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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 xyplane is an ellipse. ; 49. Graph the surfaces z y y 3x 5E13(pp 868877) 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 threedimensional space is represented
by the ordered triple r, , z , where r and are polar coordinates of the projection of P
onto the xyplane and z is the directed distance from the xyplane 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 5E13(pp 868877) 876 ❙❙❙❙ 1/18/06 11:37 AM Page 8...
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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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