustrating the underlying fre quency distribution. The differences between such histograms constitute what we call sampling variation, and this becomes more prominent at the size of sample decreases.VocabularySocio-medical survey 社会医疗调查表 visual 可见的。

数学专业英语(Doc版).21

学专业英语－StatisticsThe term statistics is used in either of two senses.In common parlance it is ge nerally employed synonymously with the word data.Thus someone may say tha t he has seen”statistics of industrial accidents in the United States.”It would be conducive to greater precision of meaning if we were not to use statistics i n this sense,but rather to say “data (or figures ) of industrial accidents in the United States.”“Statistics”also refers to the statistical principles and methods which have be en developed for handling numerical data and which form the subject matter o f this text.Statistical methods,or statistics, range form the most elementary descr iptive devices, which may be understood by anyone , to those extremely compl icated mathematical procedures which are comprehended by only the most expe rt theoreticians.It is the purpose of this volume not to enter into the highly ma thematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.Statistics may be defined as the collection, presentation, analysis, and interpreta tion of numerical data.The facts which are dealt with must be capable of num erical expression.We can make little use statistically of the information that dw ellings are built of brick, stone, wood, and other materials; however, if we are able to determine how many or what proportion of,dwellings are constructed of each type of material, we have numerical data suitable for statistical analysi s.Statistics should not be thought of as a subject correlative with physics, chemi stry, economics, and sociology. Statistics is not a science; it is a scientific met hod. The methods and procedures which we are about to examine constitute a useful and often indispensable tool for the research worker. Without an adequat e understanding of statistics, the investigator in the social sciences may frequen tly be like the blind man groping in a dark closet for a black cat that isn’t t here. The methods of statistics are useful in an ever---widening range of huma n activities, in any field of thought in which numerical data may be had.In defining statistics it was pointed out that the numerical data are collected, p resented, analyzed, and interpreted. Let us briefly examine each of these four p rocedures.COLLECTION Statistical data may be obtained from existing published or un published sources, such as government agencies, trade associations, research bur eaus, magazines, newspapers, individual research workers, and elsewhere. On th e other hand, the investigator may collect his own information, going perhaps f rom house to house or from firm to firm to obtain his data. The first-hand col lection of statistical data is one of the most difficult and important tasks whicha statistician must face. The soundness of his procedure determines in an ove rwhelming degree the usefulness of the data which he obtains.It should be emphasized, however, that the investigator who has experience an d good common sense is at a distinct advantage if original data must be colle cted. There is much which may be taught about this phase of statistics, but th ere is much more which can be learned only through experience. Although a p erson may never collect statistical data for his own use and may always use p ublished sources, it is essential that he have a working knowledge of the proce sses of collection and that he be able to evaluate the reliability of the data he proposes to use. Untrustworthy data do not constitute a satisfactory base upon which to rest a conclusion.It is to be regretted that many people have a tendency to accept statistical dat a without question. To them, any statement which is presented in numerical ter ms is correct and its authenticity is automatically established.PRESENTATION Either for one’s own use or for the use of others, the dat a must be presented in some suitable form. Usually the figures are arranged in tables or presented by graphic devices.ANALYSIS In the process of analysis, data must be classified into useful and logical categories. The possible categories must be considered when plans are made for collecting the data, and the data must be classified as they are tabu lated and before they can be shown graphically. Thus the process of analysis i s partially concurrent with collection and presentation.There are four important bases of classification of statistical data: (1) qualitativ e, (2) quantitative, (3) chronological, and (4) geographical, each of which will be examined in turn.Qualitative When, for example, employees are classified as union or non—uni on, we have a qualitative differentiation. The distinction is one of kind rather t han of amount. Individuals may be classified concerning marital status, as singl e, married, widowed, divorced, and separated. Farm operators may be classified as full owners, part owners, managers, and tenants. Natural rubber may be de signated as plantation or wild according to its source.Quantitative When items vary in respect to some measurable characteristics, a quantitative classification is appropriate. Families may be classified according t o the number of children. Manufacturing concerns may be classified according to the number of workers employed, and also according to the values of goods produced. Individuals may be classified according to the amount of income ta x paid.Chronological Chronological data or time series show figures concerning a par ticular phenomenon at various specified times. For example, the closing price o f a certain stock may be shown for each day over a period of months of year s; the birth rate in the United States may be listed for each of a number of y ears; production of coal may be shown monthly for a span of years. The anal ysis of time series, involving a consideration of trend, cyclical period (seasonal ), and irregular movements, will be discussed.In a certain sense, time series are somewhat akin to quantitative distributions i n that each succeeding year or month of a series is one year or one month fu rther removed from some earlier point of reference. However, periods of time —or, rather, the events occurring within these periods—differ qualitatively from each other also. The essential arrangement of the figures in a time sequence i s inherent in the nature of the data under consideration.Geographical The geographical distribution is essentially a type of qualitative distribution, but is generally considered as a distinct classification. When the p opulation is shown for each of the states in the United States, we have data which are classified geographically. Although there is a qualitative difference b etween any two states, the distinction that is being made is not so much of ki nd as of location.The presentation of classified data in tabular and graphic form is but one elem entary step in the analysis of statistical data. Many other processes are describ ed in the following passages of this book. Statistical investigation frequently en deavors to ascertain what is typical in a given situation. Hence all type of occ urrences must be considered, both the usual and the unusual.In forming an opinion, most individuals are apt to be unduly influenced by un usual occurrences and to disregard the ordinary happenings. In any sort or inve stigation, statistical or otherwise, the unusual cases must not exert undue influe nce. Many people are of the opinion that to break a mirror brings bad luck. H aving broken a mirror, a person is apt to be on the lookout for the unexpecte d”bad luck “and to attribute any untoward event to the breaking of the mir ror. If nothing happens after the mirror has been broken, there is nothing to re member and this result (perhaps the usual result )is disregarded. If bad luck oc curs, it is so unusual that it is remembered, and consequently the belief is rein forced. The scienticfic procedure would include all happenings following the br eaking of the mirror, and would compare the “resulting”bad luck to the am ount of bad luck occurring when a mirror has not been broken.Statistics, then, must include in its analysis all sorts of happenings. If we are studying the duration of cases of pncumonia, we may study what is typical by determining the average length and possibly also the divergence below and ab ove the average. When considering a time series showing steel—mill activity,we may give attention to the typical seasonal pattern of the series, to the gro wth factor( trend) present, and to the cyclical behaviour. Sometimes it is found that two sets of statistical data tend to be associated.Occasionlly a statistical investigation may be exhaustive and include all possibl e occurrences. More frequently, however, it is necessary to study a small grou p or sample. If we desire to study the expenditures of lawyers for life insuran ce, it would hardly be possible to include all lawyers in the United States. Re sort must be had to a sample;and it is essential that the sample be as nearly r epresentative as possible of the entire group, so that we may be able to make a reasonable inference as to the results to be expected for an entire populatio n. The problem of selecting a sample is discussed in the following chapter.Sometimes the statistician is faced with the task of forecasting. He may be req uired to prognosticate the sales of automobile tires a year hence, or to forecast the population some years in advance. Several years ago a student appeared i n summer session class of one of the writers. In a private talk he announced t hat he had come to the course for a single purpose: to get a formula which w ould enable him to forecast the price of cotton. It was important to him and h is employers to have some advance information on cotton prices, since the con cern purchased enormous quantities of cotton. Regrettably, the young man had to be disillusioned. To our knowledge, there are no magic formulae for forecas ting. This does not mean that forecasting is impossible; rather it means that fo recasting is a complicated process of which a formula is but a small part. And forecasting is uncertain and dangerous. To attempt to say what will happen in the future requires a thorough grasp of the subject to be forecast, up-to-the-m inute knowledge of developments in allied fields, and recognition of the limitat ions of any mechanica forecasting device.INTERPRETATION The final step in an investigation consists of interpreting the data which have been obtained. What are the conclusions growing out of t he analysis? What do the figures tell us that is new or that reinforces or casts doubt upon previous hypotheses? The results must be interpreted in the light of the limitations of the original material. Too exact conclusions must not be drawn from data which themselves are but approximations. It is essential, howe ver, that the investigator discover and clarify all the useful and applicable mea ning which is present in his data.VocabularyStatistics统计学in tables 列成表Statistical 统计的tabular列成表的Statistical data统计数据sample样本Statistical method统计方法 inference推理，推断Original data原始数据 reliance信赖Qualitative定性的forecasting预测Quantitative定量的 in common parlance按一般说法Chronological年代学的 conducive有帮助的Time series时间序列grope摸索Cyclical循环的 akin to类似Period周期apt to易于Periodic周期的 undue不适当的Prognosticate预测 sociology社会学Authenticity可信性，真实性phase相位；方面Synonymously同义的categories范畴，类型Correlative相互关系的，相依的 concurrent会合的，一致的，同时发生的Notes1. It is the purpose of this volume not to enter into the highly mathematical and theoretical a spects of the subject but rather to treat of its more elementary and more frequently used phases.意思是：本书的目的并不是要深入到这个论题的有关高深的数学与理论的方面，而是要讨论它的更为初等和更为常用的方面，not…but rather 意思是“不是…而是”，而rather than意思是“宁愿…而不”，两者意思相近但有差别（主要表现为强调哪方面的差别）。

数学专业英语45753

Therefore, the difference quotient represents the trigonometric tangent of the angle a that PQ makes with the horizontal.
因此，差商表示PQ与水平线的夹角a的正切。
The real number tana is called the slope of the line through P and Q and it provides a way of measuring the “steepness” of the line.
上一节描述的例子指出了介绍导数概念的方法。
We begin with a function f defined at least on some open interval (a,b) on the x-axis. 我们从一个至少定义在x轴的开区间(a,b)上的函数入手。
Then we choose a fixed point x in this interval and introduce the difference quotient
一般地, 由f(x)产生f'(x)的极限的过程提供了一种方法, 从一个给定的函数 f 得到一个新的函数f'。这个过程叫做微分法。 f'叫做f的一阶导数。
If f' , in turn, is defined on an open interval, we can try to compute its first derivative, denoted by f″ and called the second derivative of f.
(P67 第四段第一句话)
在直线运动中，速度的一阶导数称为加速度。

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(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.。

数学专业英语(Doc版).15

数学专业英语－Notations and Abbreviations (I) Learn to understand N set of natural numbersZ set of integersR set of real numbersC set of complex numbers+ plus; positive－minus; negative×multiplied by; times÷divided by＝equals; is equal to≡identically equal to≈,≌approximately equal to＞greater than≥greater than or equal to＜less than≤less than or equal to》much greater than《much less thansquare rootcube rootnth root│a│ absolute value of an! n factoriala to the power n ; the nth power of a[a] the greatest integer≤athe reciprocal of aLet A, B be sets∈ belongs to ; be a member ofnot belongs tox∈A x os amember of A∪ unionA∪B A union B∩ intersectionA∩B A intersection BA B A is a subset of B;A is contained in B A B A contains Bcomplement of Athe closure of Aempty set( ) i=1,2,…,r j=1,2,…,s r-by-s（r×s）matrix││I,j=1,2,…,n determinant of order ndet( ) the determinant of the matrix ( )vector Fx=( , ,…, ) x is an n-tuple of‖‖the norm of …‖ parallel to┴ perpendicular tothe exponential function of xlin x the logarithmic function of xsie sinecos cosinetan tangentsinh hyperbolic sinecosh hyperbolic cosinethe inverse of ff is the composite or the composition of u and vthe limit of …as n approaches ∞(as x approaches )x a x approaches a, the differential coefficient of y; the 1st derivative of y , the nth derivative of ythe partial derivative of f with respect to xthe partial derivative of f with respect to ythe indefinite integral of fthe definite integral of f between a and b (from a to b) the increment of xdifferential xsummation of …the sum of the terms indicated∏the product of the terms indicated=> impliesis equivalent to（）round brackets; parantheses[ ] square brackets{ } braces。

数学专业英语(只是部分,不是很完整)

Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition “+”, subtraction “-” , multiplication “×”, division “÷” and equality “=”. 数学家研究的是概念和命题，公理，公设，定义和定理都是命题。符号是数学中一个特殊而有用的工具，常用于表达概念和命题。公式，图形和图表都是不同的符号……..
1－B Equation
An equation is a statement of the equality between two equal numbers or number symbols. Equations are of two kinds---- identities and equations of condition. An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation. 等式是关于两个数或者数的符号相等的一种描述。等式有两种－恒等式和条件等式。算术或者代数恒等式都是等式。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。

数学专业英语课后答案

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

2023大学_数学专业英语试题及参考答案

2023数学专业英语试题及参考答案数学专业英语试题一、词汇及短语1. For a long period of the history of mathematics, the centric place of mathematical methods was occupied by the logical deductions “在数学史的很长的时期内，是逻辑推理一直占据数学方法的中心地位”2. An equation is a statement of the equality between two equal numbers or number symbols.equation ：“方程”“等式”等式是关于两个数或数的符号相等的'一种陈述3. In such an equation either the two members are alike, or become alike on performance of the indicated operation. 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。

注“two members”表等号的两端alike 相同的一样的On the performance of …中的“on”引导一个介词短语做状语Either…or…4. is true “成立”5. to more and change the terms移次和变形without making the equation untrue 保持方程同解数学专业英语试题二、句型及典型翻译1. change the terms about 变形2. full of ：有许多的充满的例 The streets are full of people as on a holiday(像假日一样，街上行人川流不息)3. in groups of ten…4. match something against sb. “匹配”例 Long ago ,when people had to count many things ,they matched them against their fingers. 古时候，当人们必须数东西时，在那些东西和自己的手指之间配对。

数学专业英语课后部分习题答案

数学专业英语课后部分习题答案2.1 数学、方程与比例（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。

Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.（2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。

No modern scientific and technological branches could be regularly developed without the application of mathematics.（3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。

Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.（4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。

Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。

Equation is different from arithmetic identity in that it contains unknown quantity which can join operations.（6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。

你的数学作业写作文英语

My Math Homework in EnglishMathematics is a subject that often requires a lot of practice and understanding of various concepts.When it comes to math homework,it can be both challenging and rewarding. Heres a brief account of my experience with math homework in English.1.Understanding the Assignment:The first step in tackling any math homework is to understand the assignment.This involves reading the questions carefully and identifying the key concepts that are being tested.2.Gathering Materials:Before starting,I gather all the necessary materials such as textbooks,notebooks,and any reference materials that might be helpful.This also includes having a calculator handy for any complex calculations.3.Breaking Down the Problems:I find it helpful to break down complex problems into smaller,more manageable parts.This allows me to focus on one aspect at a time and ensures that I dont miss any steps.4.Working Through the Problems:I start working on the problems,making sure to show all my work.This is important for two reasons:it helps me keep track of my thought process,and it allows my teacher to see where I might be going wrong if I make a mistake.5.Checking My Work:After completing a problem,I go back and check my work to ensure that I havent made any arithmetic errors or misinterpreted the question.This is a crucial step that can save time and frustration later on.6.Seeking Help When Needed:If Im stuck on a particular problem,I dont hesitate to seek help.This could be from a classmate,a teacher,or even online resources.Its better to ask for help than to submit incorrect work.7.Reviewing and Learning:After completing the homework,I review the problems, especially the ones I found difficult.Understanding why a particular method works helps me learn and retain the information better.8.Submitting the Homework:Once Im satisfied with my work,I organize it neatly and submit it on time.Being punctual with homework submissions is important for maintaining good academic standing.9.Reflecting on the Process:After submitting,I reflect on the process.Did I understandthe concepts?Were there any areas where I struggled?This reflection helps me identify areas for improvement.10.Preparing for the Next Assignment:Finally,I prepare for the next math homework by reviewing the concepts covered in the current assignment and anticipating what might be covered next.In conclusion,doing math homework in English,or any language,is about more than just solving problems.Its about developing a deep understanding of mathematical concepts, practicing problemsolving skills,and learning from mistakes.Its a process that requires patience,persistence,and a willingness to learn.。

有数学作业英语作文初一

As a high school student, Ive always found math to be one of the most challenging subjects. Its not just about solving problems its about understanding the logic and the reasoning behind each step. One particular incident with a math homework assignment in my first year of junior high school stands out as a pivotal moment in my academic journey.It was a typical Tuesday evening, and I was sitting at my desk, staring at a page filled with algebraic equations that seemed to be dancing before my eyes. The assignment was to solve a set of problems that involved complex algebraic manipulations, and I was struggling. My mind was a whirlwind of numbers and symbols, and I felt like I was in a maze with no clear path to the exit.The first problem was a simple equation, but as I moved on, the complexity increased. I tried to apply the methods I had learned in class, but it felt like I was hitting a wall. Frustration began to build up inside me, and I could feel the pressure mounting. I glanced at the clock it was getting late, and I still had other subjects to study for.Determined not to give up, I took a deep breath and started over. I broke down each problem into smaller parts, analyzing the equations step by step. I remembered my teachers advice: Dont rush through the problems take your time to understand the structure of each equation. I began to visualize the problems as puzzles, each with a unique solution waiting to be discovered.Slowly but surely, I started to see patterns and connections between theequations. The numbers and symbols that once seemed like an impenetrable fortress now appeared as a series of logical steps. Each correct solution brought a sense of accomplishment and boosted my confidence.As I delved deeper into the assignment, I discovered the beauty of mathematics. It wasnt just about getting the right answer it was about the journey of discovery and the satisfaction of overcoming a challenge. The problems that once seemed insurmountable now felt like a puzzle that I was piecing together.By the time I finished the last problem, it was well past midnight. My eyes were heavy, and my hand ached from writing, but I felt a sense of triumph.I had conquered the math homework, not just by solving the problems but by gaining a deeper understanding of the subject.The next day, when I handed in my homework, I felt a sense of pride. My teachers feedback was positive, and my classmates were impressed by my perseverance. This experience taught me that with patience, determination, and a willingness to break down complex problems, I could overcome any challenge that came my way.Looking back, that math homework assignment wasnt just about solving equations it was a lesson in resilience and problemsolving. It showed me that even when faced with difficulties, theres always a way to find a solution. Its a lesson that I carry with me to this day, not just in math but in all aspects of life.。

数学的英语怎么写作业

数学的英语怎么写作业Certainly! Here's an example of how to write an assignment in English on the topic of mathematics:Introduction:Mathematics is a universal language that transcends cultural and geographical boundaries. It is the foundation upon which many scientific disciplines are built. In this assignment, we will delve into the basic principles of mathematics and explore how they are applied in various contexts.Section 1: Numbers and Operations- Subsection 1.1: Integers- Definition and properties of integers- Examples of integer operations (addition, subtraction, multiplication, division)- Subsection 1.2: Fractions and Decimals- Understanding fractions and their equivalence to decimals - Conversion between fractions and decimals- Basic arithmetic with fractionsSection 2: Algebra- Subsection 2.1: Variables and Expressions- Introduction to variables and algebraic expressions- Simplifying expressions- Subsection 2.2: Equations and Inequalities- Solving linear equations- Handling inequalities- Applications of algebra in real-life scenariosSection 3: Geometry- Subsection 3.1: Shapes and Angles- Classification of geometric shapes (triangles, quadrilaterals, circles)- Properties of angles and their types (acute, obtuse, right)- Subsection 3.2: Area and Perimeter- Formulas for calculating the area and perimeter of common shapes- Practical applications in everyday lifeSection 4: Trigonometry- Subsection 4.1: Basic Trigonometric Ratios- Sine, cosine, and tangent in right-angled triangles- Relationships between these ratios- Subsection 4.2: Trigonometric Functions- Graphs of trigonometric functions- Applications in physics and engineeringConclusion:Mathematics is not just a subject confined to textbooks; it is a tool that helps us understand the world around us. By mastering the basics, we can unlock the door to more complex and fascinating areas of study.References:- [List of academic references or textbooks used for research]Remember to follow the guidelines provided by your teacher or institution for formatting, citation, and structure. This isa basic template and can be expanded or modified according to the specific requirements of your assignment.。

数学专业英语课后答案

数学专业英语作业

作业一、汉译英1．唯一性：若数列{n x }收敛，则它只有一个极限.证明. 设lim ,n n x A →∞= 对于给定的正数ε，存在正整数1,N 对于一切 1,n N >有||.2n x A ε-< （1）又设lim ,n n x B →∞= 对于上述的正数ε，存在正整数2,N 对于一切2,n N >有||.2n x B ε-< （2）取12max{,},N N N =当,n N >（1）与（2）式都成立，此时有|||()()|||||,22n n n n A B x B x A x A x B εεε-=---≤-+-<+=即||,A B ε-<由于A ，B 均为常数，而ε是任意正数，故只能.A B =2．基于学生数学能力结构的分析提出的培养建议摘要公民数学素养的提高无论对于社会发展还是其个人发展都具有重要意义。

为了提高中小学数学教育质量，必须明确中小学数学能力结构。

中小学生的数学能力可以分为两个层次，运算能力、空间想象能力、信息处理能力是第一个层次，逻辑思维能力和问题解决能力是第二个层次；模式能力在这两个层次之间起着非常重要的桥梁作用。

据此，本文提出了中小学生数学能力培养的几点建议。

关键词中小学生；数学能力结构；模式能力二、英译汉1．The Role of Mathematics in Economics. Economics is a mathematical discipline. This assertion may seem strange to the traditional political economist, but mathematical methods were introduced at an early stage (Cournot,1838) in the two-hundred-year history of our subject and have been steadily growing in significance. At the present time and, essentially, since the end of World War II, mathematical methods have become predominant in American economics. The mathematical approach was originally inspired in Europe and England but it has flowered in America, with no little stimulus from European immigrants. The mathematical approach is steadily gaining favor throughout the world, especially because the younger generation in developing economics is embracing the new methods and because the socialist countries have shed a previous bias against the use of mathematical methods in economics. It is clear that the future development of economics will see continued and increasing use of mathematics, although it would be rash to assume that the future course of economic analysis will be predominantly mathematical as it has been in the last twenty years.2. The first paragraph of the introduction should be comprehensible to any mathematician, and it should pinpoint the location of the subject matter. The main purpose of the introduction is to present a rough statement of the principal results; include this statement as soon as it isfeasible to do so, although it is sometimes well to set the stage with a preliminary paragraph. The remainder of the introduction can discuss the connections with other results.3．The values of the constructed function should not exceed the maximum permissible.4．Rational number can be represented by the quotient of two integers, which is different from irrational one.5．It should be noted that equation (1) and (2) are merely two different ways of expressing precisely the same relations.。

数学专业英语(10)

Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@§2.10 Dependent and Independent SetsTEXT A Dependent and independent sets in a linear spaceDEFINITION A set of S of elements in a linear space V is called dependent if there is a finite set of distinct elements in S, say x 1, …, x k, and corresponding set of scalars c 1, …, c k , not all zero, such that10.k i i i c x==∑The set S is called independent if it is not dependent. In this case, for all choices of distinct elements x 1, …, x k in S and scalars c 1, …, c k ,10k i i i c x==∑ implies c 1=c 2=⋯=c k =0Although dependence and independence are properties of sets ofelements, we also apply these terms to the elements themselves. For example, the elements in an independent set are called independent elements.If S is a finite set, the foregoing definition agrees with that given in Chapter 12 for the space V n (i.e. R n). However, the present definition is not restricted to finite sets.EXAMPLE 1 If a subset T of a set S is dependent, then S itself is dependent. This is logically equivalent to the statement that every subset of an independent set is independent.EXAMPLE 2 If one element in S is a scalar multiple of another, then S is dependent.EXAMPLE 3 If 0 S, then S dependent.EXAMPLE 4 The empty set is independent.Many examples of dependent and independent sets of vectors in V nwere discussed in Chapter 8. The following examples illustrate these concepts in function spaces. In each case the underlying linear space V is the set of all real-valued functions defined on the real line.EXAMPLE 5 Let u 1(t) =cos 2t, u 2(t)=sin 2t, u 3(t)=1 for all real t . The Pythagorean identity shows that u 1+u 2-u 3=0, so the three functions u 1, u 2, u 3 are dependent.EXAMPLE 6 Let u k (t)=t k for k=0,1 ,2, …, and t real. The set S={u 0, u 1, u 2,…} is independent. To prove this , it suffices to show that for each n the n+1 polynomials u 0, u 1, …, u n are independent. A relation of the form 0k k c u =∑ means that(10.1) 0k k c t =∑for all real t . When t=0, this gives c 0=0. Differentiating (10.1) and setting t=0, we find that c 1=0. Repeating the process, we find that eachcoefficient c k is zero.EXAMPLE 7 If a 1, …, a n are distinct real numbers, the n exponential functionsu 1(x)=e a 1x , …, u n (x)=e a n xare independent. We can prove this by induction on n . The result holds trivially when n=1. Therefore, assume it is true for n-1 exponential functions and consider scalars c 1,…,c n such that(10.2) 10a x k n kk c e ==∑.Let a M be the largest of the n numbers a 1,…,a n . Multiplying bothmembers of (10.2) by e -a Mx , we obtain (10.3) ()10k M n a a x kk c e -==∑.If k≠M, the number a k-a M is negative. Therefore, when x→∞ in Equation (10.3), each term with k≠M tends to zero and we find that c M=0. Deleting the M th term from (10.2) and applying the induction hypothesis, we find that each of the remaining n-1 coefficients c k is zero.THEOREM 10.5 Let S be an independent set consisting of k elements in a linear space V and let L(S) be the subspace spanned by S. Then every set of k+1 elements in L(S) is dependent.Proof. When V=V n, Theorem 10.5 reduces to Theorem 8.8. If we examine the proof of Theorem 8.8, we find that it is based only on the fact that V n is a linear space and not on any other special property of V n. Therefore the proof given for Theorem 8.8 is valid for any linear space V.Notationslinear space The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties. Spaces which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to names which give very little insight into the relevant properties of a given space. One of the most general type of mathematical spaces is the topological space.linear space, also vector space, is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R n, where every element is represented by a list of n real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.For a general vector space, the scalars are members of a field F, in which case V is called a vector space over F.Euclidean n-space R n is called a real vector space, and C n is called a complex vector space.In order for V to be a vector space, the following conditions must hold for all elements X, Y, Z∈V and any scalars r, s∈F:1. Commutativity:2. Associativity of vector addition:3. Additive identity: For all X,4. Existence of additive inverse: For any X, there exists a -X such that5. Associativity of scalar multiplication:6. Distributivity of scalar sums:7. Distributivity of vector sums:8. Scalar multiplication identity:TEXT B Basis and dimensionDEFINITION A finite set S of elements in a linear space V is called a finite basis for V if S is independent and spans V. The space V is called finite dimensional if it has a finite basis, or if V consists of 0 alone. Otherwise V is called infinite dimensional.THEOREM 10.6 Let V be a finite-dimensional linear space. Then every finite basis for V has the same number of elements.Proof Let S and T be two finite bases for V. Suppose S consists of k elements and T consists of m elements. Since S is independent and spans V, Theorem 10.5 tells us that every set of k+1 elements in V is dependent. Therefore, every 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. Therefore k=m.DEFINITION If a linear space V has a basis of n elements, the integer n is called the dimension of V. We write n=dim V. if V={0}, we say V has dimension 0.EXAMPLE 1 The space V n has dimension n. One basis is the set of n unit coordinate vectors.EXAMPLE 2 The space of all polynomials p(t)of degree ≤n has dimension n+1. One basis is the set of n+1 polynomials {1, t, t2, …, t n}. Every polynomial of degree ≤n is a linear combination of these n+1 polynomials.EXAMPLE 3 The space of solutions of the differential equation y"-2y'-3y=0 has dimension 2. One basis consists of the two functions u1(x)=e-x, u2(x)=e3x. Every solution is a linear combination of these two.EXAMPLE 4 The space of all polynomials p(t)is infinite-dimensional. Although the infinite set {1, t, t2,…}spans this space, no finite set ofpolynomials spans the space. Another famous infinite-dimensional space is L p-space.THEOREM 10.7 Let V be a finite-dimensional linear space with dim V=n. Then we have the following.(a) Any set of independent elements in V is a subset of some basis forV.(b) Any set of n independent elements is a basis for V.Proof. The proof of (a) is identical to that of part(b) of Theorem 8.10. The proof of (b) is identical to that of part (c) of Theorem 8.10.Let V be a linear space of dimension n and consider a basis whose elements e1, …, e n are taken in a given order. We denote such an ordered basis as an n-tuple (e1, …, e n). If x V, we can express x as a linear combination of these basis elements:(10.4) 1ni i i x c e ==∑. The coefficients in this equation determine an n -tuple of numbers (c 1, … c n ) that is uniquely determined by x . In fact, if we have another representation of x as a linear combination of e 1, … ,e n , say1ni i i x d e ==∑, then by subtraction from (10.4), we find that 1()0n i i i i c d e =-=∑. But since the basis elements are independent, this implies c i =d i , for each i , so we have (c 1, …, c n )=(d 1, …, d n ).The components of the ordered n -tuple (c 1, …, c n ) determined by Equation (10.4) are called the components of x relative to the ordered basis (e 1, …, e n ).NotationsL p-space On a measure space X, the set of p-integrable functions is an L p-space (p>0). The set of L p-functions generalizes L2-space. Instead of square integrable, the measurable function f must be p-integrable for f to be in L p.On a measure space X, the L p norm of a function f isThe L p-functions are the functions for which this integral converges. For p 2, the space of L p-functions is a Banach space which is not a Hilbert space.As in the case of an L2-space, an L p-function is really an equivalence class of functions which agree almost everywhere.For p>1, the dual space to L p is given by integrating against functions in L q, where 1/p+1/q=1. This makes sense because of Hölder's inequality for integrals. In particular, the only L p-space which is self-dual is L2.While the use of L p-functions is not as common as L2, they are very important in analysis and partial differential equations. For instance, some operators are only bounded in L p for some p>2.SUPPLEMENT MatrixA matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix, and its close relative the determinant, are extremely important concepts in linear algebra, and were first formulated by Sylvester (1850) and Cayley.In his 1850 paper, Sylvester wrote, "For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding of p thorder." Because Sylvester was interested in the determinant formed from the rectangular array of number and not the array itself, Sylvester used the term "matrix" in its conventional usage to mean 'the place from which something else originates'. Sylvester (1851) subsequently used the term matrix informally, stating "Form the rectangular matrix consisting of n rows and (n+1) columns.... Then all the n+1 determinants that can be formed by rejecting any one column at pleasure out of this matrix are identically zero." However, it remained up to Sylvester's collaborator Cayley to use the terminology in its modern form in papers of 1855 and 1858.In his 1867 treatise on determinants, C.L. Dodgson objected to the use of the term 'matrix,' stating, "I am aware that the word 'Matrix' is already in use to express the very meaning for which I use the word 'Block'; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage ofsuch quantities...." However, Dodgson's objections have passed unheeded and the term 'matrix' has stuck.The transformation given by the system of equationsis represented as a matrix equation bywhere the a ij are called matrix elements.An m⨯n matrix consists of m rows and n columns, and the set of m⨯n matrices with real coefficients is sometimes denoted R m⨯n. To remember which index refers to which direction, identify the indices of the last (i.e., lower right) term, so the indices m, n of the last element in the above matrix identifies it as an m⨯n matrix.A matrix is said to be square if m=n, and rectangular if m≠n. An m⨯1 matrix is called a column vector, and a 1⨯n matrix is called a row vector. Special types of square matrices include the identity matrix I, with a ij=δij(where δij is the Kronecker delta) and the diagonal matrix a ij=c iδij(where c i are a set of constants).In this work, matrices are represented using square brackets as delimiters, but in the general literature, they are more commonly delimited using parentheses.The transformation given in the above equation can be written X'=AX, where X'and X are vectors and A is a matrix.It is sometimes convenient to represent an entire matrix in terms of its matrix elements. Therefore, the (i, j)th element of the matrix A could be written a ij, and the matrix composed of entries a ij could be written as (a)ij for short.Two matrices may be added (matrix addition) or multiplied (matrix multiplication) together to yield a new matrix. Other common operationson a single matrix are matrix diagonalization, matrix inversion, and transposition.The determinant det(A) or |A| of a matrix A is a very important quantity which appears in many diverse applications. The sum of the diagonal elements of a square matrix is known as the matrix trace Tr(A) and is also an important quantity in many sorts of computations.Problem 1 If matrix A has m rows and l columns, B has n rows and m columns, C has k rows and n columns then matrix CBA has ____ rows and ____ columns.A k, lB m, nC n, lD k, mProblem 2 A square matrix A n⨯n, n>3 is tridiagonal if a ij=0 when i>j+1 or j>i+1, the other entries are unrestricted. In the following four matrices, which one is tridiagonal?A1200323004340054⎛⎫⎪⎪⎪⎪⎪⎝⎭B1234012300120001⎛⎫⎪⎪⎪⎪⎪⎝⎭C1000210032100321⎛⎫⎪⎪⎪⎪⎪⎝⎭D100010001⎛⎫⎪⎪⎪⎝⎭。

高数一作业英文版

x
and f1 (x) = log 1 (x). 2
6. Using the needed transformations, plot the graph of the following functions and ﬁnd their domain, range and zeroes: −x x−1 1 4. f (x) = 2 1. f (x) = 2x+1 + 1 2. f (x) = 1 − 2x/2 3. f (x) = − 1 2 1 (x − 2) − 1 5. f (x) = log2 (x + 4) 6. f (x) = log2 (−x) + 1 7. f (x) = log 2 8. f (x) = log 1 (−x − 2) 2 7. Find the domain of the following functions. Using the properties of logs, write them in an easier form, then plot the graph: 1. f (x) = log2 (x2 − 1) − log2 (x + 1) 2. f3 (x) = f2 x −
1 2x π 3
2. f3 (x) = f2 x +
3.
f4 (x) = f3 (x) − 1
10. Draw the graph of the following functions and determine the minimum period of each of them. 1. 5. f (x) = 1 + sin(−x) f (x) = 1 − sin(−x + π ) 2. f (x) = cos 1 2x 6. f (x) = − tan x +

数学专业英语作业

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数学专业英语(只是部分,不是很完整)

数学专业英语课后答案

2023大学_数学专业英语试题及参考答案

数学专业英语课后部分习题答案

你的数学作业写作文英语

有数学作业英语作文初一

数学的英语怎么写作业

数学专业英语课后答案

数学专业英语作业

数学专业英语(10)

高数一作业 英文版

高数一作业英文版