Types of Conditional Sentences
| If-clause (hypothesis) | Main clause (result clause) | Use | |
| Type 1 | If +any present form (Present Simple, Present Continuous or Present Perfect) | Future /Imperative, can, may, might, must, should+ Present Simple | Real Present True or likely to happen in the present or the future |
| If we catch the 10 o’clock train, we shall (can, may,etc.) get there by lunch time. Если мы сядем на поезд в 10 часов, мы приедем (можем приехать) туда к обеду. If you wake up before me, give me a call. Если вы проснетесь раньше меня, позвоните мне. We can use should after if to express something which is possible, but not likely to happen. If he should come earlier, tell him to wait for me. Eсли он все же придет раньше, скажите ему подождать меня. | |||
| Type 2 | If + Past Simple or Past Continuous | would/could, might+ Infinitive without to | Unreal Present Untrue in the present; also is used to give advice |
| If we caught the 10 o’clock train, we would (could, might, etc.) get there by lunch time. Если бы мы сели на поезд в 10 часов, мы бы приехали (могли бы приехать) туда к обеду. If I knew how the device worked, I could tell you what to do. Если бы я знал, как это устройство работает, я мог бы сказать вам, что делать. If I were you, I would follow his advice. На вашем месте я бы последовал его совету. After if we normally use were instead of wasin all persons in Type 2 conditionals. | |||
| Type 3 | If + Past Perfect or Past Perfect Continuous | would/could/might+ have+Past Participle | Unreal Past Imaginary situation contrary to the fact in the past; also is used to express regret or criticism |
| If we had caught the 10 o’clock train, we would (could, might, etc.) have got there by lunch time. Если бы мы сели на 10-ти часовой поезд, мы бы приехали (могли бы приехать) туда к обеду. If he had been studying hard, hecould have passedthe exam. Если бы он учился хорошо, он смог бы сдать экзамен. |
Ex. 1. State the type of the conditional sentences and translate them.
1. If you work hard, you’ll be able to finish your work in time.
2. If it is not raining, we shall play football.
3. If he had time, he would do the work.
4. If she were more attentive, she would not make so many mistakes.
5. If I had known your telephone number, I should have phoned you.
6. If he had gone to the station an hour ago, he wouldn’t have missed the train.
7. You will not solve this problem unless you know the Viet’s theorem.
8. If it were not so late, we would continue our debates.
9. If you follow the advice of the teacher, you will save a lot of time.
10. If I had understood the importance of learning English earlier, I should have taken part in the international scientific conference for young scientists.
Ex. 2. Complete these sentences following one of the patterns for conditionals of Type 1.
1. We’ll just manage to find the correct solution if …
2. If I see him again, I …
3. I will accept your explanation only if …
4. If you tell me the truth, I …
5. What will happen if …
6. If your work harder, …
7. If you don’t reserve the ticket, …
8. If it rains, …
9. If you are hungry, …
Ex. 3. Complete the sentences following the pattern of Type 2 conditionals.
1. If you explained the situation to your friend, he …
2. Perhaps he … if you spoke to him.
3. If you changed your job, you …
4. If they came to see us in London, we …
5. If you read the book a second time, you …
6. If I were you, I…
7. If they had more money, …
Ex. 4. Write sentences following the pattern of Type 3 conditional based on the given facts.
Model. The driver was not careful enough last Sunday, the accident happened.
If the driver had been more careful last Sunday, the accident wouldn’t have happened.
1. As you didn’t explain your problem to me, I wasn’t able to help you.
2. She didn’t read the book, she couldn’t discuss a new novel.
3. We didn’t take a map, we didn’t find the hotel quickly.
4. You didn’t invite him, he didn’t come to the party.
5. I didn’t know you were arriving on the train, I didn’t meet you.
6. He missed the seminar, he was not told about it.
Ex. 5. Translate into English.
1. Если мы определим кривую, мы найдем уравнение геометрического места точек.
2. Он бы помог вам, если бы он был в городе сейчас.
3. Если бы я знала ее электронный адрес, я бы написала ей немедленно.
4. Если бы наша студенческая группа приняла участие в спортивных соревнованиях вчера, мы бы заняли первое место.
5. Я бы взял такси, если бы знал, что у нас мало времени.
6. Если бы я хорошо знал английский, я бы читал всю новую научную литературу по-английски.
7. Ваша команда обязательно выиграет следующий матч, если вы будете много тренироваться.
8. Если бы я был на вашем месте, я бы не просил его помочь, а постарался бы справиться с трудным заданием сам.
9. Если она не сможет прийти сама, она позвонит по телефону.
10. Если бы он мог получить необходимую информацию вчера, он бы сказал нам о своих планах на выходной день.
Mixed Conditionals.
All types of conditionals can be mixed. Any tense combination is possible if the context permits it.
| If clause | Main clause |
| If they were playing all day,(Type 2) Если они играли весь день, | they will be tired out now.(Type 1) они будут усталыми сейчас. |
| If I were you, (Type 2) На вашем месте, | I would have visited them. (Type 3) я бы навестил их тогда. |
| If we had brought a map with us, (Type 3) Если бы мы тогда взяли карту с собой, | we would know which road to take now. (Type 2) мы бы сейчас знали по какой дороге ехать. |
Ex. 6. Write mixed sentences based on the given facts, use the table given above.
1. He failed his examination last year, so he is taking it again in June.
2. Since you didn’t take my advice, you’re in a difficult position now.
3. There was a sharp frost last night, so we are able to go skating now.
4. She isn’t at the meeting because she wasn’t told about it.
5. I didn’t apply for the job as I don’t want to work there.
6. He is not a fast runner, so he didn’t win the race.
7. She didn’t save her money, so she isn’t going on holiday.
Inversion
We can omit if in conditional sentences. When we do that, should, were, had (Past Perfect) and could come before the subject.
| If he were here, he would help us. | Were he here, he would help us. |
| If I should see him today, I’ll tell him to call you. If he were here now, we could work together. If I had known about that matter, I would have told him yesterday. | Should I see him today, I’ll tell him to call you. Were he here now, we could work together. Had I known about that matter, I would have told him yesterday. |
Ex. 7. Rewrite the sentences making an inversion in the conditional clauses.
Model.If he should come, give him my letter.
Should he come, give him my letter.
1. The talks will continue if the need should arise.
2. If you should be late again, you’ll lose your job.
3. If he had taken a little more time to think, he might have acted more sensibly.
4. If it were not for the price of the ticket, I would go there by plane.
5. If he had known the news, he would have told us.
6. If I were you, I wouldn’t buy such an expensive iPhone.
7. If you should drink too much coffee, you won’t be able to sleep.
Ex. 8. Translate the following sentences.
1. Were these words synonyms, you could use either of them.
2. Had I known the facts better, I should have made a new test.
3. Were he not so tired, he would continue his work.
4. Had you taken part in our experiment, you would have helped us a lot.
5. Were she good at mathematical analysis, she would be able to solve some of these problems.
6. Should he come to the laboratory, tell him to leave his notes there.
7. Had the students of our group attended all lectures and seminars, the results of the examination in functional analysis wouldn’t have been so bad.
Ex. 9. Answer the questions.
1. What will you do next Sunday if the weather is fine? 2. Where would you go if you were free now? 3. Would your favourite football team have won the last match if the football players trained more? 4. Will you study German if you have enough time? 5. Will you speak English better if you watch a lot of foreign films in English? 6. Who will you ask to help you if you can’t translate the article yourself? 7. What places of interest would you visit in London if you had an opportunity to go there? 8. How long can you stay in the south if you go there in summer? 9. What present would you buy to your mother if it were her birthday tomorrow? 10. Will you go to the station by underground or will you take a taxi if you have little time? 11. Will he improve his health if he goes in for sport?
Pre-Reading Activity
Guess the meaning of the following words:
interval [´Intqv(q)l] function [´fANkS(q)n]
system [´sIstIm] form [fO:m]
distance [´dIst(q)ns] contrary [´kOntrqrI]
portion [´pO:Sqn] section [´sekS(q)n]
reserve [rI´zq:v] result [rI´zAlt]
special [´speS(q)l ] identity [aI'dentItI ]
figure [´fIgq] complex [´kOmpleks]
family [´fxmIlI] fix [´fIks]
real [´rIql] ordinary [´O:dnrI]
hypothesis [haI´pOTIsIs ] linear [´lInIq]
proportional [prq´pO:Sqnl] projection [ prq´GekS(q)n]
term [tq:m]
Read and learn the following words:
| curve (n) | [kq:v] | кривая |
| dimension(n) | [dI'menS(q)n] | размерность, измерение |
| establish (v) | [Is¢txblIS] | устанавливать, основывать, создавать |
| describe (v) | [dIs¢kraIb] | описывать, вычерчивать |
| locus (loci) (n) | [¢loukqs] | геометрическое место точек |
| single (adj) | ['sINgl] | единственный, одиночный, отдельный, единый |
| infinity (n) | [In'fInItI] | бесконечность |
| vary (v) | ['vFqrI] | менять, изменять |
| variable (n) | ['vFqrIqbl] | переменная |
| proper (adj) | ['propq] | правильный, собственный |
| permit (v) | [pq'mIt] | позволять, допускать, разрешать |
| take on (v) | ['teIk'Ln] | принимать, приобретать (форму, качество и т. д.) |
| condition (n) | [kqn'dIS(q)n] | условие |
| imaginary (adj) | [I'mxdZInqrI] | мнимый |
| reduce (v) | [rI'djHs] | уменьшать, превращать, приводить к, сокращать |
| occur (v) | [q'kW] | случаться, происходить, встречаться |
| singular (adj) | ['siNgjulq] | особый |
| assume (v) | [q'sjHm] | принимать, допускать |
| expand (v) | [Iks'pxnd] | расширять, разлагать |
| power series | ['pauq 'sIqrI:z] | степенные ряды |
| converge (v) | [kqn'vq:dZ] | сходиться в одной точке, сводить в одну точку |
| adduce (v) | [q'dju:s] | приводить (в качестве доказательства) |
| revolve (v) | [rI'vOlv] | вращаться |
| revolution (n) | ["revq'lu:S(q)n] | вращение |
| implicit (adj) | [Im'plIsIt] | неявный |
| explicit (adj) | [Iks'plIsIt] | явный |
| point of view | ['pOInt qv"vjH] | точка зрения |
| simultaneous (adj) | ["sIm(q)l'teInjqs] | одновременный, совместный |
| helix (n) | ['hJlIks] | спираль, винтообразная линия |
| twisted cubic (n) | ['twIstId'kjubIk] | неплоская кривая 3-го порядка |
| residual (adj) | [rI'zIdjuql] | остаточный |
| screw (n) | [skrH] | винт, шуруп |
TEXT A
CURVES
Definition and equations of a curve. In ordinary three-dimensional space let us establish a left-handed orthogonal cartesian coordinate system with the same unit of distance for all three axes. In this system any point P has coordinates x, y, z.
A curve may be described qualitatively as the locus of a point moving with one degree of freedom. A curve is also sometimes said to be the locus of a one-parameter family of points or the locus of a single infinity of points.
Definition 1. Let the coordinates x, y, z of а point P be given as single-valued real-valued analytic functions of a real independent variable t on an interval T of t-axisby equations of the form:
x = x (t), у = y (t), z = z (t).(1.1)
Further suppose that the functions x(t), y(t), z(t) are not all constant on T. Then the locus of the point P, as t varies on the interval T, is a real proper analytic curve C.
Some comments on the foregoing definition will perhaps clarify its meaning. Equations (1.1) аге called the parametric equations of the curve C, the parameter being the variable t. We reserve the right to permit the parameter t to take on complex values. Moreover, one or more of the coordinates x, y, z may, under suitable conditions, be allowed to be complex. The curve С would in this case be called complex, or perhaps, on suitable conditions, imaginary. To say that a curve is proper means that it does not reduce to a single fixed point, as it would do if the coordinates x, y, z were all constant. It is clear that at an ordinary point of a real proper analytic curve, i. e. a point where nothing exceptional occurs, the inequality
x'2+ y'2+ z'2 > 0(x 
) (1.2)
holds. Any point of such a curve where this inequality fails to hold is called singular, although the singularity may belong to the parametric representation being used for the curve defined as a point-locus, or may belong to the curve itself. A curve, or portion of a curve, which is free of singular points may be called nonsingular. Furthermore, we assume that the interval T is so small that
values of the parameter t on the interval T and points (x, y, z) on the curve С are in one-to-one correspondence, so that the parameter t is a coordinate of the corresponding point (x, y, z) on the curve С.
To say that the functions are analytic means, roughly, that they can be expanded into power series. More precisely, this statement means that, at each point t0 within the interval T,each of these functions can be expanded into a Taylor's series of power of the difference t-t0 which converges when the absolute value t-t0 is sufficiently small. It would be possible to study differential geometry under the hypothesis that the functions considered possess only a definite, and rather small number of derivatives; but weassume analyticity in the interests of simplicity. So the word "function" will mean for us "analytic function", and the word "curve" will mean a real proper nonsingular analytic curve unless the contrary is indicated.
Some examples of parametric equations of curves will now be adduced. First of all, the equations (1.1) may be linear, of the form
x = a + lt, y = b + mt, z = c + nt (1.3)
in which a, b, с and l, m, n are constants. Then the curve С is a straight line through the fixed points (a, b, c) and with direction cosines proportional to
l, m, n. If t is the algebraic distance from the fixed point (a, b, c) to the variable point (x, y, z) on the line then l, m, n are the direction cosines of the line and satisfy the equation
l2+ m2+ n2= l (1.4)
As a second example, equations (1.1) may take the form
x = t, у = t2, x = t3 (1.5)
The curve С is then a cubical parabola. This is one form of a twisted cubic which can be defined as the residual intersection of two quadric surfaces that intersect elsewhere in a straight line. Finally, if equations (1.1) have the form
x = a cos t , y = a sin t, z = bt (a> 0, b <O) (1.6)
The curve С is a left-handed circular helix, or machine screw. This may be described as the locus of a point which revolves around the z - axis at a constant distance a from it and at the same time moves parallel to the z - axis at a rate proportional to the angle t of revolution. If we had supposed b < 0, then the helix would have been right-handed.
A curve can be represented analytically in other ways than by its parametric equations. For example, it is known that one equation in x, y, z represents a surface, and that two independent simultaneous equations in
x, y, z, say
F(x, y, z) = 0, C(x, y, z) = 0 (1.7)
represent the intersection of two surfaces, which is a curve. Equations (1.7)are called implicit equations of this curve. Sometimes it is convenient to represent a curve by implicit equations, when really the curve under consideration is only part of the intersection of the two surfaces represented by the individual equations.
If the implicit equations (1.7) be solved for two of the variables in terms of the third, say for у and z in terms of x, the result can be written in the form
y = y(x), z=z(x). (1.8)
These equations represent the same curve as equations (1.7), and they, or the equations, which similarly express any two of the coordinates of a variable point on the curve as functions of the third coordinate, are called explicit equations of the curve. Each of equations (1.8) separately represents a cylinder projecting the curve onto one of the coordinate planes. So equations (1.8) are a special form of equations (1.7) for which the two surfaces are projecting cylinders.
If the first of the parametric equations(1.1) of a curve С be solved for t as a function of x, and if the result is substituted in the remaining two of these equations, the explicit equations (1.8) of the curve С are obtained. From one point of view the explicit equations (1.8) of a curve, when supplemented by identity, x = x, are parametric equations
x=x, y=y(x), z=z(x). (1.9)
of the curve, the parameter now being the coordinate x.
Post-Reading Activity
Ex. 10. Match the English words and word combinations with their Russian equivalents.
| 1. one-to-one correspondence | a) соответствующая точка |
| 2. a real proper analytic function | b) взаимнооднозначное соответствие |
| 3. to describe qualitatively | c) разлагать в степенные ряды |
| 4. the inequality doesn’t hold | d) направляющие косинусы |
| 5. the above definition | e) действительная правильная аналитическая кривая |
| 6. a suitable condition | f) параметрические уравнения |
| 7. to adduce the examples | g) действительная независимая переменная |
| 8. a point-locus | h) описывать качественно |
| 9. the corresponding point | i) вышеуказанное определение |
| 10. direction cosines | j) подходящее условие |
| 11. parametric equations | k) неравенство не выполняется |
| 12. to expand into power series | l) представить примеры |
| 13. a real independent variable 14. to satisfy the equations | m) удовлетворять уравнениям |
| 15. a left-handed helix | n) пересекаться на прямой линии |
| 16. intersection of two surfaces | o) график точки |
| 17. to revolve around the axis | p) левосторонняя спираль |
| 18. to substitute the result into the equations | q) независимая система уравнений |
| 19. to intersect in a straight line | r) пересечение двух поверхностей |
| 20. independent simultaneous equations | s) подставить результат в уравнения |
| t) вращаться вокруг оси |
Ex. 11. Find out whether the statements are true or false. Use introductory phrases.
| Exactly. Quite so. I fully agree to it. I don’t think this is the case. | Quite the contrary. Not quite. It’s unlikely. Just the reverse. |
1. A curve can be described qualitatively and quantitatively as the locus of a point moving with one degree of freedom.
2. In the parametric equations of the curve C the parameter is the variable t.
3. If the coordinate x, y, z were all constant, the proper curve would reduce to a single fixed point.
4. Analytic functions can be expanded into power series.
5. A cubical parabola is the residual intersection of two plane surfaces that intersect elsewhere in a straight line.
6. A curve is represented by the intersection of two surfaces, if we have two independent simultaneous equations in x, y, z.
7. If any two of the coordinates of a variable point on the curve are expressed as functions of the third coordinate, the equations are called the explicit equations of the curve.
Ex. 12. Learn the following word combinations.
a) Use “under consideration” or “in question” instead of “considered” in order to express the same idea “рассматриваемый”:
The theorem considered, the figure considered, the problem considered, the function considered, the equation considered, the point considered, the curve considered.
b) Use “to hold” instead of “to be valid”, “to be true” meaning “иметь силу, выполняться”:
1. This inequality is valid for all cases. 2. This theorem is valid in the case of the uniform convergence. 3. This formula is valid for a single-valued analytic function too. 4. For a = b= 1 the given property is true. 5. These relations are true under suitable conditions.
c) Use “to fail to do something ” instead of “do not” meaning “не суметь, не быть в состоянии, оказаться неспособным сделать что-то”
1. I did not solve the problem given by the professor. 2. These properties do not hold for real numbers. 3. We did not expand these functions into power series. 4. He did not prove the theorem correctly. 5. We do not represent this curve by an implicit equation. 6. I did not understand your question. 7. The boy did not add these two numbers correctly.
Ex 13. Replace the Russian words by their English equivalents according to the text:
1. A curve may be defined as геометрическое место точек of a one-parameter family of points.
2. If one or more of the coordinates x, y, z are allowed to be complex, the curve C will be called complex or мнимая.
3. Any point of such a curve where this inequality не выполняется is called singular.
4. The functions are analytic if they can be expanded into степенной ряд.
5. We представим some examples of parametric equations of curves.
6. If we have supposed b<0, then спираль would have been right-handed.
7. When the curve under consideration is only part of the intersection of two surfaces represented by the individual equations, we represent a curve by неявные equations.
8. In differential geometry we assume the functions considered possess only a definite number of производные.
Ex. 14. Analyze the use of the conjunctions introducing adverbial clauses of condition: in case, provided, suppose, unless, on condition, as long as. Translate the sentences into Russian.
1. Suppose you told him the truth, what could he do about it?
2. You can borrow my notes on condition (provided) you give them back to me tomorrow.
3. You can come with us as long as you don’t make too much noise.
4. He wouldn’t have come unless you had invited him.
5. Unless you read the text on graphs, you won’t be able to discuss it.
6. I’m taking an umbrella in case it rains later on.
Ex. 15. State the type of these conditional sentences and translate them.
1. A curve is called nonsingular if it is free of singular points. 2. The curve would reduce to a single fixed point if the coordinates x, y, z were all constants.
3. The curve is called complex if one or more of the coordinates x, y, z are complex. 4. If t is the algebraic distance from the fixed point (x, y, z) on the line, then 1, m, n in the given equations are direction cosines. 5. If the endpoints are included, the interval is called closed. 6. The result would have been written in the form y = у(x), z = z(x) if the implicit equations had been solved for the two of the variables in terms of the third. 7. If these implicit equations were solved for two of the variables in terms of the third, they could be written in another form. 8. If the equations x = x (t), у = у (t), z = z (t) had taken the form
x = t, у = t2, z= t2, then the curve С would have been a cubic parabola
Ex. 16. Answer the following questions.
1. In what way may a curve be described? 2. Can you give the definition of a real proper analytic curve? 3. What do we call the equations of the form
x = x(t), у = у(t), z = z (t)? 4. What does the letter t denote in these equations? 6. May the parameter t take on complex values? 7. What do the letters x, y, z denote in these equations? 8. In what case is the curve С called complex or imaginary? 9. When do we call the curve С proper? 10. When does the curve С reduce to a single point? 11. What point is called singular? 12. What curve is called nonsingular? 13. When are the coordinates x, y, z analytic? 14. What forms can the equations x = x(t), у = y(t), z = z(t) take? 15. In which case is the curve С a straight line, a cubic parabola and a left-handed helix? 16. Can a curve be represented analytically? 17. By what equations can it be represented analytically? 18. What equations are called implicit or explicit equations?
Ex. 17. Translate into English.
1. Установим в трехмерном пространстве левостороннюю ортогональную декартову систему координат для того, чтобы вывести уравнение кривой. 2. Любая точка этой системы имеет три координаты по осям (x, y, z), имеющим одинаковый масштаб. 3. Уравнения x = x(t), у = у(t), z = z (t) называются параметрическими уравнениями кривой С, при этом параметром является переменная t. 4. Кривую можно описать как траекторию точки, движущейся с одной степенью свободы. 5. Кривая является собственной, когда она не сводится к отдельной фиксированной точке. 6. Аналитические формулы могут быть разложены в степенные ряды. 7. Кривая может быть представлена аналитически не только параметрическими уравнениями, иногда удобно представить кривую неявными и явными уравнениями. 8. Кривая может представлять прямую линию, кубическую параболу (неплоскую кривую третьего порядка) или левостороннюю круговую спираль.
Ex. 18. Topics for discussion.
1. Give the definition and equations of a curve.
2. Dwell on the analytic representation of a curve.
3. Speak on the implicit and explicit equations of a curve.
Ex. 19. Read the text and find the answers to the following questions.
1. What curves are of interest in mathematics?
2. What does the precise meaning of the term “curve” depend on?
3. When is an arc called a line segment?
4. What can you say about algebraic curves?
TEXT B
CURVES
In mathematics, a curve(also called a curved linein older texts) is, generally speaking, an object similar to a line but which is not required to be straight. This means that a line is a special case of a curve, namely a curve with null curvature. Often curves in two-dimensional (plane curves) or three-dimensional (space curves) Euclidean space are of interest.
Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right.
A large number of other curves have been studied in multiple mathematical fields.
The term curve has several meanings in non-mathematical language as well. For example, it can be almost synonymous with mathematical function, or graph of a function.
An arc or segment of a curve is a part of a curve that is bounded by two distinct end points and contains every point on the curve between its end points. Depending on how the arc is defined, either of the two end points may or may not be part of it. When the arc is straight, it is typically called a line segment.
Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the locus of the points of coordinates x, y such that
f(x ,y) = 0, where f is a polynomial in two variables defined over some field F. Algebraic geometry normally looks not only on points with coordinates in F but on all the points with coordinates in an algebraically closed field K. If С is a curve defined by a polynomial f with coefficients in F, the curve is said to be defined over F. The points of the curve С with the coordinates in a field G are said to be rational over G and can be denoted C(G); thus the full curve С=C(K).
Algebraic curves can also be space curves, or curves in even higher dimension, obtained as the intersection (common solution set) of more than one polynomial equation in more than two variables. By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce singularities such as cusps (точка пересечения двух прямых) or double points.
UNIT VII