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Unformatted text preview: S E C T I O N 13.7 Cylindrical and Spherical Coordinates (ET Section 12.7) 411 SOLUTION A point P on the parabola C has the form P = x , ax 2 , c , hence the parametric equations of the line through the origin and P are x = t x , y = tax 2 , z = tc To find a direct relation between x y and z we notice that yz = tax 2 ct = ac ( t x ) 2 = acx 2 Now, defining new variables z = u- v and y = u + v . This equation becomes ( u + v)( u- v) = acx 2 u 2- v 2 = acx 2 u 2 = acx 2 + v 2 This is the equation of an elliptic cone in the variables x , v , u . We, thus, showed that the cone on the parabola C is transformed to an elliptic cone by the transformation (change of variables) y = u + v , z = u- v , x = x . 13.7 Cylindrical and Spherical Coordinates (ET Section 12.7) Preliminary Questions 1. Describe the surfaces r = R in cylindrical coordinates and = R in spherical coordinates. SOLUTION The surface r = R consists of all points located at a distance R from the z-axis. This surface is the cylinder of radius R whose axis is the z-axis. The surface = R consists of all points located at a distance R from the origin. This is the sphere of radius R centered at the origin. 2. Which statement about the cylindrical coordinates is correct? (a) If = 0, then P lies on the z-axis. (b) If = 0, then P lies in the x z-plane. SOLUTION The equation = 0 defines the half-plane of all points that project onto the ray = 0, that is, onto the nonnegative x-axis. This half plane is part of the ( x , z )-plane, therefore if = 0, then P lies in the ( x , z )-plane. z y x The half-plane q = For instance, the point P = ( 1 , , 1 ) satisfies = 0, but it does not lie on the z-axis. We conclude that statement (b) is correct and statement (a) is false. 3. Which statement about spherical coordinates is correct? (a) If = 0, then P lies on the z-axis. (b) If = 0, then P lies in the x y-plane. SOLUTION The equation = 0 describes the nonnegative z-axis. Therefore, if = 0, P lies on the z-axis as stated in (a). Statement (b) is false, since the point ( , , 1 ) satisfies = 0, but it does not lie in the ( x , y )-plane. 4. The level surface = in spherical coordinates, usually a cone, reduces to a half-line for two values of . Which two values? SOLUTION For = 0, the level surface = 0 is the upper part of the z-axis. For = , the level surface = is the lower part of the z-axis. These are the two values of where the level surface = reduces to a half-line. 5. For which value of is = a plane? Which plane? SOLUTION For = 2 , the level surface = 2 is the x y-plane. 412 C H A P T E R 13 VECTOR GEOMETRY (ET CHAPTER 12) z y P P x 2 2 Exercises In Exercises 14, convert from cylindrical to rectangular coordinates....
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