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# 1000 2000 3000 4000 5000 5000 fig 17 6 draw a series

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* 1000 2000 3000 4000 5000 5000 Fig. 17. 6. Draw a series of contour lines to illustrate the form of the surface 2 z = 3 xy . 7. Right circular cones. Take the origin of coordinates at the vertex of * We assume that the effects of the earth’s curvature may be neglected.

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[II : 33] FUNCTIONS OF REAL VARIABLES 74 the cone and the axis of z along the axis of the cone; and let α be the semi- vertical angle of the cone. The equation of the cone (which must be regarded as extending both ways from its vertex) is x 2 + y 2 - z 2 tan 2 α = 0. 8. Surfaces of revolution in general. The cone of Ex. 7 cuts ZOX in two lines whose equations may be combined in the equation x 2 = z 2 tan 2 α . That is to say, the equation of the surface generated by the revolution of the curve y = 0, x 2 = z 2 tan 2 α round the axis of z is derived from the second of these equations by changing x 2 into x 2 + y 2 . Show generally that the equation of the surface generated by the revolution of the curve y = 0, x = f ( z ), round the axis of z , is p x 2 + y 2 = f ( z ) . 9. Cones in general. A surface formed by straight lines passing through a fixed point is called a cone : the point is called the vertex . A particular case is given by the right circular cone of Ex. 7. Show that the equation of a cone whose vertex is O is of the form f ( z/x, z/y ) = 0, and that any equation of this form represents a cone. [If ( x, y, z ) lies on the cone, so must ( λx, λy, λz ), for any value of λ .] 10. Ruled surfaces. Cylinders and cones are special cases of surfaces composed of straight lines . Such surfaces are called ruled surfaces . The two equations x = az + b, y = cz + d, (1) represent the intersection of two planes, i.e. a straight line. Now suppose that a , b , c , d instead of being fixed are functions of an auxiliary variable t . For any particular value of t the equations (1) give a line. As t varies, this line moves and generates a surface, whose equation may be found by eliminating t between the two equations (1). For instance, in Ex. 7 the equations of the line which generates the cone are x = z tan α cos t, y = z tan α sin t, where t is the angle between the plane XOZ and a plane through the line and the axis of z . Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance l apart ( Fig. 18a ). We can imagine the surface of the cylinder to be
[II : 33] FUNCTIONS OF REAL VARIABLES 75 made up of a number of thin parallel rigid rods of length l , such as PQ , the ends of the rods being fastened to two circular rods of radius a . Now let us take a third circular rod of the same radius and place it round the surface of the cylinder at a distance h from one of the first two rods (see Fig. 18a , where Pq = h ). Unfasten the end Q of the rod PQ and turn PQ about P until Q can be fastened to the third circular rod in the position Q 0 .

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