2015-04-01
598
EXERCISES:
I. Define the type of the Conditional Sentences and translate them:
1. It is clear that every horizontal plane section of the cone is a circle, unless it is the vertex itself. 2. If we want the range of the function to contain only real numbers, we must restrict the range of x to the set of non-negative real numbers, for if x were permitted to have a negative value, y would be imaginary. 3. If we wish to express that a number n1 is as large as n2, we write n1 £ n2. 4. If we add ½ + 3/2, we obtain ½. 5. If an analytic function f (x) could be represented asymptotically by a divergent series S (x) in the complete neighborhood, it wold be analytic there. 6. If a rectangular coordinate system is chosen, then to every point of space there corresponds uniquely a triple of real numbers. 7. If ad were not greater than bc, then ad - bc would be a natural number and would not be a function.
II. 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 not so tired, he would continue his work. 4. Had you taken part in our experiment, you have helped us much. 5. Had they spent more time in Moscow, they could have visited much more places of interest. 6. Were she good at mathematical analysis, she would be able to solve some of these problems. 7. Were this dress little longer. I could wear it. Had the students of your group known mathematical analysis better, the results of the examination in functional analysis would not have been so had. 9. Could you spend more time in the language laboratory, you would improve your English greatly. 10. Had he better knowledge of the English grammar, he would not make so many mistakes. 11. Were I in your place, I should translate this text again and correct all the mistakes.
III. Learn these words:
surface ['sq:fIs] поверхность
double [dAbl] двойной
consist [kqn'sIst] состоять
unit ['ju:nIt] единица, единичный
establish [Is'txblIS] устанавливать
impose [Im'pouz] налагать (условие)
homeogeneous ["hOmq'Gi:njqs] однородный
miss [mIs] отсутствовать
generator [GenqreItq]///// образующая
exhibit [Ig'zIbIt] показывать, проявлять
transpose [trxns'pouz] переставлять, переносить
vanish ['vxnIS] исчезать, превращаться в нуль
degenerate [dI'GenqrIt] вырожденный
rule out ['ru:l qut] исключать
furthermore ['fq:DqmO:] кроме того
imply []im'plaI] означать
arbitrary [a'bItrqrI] произвольный
simultaneous ["sImql'teInjqs] одновременный
elimination [I'lImIneISn] исключение (неизвестного)
vice versa ['vaIsI'vq^sq] наоборот
jacobian [Gq'koubIqn] якобиан
desire [dI'zqIq] желать
infinite ['InfInIt] бесконечный
infinity [In'fInIt] бесконечность
to set equal ['set i^kwOl] положить равным
respectively [rIs'pektivlI] соответственно
IV. Read the text.
SURFACES
A surface can be described as a two-parameter family, or double infinity, of points. A surface can also be said to be the locus of a point moving with two degrees of freedom.
One method of representing a surface analytically consists in first establishing the usual left-handed orthogonal cartesian coordinate system with the same unit of distance on all three axes and them imposing one condition on a variable point P (x, y, z) by an equation of the form:
F (x, y, z) = 0 (1.1)
Such an equation is called the implicit equation of the surface represented by it.
Certain very simple types of surfaces are already familiar. For example, if the equation (1.1) is linear in the variables x, y, z the surface represented by it is a plane, which is the simplest surface of all. Perhaps the next simplest surface is the sphere. If the equation (1.1) is homogeneous in x, y, z it represents a cone which vertex is at the origin. Finally, if one of the variables is missing from the implicit equation of a surface, the surface is a cylinder whose generators are parallel to the axis of the missing variable.
If the implicit equation (1.1) be solved for one of the variables as a function of the other two, say for z as a function of x, y, the resulting equation
z = f (x, y), (1.2)
represent the same surface as before. Such an equation is called the explicit equation of the surface represented by it. The explicit equation (1.2) can be exhibited as a special case of the implicit equation (1.1) by transposing z to the right member and placing
F (x, y, z) = f (x, y) – z
Although for some purposes the implicit and explicit equations of surfaces are very useful, the definition of a real proper analytic surface will be based on a parametric representation.
Definition 1. Let the coordinates x, y, z of a point P be given as a single-valued analytic function of two real independent variables u, v on a rectangle T in a uv-plane, by equations of the form:
x = x (u, v), y = y (u, v), z = z (u, v) (1.3)
Further, let the jacobians of x, y, z with respect to u, v, be denoted by J1, J2, J3 so that
J1 = yu zv – yv zu, J2 = su xv – sv xu, J3 = xu yv – xv yu (xu =,...) (1.4)
and suppose that not all of by J1, J2, J3 vanish identically on the rectangle T. Then the locus of the point P, as u, v vary on T, is a real proper analytic surface S.
Equations (1.3) are called parametric equations of the surface S, the parameters being the variables u, v. We reserve the right to permit the parameters to take on complex values. Moreover, one or more of the coordinates x, y, z may, under suitable conditions, be allowed to be complex.
To say that a surface is proper means that it does not reduce to a curve. Both of these degenerate cases are ruled out by the hypothesis that the jacobians J1= (1, 2, 3) do not all vanish identically. For, if the locus S were to reduce to a fixed point P, the coordinates x, y, z of P would all be constant, and the jacobians J1 would all vanish identically. Furthermore, if the locus S were to reduce to a curve, this curve could be represented parametrically by equations of the form (1.2). If in these equations the parameter t is set equal to any function of u, v, the result is three equations of the form (1.3), for which the jacobians J1 are easily proved, by actual calculation, to vanish identically. Conversely, the identical vanishing the jacobians J1 would imply that the locus of the point P was not a proper surface. For, if the jacobians all vanish identically, then the functions x, y, z are three solutions of a linear homogeneous partial differential equation of the form
a θ u + b θ v = 0 (1.5)
in which the coefficients a, b are functions of u, v. The theory of linear partial differential equations of the first order tells us how to integrate this equation.
First form the associated ordinary differential equation
b d u – a d v = 0 (1.6)
This equation has an integral
t (u, v) = const., (1.7)
and the most general solution of equation (1.5) is given by the formula
θ = F (t(u, v)), (1.8)
the function F being arbitrary. Consequently the coordinates x, y, z are either all constant or are, at most, functions of a single parameter t, so that either P is a fixed point or else has for its locus a curve.
Even if the jacobians J1, J2, J3 do not all vanish idetically on the rectangle T., it may happen that they vanish simultaneously for one or more isolated paire of values of u, v or perhaps they vanish simultaneously along a curve v = v(u) in T. Any point of real proper analytic surface at which the jacobians J1, J2, J3 vanish simultaneously is called singular, although the singularity may belong to the parametric reoresentation being used for the surface defined as a point-locus, as in the case of the sphere, or else the singularity may belong to the surface itself. A surface, or portion of a surface, which is free of singular points may be called nonsingular.
Elimination of u, v from the parametric equations (1.3) of a surface S would lead to the implicit equation (1.1), of S. Vice versa, if the implicit equation (1.1) of a surface is desired, let two of the variable, say x and y, be arbitrary functions of two parameters u, v, and then solve (1.1) for z as a function (1.1) for a would lead to the explicit equation (1.2) of the surface, except that u and v would occur in place of x and y, respectively. Indeed, the explicit equation (1.2) of a surface, when supplement by the identities x=x, y=y, becomes the parameter equations
x = x, y = y, z = f(x,y)
of the same surface, the parameters now being now being the coordinates x, y.
