Chapter 13

# The graph of the equation z z z c 0 y x x 0 0 0 y x y

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Unformatted text preview: is placed at this point. For example, the sphere with center the c (see Figure 6); this is the reason for origin and radius c has the simple equation c is a vertical half-plane the name “spherical” coordinates. The graph of the equation z z z z c 0 0 0 c y 0 y c y y x x x F IGURE 6 ∏=c , a sphere FIGURE 7 ¨=c , a half-plane FIGURE 8 ˙=c , a half-cone x 0<c<π/2 π/2<c<π 5E-13(pp 868-877) 1/18/06 11:38 AM Page 877 SECTION 13.7 CYLINDRICAL AND SPHERICAL COORDINATES ❙❙❙❙ 877 (see Figure 7), and the equation c represents a half-cone with the z-axis as its axis (see Figure 8). The relationship between rectangular and spherical coordinates can be seen from Figure 9. From triangles OPQ and OPP we have z Q z P (x, y, z) P (∏, ¨, ˙) z ∏ But x r cos and y we use the equations ˙ ˙ cos r sin r sin , so to convert from spherical to rectangular coordinates, O x x 3 r ¨ y x sin cos y sin y2 z sin cos z2 y P ª(x, y, 0) Also, the distance formula shows that FIGURE 9 2 4 x2 We use this equation in converting from rectangular to spherical coordinates. EXAMPLE 4 The point 2, 4, 3 is given in spherical coordinates. Plot the point and ﬁnd its rectangular coordinates. SOLUTION We plot the point in Figure 10. From Equations 3 we have z x sin cos 2 sin y sin sin 2 sin z cos 3 cos 2 4 s3 2 1 s2 3 2 (2, π/4, π/3) π 3 2 O x π 4 FIGURE 10 y Thus, the point 2, 4, 2 cos 3 sin 4 2( 1 ) 2 3 2 s3 2 1 s2 3 2 1 3 is (s3 2, s3 2, 1) in rectangular coordinates. EXAMPLE 5 The point (0, 2 s3, coordinates for this point. 2) is given in rectangular coordinates. Find spherical SOLUTION From Equation 4 we have sx 2 z2 s0 2 4 y2 1 2 12 4 4 and so Equations 3 give cos cos (Note that 3 2 because y given point are 4, 2, 2 3 . z x sin 2 s3 0 2 3 2 0.) Therefore, spherical coordinates of the 5E-13(pp 878-883) 878 ❙❙❙❙ 1/18/06 11:45 AM Page 878 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 6 Find an equation in spherical coordinates for the hyperboloid of two sheets with equation x 2 In Module 13.7 you can investigate families of surfaces in cylindrical and spherical coordi...
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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.

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