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Chapter 13

# 5e 13pp 868 877 11806 1137 am page 875 s ection 137

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Unformatted text preview: mmon screen using the domain x 1.2, y 1.2 and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the xy-plane is an ellipse. ; 49. Graph the surfaces z y y 3x 5E-13(pp 868-877) 1/18/06 11:37 AM Page 875 S ECTION 13.7 CYLINDRICAL AND SPHERICAL COORDINATES |||| 13.7 ❙❙❙❙ 875 Cylindrical and Spherical Coordinates Recall that in plane geometry we introduced the polar coordinate system in order to give a convenient description of certain curves and regions. (See Section 11.3.) In three dimensions there are two coordinate systems that are similar to polar coordinates and give convenient descriptions of some commonly occurring surfaces and solids. They will be especially useful in Chapter 16 when we compute volumes and triple integrals. Cylindrical Coordinates z In the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple r, , z , where r and are polar coordinates of the projection of P onto the xy-plane and z is the directed distance from the xy-plane to P (see Figure 1). To convert from cylindrical to rectangular coordinates, we use the equations P (r, ¨, z) z O x x 1 r ¨ r cos y z r sin z y (r, ¨, 0) whereas to convert from rectangular to cylindrical coordinates, we use FIGURE 1 The cylindrical coordinates of a point r2 2 x2 y2 y x tan z z These equations follow from Equations 11.3.1 and 11.3.2. EXAMPLE 1 (a) Plot the point with cylindrical coordinates 2, 2 3, 1 and ﬁnd its rectangular coordinates. (b) Find cylindrical coordinates of the point with rectangular coordinates 3, 3, 7 . SOLUTION z (a) The point with cylindrical coordinates 2, 2 Equations 1, its rectangular coordinates are 2π ”2, , 1’ 3 x 2 cos 2 3 2 y 2 sin 2 3 2 z 1 1 2 0 2π 3 y x FIGURE 2 3, 1 is plotted in Figure 2. From 1 2 s3 2 1 s3 Thus, the point is ( 1, s3, 1) in rectangular coordinates. (b) From Equations 2 we have tan z s3 2 3 3 3 r 1 7 2 3 s2 so 7 4 2n 5E-13(pp 868-877) 876 ❙❙❙❙ 1/18/06 11:37 AM Page 8...
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