Engineering Calculus Notes 284

Engineering Calculus Notes 284 - xz-plane y = c(again with...

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272 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION These slicing techniques can also be used to study surfaces given by equations in x , y and z which are not explicitly graphs of functions. We consider three examples. The Frst is given by the equation x 2 4 + y 2 + z 2 = 1 . The intersection of this with the xy -plane z = 0 is the ellipse x 2 4 + y 2 = 1 centered at the origin and with the ends of the axes at ( ± 2 , 0 , 0) and (0 , ± 1 , 0); the intersection with any other horizontal plane z = c for which | c | < 1 is an ellipse similar to this and with the same center, but scaled down: x 2 4 + y 2 = 1 c 2 or x 2 4(1 c 2 ) + y 2 1 c 2 = 1 . There are no points on this surface with | z | > 1. Similarly, the intersection with a vertical plane parallel to the
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Unformatted text preview: xz-plane, y = c (again with | c | < 1) is a scaled version of the same ellipse, but in the xz-plane x 2 4 + z 2 = 1 − c 2 and again no points with | y | > 1. ±inally, the intersection with a plane parallel to the yz-plane, x = c , is nonempty provided v v x 2 v v < 1 or | x | < 2, and in that case is a circle centered at the origin in the yz-plane of radius r = r 1 − c 2 4 y 2 + z 2 = 1 − c 2 4 . This surface is like a sphere, but “elongated” in the direction of the x-axis by a factor of 2 (see ±igure 3.8 ); it is called an ellipsoid ....
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