Grammar and vocabulary exercises

I. Find all Conditional Sentences in the text, define their type and translate them.

II. Put each verb in brackets into a suitable tense:

a) Why didn’t you phone? If I (know) you were coming, I (meet) you at the airport.

2. It’s a pity you missed the party. If you (come) … you (meet) my friends from Hungary.

3. If we (have) some tools, we (be able) to repair the car, but we haven’t got any with us.

4. Thank you for your help. If you (not help) me, I (not pass) the examination.

5. Mark isn’t a serious athlete. If he (train) harder, he (be) … quite a good runner.

6. It rained every day on our holiday. If we (not take)… the television with us, we (not have) anything to do.

III. Translate these sentences:

1. Were there no computers, we would not be able to do much of what we are capable to do today. 2. Had another initial point been chosen, then another curve would have been obtained. 3. Were two triangles congruent, then an angle of one triangle would be congruent to its corresponding angle in the other triangle. 4. Were a point equidistant from two fixed points, then it would lie on the perpendicular bisector of the line segment joining these two points. 5. The equation x² – 2 x y + ky² + 2x – y + 1 = 0 represents a one – parameter family of curves. Were k equal to 1, the equation would represent a parabola, elipses were k greater than 1 and hyperbolas were k less than 1.

IV. Translate the following sentences paying attention to the underlined constructions.

1. in this case the resulting equations are parametric equations of the curve c, the parameter now being u. 2. A plane determined by two intersecting lines will be written in the form (ab), the brackets being used to avoid the confusion with the point ab. 3. The given prism can be divided into a set of triangular prisms, the distance between the bases being the same for every prism of the set. 4.Suppose that f (x) = p (x) / Q (x) where P and Q are polynomials, and that f (x) = f (x + a), each of these equations holding for all values of x.

V. Find the sentences with the Nominative Absolute Construction in the text and translate them.

VI. Give the Russian equivalents for:

double infinity of points, the locus of a point, two degrees of freedom, to impose a condition, a variable point, the missing variable, the resulting equation, a real proper analytic surface, with respect to, to vanish identically, to reserve the right, under suitable conditions, to reduce to a curve, a linear homogeneous partial differential equation, vice versa, arbitrary functions.

VII. Arrange the given words according to the parts of speech they belong to:

system, infinite, differential, analytically, systematic, differentiation, consequently, singularity, elimination, simultaneously, special, homogeneous, particular, respectively, useful, generator, identical, familiar, simplest, definition, finally.

VIII. Complete the sentences using the given word-groups;

at the origin, the variables, the implicit equation of the surface, a plane, the parametric equations of the surface, arbitrary, a curve, singular, nonsingular, to vanish simultaneously.

1. The equation F (x, y, z) = 0 is called … 2. If this equation is linear in the variables x, y, z, the surface which it represent is …, and if it is homogeneous in x, y, z, it represents a cone which vertex is … . 3. Equations x = x (u, v); y = y (u, v), z = z (u, v) are called … , whose parameters are … . 4. A surface is proper if it does not reduce to … . 5. The function F in the equation θ = F (t (u, v)) is … . 6. Any point of a real proper analytic surface at which the jacobians … is called … . 7. A surface, or a portion of a surface, which is free of singular points may be called … .

IX. Agree or disagree with the statements:

1. To represent a surface analytically we establish a polar coordinate system. 2. The equation f (x, y, z) = 0 is called an implicit equation of the surface. The simplest surface of all types of surfaces is a cone. 4. In the parametric equations of the surface x = x (u, v); y = y (u, v); z =z (u, v) the letters x, y, z denote the parameters. 5. A surface or a portion of a surface, which is free of singular points may be called singular. 6. A surface is proper if it does not reduce to a curve. 7. If the equation f (x, y, z) = 0 is homogeneous in x, y, z it represents a plane. 8. The explicit equation of the surface Z = f (x, y) is a special case of the implicit equation of the form F (x, y, z) = 0, if we transpose z to the right member.

X. Answer the questions:

1. What does this text consider? 2. What must be established in order to represent a surface analytically? 3. What types of surfaces do you know? 4. A plane and a sphere are the simplest types of surfaces, aren’t they? 5. Is the equation F (x, y, z) = 0 an implicit or an explicit equation of a sur- face? 6. When does the equation F (x, y, z) = 0 represent a plane, a cone and a cylinder? 7. What can you say about the equation x = x (u, v); y = y (u, v); z = z (u, v)? 8. What does it mean to say that a surface is proper? 9. What is a jacobian? 10. What do we call any point of a real proper analytic surface at which the jacobians vanish simultaneously? 11. What surface is called nonsingular?

SPEECH EXERCISES

I. Topics for discussion:

1. Speak on the definition of a surface given in this text and the definitions given in the lectures on differential geometry.

2. Discuss implicit and explicit equation of a surface.

3. Speak about the cases when a surface represented by the equations x = x (u, v), y = y (u, v), z = z (u, v) reduces to a point or to a curve.

II. Say it in English:

В этом тексте даются определение поверхности и ее аналитическое представление. Для того чтобы представить поверхность аналитически, необходимо установить левостороннюю ортогональную декартову систему координат. Эта система координат имеет одинаковую единицу расстояния для всех трех осей. Затем мы налагаем одно условие на точку Р (x, y, z) уравнением вида F (x, y, z) = 0. Это уравнение называется неявным уравнением поверхности. Нам известны простые типы поверхностей, такие, как плоскость, сфера, цилиндр, конус. Уравнение

F (x, y, z) = 0 при определенных условиях может представлять или плоскость, или конус, или цилиндр.

Уравнения (1.3) называются параметрическими уравнениями поверхности.

Поверхность является собственно поверхностью, если она не превращается в кривую. Любая точка поверхности, в которой якобианы одновременно равны нулю, называется вырожденной, а поверхность, не имеющая вырожденных точек, называется невырожденной.

III. Read the text and reproduce it.

Parametric curves on a surface are defined as follows: the parametric curves on a surface S, which is represented by parametric equations of the form (1.3), are defined to be those curves on A along each of which one of the parameters varies while the other is constant.

If the parameter y is held fixed while u varies, the locus of the variable point P (x, y, z) is a curve on the surface S. This curve , which sometimes is denoted by C^u, is called a u – curve because its parametr is u. If v is given a different value and is again held fixed while u varies, the locus of the point P is another u – curve on the surface S. In this way, by placing v = const., a one – parameter family of u – curves is defied. These cover the surface S and are completely described by the differential equation d v = 0.

Simalarly, interchanging the roles of the parameters u and v, we define a one – parameter family of v – curves on the surface S, along any one of which, denoted by C^v, the parameter v varies and u = const., so that du = 0. The familly of u - curves and the family of v – curves together constitute the parametric curves, which are completely represented analytically by the differential equation

dudv = 0. (2.1)

tangent lines of parametric curves are called parametric tangents, those of

u – curves and of v – curves being named u – tangents and v – tangents, respectively. Sometimes the valuesof a pair u, v which locates a point P on a surface S are spoken of as ourvilinear coordinates of F. Since on a surface S point P and pairs of values of u, v are in continuous oneto – one coorespondence when the range T of the variables u, v is sufficiently small, the terminology is then justified. Parametric curves are sometimescalled coordinate curves.