Speech exercises

I. Say these sentences in English.

1.Численнозначная функция ставит в соответствие каждой точке р в ее области определения одно действительное число f (p). 2. Действительные функции часто подразделяются в соответствии с их областями определения. 3. Функция f может быть определена только для точек p, которые лежат на определенной кривой С в пространстве. 4. Говорят, что функция f, определенная на линии, будет монотонно возрастающей, если f (x) < (x'), всякий раз как x < x'. 5. Рядом с понятием функции, как соответствием между двумя множествами, стоит понятие графа. 6. Многие особые свойства функции отражаются в простых геометрических свойствах ее графа. 7. Функция двух переменных, как говорят, будет выпуклой, если она подчиняется вышеупомянутому условию. 8. Мы должны видоизменить исходное утверждение. 9. Реше- ние уравнения через у редко дает однозначный ответ. 10. Очень важ-

но сделать четкое различие между самой функцией и ее значениями. 11. Уравнение x² + y² - 16 дает две функции. 12. Полагают, что самым удовлетворительным решением является то, которое определяет функцию через множества и приведенные выше понятия.

II. The topic for discussion:

Functions and their graphical representation.

T E X T VI

Grammar: condional sentences revision: the nominative absolute participal constructions exercises

I. Read, analyse and translate the following sentences:

1. expressing a real condition referring to present, past and future (Type I)

1. If you work hard, you will be able to finish the work in time. 2. If you are right, then I am wrong. 3. If it is not raining, we shall play football. 4. I shall not help him, unless he asks me. 5. If the water in the sea was not cold, we always went for a swim. 6. You will see the Kremlin if you go to Moscow. 7. If you have time, help me. 8. She will enjoy the music, if she hears the new opera of this famous composer.

b) expressing an unreal condition referring to present and future (Type II)

1. Helen would fly to Moscow tomorrow if she got a ticket.2. If he had time, he would do the work. 3. If she were more attentive, she would not make so many mistakes. 4. If the rain stopped, the children could go for a walk. 5. If you had a chance of speaking English you would improve it. 6. You would be cold if you did not wear a warm coat.

c) expressing an unreal condition referring to past (Type III)

1. If he had gone to the station an hour ago, he would not have missed the train. 2. The teacher would have helped you if you had any doubts about the exercise. 3. If she had worked hard, she would have learned grammar. 4. I should have spoken if I had been sure of the answer. 5. If I had known your telephone number, I should have phoned you. 6. I should have done a much better translation if I had had a better dictionary.

II. State the type of the conditional sentences and translate them:

1. I would buy this coat if I had had more money. 2. If I had come home much earlier yesterday, we should have finished the work. 3. If she goes to Moscow, she always visits the Tretyakov Gallery. 4. You will not solve this problem unless you know the Viet’s theorem. 5. If it were not so late, we would continue our discussion. 6. If you follow the advice of the teacher, you will save a lot of time. 7. If I had understood the importance of learning English at school, I should have known it quite well. 8. We may discuss this problem later, if you are busy now. 9. If I go to St. Peterburg, I shall try to see a wonderful collection of pictures in the Hermitage. 10. We could read the sentences written on the board, unless it were so dark in the classroom. 11. They would improve their English if they spent more time in the language laboratory.

III. Complete the following sentences:

1. If it were not so late … . 2. If he works hard … . 3. If I offer you my help … . 4. If the rain stops … . 5. If I had known the answer … . 6. If you go on smoking … . 7. I will stay at home tomorrow, if … . 8. If the students read more English books … . 9. He would have visited his parents last Sunday, unless … . 10. You will enjoy the fresh air if … . 11. If this book were more interesting … .

IV. Answer the questions.

1. What will you do next Sunday if the weather is good? 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 had trained more? 4. Will you speak English much better if you work with a tape-recorder? 5. Will you study French if you have enough time? 6. What foreign language will you study if you have time? 7. Whom will you ask to help you, if you can’t translate this article yourself? 8. What would you visit in London if you had an opportunity to go there? 9. How long can you stay in the South, if you go there in summer? 10. What present would you buy for your mother if it were her birthday tomorrow? 11. Will you go to the station by bus or by the underground if you have little time? 12. Will he improve his health if he goes in for sport?

V. Translate into English:

1. Если вы пойдете в библиотеку, вы найдете там нужную вам книгу. 2. Он бы помог вам, если бы он был в городе сейчас. 3. Если бы я знала ее адрес, я бы написала ей немедленно. 4. Если бы Анна приняла участие в спортивных соревнованиях вчера, она бы заняла первое место. 5. Я бы взял такси, если бы знал, что у нас так мало времени. 6. Если бы я хорошо знала английский язык, я бы читала произведения английских и американских авторов только по-английски. 7. Я бы купила этот англо-русский словарь, если бы у меня было достаточно денег. 8. Ваша команда обязательно выиграет следующий матч, если вы будете много тренироваться. 9. Если бы я была на вашем месте, я бы не обратилась к нему за советом. 10. Если она не сможет прийти к вам сама, она позвонит вам по телефону.

VI. Guess the meaning of these words:

coordinate figure function constant

system parameter analytic complex

orthogonal family interval fixed

distance real form ordinary

portion hypothesis contrary linear

reserve proportional revolution implicit

explicit term result projection

special cylinder identity section

VII. Learn the following words:

curve (n) ['kWv] кривая

dimension (n) dI'menSqn] размерность

establish (v) [Is'txblIS] устанавливать

describe (v) [dIs'kraIb] описывать

locus (loci) (n) ['loukqs] [lou'saI] геометрическое место

single (adj) [sINgl] единственный, одиночный, единый

infinity (n) [In'fInItI] бесконечность

vary (v) ['vFqrI] менять(ся), изменять(ся)

variable (n) ['vFrIqbl] переменная

proper (adj) ['prOpq] собственный, правильный

permit (v) [pq'mIt] позволять, допускать, разрешать

take on (v) ['teIk'On] принимать

condition (n) [kqndISqn] условие

imaginary (adj) [ImxdInqrI] мнимый, воображаемый

reduce (v) [rI'dju:s] приводить, уменьшать, превращать

occur (v) [q'kq] случаться, происходить

singular (adj) [sINgjulq] особый

assume (v) [q'sju:m] допускать , принимать

expand (v) [Iks'pxnd] разлагать

power series ['pauq'sIqrI:z] степенной ряд

converge (v) [kqn'vq:G] сходиться

adduce (v) [q'dju:s] представлять

revolve (v) [rI'vOlv] вращать(ся), поворачивать(ся)

revolution (n) ["revq'lu:Sn] вращение

implicit (adj) [Im'lIsIt] неявный

explicit (adj) [IksIt] явный

point of view ['pOInt qv vju:] точка зрения

simultaneous (adj) ["sImql'te'njes] одновременный, совместный

helix (n) ['hI:lIks] спираль, спиральная линия

twisted cubic [twIstId 'kjubIk] неплоская кривая 3-го порядка

residual (adj) [rI'zIdjuql] остаточный, оставшийся

screw (n) [skru:] винт, шуруп

C U R V E S

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 a point P be given as single-valued real-valued analytic functions of a real independent variable t on an interval T of t-axis, by equations of the form:

x = x (t), y = y (t), z = z (t). (2.1)

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 (2.1) are called the parametric equations of the curve C, the parameter being the variable t. We reserve the right to permit the parame-ter 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 C would in this case be called complex, or perhaps, or 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' ; … ) (2.2)

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 C are in one-to-one correspondence, so that the parameter t is a coordinate of the corresponding point (x, y, z) on the curve C.

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 t_o within the interval T, each of these functions can be expanded into a Taylor’s series of power of the difference t – t_o which converges when the absolute value t – t_o 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 we assume 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 (2.1) may be linear, of the form

x = a + Lt, y = b + mt, z = c + nt (2.3)

in which a, b, c and l, m, n are constants. Then the curve C 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 = 1 (2.4)

As a second example, equations (2.1) may take the form

x = t, y = t², x = t³ (2.5)

The curve C 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 (2.1) have the form

x = a cos t, y = a sin t, z = bt (a> 0, b <C) (2.6)

the curve C is a left-handed circular helix, or machine sorew. 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 (2.7)

represent the intersection of two surfaces, which is a curve. Equations (2.7) 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 (2.7) be solved for two of the variables in terms of the third, say for y and z in terms of x, the result can be written in the form

y = y (x), z = z (x). (2.8)

These equations represent the same curve as equations (2.7); 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 (2.8) separately represents a cylinder projecting the curve onto one of the coordinate planes. So equations (2.8) are a special form of equations (2.7) for which the two surfaces are projecting cylinders.

If the first of the parametric equations (2.1) of a curve C 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 (2.8) of the curve C are obtained. From one point of view the explicit equations (2.8) of a curve, when supplemented by identity x = x, are parametric equations

x = x, y = y (x), z = z (x). (2.9)

of the curve, the parameter now being the coordinate x.

I. Learn the following word combination:

Ordinary three-dimensional space, a left-handed orthogonal coordinate system, the same unit of distance, one degree of freedom, the locus of a point, a one-parameter family of curves, a single infinity of points, a real proper analytic space curve, a single-valued real-valued analytic function, a real independent variable, under a suitable condition, to reduce to a single fixed point, the function considered (under consideration), at the same time, in other ways, in terms of, from one point of view, in one-to-one correspondence, fail to hold, a point-locus, under the hypothesis, unless the contrary is indicated, direction cosines, a twisted cubic, a left-handed circular helix, independent simultaneous equations.

II. Form nouns from the verbs using the suffixes:

-ation, -tion, -ion -ment -ence

to direct to establish to differ

to represent to state to occur

to substitute to develop to converge

to intersect to improve to correspond

to consider to move to depend

to vary

to define

III. Form adverbs from the adjectives, using the suffix–ly:

Clear, real, final, absolute, individual, indefinite, independent, convenient, implicit, explicit, proportional, proper, sufficient, simultaneous, separate, special.

IV. Learn the following nouns:

identity, difficulty, reality, simplicity, analyticity, singularity, inequality, infinity.

V. Use “under consideration” or “in question” instead of “considered” in order to express the same idea:

The theorem considered, the figure considered, the problem considered, the function considered, the equation considered, the point considered, the curve considered.

VI. Use “to hold” instead of “to be valid”, “to be true”:

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. These relations are true under suitable conditions. 5. For a = b = 1 the given property is true.

VII. Use “to fail to” instead of “do not”:

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.

VIII. Read and translate these sentences, paying attention to the new worlds and word combinations from the text.

1. In a left-handed orthogonal cartesian coordinate system any point has three coordinates. 2. A curve may be defined as the locus of a one-parameter family of points. 3. The coordinates x, y, z are given here as single-valued real-valued analytic functions by the equations of the form x = x (t), y = y (t), z = z (t). 4. These equations are called parametric equations. 5. The parameter may take on complex values. 6. The curve is proper when it does not reduce to a single fixed point. 7. A curve which is free of singular points is called nonsingular. 8. The functions are analytic if they can be expanded into power series. 9. Sometimes a curve may be represented be implicit equations. 10. Two variables in this implicit equation may be solved in terms of the third. 11. The curve under consideration is a real proper analytic space curve.

IX. 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 constant. 3. If the equations x = x (t), y = y (t), z = z (t) had taken the form x = t, y = t², z = t², then the curve C would have been a cubic parabola. 4. The curve is called complex, if one or more of the coordinates x, y, z are complex. 5. If “t” is the algebraic distance from the fixed point (x, y, z) on the line, then l, m, n in the given equations are direction cosines. 6. If the endpointes are included, the interval is called closed. 7. If these implicit equations were solved for two of the variables in terms of the third, they could be writtebn in another form. 8. The result would have been written in the form y = y (x), z = z (x) if the implicite quations had been solved for the two of the variables in terms of te third.

X. Change the conditional sentences of type I to those of Type II.

1. If the coordinates of the point satisfy the equation, the point lies on the curve. 2. The curve is called imaginary if one or more of the coordinates are complex. 3. If a is less than 0, then this formula holds. 4. If the equation F (x, y, z) = 0 is homogeneous in x, y, z, it represents a cone. 5. We cannot solve the system of equations unless we eliminate the unknowns. 6. If we suppose that b<0, the helix is right-handed. 7. We obtain the explicit equations of the curve C if we solve one of the parametric equations of the curve for t as a function of x.

XI. Analyze the following sentences and translate them:

1. One factor of the product being equal to 0, the product must be equal to 0. 2. A function being continuous at every point of the set, it is continuous throught the set. 3. The resulting equations are parametric equations of the curve C, the parameter being t. 4. We call such an equation linear, any other type being called non-linear. 5. The equation x – y = 0 being given, we can rewrite it in the for y = x. 6. We study polygons now, this type of geometric figures being very important in studying geometry.

XII. Answer the following questions:

1. What is this text about? 2. In what way may a curve be described? 3. Can you give the definition of a real proper analytic curve? 4. What do we call the equations of the form x = x (t), y = y (t), z = z (t)? 5. 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 C called complex or imaginary? 9. When do we call the curve C pro- per? 10. When does the curve C reduce to a single point? 11. What point is called singlar? 12. What curve is called nonsinglar? 13. When are coordinates x, y, z analytic? 14. What forms can the equations x = x (t), y = y (t), z = z (t) take? 15. In which case is the curve C a atraight 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?