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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online