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Question 1. Define o: R² → R³ by o(u, v) = (v + cos u, sin u, v), and let S be the image of o. Then S is a smooth surface (and you can assume this). (1) Show that S is a ruled surface.

Question 1. Define o: R² → R³ by o(u, v) = (v + cos u, sin u, v), and let S be the image of o. Then S is a smooth surface (and you can assume this). (1) Show that S is a ruled surface.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1. Define o: R2 → R³ by o(u, v) = (v + cos u, sin u, v), and let S be the image of o. Then S is a smooth surface (and you can assume this). (1) Show that S is a ruled surface. (2) Give a quadratic equation for S. That is, show that there is a quadratic equation in the Cartesian coordinates x, y, z for R³ which is satisfied by all points of S. (3) Show, conversely, that all points satisfying your quadratic equation lie on S, and hence that S is a quadric. (4) Show that S is an elliptic cylinder, so that a cross section of S perpendicular to the rulings is an ellipse. What are the lengths of its axes?
Transcribed Image Text:Question 1. Define o: R2 → R³ by o(u, v) = (v + cos u, sin u, v), and let S be the image of o. Then S is a smooth surface (and you can assume this). (1) Show that S is a ruled surface. (2) Give a quadratic equation for S. That is, show that there is a quadratic equation in the Cartesian coordinates x, y, z for R³ which is satisfied by all points of S. (3) Show, conversely, that all points satisfying your quadratic equation lie on S, and hence that S is a quadric. (4) Show that S is an elliptic cylinder, so that a cross section of S perpendicular to the rulings is an ellipse. What are the lengths of its axes?
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