Speech exercises. I. Topics for discussion

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.

CONTENS

I THE PARTICIPLE. FORMS OF THE PARTICIPLE 3

II THE ABSOLUTE PARTICIPLE CONSTRUCTION 12

III ORDINARY DIFFERENTIAL EQUATIONS 19

IV THE INFINITIVE, ITS FORMS AND FUNCTIONS 31

V THE INFINITIVE CONSTRUCTIONS (COMPLEX SUBJECT,

COMPLEX OBJECT AND FOR-PHRASES) 40

VI CONDIONAL SENTENCES 50

VII SURFACES 59