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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 zaxis 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 zaxis. 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 zvalue, or height, of each point on the surface is the
same as r, the distance from the point to the zaxis. 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 zaxis (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 zaxis 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.
 Fall '09
 hamrick

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