Chapter 13

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Unformatted text preview: 76 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z Therefore, one set of cylindrical coordinates is (3 s2, 7 4, 7). Another is (3 s2, 4, 7). As with polar coordinates, there are infinitely many choices. 0 (0, c, 0) y (c, 0, 0) x Cylindrical coordinates are useful in problems that involve symmetry about an axis, and the z-axis is chosen to coincide with this axis of symmetry. For instance, the axis of the circular cylinder with Cartesian equation x 2 y 2 c 2 is the z-axis. In cylindrical coordinates this cylinder has the very simple equation r c. (See Figure 3.) This is the reason for the name “cylindrical” coordinates. EXAMPLE 2 Describe the surface whose equation in cylindrical coordinates is z FIGURE 3 r. SOLUTION The equation says that the z-value, or height, of each point on the surface is the same as r, the distance from the point to the z-axis. Because doesn’t appear, it can vary. So any horizontal trace in the plane z k k 0 is a circle of radius k. These traces suggest that the surface is a cone. This prediction can be confirmed by converting the equation into rectangular coordinates. From the first equation in (2) we have r=c, a cylinder z z2 r2 x2 y2 We recognize the equation z 2 x 2 y 2 (by comparison with Table 1 in Section 13.6) as being a circular cone whose axis is the z-axis (see Figure 4). 0 y EXAMPLE 3 Find an equation in cylindrical coordinates for the ellipsoid x 4x 2 FIGURE 4 4y 2 z2 SOLUTION Since r 2 1. x2 y 2 from Equations 2, we have z=r , a cone z2 4 x2 1 y2 1 4r 2 So an equation of the ellipsoid in cylindrical coordinates is z 2 z 4r 2. 1 P ( ∏, ¨, ˙) Spherical Coordinates ∏ ˙ The spherical coordinates , , of a point P in space are shown in Figure 5, where OP is the distance from the origin to P, is the same angle as in cylindrical coordinates, and is the angle between the positive z-axis and the line segment OP. Note that O ¨ y x FIGURE 5 The spherical coordinates of a point 0 0 The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.

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